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Supersymmetric solutions with SU(3), G2 and Spin(7)
structures
（SU(3),G2及び Spin(7)構造を持つ超対称性解）
理学研究科
数物系専攻
平成 26年度
Kazuki Hinoue
（樋ノ上 和貴）
Supersymmetric Solutions with SU(3), G2 and Spin(7) Structures
Graduate School of ScienceMathematics and Physics
2014
Kazuki Hinoue
Acknowledgements
I am most grateful to my advisor Professor Yukinori Yasui for his support and encouragement; Ihave had good fortune to learn from and work with him. I also express my deep appreciation to mycollaborators Professor Shun’ya Mizoguchi and Doctor Tsuyoshi Houri for their attentive guidanceand support. My thanks also go to my collaborator Doctor Christina Rugina. I heartily thankDoctor Kaname Hashimoto for his instruction on mathematical things, especially, G-structures.Due to the generous efforts of the members of the research group consisting Mathematical Physics,Theoretical Astrophysics and Particle Theory in Osaka City University, I have been able toproduce this thesis. Without their support and encouragement, the thesis would not have beencompleted. Especially, I express deeply my thankfulness to Professor Hiroshi Itoyama for hisgreat kindness.
1
Preface
This thesis is based on the following three papers:
1. “Supersymmetric heterotic solutions via non-SU(3) standard embedding” [1],
2. “Heterotic solutions with G2 and Spin(7) structures” [2],
3. “General Wahlquist solutions in all dimensions” [3].
The main bodies of this thesis are based on the papers 1(Chapter 4) and 2(Chapter 3), which aredescribed on constructing supersymmetric solutions with SU(3)-, G2-, and Spin(7)-structures inheterotic supergravity theory. Appendix F is based on the paper 3 in the view of torsion geometry,which is described on Killing-Yano symmetry of the Wahlquist spacetime and generalizing theWahlquist metric to higher dimension in General relativity.
2
Abstract
In this thesis, we study supersymmetric solutions in 6-, 7-, and 8-dimensional supergravitytheories, where an SU(3)-structure on 6-dimensional manifolds, a G2-structure on 7-dimensionalmanifolds, and an Spin(7)-structure on 8-dimensional manifolds play an important role. Indimension 6, we obtain an intersecting metric by superposing two Gibbons-Hawking metrics withconformal factors. Then, the metric, the fundamental 2-form and the complex (3, 0)-form satisfythe defining equations of the SU(3)-structure associated with type II supergravity whose geometryconsists of an exact Lee form and a closed Bismut torsion, which are identified with the dilatonand the 3-form flux respectively. It is shown that the corresponding six dimensional manifold is aCalabi–Yau manifold with torsion describing a supersymmetric solution of the theory. By usingU(1) isometries the solution is twice T-dualized, which leads to the supersymmetric solution withSO(6) holonomy. In dimension 7, a G2-structure associated with Abelian heterotic supergravitytheory is introduced. Since the torsion 3-form is closed and the Lee form is exact, they areidentified with the 3-form flux and the dilaton as in the case of the SU(3)-structure. The G2-structure is determined by the fundamental 3-form, it’s Hodge dual 4-form and the field strengthof U(1) gauge field. For the 7-dimensional manifold, we assume the cohomogeneity one manifoldof the form R+ × S3 × S3. We obtain 2 types of regular supersymmetric solutions, i.e., S3-boltand T 1,1-bolt solutions. In dimension 8, a Spin(7)-structure associated with Abelian heteroticsupergravity theory is introduced. Then, the Spin(7)-structure is determined by a fundamental4-form and a field strength of U(1) gauge field. Using an ansatz of 3-Sasakian manifolds, weconstruct supersymmetric regular solutions.
3
Contents
1 Introduction 6
2 G-structures 102.1 SU(3)-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 HKT-structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 G2-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Spin(7)-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Heterotic solutions with G2 and Spin(7) structures 293.1 G2-structure associated with supergravity . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.1 Abelian heterotic supergravity . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.2 G2-structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 G2 solutions in 7-dimensional Abelian heterotic supergravity . . . . . . . . . . . . 323.2.1 Cohomogeneity one ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.2 G2T solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Spin(7) solutions in 8-dimensional Abelian heterotic supergravity . . . . . . . . . . 413.3.1 Spin(7)-structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.2 3-Sasakian ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Supersymmetric heterotic solutions with SU(3)-structure 474.1 HKT geometry as a conformal transform . . . . . . . . . . . . . . . . . . . . . . . . 484.2 Intersecting HKT metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3 Embedding into heterotic string theory and T-duality . . . . . . . . . . . . . . . . 534.4 SUSY domain wall metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 Summary 67
A Nijenhuis tensor associated with J 70
B (1, 0)-form 74
4
C The SU(3)-structure on the superposed HKT metrics 77C.1 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
D SU(3) structure of the T-dualized solution along ∂3 85D.1 Bismut curvature and Hull curvature . . . . . . . . . . . . . . . . . . . . . . . . . . 92
E SU(3) structure of the T-dualized solution along ∂5 96E.1 Bismut curvature and Hull curvature . . . . . . . . . . . . . . . . . . . . . . . . . . 103
F General Wahlquist metrics in all dimensions 109F.1 Killing-Yano symmetry of the Wahlquist spacetime . . . . . . . . . . . . . . . . . . 109
F.1.1 Killing-Yano symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110F.2 Higher-dimensional Wahlquist spacetimes . . . . . . . . . . . . . . . . . . . . . . . 112
F.2.1 Even dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112F.2.2 Odd dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5
Chapter 1
Introduction
String theory has been thought of as an important candidate of theories that unify grav-ity and the other fundamental forces. One of the most exciting predictions is that our worldmight have extra dimensions because the theory is required to be defined in ten dimensions.However, as our “visible” spacetime is in four dimensions, one needs to explain why we see thefour-dimensional spacetime. One possible solution is the idea of compactification, where ten-dimensional spacetime M10 is given by a product of our four-dimensional spacetime R1,3 and acompactified six-dimensional space M6, M10 = R1,3 ×M6. The “invisible” six-dimensional spaceis called an internal space.
In superstring theory, one requires invariance of the theory under supersymmetry transfor-mation in ten dimensions, which does not necessarily require invariance in four dimensions. Thepresence of supersymmetry is helpful in solving equations of motion with a special geometricstructure of the internal space. In heterotic string theory, for example, necessary and sufficientconditions for preserving supersymmetry in R1,3 are equivalent to the vanishing of supersym-metry variations for the gravitino ψ, the dilatino λ, and the gaugino χ on the internal spaceM6,
δψ ∝(∇µ +
18Hµνργ
µνρ
)ε = 0 , (1.1)
δχ ∝(
(∂µϕ)γµ +112
Hµνργµνρ
)ε = 0 , (1.2)
δλ ∝ Fµνγµνε = 0 , (1.3)
where Hµνρ, ϕ, and Fµν denotes the 3-form flux, the dilaton, and the field strength of the gaugefield, respectively. The parameter of supersymmetry transformations, denoted by ε, is called aKilling spinor on M6. Once at least one Killing spinor exists, the internal space M6 admits aspecial geometric structure, called a G-structure, and the holonomy group shrinks to a group Gthat is characterized by special differential forms, called G-invariant forms or fundamental forms,associated with the G-structure. Thus, supersymmetry conditions that are expressed in terms ofKilling spinors are encoded to those in terms of the G-invariant forms, which provides us one wayto study supersymmetric solutions, specifically, of the Neveu–Schwarz (NS) sector in heterotic
6
string theory and of the NS-NS sector in type II string theory [4, 5].
In superstring theory, one often considers the situation in which the matter fields are absent(Hµνρ = ϕ = Fµν = 0) because the supersymmetry variation equation for the gravitino simplifiesto a parallel spinor equation of the Levi-Civita connection, and a Ricci-flat Kahler manifold,known as a Calabi–Yau manifold, is chosen as the internal space M6. In complex geometry itis known that holonomy of a Kahler manifold is contained in U(n). In particular, if a Kahlermanifold admits a Ricci-flat metric, the holonomy shrinks to SU(n). It is also known that aKahler manifold with SU(n) holonomy has a parallel spinor. Conversely, if a parallel spinorexists on a Kahler manifold, the Kahler manifold admits the SU(n) holonomy. Furthermore, theCalabi conjecture and its proof by Yau states that the Kahler manifold admits a unique Ricci-flatKahler metric if the first Chern class vanishes [6, 7].
U(n) and SU(n) holonomy groups on Riemannian manifolds are examples of the specialholonomy groups classified by Berger [8] (see also [6]). If there exists at least one parallel spinoron an n-dimensional Riemannian manifold, the holonomy groups and the number of independentparallel spinors are provided by the following table [6] (Theorem. 3.6.1),
Name dimension n Hol(∇) # of spinor (N+, N−)Calabi–Yau 4m SU(2m) (2,0)Calabi–Yau 4m + 2 SU(2m + 1) (1,1)
Hyper Kahler 4m Sp(m) (m+1,0)Parallel G2 7 G2 1
Parallel Spin(7) 8 Spin(7) (1,0)
where N± denotes the number of positive or negative spinors.In particular, every Riemannianmanifold with one of special holonomy groups other than U(n) admits at least one parallel spinor.Metrics on such manifolds are Ricci-flat metrics. Ricci-flat metrics on a Riemannian manifoldwith a special holonomy group have been investigated actively. If there exist G-invariant formsof a G-structure on an n-dimensional Riemannian manifold, the holonomy group is containedin G. The corresponding Ricci-flat metrics are constructed by the G-invariant forms, which aredetermined by first-order differential equations of the G-invariant forms. Consequently, a questionof construction of a Ricci-flat metric on a Riemannian manifold admitting a special holonomygroup arrives at a problem with solving the first-order differential equations of G-invariant forms.
G-structures can be also applied to constructing not only Ricci-flat metrics but also solutionsof supergravity theories. A G-structure related to special holonomy is classified into some classesby first-order differential equations of the corresponding G-invariant forms. Some specific classesgive necessary conditions for supersymmetry preservation in supergravity theories [9]. Ricci-flatmetrics are obtained when the first-order differential equations of G-invariant forms takes themost simple form. In other class, the equations of G-invariant forms give rise to a gravitationalpart of equations of motion in supergravity theories. In Refs. [10, 11], it was shown that equationsof motion in supergravity theory can be solved with the G-invariant forms if equations of motionfor the dilaton and Bianchi identity hold.
7
An SU(n)-structure on a 2n-dimensional almost Hermitian manifold (M2n, J) is determined bya fundamental 2-form, κ, and a complex (n, 0)-form, Υ. If and only if a 2-form κ and a (n, 0)-formΥ with a volume-matching condition satisfy
dκ = 0 , dΥ = 0 , (1.4)
the manifold, (M2n, J, κ,Υ), is said to be a Calabi–Yau manifold. A Calabi–Yau manifold isalso a Kahler manifold with ∇Υ = 0, and the holonomy is contained in SU(n). The associatedmetric is Ricci-flat. The many explicit examples of Calabi–Yau manifolds have been constructedas hypersurfaces in complex projective spaces[6]. In the case where n = 2, a famous example of aCalabi–Yau 2-fold is the K3 surface, which is a hypersurface in 3-dimensional complex projectivespace CP 3. A K3 surface is a compact manifold that has a self-dual metric with self-dualRiemannian curvature. In the case of n = 3, an example of a Calabi-Yau metric is the resolvedcone metric from the Sasaki Einstein metric [12].
When one considers G2-structure, it is characterised by the G2-invariant 3-form Ω satisfyingthe condition
dΩ = 0 , d ∗ Ω = 0 . (1.5)
The corresponding G2 metric is a Ricci-flat metric with G2 holonomy [13], which is obtained bysolving the first-order equations (1.5). G2 metrics for the principal orbits CP (3) or S3 ×S3 havebeen studied in [5, 14, 15, 16, 17]. Under a cohomogeneity one ansatz, equations (1.5) reduceto ordinary differential equations. These equations were explicitly solved with asymptoticallyconical (AC) [14, 15] and asymptotically locally conical (ALC) [5, 16, 17] metrics. Similarly,Spin(7)-structure is given by the condition
dΨ = 0 , (1.6)
where Ψ is a Spin(7)-invariant 4-form satisfying ∗Ψ = Ψ [18]. The corresponding metricswith Spin(7) holonomy were obtained in the similar manner to G2 metrics. For Sp(2)/Sp(1)or SU(3)/U(1) principal orbits, Spin(7) metrics have been studied in [16, 17, 19, 20, 21, 22].Examples of compact G2- and Spin(7)-manifolds were constructed by Joyce [6].
In this thesis, we study supersymmetric solutions in 6-, 7-, and 8-dimensional supergravitytheories, where an SU(3)-structure on 6-dimensional manifolds, a G2-structure on 7-dimensionalmanifolds, and a Spin(7)-structure on 8-dimensional manifolds play important roles. In dimen-sion 6, we construct an intersecting metric g6 by superposing two Gibbons–Hawking metrics withconformal factors. The metric g6, the fundamental 2-form κ, and the complex (3, 0)-form Υ sat-isfy the defining equations of the SU(3)-structure associated with type II supergravity, where theLee form Θ6 = 2dϕ6 is an exact 1-form and the Bismut torsion H6 is a closed 3-form. We showthat the manifold (M6, g6, κ, Υ) is a Calabi–Yau with torsion manifold and thus (g6,H6, ϕ6) aresupersymmetric solutions in the theory [11]. In dimension 7, we introduce a class of G2-structuresassociated with Abelian heterotic supergravity theory together with a class having a closed 3-form torsion H7 and an exact Lee form Θ7 = 2dϕ7. This class is determined by the relevant(g7,Ω, F7), where g7 is the metric associated with the fundamental 3-form Ω and F7 is the field
8
strength of U(1) gauge field satisfying Bianchi identity. Then, we solve the defining equations ofthe class on cohomogeneity one manifold of the form R+ × S3 × S3. The quartet (g7, ϕ7, H7, F7)solves the equations of motion automatically in the theory [23]. In dimension 8, similar to thecase in dimension 7, we introduce a class of Spin(7)-structure associated with Abelian heteroticsupergravity theory. The class is determined by the proper (g8, Ψ, F8), where g8 is the metricassociated with the fundamental 4-form Ψ and F8 is the field strength of U(1) gauge field. Weassume the form of manifolds R+ × M3−Sasaki, where M3−Sasaki is a manifold with 3-Sasakianstructure. Then, we construct supersymmetric solutions in the theory.
The remainder of this thesis is organized as follows. In chapter 2, we briefly review the SU(3)-,HKT(Hyper Kahler with torsion), G2-, and Spin(7)-structure. In chapter 3, we construct super-symmetric solutions in 7-, and 8- dimensional Abelian heterotic supergravity theory by solvingthe defining equations of the specific classes. In chapter 4, we construct the supersymmetric solu-tion in heterotic supergravity theory with SU(3) holonomy of the Bismut connection and SO(6)holonomy of the Hull connection.
9
Chapter 2
G-structures
We review a general method of characterizing a geometric G-structure admitting a G-connectionwith a 3-form torsion in accordance with Ref. [24]. Let (Mn, g) be an oriented Riemannianmanifold and denoted by L(Mn) its frame bundle. The Levi-Civita connection on the framebundle L(Mn) is a 1-form
ZLC : T (L(Mn)) −→ so(n) (2.1)
with values in the Lie algebra so(n). We fix a closed subgroup G of the special orthogonal groupSO(n), where G’s action is simply transitive.A G-structure on Mn is a G-subbundle Q ⊂ L(Mn).We decompose the Lie algebra so(n) into the subalgebra g and its orthogonal complement m,
so(n) = g ⊕ m , (2.2)
where g is the Lie algebra to Lie group G. Restricting ZLC to the tangent bundle T (Q) bringsabout the decomposition of ZLC |T (Q) corresponding to (2.2),
ZLC |T (Q) = Z ⊕ T . (2.3)
Z is a connection 1-form field on Q with values in the associated bundle Q×Ad g,
Z : T (Q) −→ Q×Ad g . (2.4)
T is 1-form field on Q with values in the associated bundle (Q× m)/G,
T : T (Q) −→ (Q× m)/G (2.5)
and this is known as an intrinsic torsion.
[supplementation]Q×Ad g is the vector bundle associated with G-structure Q constructed by adjoint representationand its fiber is the Lie algebra g. Q×Ad g ≡ (Q× g)/G means
s : (u,A) 7−→ (us, s−1As) (s ∈ G , (u,A) ∈ Q× g)
10
(u,A) is identified with (u, s−1As).Q× ρ m is the vector bundle associated with G-structure Q constructed by a representation ρ andits fiber is m. Q× ρ m ≡ (Q× m)/G means
s : (u, y) 7−→ (us, ρ(s)−1y) (s ∈ G , (u, y) ∈ Q× m)
(u, y) is identified with (u, ρ(s)−1y).
An arbitrary G-connection Z on Q differs from Z by an 1-form Σ with values in the associatedbundle Q×Ad g,
Z = Z + Σ = ZLC |T (Q) − T + Σ . (2.6)
From now on, we consider the corresponding connection ∇T to (2.6) on the associated bundleQ ×G Rn.Hence the connection 1-form field ωT of ∇T associated with Z belongs to T ∗(Mn) ⊗(Q×Ad g),
ωT ∈ T ∗(Mn) ⊗ (Q×Ad g) ∼= (Q×G Rn) ⊗ (Q×Ad g) (∵ T ∗p Mn
∼= Rn) . (2.7)
On the associated bundle Q×G Rn the intrinsic torsion T on Mn belongs to T ∗(Mn)⊗ (Q×G m),
T ∈ T ∗(Mn) ⊗ (Q×G m) ∼= (Q×G Rn) ⊗ (Q×G m) (∵ T ∗p Mn
∼= Rn) . (2.8)
At local ωT and T belong to Rn ⊗ g and Rn ⊗ m respectively:
ωT (p) ∈ Rn ⊗ g (p ∈ Mn) , (2.9)
T (p) ∈ Rn ⊗ m (p ∈ Mn) . (2.10)
We identify 1-forms ωT (p) and T (p) with 1-form fields ωT and T respectively. The correspondingconnection is given by
∇TXY = ∇LC
X Y − T (X)(Y ) + Σ(X)(Y ) (X,Y ∈ Ep(Q×G Rn) = Rn ∼= Tp(Mn)) , (2.11)
where ∇LC is the Levi-Civita connection on the associated bundle Q×G Rn.Of course, ∇LCX Y ∈
Rn⊗ so(n), T (X)(Y ) ∈ Rn⊗m , Σ(X)(Y ) ∈ Rn⊗g and ∇TXY ∈ Rn⊗ so(n). The torsion tensor
of affine connection ∇T is defined by
T (X,Y ) =12(∇T
XY −∇TY X − [X,Y ]) (2.12)
and then by using (2.11) this is rewritten as follows:
T (X,Y ) =12(∇LC
X Y − T (X)(Y ) + Σ(X)(Y ) −∇LCY X + T (Y )(X) − Σ(Y )(X) − [X,Y ] )
=12(−T (X)(Y ) + Σ(X)(Y ) + T (Y )(X) − Σ(Y )(X) ) +
12(∇LC
X Y −∇LCY X − [X,Y ] )
=12(−T (X)(Y ) + Σ(X)(Y ) + T (Y )(X) − Σ(Y )(X) ) + TLC(X,Y )
=12(−T (X)(Y ) + Σ(X)(Y ) + T (Y )(X) − Σ(Y )(X) ) ,
11
where in the last line it is used that the Levi-Civita connection ∇LC is torsion free. Thus thetorsion tensor of affine connection ∇T depends on T and Σ,
T (X,Y, Z) = −g(T (X)(Y ), Z) + g(T (Y )(X), Z) + g(Σ(X)(Y ), Z) − g(Σ(Y )(X), Z) , (2.13)
where T (X,Y, Z) = −T (Y,X,Z) apparently. By T (X,Y, Z) = −T (X,Z, Y ), T is a 3-form if andonly if
g(T (Z)(X), Y ) + g(T (Y )(X), Z) = g(Σ(Y )(X), Z) + g(Σ(Z)(X), Y ) (2.14)
holds. Actually, comparing
−T (X,Z, Y ) = g(T (X)(Z), Y ) − g(T (Z)(X), Y ) − g(Σ(X)(Z), Y ) + g(Σ(Z)(X), Y )
to (2.13) give a equation
g(T (Z)(X), Y ) + g(T (Y )(X), Z) − g(T (X)(Y ), Z) − g(T (X)(Z), Y )= g(Σ(Y )(X), Z) + g(Σ(Z)(X), Y ) − g(Σ(X)(Y ), Z) − g(Σ(X)(Z), Y ) .
If (2.14) is valid, then we have
g(T (X)(Y ), Z) + g(T (X)(Z), Y ) = g(Σ(X)(Y ), Z) + g(Σ(X)(Z), Y ) .
Now we see that
T (Z, Y,X) = −g(T (Z)(Y ), X) + g(T (Y )(Z), X) + g(Σ(Z)(Y ), X) − g(Σ(Y )(Z), X)= g(T (Z)(X), Y ) − g(T (X)(Z), Y ) − g(Σ(Z)(X), Y ) + g(Σ(X)(Z), Y ) (∵ X ↔ Y )= −g(T (Y )(X), Z) + g(Σ(Y )(X), Z) − g(T (X)(Z), Y ) + g(Σ(X)(Z), Y ) (∵ (2.14))= −g(T (Y )(X), Z) + g(Σ(Y )(X), Z) + g(T (X)(Y ), Z) − g(Σ(X)(Y ), Z) (∵ (2))= −T (X,Y, Z)
Indeed, when (2.14) is valid, torsion tensor (2.13) is totally skew symmetric tensor. Also, themetric g on Mn is given by a G-invariant form Ω associated with the G-structure Q and thus themetric has G-invariance.
On associated bundle Q×G Rn, Σ and T belong to Rn ⊗ g and Rn ⊗ m respectively:
Σ ∈ Rn ⊗ g , T ∈ Rn ⊗ m .
Because TpMn∼= Rn (p ∈ Mn) as vector space, Σ ∈ TpMn ⊗ g and T ∈ TpMn ⊗m. The metric g
on Mn is given by a G-invariant form Ω associated with the G-structure Q and thus the metrichas G-invariance. Since the metric g is a G-invariant metric, we can define G-invariant maps asfollows:
Φ : Rn ⊗ g −→ Rn ⊗ S2(Rn)
∈ ∈
Σ 7−→ Φ(Σ) (2.15)
12
Ψ : Rn ⊗ m −→ Rn ⊗ S2(Rn)
∈ ∈
T 7−→ Ψ(T ) , (2.16)
where S2(Rn) is a rank 2 symmetric space on Rn and Φ(Σ) and Ψ(T ) are defined by
Φ(Σ)(X,Y, Z) := g(Σ(Z)(X), Y ) + g(Σ(Y )(X), Z) (2.17)
Ψ(T )(X,Y, Z) := g(T (Z)(X), Y ) + g(T (Y )(X), Z) . (2.18)
It should be notice that Φ(Σ)(X,Y, Z) and Ψ(T )(X,Y, Z) are symmetric for interchange Y withZ. If we assume the 3-form condition (2.14), then we have
Ψ(T )(X,Y, Z) = g(T (Z)(X), Y ) + g(T (Y )(X), Z)= g(Σ(Z)(X), Y ) + g(Σ(Y )(X), Z) (∵ 3-form torsion condition (2.14))= Φ(Σ)(X,Y, Z) .
Z = ZLC − T + Σ is an affine connection on a G-structure Q and the torsion T is givenby T and Σ. The condition that the torsion tensor is a totally skew symmetric is equivalent toΨ(T ) = Φ(Σ). Since dim g > dimm, K ⊆ L, where Ψ(T ) ∈ K and Φ(Σ) ∈ L, if the torsion T is3-form. It is known that Q admits Z with a 3-form torsion T if and only if K ⊆ L [24].
Associated vector bundles (Q×ρRn)⊗(Q×ρm) have variety by a chosen representation ρ of thegroup G. A representation ρ of the group G can be decomposed into irreducible representationsof G and then the representation space also can be decomposed into irreducible components.Accordingly the vector bundle (Q×ρ Rn)⊗ (Q×ρ m) can be splited into irreducible components.Hence on the neighbourhood around a point in Mn the space Rn⊗m is decomposed into irreduciblecomponents and T is decomposed into associated components,
T ∈ Rn ⊗ m =⊕ρa
Λa
T =⊕
a
Ta , (2.19)
where ρa is a irreducible representation and Λa is the associated a-dimensional irreducible com-ponents.By decomposing Rn⊗m into irreducible components of the group G, we can characterizethe geometric G-structure admitting the G-connection Z with a 3-form torsion T . The geomet-ric G-structure is also called parallel G-structure and parallel G-structure means ∇T Ω = 0 fortorsion connection ∇T . The usage of the parallel G-structure is varied from the standard mean:∇LCΩ = 0.
13
2.1 SU(3)-structures
An almost complex manifold (M2n, J) is an 2n-dimensional real differentiable manifold M2n
with an almost complex structure J , which is a tensor field at every point p of M2n such that
J : TpM2n −→ TpM2n (J is an endomorphism on TpM2n) , J2 = −1 . (2.20)
A Nijenhuis tensor NJ is a type (1,2) tensor and it is defined by
NJ(X,Y ) = [JX, JY ] − [X,Y ] − J [JX, Y ] − J [X,JY ] , (2.21)
where [· , ·] denotes a commutation relation between vector fields. A condition that a Nijenhuistensor vanishes is known as an integrability condition. If an integrability condition is satisfied,an almost complex manifold is a complex manifold. An almost Hermitian manifold (M2n, g, J) isan almost complex manifold equipped with an almost Hermitian metric g,
g(JX, JY ) = g(X,Y ) . (2.22)
If the associated Nijenhuis tensor vanishes, the almost Hermitian manifold is said to be a Hermi-tian manifold. A fundamental 2-form on an almost Hermitian manifold is defined by
κ(X,Y ) = g(X,JY ) (2.23)
and the 2-form is known as a Kahler form if the Nijenhuis tensor vanishes and κ is a closed form.Then, the Hermitian manifold is said to be Kahler manifold. Since g is invariant by J , κ is alsoinvariant by J ,
κ(JX, JY ) = κ(X,Y ) . (2.24)
Especially, if the holonomy group of the Levi-Civita connection ∇ on a Kahler manifold is con-tained in SU(n), the manifold is known as a Calabi–Yau manifold. According to Ref. [25], if andonly if a triplet (κ, J,Υ) satisfies
dκ = 0 , dΥ = 0 , (2.25)
i.e. ∇κ = 0, ∇J = 0 and ∇Υ = 0, a manifold (M2n, κ, J,Υ) is a Calabi–Yau manifold and thusthe condition dΥ = 0 implies that NJ = 0. A Kahler with torsion (KT) manifold (M2n, g, J) isan almost Hermitian manifold equipped with a 3-form Nijenhuis tensor and the conditions
∇T g = 0 , ∇T κ = 0 , (2.26)
where T is a 3-form torsion and it is given by [24]
T = −J∗dκ + N . (2.27)
In N = 0 case, the 3-form torsion T is said to be the Bismut torsion [23]. If the holonomy of aconnection on a KT manifold with vanishing Nijenhuis tensor is contained in SU(n), the manifoldis called as a Calabi–Yau with torsion (CYT) manifold. In the case T = N , the manifold is alsoknown as the almost Kahler manifold.
14
A SU(n)-structure Q2n is a principal subbundle of a frame bundle L(M2n) over a 2n-dimensionalalmost Hermitian manifold M2n. A SU(n)-structure is determined by the choice of an almostcomplex structure J , a fundamental 2-form κ, and a complex (n, 0)-form Υ, where κ and Υ satisfythe volume matching condition,
Υ ∧ Υ =43i κn . (2.28)
The subgroup SU(n) is defined by a stabilizer subgroup of fundamental forms κ and Υ. Then,the structure Q2n has a structure group SU(n) and the group is a reduced subgroup SO(2n),which is a structure group of L(M2n). From now on, we consider the case n = 3. We decomposethe Lie algebra so(6) into su(3) and the orthogonal complement m, so(6) = su(3) ⊕ m. A gapof the holonomy group by shrinking from SO(6) to SU(3) is measured by an intrinsic torsionT , which is identified with ∇κ, where ∇ denotes Levi-Civita connection. The torsion T belongsto R6 ⊗ m at local and the space of the intrinsic torsion is decomposed into five components byirreducible representations of SU(3),
R6 ⊗ m = W1 ⊕W2 ⊕W3 ⊕W4 ⊕W5 , (2.29)
where dimR W1 = 2 , dimR W2 = 16 , dimR W3 = 12 , dimR W4 = 6 , dimR W5 = 6 [26](see also[23]). Here W1 splits into W+
1 ⊕W−1 = R⊕R and W2 decomposes into W+
2 ⊕W−2 = su(3)⊕su(3)
[27, 28]. According to Ref. [23], the classification of SU(3)-structures is the following:
name type conditionNearly Kahler W1 dκ is (3, 0) + (0, 3)-formalmost Kahler W2 dκ = 0
Balanced Hermitian W3 N = 0 and δκ = 0Conformally Kahler W4 dκ = Θ ∧ κ
KT W1 ⊕W3 ⊕W4 ∇T g = 0,∇T κ = 0 and T and N are 3-formHalf-flat W−
1 ⊕W−2 ⊕W3 dImΥ = 0, d(κ ∧ κ) = 0
Here Θ denotes the Lee form and it is defined by
Θ = −J∗δκ , (2.30)
where δ is a co-derivative, δ = (−1)np−p2+p ∗ d∗ for p-form on Mn=6.
We consider a six-dimensional real manifold M6 admitting a Killing spinor χ satisfyingsupersymmetric equations in NS-NS sector of type II supergravity theory,
∇+a χ = 0 , (2.31)
( (∂aϕ)γa − 112
Habcγabc )χ = 0 , (2.32)
where a 3-form flux H should be closed form. Here ∇+χ is defined by
∇+a χ = ∇aχ − 1
4ιeaH · χ = ∇aχ − 1
8Habcγ
bcχ . (2.33)
15
When differential forms κab and Υabc are introduced as bilinear form of the Killing spinor,
κab = iχ†γabχ , Υabc = χtγabcχ , (2.34)
the supersymmetric equations (3.23) and (3.24) lead to the equations [9]
d(e−2dϕ ∗ κ) = 0 , (2.35)d(e−2dϕΥ) = 0 , (2.36)∗H = e2dϕd(e−2dϕκ) . (2.37)
On the other hand, a six-dimensional CYT manifold permits the SU(3)-structure which is definedby
d ∗ κ − Θ ∧ ∗κ = 0 , (2.38)dΥ − Θ ∧ Υ = 0 , (2.39)dκ − Θ ∧ κ = ∗T , (2.40)
where T denotes the Bismut torsion and Θ is the Lee form.Under the identification H = T and2dϕ = Θ, together with the strong KT condition dT = 0, (2.35), (2.36), and (2.37) are clearlyequivalent to (2.38), (2.39), and (2.40) Therefore (2.35), (2.36), and (2.37) are defining equationsof the SU(3)-structure associated with type II supergravity.
A CYT manifold with a closed Bismut torsion and an exact Lee form is a supersymmetricsolution of type II supergravity theory [11]. Hence whether the given geometric background hasa CYT structure admitting a closed Bismut torsion and an exact Lee form or not is significant onconstructing supersymmetric solutions of type II supergravity theory. Especially, the holonomyof a Bismut connection of a KT manifold with a closed Bismut torsion and an exact Lee formshould be investigated that it is included in SU(3). According to Ref. [11], if the restrictedholonomy is contained in SU(3), then the Ricci form of the Bismut connection vanishes. A Ricciform ρ associated with Bismut connection ∇+ on a CYT manifold is defined by (see [29])
ρ(X,Y ) :=12
6∑µ=1
g(R+#(X,Y )eµ, Jeµ) (∀X,Y ∈ TpM6) , (2.41)
where R+# is curvature as type (1, 3) tensor field,
R+#(X,Y ) : eµ 7−→ R+
#(X,Y )eµ = R+#(eµ, X, Y ) . (2.42)
From now on, we will rewrite the condition ρ = 0 as easier condition.
By using definition of a fundamental 2-form κ, we rewrite (2.41) as follows,
ρ(X,Y ) =12
6∑µ=1
κ(R+#(X,Y )eµ, eµ) . (2.43)
16
In addition, the right hand side is calculated as follows,
6∑µ=1
κ(R+#(X,Y )eµ, eµ) =
6∑µ=1
κ(XνY ρR+#(eν , eρ)eµ, eµ) = XνY ρ
6∑µ=1
κ(R+#(eµ, eν , eρ), eµ)
= XνY ρ6∑
µ=1
κ(R+ σµνρeσ, eµ) = XνY ρ
6∑µ=1
R+ σµνρκ(eσ, eµ)
=6∑
µ,σ=1
XνY ρR+σµνρκσµ =
6∑µ,σ=1
XνY ρR+σµ(eν , eρ)κσµ
=6∑
µ,σ=1
R+σµκσµ(X,Y ) .
Hence we obtain
ρ(X,Y ) =12
6∑µ,σ=1
R+σµκσµ(X,Y ) . (2.44)
Here RTµν is curvature 2-form of a 3-form torsion connection ∇T and it is defined by
RTµν = dωT
µν + σρωTµρ ∧ ωT
ρν , (2.45)
where ωTµν denotes spin connection with 3-form torsion, which is given by
ωTµν = ωµν +
12ιeµιeνT . (2.46)
If ρ(X,Y ) vanishes for all X,Y , then∑6
µ,σ=1 R+σµκσµ(X,Y ) vanishes for all X,Y . Thus ρ = 0
is equivalent to6∑
µ,ν=1
R+µνκµν = 0 . (2.47)
This condition is automatically satisfied if the following equation is valid,
12
∑µ,ν
ω+µνJµν = 0 . (2.48)
A complex structure J is type (1, 1) tensor while it can be also written as type (0, 2) tensor J#,
g(JY,X) = J#(X,Y ) ≡ κ(X,Y ) (2.49)
and thus Jµν = κµν . A complex structure J preserves a fundamental 2-form κ,
J∗κ = κ . (2.50)
The left hand side is calculated as follows in the view of tensor field,
J∗κ = Jρµeρ ⊗ eµ(κσνe
σ ⊗ eν) = Jρµκσνδρ
σeµ ⊗ eν = Jρµκρνe
µ ⊗ eν .
17
Then, we obtainJρ
µκρν = κµν , i.e. JµρJρν = Jµν . (2.51)
Now, we can show that (2.48) automatically leads (2.47) as follows:
6∑µ,ν=1
R+µνκµν =
6∑µ,ν=1
R+µνJµν =
6∑µ,ν=1
(dω+
µν +∑
ρ
ω+µρ ∧ ω+
ρν
)Jµν
=6∑
µ,ν=1
d(ω+
µνJµν) +∑
ρ
ω+µρJ
σµ ∧ Jσνω
+ρν
=6∑
µ,ν,ρ,σ=1
ω+µρJσµ ∧ ω+
ρνJσν = −6∑
µ,ν,ρ,σ=1
ω+µρJσµ ∧ ω+
νρJσν
= −12
6∑µ,ν,ρ,σ=1
ω+µρJσµ ∧ ω+
νρJσν +6∑
µ,ν,ρ,σ=1
ω+µρJσµ ∧ ω+
νρJσν
= −1
2
6∑µ,ν,ρ,σ=1
ω+µρJσµ ∧ ω+
νρJσν −6∑
µ,ν,ρ,σ=1
ω+νρJσν ∧ ω+
µρJσµ
= −1
2
6∑µ,ν,ρ,σ=1
ω+µρJσµ ∧ ω+
νρJσν −6∑
µ,ν,ρ,σ=1
ω+µρJσµ ∧ ω+
νρJσν
= 0 .
Thus (2.47) is trivially assured when (2.48) is valid.
2.2 HKT-structure
We consider an almost Hermitian manifold (M4n, g, J1, J2, J3) which has three almost complexstructure J1, J2, and J3 where the almost complex structures satisfy the conditions
J1J2 = J3 , J3J1 = J2 , J2J3 = J1 , Ja2 = −1 (a = 1, 2, 3) (2.52)
and Hermitian conditionsg(JaX , JaY ) = g(X,Y ) . (2.53)
Then, the fundamental 2-forms κa (a = 1, 2, 3) are defined by
κa(X,Y ) = g(X,JaY ) . (2.54)
A hyper Kahler manifold is the manifold (M4n, g, J1, J2, J3) admitting the conditions thatNijenhuis tensors associated with three almost complex structures vanish
NJ1 = 0 , NJ2 = 0 , NJ3 = 0 (2.55)
18
and the complex structures are parallel for a covariant derivative of Levi-Civita connection,
∇J1 = 0 , ∇J2 = 0 , ∇J3 = 0 . (2.56)
The examples of 4-dimensional hyper Kahler metrics are known as Euclidean Taub-NUT metric,Eguchi–Hanson metric, Gibbons–Hawking metric and Atiyah–Hitchin metric.
A hyper Kahler with torsion (HKT) manifold is the manifold (M4n, g, J1, J2, J3) obeying theconditions that Nijenhuis tensors NJa (a = 1, 2, 3) vanish and there exist a unique connectionwith 3-form torsion T such that
∇T J1 = 0 , ∇T J2 = 0 , ∇T J3 = 0 , (2.57)
where the torsion is given byT = J1dκ1 = J2dκ2 = J3dκ3 . (2.58)
4-dimensional HKT metrics are constructed by conformal transformation of hyper Kahler metrics,gHKT = ΦgHK .
2.3 G2-structures
We consider R7 equipped with an orientation and inner product. The space of 2-forms on R4
∧2R4 can be separated by the eigenvalue as follows,
∧2R4 = ∧2+R4 ⊕ ∧2
−R4 . (2.59)
Then, the space R7 is isomorphic to the space ∧2±R4 as vector space. Namely, ∧2
±R4 is the totalspace over the base space R4 with fiber Γ(∧2
±R4), where Γ(∧2±R4) is a space of 2-forms,
Γ(∧2±R4) = α ∈ ∧2R4 : ∗α = −α . (2.60)
Let us introduce a coordinate (y1, y2, y3, y4) on R4 and 2-forms under the volume form dy1234
β1 = dy12 − dy34 , β2 = dy42 − dy13 , β3 = dy14 − dy23 , (2.61)
where dyij means dyi ∧ dyj for short. There exist a projection π1 between ∧2R4 and R4,
π1 : ∧2−R4 −→ R4 .
The fiber of ∧2R4 is given by
π−11 (p) = αp ∈ ∧2
−R4 : ∗αp = −αp (2.62)
and we define a fiber coordinate (a1, a2, a3) on the total space ∧2−R4 as follows,
β = a1β1 + a2β2 + a3β3 . (2.63)
19
Now, since R7 is isomorphic to ∧2−R4, we can define a 3-form Ω0 on R7 by
Ω0 = da123 −3∑
i=1
dai ∧ βi . (2.64)
The written down form is given by
Ω0 = da123 − da1 ∧ dy12 + da1 ∧ dy34 − da2 ∧ dy42 + da2 ∧ dy13 − da3 ∧ dy14 + da3 ∧ dy23 . (2.65)
When we define an another coordinate (x1, · · · , x7) as
x1 = a1 , x2 = a2 , x3 = a3 , x4 = y1 , x5 = y4 , x6 = y3 , x7 = −y2 , (2.66)
we obtainΩ0 = dx123 + dx147 + dx165 + dx257 + dx246 + dx354 + dx367 . (2.67)
The 3-form Ω0 define the exceptional Lie group G2 as stabilizer subgroup of GL(7;R),
G2 = s ∈ GL(7; R) : ρ∗(s)Ω0 = Ω0 , (2.68)
where the 3-form Ω0 is said to be the fundamental 3-form.
We can lift up this to a 7-dimensional Riemannian manifold M7. Namely, the group G2 is theisotropy group of the fundamental 3-form Ω,
G2 = s ∈ SO(7) : ρ∗(s)Ω = Ω , (2.69)
where by using orthonormal basis e1, · · · , e7 on TpM7, the 3-form Ω is expressed locally asfollows:
Ω = e123 + e147 + e165 + e246 + e257 + e354 + e367 . (2.70)
Given the volume form of M7 by Vol.(M7) = e1234567, the Hodge dual of Ω is
∗Ω = e4567 + e2356 − e2347 + e1357 + e1346 − e1267 + e1245 . (2.71)
In the view of a vector space, we identify the Lie algebra so(7) with the space of all 2-form,
so(7) ∼= ∧2(R7) =
A =
∑µ<ν
βµνeµν
. (2.72)
The Lie algebra g2 of the Lie group G2 is given by
g2 = a ∈ so(7) : ϱ(a)Ω = 0 =
A =
∑µ<ν
βµνeµν ∈ ∧2(R7) : A ∧ ∗Ω = 0
, (2.73)
20
where the map ϱ is a faithful representation, ϱ : so(7) −→ so(V ),
ϱ(a)Ω = A ∧ ∗Ω =∑µ<ν
βµνeµν ∧ (e4567 + e2356 − e2347 + e1357 + e1346 − e1267 + e1245)
= (β23 + β47 − β56)e234567 + (β13 − β46 − β57)e134567 + (β12 − β45 + β67)e124567
+(β17 + β26 − β35)e123567 + (β16 − β27 − β34)e123467 + (β15 − β24 + β37)e123457
+(β14 + β25 + β36)e123456 .
Therefore we can restate the definition of the Lie algebra g2
β23 + β47 − β56 = 0 , β13 − β46 − β57 = 0 , β12 − β45 + β67 = 0 , β17 + β26 − β35 = 0 ,
β16 − β27 − β34 = 0 , β15 − β24 + β37 = 0 , β14 + β25 + β36 = 0 . (2.74)
Here the condition A ∧ ∗Ω = 0 is equivalent to ∗(A ∧ Ω) = A. Actually, ∗(A ∧ Ω) is given by
∗(A ∧ Ω) = (β45 − β12)e67 − (β46 − β13)e57 + (β47 + β23)e56 + (β56 − β23)e47 − (β57 − β13)e46
+(β67 + β12)e45 − (−β24 + β15)e37 + (−β25 − β14)e36 − (−β26 − β17)e35
+(−β27 + β16)e34 + (−β34 + β16)e27 − (−β35 + β17)e26 + (−β36 − β14)e25
−(−β37 − β15)e24 + (β56 − β47)e23 − (−β35 + β26)e17 + (β34 + β27)e16
−(β37 − β24)e15 + (−β36 − β25)e14 − (−β57 − β46)e13 + (−β67 + β45)e12 .
Hence we see that the condition ∗(A ∧ Ω) = A lead us to (2.74). Thus (2.73) is rewritten asfollows,
g2 =
A =
∑µ<ν
βµνeµν ∈ ∧2(R7) : A ∧ ∗Ω = 0
=
A =
∑µ<ν
βµνeµν ∈ ∧2(R7) : ∗(A ∧ Ω) = A
. (2.75)
A G2-structure Q7 is a principal subbundle of a frame bundle L(M7) over a seven-dimensionaloriented Riemannian manifold M7. A G2-structure is determined by the choice of a fundamental3-form Ω. As was already mentioned, the subgroup G2 is defined by a stabilizer subgroup ofΩ. Then, the structure Q7 has a structure group G2 and the group is a reduced subgroup ofSO(7), which is a structure group L(M7). A seven dimensional Riemannian manifold admittingG2-structure is called a G2-manifold. If the fundamental 3-form Ω is parallel with respect toLevi-Civita connection, ∇Ω = 0, then the holonomy is contained in G2 and also the G2-manifoldcalled parallel G2-manifold. In addition, the condition ∇Ω = 0 is equivalent to dΩ = d ∗ Ω = 0.
We decompose the Lie algebra so(7) into g2 and the orthogonal m, so(7) = g2⊕m. A failure ofthe holonomy group by shrinking from SO(7) to G2 is measured by an intrinsic torsion T , whichis identified with ∇Ω, where ∇ denotes Levi-Civita connection. The torsion T belongs to R7 ⊗m
21
locally and the space of the intrinsic torsion is decomposed into four irreducible components byirreducible representations of G2,
R7 ⊗ m = W1 ⊕W2 ⊕W3 ⊕W4 , (2.76)
where dimW1 = 1 , dimW2 = 14 , dimW3 = 27 , dimW4 = 7 [13](see also Ref. [30]). Since Tbelongs to R7 ⊗ m, we have
∇Ω = ϱ(T )(Ω) ∈ W1 ⊕W2 ⊕W3 ⊕W4 . (2.77)
On a G2-manifold, the condition that the W2 component of the intrinsic torsion T vanishes isequivalent to the condition that there exist an unique affine connection ∇T with 3-form torsionpreserving the G2-structure, ∇T Ω = 0 [24]. We consider the G2-structure of class W1⊕W3⊕W4,which is characterized by the condition T ∈ W1 ⊕W4 ⊕W3. Then, we decompose T into threeparts,
T = λ Id ⊕ Θ ⊕ T 327 , (2.78)
where λ is a function on M7. The W1 component λ Id is the map λ Id : R7 → m, which is definedby
λ Id(X) :=λ
12ιXΩ (2.79)
where X ∈ R7. Let us introduce the orthonormal projection prm : so(7) = ∧2(R7) → m by theformula
prm(α2) =13
7∑i=1
(α2, ιeiΩ) · ιeiΩ , (2.80)
where(
1√3ιe1Ω, · · · , 1√
3ιe7Ω
)is an orthonormal basis of m ⊂ so(7). The W4 component Θ is the
1-form Θ : R7 → m, which is defined by
Θ(X) :=14prm(Θ ∧ X∗) =
112
(Θ ∧ X∗ , ιeiΩ) · ιeiΩ . (2.81)
The W3 component T 327 is the 3-form T 3
27 : R7 → m, which is defined by
T 327(X) :=
12prm(ιXT 3
27) =16
7∑i=1
(ιXT 327, ιeiΩ) · ιeiΩ . (2.82)
The equation (2.80) is rewritten as follows,
prm(α2) = ιY Ω , Y =7∑
i=1
Yiei ≡13
7∑i=1
(α2, ιeiΩ)ei (2.83)
and thus we have α2 = ιY Ω. According to Ref. [31], ϱ(α2) is given by
ϱ(α2)(Ω) = −3ιY ∗ Ω . (2.84)
22
Thus (2.77) without a W2 component is given by
∇XΩ = −λ
4(ιX ∗ Ω) −
7∑i=1
(14Θ ∧ X∗ +
12ιXT 3
27 , ιeiΩ)· (ιei ∗ Ω) . (2.85)
Spaces of differential forms are splited into irreducible components of G2 representations andthe spaces are given by [24, 30]
∧27 = α2 ∈ ∧2(R7) : ∗(Ω ∧ α2) = −2α2 = ιXΩ : X ∈ R7 = ∗(∗Ω ∧ α1) : α1 ∈ ∧1
7 ,
∧214 = α2 ∈ ∧2(R7) : ∗(Ω ∧ α2) = α2 , (2.86)
∧31 = tΩ : t ∈ R ,
∧37 = ∗(Ω ∧ α1) : α1 ∈ ∧1(R7) = ιX ∗ Ω : X ∈ R7 ,
∧327 = α3 ∈ ∧3(R7) : α3 ∧ Ω = 0 , α3 ∧ ∗Ω = 0 , (2.87)
∧41 = t ∗ Ω : t ∈ R ,
∧47 = α1 ∧ Ω | α1 ∈ ∧1(R7) ,
∧427 = α4 ∈ ∧3(R7) : α4 ∧ Ω = 0 . (2.88)
Then, on the components W1, W2, W3, and W4 they are isomorphic to ∧01, ∧2
14, ∧327, and ∧1
7
as vector spaces respectively, W1∼= ∧0
1, W2∼= ∧2
14, W3∼= ∧3
27, and W4∼= ∧1
7. We considercoderivative of Ω and it is given by
δΩ := −7∑
i=1
(ιej∇ejΩ) =7∑
i,j=1
(14Θ ∧ ej +
12(ιejT 3
27) , ιeiΩ)· ιej ιei ∗ Ω , (2.89)
where in the second equal sigh, we used the equation
7∑j=1
ιej ιej ∗ Ω = 0 . (2.90)
The linear map form the vector space ∧327 to vector space ∧2 = ∧2
7 ⊕ ∧214, i.e.
T 327 ∈ ∧3
27 7−→7∑
i=1
(ιejT 327 , ιeiΩ) · ιej ιei ∗ Ω ∈ ∧2
7 ⊕ ∧214 ,
is the zero-map on the vector spaces and thus we have
7∑i=1
(ιejT 327 , ιeiΩ) · ιej ιei ∗ Ω = 0 . (2.91)
23
Also, the linear map
Θ ∈ ∧17 = R7 7−→
7∑i,j=1
(Θ ∧ ej , ιeiΩ) · ιej ιei ∗ Ω ∈ ∧2 = ∧27 ⊕ ∧2
14
is the identity map between the vector spaces and thus the image of the map contained in ∧27,
Θ 7−→ ιΘ∗Ω .
Hence we obtain7∑
i,j=1
(Θ ∧ ej , ιeiΩ) · ιej ιei ∗ Ω = −4ιΘ∗Ω . (2.92)
Consequently, δΩ is given byδΩ = −ιΘΩ = − ∗ (Θ ∧ ∗Ω) . (2.93)
We consider exterior derivative of Ω and it is given by
dΩ :=7∑
i=1
ei∧∇eiΩ = −λ
4
7∑i=1
ei∧ (ιei ∗Ω)−7∑
i=1
(14Θ ∧ ei +
12ιeiT 3
27 , ιejΩ)· (ei∧ ιej ∗Ω) . (2.94)
On the 1-st term of dΩ, it is in proportion to ∗Ω and the constant of the proportionality is equalto 4 because there are four each ei (i = 1, · · · , 7) in ∗Ω. Thus we obtain
−λ
4
7∑i=1
ei ∧ (ιei ∗ Ω) = −λ ∗ Ω . (2.95)
The linear map from ∧327 to ∧4 = ∧4
1 ⊕ ∧47 ⊕ ∧4
27, which is defined by
T 327 ∈ ∧3
27 7−→7∑
i=1
(12ιeiT 3
27 , ιejΩ)· (ei ∧ ιej ∗ Ω) ∈ ∧4
1 ⊕ ∧47 ⊕ ∧4
27 ,
is the identity map between the vector spaces ∧327 and ∧4
27, where ∗T 327 ∈ ∧4
27. Thus we have
7∑i=1
(12ιeiT 3
27 , ιejΩ)· (ei ∧ ιej ∗ Ω) = −2 ∗ T 3
27 . (2.96)
The linear map from ∧17 to ∧4
1 ⊕ ∧47 ⊕ ∧4
27, i.e.,
Θ ∈ ∧17 = R7 7−→
7∑i=1
(14Θ ∧ ei , ιejΩ
)· (ei ∧ ιej ∗ Ω) ∈ ∧4
1 ⊕ ∧47 ⊕ ∧4
27 ,
is the identity map from ∧17 = ∧3
7 to ∧47 and the image of Θ, i.e., Θ ∧ Ω, is in ∧4
7. Therefore weobtain
7∑i=1
(14Θ ∧ ei , ιejΩ
)· (ei ∧ ιej ∗ Ω) = −3Θ ∧ Ω . (2.97)
24
Consequently, dΩ is given by
dΩ = −λ ∗ Ω + ∗T 327 +
34(Θ ∧ Ω) . (2.98)
We rewrite 3-form torsion T , (2.13) with the condition (2.14) by fundamental 3-form Ω. Let usintroduce the map ς : ∧1
7 ⊕ ∧327 → R7 ⊗ g2 by the formula
ς(T )(X) =12prg2
(ιXT 3
27 −14Θ ∧ X∗
), (2.99)
where prg2(α2) = α2 − prm(α2). Also, we define the 1-form T ∗ as follows,
T ∗(X) = T (X) − ς(T )(X) (2.100)
and it takes the following explicit form,
T ∗(X) =λ
12ιXΩ +
12ιXT 3
27 +38prm(Θ ∧ X∗) − 1
8Θ ∧ X∗ . (2.101)
Then, the 3-form torsion T is given by
T (X,Y, Z) = −g(T ∗(X)(Y ) , Z) + g(T ∗(Y )(X) , Z) (2.102)
and the direct calculation lead
T = −λ
6Ω − T 3
27 −14ιΘ∗ ∗ Ω . (2.103)
When we do inner product with ∗Ω in (2.98), the each terms in the right hand side are given by
(∗Ω , ∗Ω) =∫
M7
∗Ω ∧ Ω =∫
M7
7Vol.(M7) · 1 = 7 ,
(∗T 327 , ∗Ω) =
∫M7
∗T 327 ∧ Ω = 0 (∵ ∗T 3
27 ∈ ∧427) ,
(Θ ∧ Ω , ∗Ω) =∫
M7
Θ ∧ Ω ∧ Ω = 0 .
Then, we obtain
λ = −17(dΩ , ∗Ω) = −1
7∗ (dΩ ∧ Ω) . (2.104)
Acting the Hodge dual operator in (2.98), we have
T 327 = ∗dΩ + λΩ − 3
4∗ (Θ ∧ Ω) . (2.105)
From the component ∧27, we obtain the relation
ιΘ∗Ω = ∗(Θ ∧ ∗Ω) . (2.106)
25
Consequently, on the G2-manifold of type W1 ⊕ W3 ⊕ W4, there exist unique affine connectionwith 3-form torsion, which is given by
T =16∗ (dΩ ∧ Ω)Ω − ∗dΩ + ∗(Θ ∧ Ω) (2.107)
and the associate G2-structure classified by
δΩ = − ∗ (Θ ∧ ∗Ω) , dΩ =17∗ (dΩ ∧ Ω) ∗ Ω + ∗T 3
27 +34(Θ ∧ Ω) . (2.108)
According to Refs. [30, 32], the classification of G2-structures is the following:
Name type conditionsParallel G2 0 dΩ = 0 , d ∗ Ω = 0
Nearly parallel G2 W1 dΩ = −λ ∗ Ω and d ∗ Ω = 0Almost parallel G2 W2 dΩ = 0
Balanced G2 W3 δΩ = 0 and λ = 0locally conformally parallel G2 W4 dΩ = −λ ∗ Ω + 3
4Θ ∧ Ωand d ∗ Ω = Θ ∧ ∗Ω
W1 ⊕W2 dΩ = −λ ∗ Ωcocalibrated G2 W1 ⊕W3 δΩ = 0
Locally conformally nearly G2 W1 ⊕W4 dΩ = −λ ∗ Ω and d ∗ Ω = Θ ∧ ∗ΩW2 ⊕W3 λ = 0 and Θ = 0
Locally conformally parallel G2 W2 ⊕W4 dΩ = 34Θ ∧ Ω
W3 ⊕W4 λ = 0 and d ∗ Ω = Θ ∧ ΩW1 ⊕W2 ⊕W3 Θ = 0W1 ⊕W2 ⊕W4 dΩ = −λ ∗ Ω + 3
4Θ ∧ ΩG2T W1 ⊕W3 ⊕W4 d ∗ Ω = Θ ∧ ∗Ω
W2 ⊕W3 ⊕W4 λ = 0W1 ⊕W2 ⊕W3 ⊕W4 no relations
Θ denotes the Lee form, which is defined by
Θ = −13∗ (∗dΩ ∧ Ω) . (2.109)
Let us rewrite the Lee form. The integration of ∗dΩ ∧ Ω on M7 is given by∫M7
∗dΩ ∧ Ω =∫
M7
∗d(∗ ∗ Ω) ∧ ∗ ∗ Ω =∫
M7
∗d ∗ (∗Ω) ∧ ∗(∗Ω)
=∫
M7
δ ∗ Ω ∧ ∗(∗Ω)
=∫
M7
∗Ω ∧ δ ∗ (∗Ω) = −∫
M7
∗Ω ∧ δΩ
=∫
M7
δΩ ∧ ∗Ω ,
26
where from the line 2 to the line 3 we use∫M7
δ(∗Ω ∧ Ω) = 7∫
M7
δVol.(M7) =∫
M7
∗d 1 = 0 . (2.110)
Thus we obtainΘ =
13∗ (δΩ ∧ ∗Ω) . (2.111)
If the Lee form Θ vanishes, i.e. δΩ = 0, and T 327 = 0 in the G2-structure of the class W1⊕W3⊕W4,
then the G2-structure reduces to the subclass W1. The G2-manifold of type W1 is known as anearly parallel G2-manifold. If the Lee form Θ vanishes and λ = 0 in class W1 ⊕ W3 ⊕ W4,the G2-structure reduce to the subclass W3, which is said to be a balanced G2-structure. Ifthe component T 3
27 is zero in the G2-manifold of type W1 ⊕W3 ⊕W4, the G2-manifold has theG2-structure of type W4. The G2-manifold of type W4 is called as locally conformally parallelG2-manifold. The condition Θ = 0 in the G2-manifold of class W1 ⊕W3 ⊕W4 gives the subclassW1⊕W3 and the G2-manifold of type W1⊕W3 is known as cocalibrated G2-manifold. When thecomponent T 3
27 vanishes and Θ∧Ω = 0 in type W1 ⊕W3 ⊕W4, we obtain the subclass W1 ⊕W4.The G2-manifold of type W1 ⊕W4 is called as locally conformally nearly G2-manifold. We willinvestigate the class W3⊕W4 because the class is associated with type II or heterotic supergravitytheory.
We consider a seven dimensional real oriented manifold admitting the type W3 ⊕ W4 G2-structure. The G2-structure is defined by the following equations [9, 24]:
d ∗ Ω − Θ ∧ ∗Ω = 0 , (2.112)dΩ ∧ Ω = 0 . (2.113)
Also, there exist an unique connection ∇T such that ∇T Ω = 0 [24] and the torsion is given by
dΩ − Θ ∧ Ω = − ∗ T . (2.114)
By the way, the G2-structure associated with type II supergravity theory is defined by
e2ϕd(e−2ϕ ∗ Ω) = 0 , (2.115)dΩ ∧ Ω = 0 , (2.116)H = −e2ϕ ∗ d(e−2ϕΩ) , (2.117)
where the H denotes the 3-form flux which is closed form and Φ denotes dilaton. If we identifythe flux H the 3-form with the torsion T in (2.114) with dT = 0 and also dilaton ϕ with theLee form Θ by 2dϕ = Θ, (2.112), (2.113), and (2.114) are equivalent to (2.115), (2.116), and(2.117). The equations (2.115), (2.116), and (2.117) are equivalent to supersymmetry preservingconditions of the theory [24, 33].
27
2.4 Spin(7)-structures
A Spin(7)-structure Q8 is a principal subbundle of a frame bundle L(M8) over an eight-dimensional real manifold M8. A Spin(7)-structure is determined by the choice of a fundamental4-form Ψ, which is satisfies the equation
∗Ψ = Ψ . (2.118)
The subgroup Spin(7) is defined by a stabilizer subgroup of Ψ. Then, the structure Q8 has astructure group Spin(7) and the group is a reduced subgroup of SO(8), which is a structuregroup L(M8). We decompose the Lie algebra so(8) into spin(7) and the orthogonal complementm, so(8) = spin(7) ⊕ m. A defect of the holonomy group by contracting from SO(8) to Spin(7)is measured by an intrinsic torsion T , which is identified with ∇Ψ, where ∇ denotes Levi-Civitaconnection. The torsion T belongs to R8 ⊗ m at local and the space of the intrinsic torsion isdecomposed into two irreducible components by irreducible representations of Spin(7),
R8 ⊗ m = W1 ⊕W2, (2.119)
where dimW1 = 48 and dimW2 = 8 [18] (see also Ref. [30]). Since T belongs to R8 ⊗m, we have
∇Ψ = ϱ(T )Ψ ∈ W1 ⊕W2 . (2.120)
According to Ref. [30], the classification of Spin(7) is the following:
Name type conditionsSpin(7) 0 dΨ = 0
W1 Θ = 0Locally conformally Spin(7) W2 dΨ = Θ ∧ Ψ
W1 ⊕W2 no relations
Here Θ is a Lee form and this is defined by
Θ = −17∗ (∗dΨ ∧ Ψ) . (2.121)
We will study the class W1 ⊕W2 because the class is associated with type II or heterotic super-gravity theory.
We consider an eight-dimensional real manifold admitting the type W1⊕W2 Spin(7)-structure.The Spin(7)-structure is defined by the condition that the Lee form should be an exact form andthe dilaton in the supergravity theory is identified with the Lee form,
2dϕ =76Θ . (2.122)
The Spin(7)-structure admits the existence of an unique connection ∇T such that ∇T Ψ = 0 andthe torsion is given by[9]
dΨ − 76Θ ∧ Ψ = − ∗ T . (2.123)
Then, the geometry preserves a single Killing spinor which satisfies supersymmetric equations inthe theory.
28
Chapter 3
Heterotic solutions with G2 andSpin(7) structures
By solving the defining equations for the G2-structure in the class W3 ⊕W4, supersymmetricsolutions of the form R1,2 × M7 in type II supergravity theory were constructed[4], where theflux H and the dilaton Φ were identified with 3-form torsion and Lee form, respectively. Similarto these solutions, we will solve the defining equations of the G2 or Spin(7)-structure in theassociated class with Abelian heterotic supergravity theory to obtain supersymmetric solutions.Unlike type II supergravity theory, Abelian heterotic supergravity theory additionally has a U(1)gauge field. However, the field strength of the U(1) gauge field is determined algebraically. Thuswe will use a similar method to derive the supersymmetric solutions in Abelian heterotic theory.This chapter is based on Ref. [2].
3.1 G2-structure associated with supergravity
In this section, we review a G2-structure related with 7-dimensional Abelian heterotic super-gravity theory. We first explain Abelian heterotic supergravity theory shortly. Next, we introducethe G2-structure and derive the fundamental equations describing supersymmetric equations.
3.1.1 Abelian heterotic supergravity
According to Ref. [10], Abelian heterotic supergravity theory is a closely related to the lowenergy effective theory with the α′ expansion of heterotic string theory, i.e., heterotic supergravitytheory, which is obtained by formal some operations [10]. This system consists of a metric g,dilaton Φ, U(1) gauge field A and B field. Then the string frame Lagrangian is given by
L = e−Φ
(R ∗ 1 + ∗dΦ ∧ dΦ − ∗F ∧ F − 1
2∗ H ∧ H
), (3.1)
where R represents a scalar curvature and F = dA. The 3-form flux H is given by H = dB +A ∧ dA, which satisfies the Bianchi identity
dH = F ∧ F. (3.2)
29
In addition, ∗ represents the Hodge dual operator associated with the metric g. The equationsof motion are written as
Rµν = −∇µ∇νΦ + FµρFνρ +
14Hµ
ρσHνρσ , (3.3)
∇2e−Φ =12e−ΦFµνF
µν +16e−ΦHµνρH
µνρ , (3.4)
d(e−Φ ∗ F ) = −e−Φ ∗ H ∧ F , (3.5)d(e−Φ ∗ H) = 0 (3.6)
and the corresponding supersymmetric equations are
∇Tµχ ≡
(∇µ +
14 · 2!
Hµνργνρ
)χ = 0 , (3.7)
(∂µΦ)γµχ +13!
Hµνργµνρχ = 0 , (3.8)
Fµνγµνχ = 0 , (3.9)
where χ denotes a Killing spinor. These equations (3.7), (3.8) and (3.9) are derived from thesupersymmetry variation of gravitino, dilatino and gaugino, respectively.
It is known that if (g,Φ,H, F ) satisfies the supersymmetric equations and the Bianchi identity,the quartet automatically solves the equations of motion [4, 23].In this chapter, we shall constructsupersymmetric solutions in the Abelian heterotic theory. For this purpose, we consider 7- and8-dimensional manifolds Mn admitting G2 and Spin(7) structures 1. The corresponding 10-dimensional spacetimes are assumed to be of the form R1,9−n × Mn (n = 7, 8) where the fieldsare non-trivial only on Mn.
3.1.2 G2-structure
A G2-structure over M7 is a principal subbundle with fiber G2 of the frame bundle F(M7) [6].There is a one to one correspondence between G2 structures and G2 invariant 3-forms Ω. Thestandard form of Ω is given by
Ω = e123 + e147 + e165 + e246 + e257 + e354 + e367 , (3.10)
where eµ (µ = 1, · · · , 7) is an orthonormal frame of the metric, g =∑7
µ=1 eµ ⊗ eµ and eµνρ =eµ ∧ eν ∧ eρ.
We consider the G2-structure Ω associated with heterotic supergravity. According to [24], werequire
d ∗ Ω = Θ ∧ ∗Ω , (3.11)
1For 6-dimensional case, supersymmetric solutions of Abelian heterotic theory were constructed in the view ofsupergravity solution generating method [10].
30
where Θ is the Lee form defined by
Θ = −13∗ (∗dΩ ∧ Ω) . (3.12)
Then we have a unique connection ∇T preserving the G2-structure with 3-form torsion[23, 24, 34]
T =16∗ (dΩ ∧ Ω)Ω − ∗ (dΩ − Θ ∧ Ω) , (3.13)
i.e., ∇T Ω = ∇T g = 0. This structure belongs to the class W1 ⊕W3 ⊕W4 in the classification byFernandez and Gray [13](see also [30]). It was shown in the papers [24], [34] that there exists anon-trivial solution to both dilatino and gravitino Killing spinor equations in dimension 7 if andonly if there exists a G2-structure satisfying (3.11) and
dΩ ∧ Ω = 0 (3.14)
together with an exact Lee form Θ. Here (3.14) yields that this G2-structure is in the classW3 ⊕ W4. If Θ = 0 and T = 0 one can find dΩ = 0 and d ∗ Ω = 0, which are equivalent to∇Ω = 0 and the corresponding metric is a Ricci-flat metric with G2 holonomy. Now, we have thefollowing identification
H = T , (3.15)dΦ = Θ . (3.16)
Thus we obtain
H = −eΦ ∗ d(e−ΦΩ) , dΦ = −13∗ (∗dΩ ∧ Ω) , (3.17)
which means vanishing of supersymmetry variations for gravitino and dilatino, respectively. Inaddition, the supersymmetry variation for gaugino requires the generalized self-dual condition onthe field strength F ,
∗(Ω ∧ F ) = F . (3.18)
In general, 2-forms on M7 are decomposed into G2 irreducible representations under action ofG2,
∧27 = α ∈ ∧2 : ∗(Ω ∧ α) = −2α , (3.19)
∧214 = α ∈ ∧2 : ∗(Ω ∧ α) = α , (3.20)
where subscripts 7, 14 of ∧2 are dimensions of the representation. Thus the field strength F in(3.18) is in ∧2
14. The Bianchi identity (3.2) gives a relation between T and F , which leads to astrong restriction for the field strength F .
In summary, supersymmetric equations associated with Abelian heterotic supergravity theory
31
are given by
eΦd(e−Φ ∗ Ω) = 0 , (3.21)dΩ ∧ Ω = 0 , (3.22)H = −eΦ ∗ d(e−ΦΩ) , (3.23)
dΦ = −13∗ (∗dΩ ∧ Ω) , (3.24)
∗(Ω ∧ F ) = F (3.25)
together with the Bianchi identitydH = F ∧ F . (3.26)
The first equation is derived from (3.11) and (3.16) easily. In the following we call these equationsG2 with torsion equations or simply G2T equations. Thus it is enough to solve G2T equations soas to construct supersymmetric solutions.
3.2 G2 solutions in 7-dimensional Abelian heterotic supergravity
In this section, we study G2T equations under cohomogeneity one ansatz. Then these equationsare reduced to first order non-linear ordinary differential equations. We solve reduced equationsnumerically and also try to construct analytically solutions for these equations.
3.2.1 Cohomogeneity one ansatz
The G2T equations can be reduced to ordinary differential equations under cohomogeneity oneansatz. A G-manifold is called cohomogeneity one manifold if the principal orbits are hypersur-faces. It is known that cohomogeneity one manifolds admitting G2-structures are classified intoseven types [27]. In this paper we will consider the case when the principal orbits are S3 × S3.Then the manifold is locally R+ × S3 × S3. Similar analysis can be done for the other orbits.
We consider the following metric with six radial functions ai(t), bi(t) (i = 1, 2, 3)
g = dt2 +3∑
i=1
ai(t)2(σi − Σi)2 +3∑
i=1
bi(t)2(σi + Σi)2 , (3.27)
where σi and Σi are left invariant 1-forms on two SU(2) group manifolds which satisfy the relations
dσ1 = −σ2 ∧ σ3 , dσ2 = −σ3 ∧ σ1 , dσ3 = −σ1 ∧ σ2 ,
dΣ1 = −Σ2 ∧ Σ3 , dΣ2 = −Σ3 ∧ Σ1 , dΣ3 = −Σ1 ∧ Σ2 . (3.28)
It is convenient to introduce an orthonormal frame eµ (µ = 1, · · · , 7) defined by
e1 = a1(t)(σ1 − Σ1) , e2 = a2(t)(σ2 − Σ2) , e3 = a3(t)(σ3 − Σ3) ,
e4 = b1(t)(σ1 + Σ1) , e5 = b2(t)(σ2 + Σ2) , e6 = b3(t)(σ3 + Σ3) , e7 = dt . (3.29)
32
Then the G2-structure Ω takes the form (3.10), which automatically satisfy (3.22). The dilatonΦ is explicitly calculated from (3.24),
dΦdt
= −13
(− 2
a1
da1
dt− 2
a2
da2
dt− 2
a3
da3
dt− 2
b1
db1
dt− 2
b2
db2
dt− 2
b3
db3
dt
+a1
2a2b3+
a1
2a3b2+
a2
2a3b1+
a2
2a1b3+
a3
2a1b2+
a3
2a2b1
+b1
2a2a3− b1
2b2b3+
b2
2a1a3− b2
2b1b3+
b3
2a1a2− b3
2b1b2
). (3.30)
The 3-form flux H = H126e126 + H456e
456 + H135e135 + H234e
234 is also calculated from (3.23),
H126 =1a3
da3
dt+
1b1
db1
dt+
1b2
db2
dt− a1
2a3b2− a2
2a3b1+
b3
2b1b2− dΦ
dt,
H456 = − 1a1
da1
dt− 1
a2
da2
dt− 1
a3
da3
dt+
b1
2a2a3+
b2
2a1a3+
b3
2a1a2+
dΦdt
,
H135 = − 1a2
da2
dt− 1
b1
db1
dt− 1
b3
db3
dt+
a1
2a2b3− b2
2b1b3+
a3
2a2b1+
dΦdt
,
H234 =1a1
da1
dt+
1b2
db2
dt+
1b3
db3
dt+
b1
2b2b3− a2
2a1b3− a3
2a1b2− dΦ
dt, (3.31)
which clearly depend on only t.
One may think that the ansatz (3.27) is artificial. However the ansatz (3.27) is reasonablefrom the view of a half-flat structure. A G2 structure and 7-dimensional metric on R+ ×S3 ×S3
are given by a one-parameter family of a half-flat structure on S3 × S3 [35]. A half-flat structureis an SU(3)-structure, which is characterized by a pair (κ, γ), where κ is a symplectic form onS3×S3 and γ is a real 3-form whose stabiliser is isomorphic to SL(3;C) and satisfies a conditionγ ∧ κ = 0. If (κ, γ) satisfies the defining equation
d(κ ∧ κ) = 0 , dγ = 0, (3.32)
then the SU(3)-structure is called a half-flat structure[25]. Indeed the following half-flat pair(κ, γ),
κ = p1σ1 ∧ Σ1 + p2σ2 ∧ Σ2 + p3σ3 ∧ Σ3 , (3.33)γ = q1σ1 ∧ σ2 ∧ σ3 + q2Σ1 ∧ Σ2 ∧ Σ3 + q3(σ2 ∧ σ3 ∧ Σ1 − σ1 ∧ Σ2 ∧ Σ3)
+q4(σ3 ∧ σ1 ∧ Σ1 − σ2 ∧ Σ3 ∧ Σ1) + q5(σ1 ∧ σ2 ∧ Σ3 − σ3 ∧ Σ1 ∧ Σ2) (3.34)
leads to the six dimensional part of the metric (3.27) under certain parameters pi and qj .
The field strength F is restricted by (3.25) and (3.26). Actually, we find F is classified intoseven types,
F12e12 + F45e
45 + F67e67 , F13e
13 + F46e46 + F57e
57 , F23e23 + F47e
47 + F56e56
F14e14 + F25e
25 + F36e36 , F15e
15 + F24e24 + F37e
37 , F16e16 + F27e
27 + F34e34 ,
F17e17 + F26e
26 + F35e35 , (3.35)
33
together with generalized self-dual relations
F13 − F46 = F57 , F56 − F23 = F47 , F45 − F12 = F67 , F36 − F14 = F25
F24 − F15 = F37 , F16 − F27 = F34 , F35 − F26 = F17 . (3.36)
Let us restrict ourselves to the case
F = F12e12 + F45e
45 + F67e67 . (3.37)
Without loss of generality this case can be selected because other cases are obtained by usingdiscrete symmetries of G2T equations 2. For examaple, G2T equations are invariant under thefollowing transformation,
F12 −→ F23 , F45 −→ F56 , F67 −→ F47 ,
a1 −→ a2 , a2 −→ a3 , a3 −→ a1 , b1 −→ b2 , b2 −→ b3 , b3 −→ b1 ,
H126 −→ H234 , H135 −→ −H126 , H456 −→ H456 , H234 −→ −H135 .
(3.38)
We have already written down (3.22), (3.23), (3.24) and (3.25) under the cohomogeneity oneansatz. The remaining G2T equation we should solve are (3.21) and (3.26). The former is givenby
1a1
da1
dt+
1a2
da2
dt+
1b1
db1
dt+
1b2
db2
dt− b3
2a1a2− a3
2a1b2− a3
2a2b1+
b3
2b1b2− dΦ
dt= 0 ,
1a1
da1
dt+
1a3
da3
dt+
1b1
db1
dt+
1b3
db3
dt− b2
2a1a3− a2
2a1b3+
b2
2b1b3− a2
2a3b1− dΦ
dt= 0 ,
1a2
da2
dt+
1a3
da3
dt+
1b2
db2
dt+
1b3
db3
dt− b1
2a2a3+
b1
2b2b3− a1
2a2b3− a1
2a3b2− dΦ
dt= 0 ,
(3.39)
and the latter is given by
dH456
dt+ H456
(1b1
db1
dt+
1b2
db2
dt+
1b3
db3
dt
)= −2F45F67 ,
dH126
dt+ H126
(1a1
da1
dt+
1a2
da2
dt+
1b3
db3
dt
)= −2F12F67 ,
dH135
dt+ H135
(1a1
da1
dt+
1a3
da3
dt+
1b2
db2
dt
)= 0 ,
dH234
dt+ H234
(1a2
da2
dt+
1a3
da3
dt+
1b1
db1
dt
)= 0 , (3.40)
2The field strength F = F14e14 +F25e
25 +F36e36 is exceptional. In this case, G2T equations lead to the solution
F = 0, which was studied in the paper[4].
34
H126a1
2a3b2− H456
b1
2a2a3− H135
a1
2a2b3− H234
b1
2b2b3= 0 ,
H126a2
2a3b1− H456
b2
2a1a3+ H135
b2
2b1b3+ H234
a2
2a1b3= 0 ,
H126b3
2b1b2+ H456
b3
2a1a2+ H135
a3
2a2b1− H234
a3
2a1b2= −2F12F45 ,
which consist of differential equations (3.40) and algebraic equations (3.41). The algebraic equa-tions (3.41) yields the following relations,
H456 =a1a2
b1b2H126 , H234 = −a1b2
a2b1H135 ,
F12 = ±
√−1
2
(b3
a1a2H126 +
a3b2
a1a22H135
),
F45 =a1a2
b1b2F12 , F67 = F45 − F12 (3.41)
and ai, bi, H126 and H135 are determined by the differential equations (3.31), (3.39) and (3.40) :
da1
dt=
−a12 + a2
2 + b32
4a2b3+
−a12 + a3
2 + b22
4a3b2− 1
2a1H126 +
12a1H135 , (3.42)
da2
dt=
−a22 + a1
2 + b32
4a1b3+
−a22 + a3
2 + b12
4a3b1− 1
2a2H126 −
12a2H234 , (3.43)
da3
dt=
−a32 + a2
2 + b12
4a2b1+
−a32 + a1
2 + b22
4a1b2+
12a3H135 −
12a3H234 , (3.44)
db1
dt=
−b12 + a2
2 + a32
4a2a3+
b12 − b2
2 − b32
4b2b3+
12b1H456 −
12b1H234 , (3.45)
db2
dt=
−b22 + a1
2 + a32
4a3a1+
b22 − b1
2 − b32
4b3b1+
12b2H456 +
12b2H135 , (3.46)
db3
dt=
−b32 + a1
2 + a22
4a1a2+
b32 − b1
2 − b22
4b1b2− 1
2b3H126 +
12b3H456 , (3.47)
dH126
dt+ H126
(1a1
da1
dt+
1a2
da2
dt+
1b3
db3
dt
)=
(a1a2
b1b2− 1
)(b3
a1a2H126 +
a3b2
a1a22H135
),
(3.48)dH135
dt+ H135
(1a1
da1
dt+
1a3
da3
dt+
1b2
db2
dt
)= 0 . (3.49)
Thus we have obtained G2T equations (3.42)–(3.49) under the cohomogeneity one ansatz withprincipal orbits S3 × S3. It should be noticed that dilaton is given by (3.30), which is equivalentto
dΦdt
= −H126 + H456 + H135 − H234 . (3.50)
Note that it is automatically assured by (3.41)–(3.49) that the field strength F is closed two-form,dF = 0.
35
3.2.2 G2T solutions
In order to construct regular metrics satisfying (3.42)–(3.49), we shall look for solutions withbolt singularities at t = 0. There exist two types of bolt singularity corresponding to a collapsingS3 or S1 at t = 0. In the torsion free case, i.e. Hµνρ = 0, these equations were studied byCvetic, Gibbons, Lu and Pope [16][17] and they found Ricci-flat metrics with G2 holonomy.These equations are also obtained from dΩ = 0 and d ∗ Ω = 0. Let us turn to non-zero flux.Although (3.42)–(3.49) admit a Taylor expansion around t = 0, numerical analysis shows thatglobal solution doesn’t exist for general six radial functions ai(t), bi(t). Hence we study thereduced case,
a1(t) = a2(t) ≡ a(t) , b1(t) = b2(t) ≡ b(t). (3.51)
Note that the Taylor expansion around t = 0 gives rise to H135 = 0.
By (3.51) and H135 = 0 the reduced system is given by
1a
da
dt=
b
4aa3+
a3
4ab− a
4a3b+
b3
4a2− 1
2H126 , (3.52)
1b
db
dt= − b
4aa3+
a3
4ab+
a
4a3b− b3
4b2+
a2
2b2H126 , (3.53)
1a3
da3
dt=
b
2aa3− a3
2ab+
a
2a3b, (3.54)
1b3
db3
dt= − b3
4a2+
b3
4b2+
12
(a2
b2− 1
)H126 , (3.55)
dH126
dt= −H126
[b2 + a2
3 − a2
2aa3b+
b3(5b2 − 3a2)4a2b2
]− 1
2
(a2
b2− 3
)H2
126 ,
(3.56)dΦdt
= H126
(a2
b2− 1
). (3.57)
Then, the algebraic equations (3.41) are written as
H456 =a2
b2H126 ,
F12 = ±
√−1
2
(b3
a2H126
),
F45 =a2
b2F12 ,
F67 = F45 − F12 . (3.58)
Now let us solve the equations (3.52)–(3.57). We put the form
b3(t) =1
h(t)β3(t) , (3.59)
36
where the radial function β3(t) is defined as the solution to the ensuing equations,
1a
da
dt=
b
4aa3+
a3
4ab− a
4a3b+
β3
4a2, (3.60)
1b
db
dt= − b
4aa3+
a3
4ab+
a
4a3b− β3
4b2, (3.61)
1a3
da3
dt=
b
2aa3− a3
2ab+
a
2a3b, (3.62)
1β3
dβ3
dt= − β3
4a2+
β3
4b2, (3.63)
which are obtained by setting H126 = 0 in (3.52)–(3.57). The function h(t) is determined asfollows. From (3.52)–(3.55), (3.59)–(3.63), we find the relation
1h
dh
dt= −dΦ
dt, (3.64)
which yields1β3
dβ3
dt=
12(h − 1)
dh
dt. (3.65)
Thus we haveh(t) = kβ3(t)2 + 1 . (3.66)
Then the relation (3.59) is given by
b3(t) =1
kβ3(t)2 + 1β3(t) (3.67)
and using (3.57), (3.64) and (3.66) leads to
H126 = − kβ33
2a2(kβ23 + 1)
. (3.68)
For the positive constant k, the algebraic equations (3.58) gives H456 and Fµν ,
H456 = − kβ33
2b2(kβ23 + 1)
, F12 = ±√
kβ32
2a2(kβ32 + 1)
,
F45 = ±√
kβ32
2b2(kβ32 + 1)
, F67 = ±√
kβ32
2(kβ32 + 1)
(− 1
a2+
1b2
). (3.69)
Thus, under the ansatz (3.59), solutions to (3.52)–(3.57) and (3.58) are written as
g = dt2 + a(t)2(σ1 − Σ1)2 + (σ2 − Σ2)2 + a3(t)2(σ3 − Σ3)2
+b(t)2(σ1 + Σ1)2 + (σ2 + Σ2)2 +1
(kβ3(t) 2 + 1)2β3(t)2(σ3 + Σ3)2 , (3.70)
Φ(t) = logm
kβ3(t)2 + 1, (3.71)
37
H = − kβ3(t)3
2a(t)2(kβ3(t) 2 + 1)e126 − kβ3(t)3
2b(t)2(kβ3(t) 2 + 1)e456 , (3.72)
F =
√kβ3
2
2a2(kβ32 + 1)
e12 +
√kβ3
2
2b2(kβ32 + 1)
e45 +
√kβ3
2
2(kβ32 + 1)
(− 1
a2+
1b2
)e67 , (3.73)
where m is an arbitrary constant 3. This formula includes additional one parameter m comparingto the formula obtained by r−1Tr transformation[10]. Now the problem of finding the solutions toG2T equations reduced to that of finding the solutions to the torsion free equations(3.60)–(3.63).It should be noticed that the torsion free equations were investigated in the study of Ricci-flatmetrics with G2 holonomy [16]. It was proved[36] that there exist the solutions with a collapsingS1 for the torsion free equations (3.60)–(3.63).
A regular solution to (3.60)–(3.63) representing a Ricci-flat metric with G2 holonomy is known[5],
a3(r) = −12r , a(r) =
14
√3(r − l)(r + 3l) ,
β3(r) = l
√r2 − 9l2
r2 − l2, b(r) = −1
4
√3(r + l)(r − 3l) , (3.74)
where r is defined by dt = 32
lβ3
dr and l is a scaling parameter (l > 0). The corresponding solutionsof G2T equations are given by
g =94
r2 − l2
r2 − 9l2dr2 +
316
(r − l)(r + 3l)(σ1 − Σ1)2 + (σ2 − Σ2)2 +14r2(σ3 − Σ3)2
+316
(r + l)(r − 3l)(σ1 + Σ1)2 + (σ2 + Σ2)2 +l2(r2 − 9l2)
kl2(r2 − 9l2) + r2 − l2(σ3 + Σ3)2 ,
H = − 8k
3(1 + k r2−9l2
r2−l2
)√r2 − 9l2
r2 − l2
[r − 3l
3(r − l)2(r + l)e126 +
r + 3l(r + l)2(r − l)
e456
],
F = − 8√
k
3(1 + k r2−9l2
r2−l2
) [r − 3l
3(r − l)2(r + l)e12 +
r + 3l(r + l)2(r − l)
e45 +2(r2 − 3l2)(r2 − l2)2
e67
],
Φ = −m log[1 + k
r2 − 9l2
r2 − l2
]. (3.75)
These solutions include two free parameters l and k 4. When k = 0, the solution reduces to theRicci-flat metric with G2 holonomy. The parameter r is larger than 3l by a regular condition andthe other parameter k represents deformation from the Ricci-flat metric.
The metric (3.75) is an ALC(Asymptotically Locally Conical) metric and the Riemannian3If we set k = tan2δ, m = 1
cos2δand hence m = k + 1, then these solutions are coincident with the solutions
obtained by r−1Tr transformation in the paper [10]. When we take a limit k → ∞, m → ∞ keeping km
→ 1, thenδ is equal to π
2, which corresponds to T-dual transformation.
4For m = k + 1, the equation (3.75) is also obtained by solution-generating method [10].
38
tensor is non-singular in the region 3l ≤ r < ∞. Further the scalar curvature R, F 2 = 12FµνF
µν
and H2 = 16HµνρH
µνρ are finite but their integrations diverge. The cause of the divergence comesfrom higher powers of r in the volume form,
√g ∼ r5.
S3-bolt solution
From now on, we explain the reason why we can put the relation (3.59). The boundary conditionrepresenting a collapsing S3 at t = 0 imposes b(t), b3(t) → 0 for t → 0. Indeed, the series solutionaround t = 0 takes the following form,
a(t) = p1 +1
16p21
t2 − 7 − 64p21p2
2560p31
t4 + · · · ,
a3(t) = −p1 −1
16p21
t2 +6 + 128p2
1p2
2560p31
t4 + · · · ,
b(t) = −14t + p2t
3 − 1 + 1344p21p2 − 98304p4
1p22
10240p41
t5 + · · · ,
b3(t) =14t +
128p21p2 + 128p4
1p3 − 164p2
1
t3
−216p21p2 − 16896p4
1p22 + 240p4
1p3 − 30720p61p2p3 − 10240p8
1p23 − 1
640p41
t5 + · · · ,
H126(t) = p3t3 +
(8p2
1p23 + 24p2p3 −
5p3
16p21
)t5 + · · · , (3.76)
where p1, p2 and p3 are free parameters. Then the metric behaves as
g → dt2 + p21[(σ1 − Σ1)2 + (σ2 − Σ2)2 + (σ3 − Σ3)2]
+t2
16[(σ1 + Σ1)2 + (σ2 + Σ2)2 + (σ3 + Σ3)2] . (3.77)
Thus the metric has a removable bolt singularity (S3-bolt solution). The parameter p1 is ascaling parameter and p1 = 3
2 l for the solutions (3.75), while p2 parametrizes a family of theregular solutions to (3.60)–(3.63). The remaining parameter p3 is essentially the deformationparameter k associated with 3-form flux (3.68),
p3 = − k
128p21
. (3.78)
Actually when p3 = 0, the flux component H126(t) vanishes and the corresponding solutionreduces to Ricci-flat metrics with a collapsing S3 at t = 0 [16]. The radial functions a(t), b(t)and a3(t) are independent on p3, which means that these functions are the same as those of theRicci-flat metrics. On the other hand, b3(t) depends on p3 so that this is different from β3(t)satisfying (3.60)–(3.63). Namely b3(t) is only deformed by the 3-form flux and hence we can takethe form (3.59).
39
It should be noticed that we can obtain all S3-bolt solutions of (3.52)–(3.57) from regularsolutions of (3.60)–(3.63). In particular, the parameters p1 = 3
2 l and p2 = − 1768p2
1correspond
to the solutions (3.75). The numerical calculation requires the inequality of the parameter p2
for each values of deformation parameter p3 in order to extend the solution in large t region.For example when p1 = 1 and p3 = −900, the parameter p2 is restricted by the inequality−7.029 × 103 ≤ p2 ≤ 1.321 × 10−3 (see Figure 3.1).
100 200 300 400
-150
-100
-50
50
100
100 200 300 400
-300
-200
-100
Figure 3.1: The both figures show S3-bolt solutions. The left figure is the solution with p1 = 1,−1.321×10−3 ≤ p2 ≤ 1.321×10−3 and p3 = −900 and the right figure is the solution with p1 = 1,−7.029 × 103 ≤ p2 ≤ −6.852 × 103 and p3 = −900. The red, green, blue, orange and magentalines represent components a(t), a3(t), b(t), b3 and H126(t), respectively.
T 1,1-bolt solution
Regular solutions with a collapsing S1 takes the following form around t = 0,
a(t) = p1 +p2 − 2p2
1p3
8p1t − 3p2
2 − 16p21 − 12p2
1p2p3 + 12p41p
23
128p31
t2 + · · · ,
a3(t) = t − 16p21 − p2
2 + 4p21p2p3 − 4p4
1p23
96p41
t3 + · · · ,
b(t) = p1 −p2 − 2p2
1p3
8p1t − 3p2
2 − 16p21 − 12p2
1p2p3 + 12p41p
23
128p31
t2 + · · · ,
b3(t) = p2 +4p3
2 − 16p2p23
64p41
t2 + · · · ,
H126(t) = p3 +2p2
1p23 − p2p3
4p21
t − −9p22p3 + 8p2
1p3 + 28p21p2p
23 − 20p4
1p33
32p41
t2 + · · · , (3.79)
40
where p1, p2 and p3 are free parameters. Then the metric behaves as
g −→ dt2 + t2(σ3 − Σ3)2 + p21(σ
21 + Σ2
1 + σ22 + Σ2
2)+p2
2(σ3 + Σ3)2 . (3.80)
When this metric has T 1,1-bolt, p1 and p2 have relation ([17])
p2 = ±√
23p1 . (3.81)
In the series (3.79), all components are deformed by the 3-form flux, which implies that therelation like (3.59) doesn’t exist in the solution. A numerical analysis indicates that the seriessolution (3.79) is extended to large t region (see Figure 3.2) 5.
200 400 600 800 1000
-300
-200
-100
100
200
300
200 400 600 800 1000
-200
200
400
600
Figure 3.2: The left figure shows a T 1,1-bolt solution with p1 =√
32p2, 0 < p2 < 12 and p3 = − 1
288 .
The right figure shows a T 1,1-bolt solution with p1 =√
32p2, −12 < p2 < 0 and p3 = 1
288 . Thered, green, blue, orange and magenta lines represent components a(t), a3(t), b(t), b3 and H126(t),respectively.
3.3 Spin(7) solutions in 8-dimensional Abelian heterotic super-gravity
3.3.1 Spin(7)-structure
A Spin(7)-structure over M8 is a principal subbundle with fiber Spin(7) of the frame bundleF(M8) [6]. There is a one to one correspondence between Spin(7) structures and Spin(7) invariant4-forms Ψ. The standard form of Ψ is given by
Ψ = e8123 + e8145 + e8167 + e8264 + e8257 + e8347 + e8356
−e4567 − e2367 − e2345 − e3157 − e1346 − e1256 − e1247, (3.82)5In the Ricci flat case, the existence of G2 metrics with a collapsing S1 was proved in [36].
41
where eµ, µ = 1, · · · , 8 is an orthonormal frame of the metric g =∑8
µ=1 eµ ⊗ eµ. The 4-form Ψis self-dual ∗Ψ = Ψ for a volume form vol = e12···8. The Lee form is defined by
Θ = −17∗ (∗dΨ ∧ Ψ). (3.83)
Then, there always exists a unique connection ∇T preserving the Spin(7) structure, ∇T Ψ =∇T g = 0, with 3-form torsion [37]
T = ∗dΨ − 76∗ (Θ ∧ Ψ). (3.84)
This is different from the case of G2, where we required an additional condition (3.11). However,except for this point we have very similar supersymmetry equations associated with Abelian het-erotic supergravity. The Lee form is an exact 1-form, Θ = (6/7)dΦ, and hence after identificationT = H the equations are written as
H = eΦ ∗ d(e−ΦΨ) , (3.85)
dΦ = −16∗ (∗dΨ ∧ Ψ) , (3.86)
∗(Ψ ∧ F ) = F , (3.87)dH = F ∧ F , (3.88)
which correspond to G2T equations. As with G2, we call these equations Spin(7) with torsionequations.
3.3.2 3-Sasakian ansatz
In order to construct explicit solutions of Spin(7) with torsion equations, we will consider a spe-cial case that gives rise to ordinary differential equations and generalizes many known examples.Let us assume the following geometrical condition for 7-dimensional manifolds. Let π : P → B bea principal SO(3)-bundle over a compact self-dual Einstein manifold (or orbifold) B. Then, it isknown that the total space P admits a 3-Sasakian structure. For details on 3-Sasakian structurewe refer the reader to [38]. The connection 1-forms φi (i = 1, 2, 3) will be chosen so that thecorresponding curvature 2-forms
ωi = dφi +12
3∑j,k=1
ϵijkφj ∧ φk (3.89)
are self-dual 2-forms on B,
ω1 = θ45 + θ67, ω2 = θ64 + θ57, ω3 = θ47 + θ56, (3.90)
which implies
ω1 ∧ ω2 = ω1 ∧ ω3 = ω2 ∧ ω3 = 0 , (3.91)ω1 ∧ ω1 = ω2 ∧ ω2 = ω3 ∧ ω3 = 2volB , (3.92)volB = θ4567 (volume form on B) (3.93)
42
and
dω1 = ω2 ∧ φ3 − φ2 ∧ ω3 ,
dω2 = ω3 ∧ φ1 − φ3 ∧ ω1 ,
dω3 = ω1 ∧ φ2 − φ1 ∧ ω2 , (3.94)
On P the metric
gS =3∑
i=1
φi ⊗ φi +7∑
a=4
θa ⊗ θa (3.95)
is a 3-Sasakian metric. Indeed, the Killing vector fields ξi (i = 1, 2, 3) dual to the 1-forms φi givethe characteristic vector fields of the 3- Sasakian structure satisfying the relations [ξi, ξj ] = ϵijk ξk.
Now we consider a Spin(7) structure on an 8-dimensional manifold M = R+ × P . Usingthe 3-Sasakian structure we consider the following metric with four radial functions ai(t) =a(t), b(t), c(t) and f(t):
g = dt2 +3∑
i=1
ai(t)2φi ⊗ φi + f(t)27∑
a=4
θa ⊗ θa . (3.96)
For this metric an orthonormal frame eµ (µ = 1, · · · , 8) is introduced by
ei = ai(t)φi, ea = f(t)θa, e8 = dt. (3.97)
Then the 1-forms eµ lead to a Spin(7) structure Ψ on M = R+ × P given by (3.82). In terms ofφi, ωi and volB we have
Ψ = abc dt ∧ φ1 ∧ φ2 ∧ φ3 − f4volB + f2(a dt ∧ φ1 − bcφ2 ∧ φ3) ∧ ω1
+f2(b dt ∧ φ2 − acφ3 ∧ φ1) ∧ ω2 + f2(c dt ∧ φ3 − abφ1 ∧ φ2) ∧ ω3. (3.98)
A straightforward computation gives
dΨ = Ψ0dt ∧ volB + Ψ1dt ∧ φ2 ∧ φ3 ∧ ω1 + Ψ2dt ∧ φ3 ∧ φ1 ∧ ω2 + Ψ3dt ∧ φ1 ∧ φ2 ∧ ω3 , (3.99)
where
Ψ0 = −4f3 df
dt− 2f2(a + b + c),
Ψ1 = −abc + f2a − f2b − f2c − 2bcfdf
dt− f2 d(bc)
dt,
Ψ2 = −abc − f2a + f2b − f2c − 2acfdf
dt− f2 d(ac)
dt,
Ψ3 = −abc − f2a − f2b + f2c − 2abfdf
dt− f2 d(ab)
dt. (3.100)
43
The Ricci-flat metrics with Spin(7) holonomy satisfy the condition dΨ = 0. In our case this isexplicitly given by the following first-order differential equations [16]:
da
dt= −a2 − (b − c)2
2bc+
a2
f2,
db
dt= −b2 − (c − a)2
2ac+
b2
f2,
dc
dt= −c2 − (a − b)2
2ab+
c2
f2,
df
dt= −a + b + c
2f. (3.101)
Now we turn to the Spin(7) with torsion equations (3.85)–(3.88). The dilaton Φ is calculatedas
dΦdt
= − 16f2
(Ψ0
f2+
2Ψ1
bc+
2Ψ2
ac+
2Ψ3
ab
)(3.102)
and the 3-form flux is given by
H = Hφφ1 ∧ φ2 ∧ φ3 +3∑
i=1
Hiφi ∧ ωi , (3.103)
where
Hφ = −abc
(Ψ0
f4+
dΦdt
), H1 = −af2
(Ψ1
bcf2+
dΦdt
),
H2 = −bf2
(Ψ2
acf2+
dΦdt
), H3 = −cf2
(Ψ3
abf2+
dΦdt
). (3.104)
If the field strength F takes the form,
F =3∑
i=1
Ftidt ∧ dφi +3∑
i=1
Fiωi +
12
3∑i,j=1
Fijφi ∧ φj , (3.105)
then the generalized self-dual equation (3.87) yields
2F1
f2= −F23
bc+
Ft1
a,
2F2
f2= −F31
ac+
Ft2
b,
2F3
f2= −F12
ab+
Ft3
c. (3.106)
From (4.26) we obtain that
dH =dHφ
dtdt ∧ φ1 ∧ φ2 ∧ φ3 +
3∑i=1
dHi
dtdt ∧ φi ∧ ωi + 2
3∑i=1
Hi volB
+(Hφ − H1 + H2 + H3)φ2 ∧ φ3 ∧ ω1 + (Hφ + H1 − H2 + H3)φ3 ∧ φ1 ∧ ω2
+(Hφ + H1 + H2 − H3)φ1 ∧ φ2 ∧ ω3 , (3.107)
44
Using (3.88) and (3.102) we find that the solutions are classified into three types:
(a) F = d(F1φ1) , H = F1φ
1 ∧ F ,
dF1
dt= aF1
(2f2
− 1bc
),
dΦdt
= −F12
a
(2f2
− 1bc
). (3.108)
(b) F = d(F2φ2) , H = F2φ
2 ∧ F ,
dF2
dt= bF2
(2f2
− 1ac
),
dΦdt
= −F22
b
(2f2
− 1ac
). (3.109)
(c) F = d(F3φ3) , H = F3φ
3 ∧ F ,
dF3
dt= cF3
(2f2
− 1ab
),
dΦdt
= −F32
c
(2f2
− 1ab
). (3.110)
Then, for case (a), the Spin(7) with torsion equation (3.85) reduces to the following differentialequations:
da
dt= −a2 − (b − c)2 − F1
2
2bc+
a2 − F12
f2,
db
dt= −b2 − (c − a)2 − F1
2
2ac+
b2
f2,
dc
dt= −c2 − (a − b)2 − F1
2
2ab+
c2
f2,
df
dt= −a + b + c
2f− F 2
1
2af, (3.111)
and (b) and (c) are given by cyclic permutations of a, b, c.
Finally, we briefly discuss the solutions to (3.101) and the Spin(7) with torsion equations givenby (3.111). A more detail about the solutions will be reported elsewhere. The regular conditionto the solutions requires the following boundary conditions at t = 0:
(I) a(0) = b(0) = c(0) = 0, f(0) = 0,
|a′(0)| = |b′(0)| = |c′(0)| =12, f ′(0) = 0, (3.112)
(II) a(0) = 0, b(0) = −c(0), f(0) = 0,
|a′(0)| = 2, b′(0) = c′(0), f ′(0) = 0. (3.113)
The equation (3.101) gives Ricci-flat Spin(7) holonomy metrics. Some explicit solutions satisfyingthe boundary conditions (I)(II) are constructed in [16, 17, 19, 20] with the help of numericalcalculations, and further the existence of the regular solutions was analytically proved [39]. TheSpin(7) with torsion case (3.111) is slightly different from the Ricci-flat case. We find that case(II) admits no solution with non-zero F1, while case (I) admits the following series solution around
45
t = 0:
a(t) = − t
2+ a3t
3 + a5t5 + · · · ,
b(t) = − t
2+ b3t
3 + b5t5 + · · · ,
c(t) = − t
2+ c3t
3 + c5t5 + · · · ,
f(t) = f0 +3
8f0t2 + f4t
4 + · · · ,
F1(t) = h2t2 + h4t
4 + · · · . (3.114)
Here, the series have four independent parameters, a3, b3, c3, f0, h2 with one constraint a3+b3+c3 = 2h2
2+1/(4f20 ), and higher coefficients are determined by these parameters. The series (3.114)
can be extended to an ALC solution of (3.108) and (3.111) when the parameters are restrictedto b3 = c3, and hence b(t) = c(t) is satisfied for all t. Specifically, the explicit ALC solution withtwo parameters ℓ, k is given by
a(r) = −ℓ(r − ℓ)
√(r − 3ℓ)(r + ℓ)
(1 + k2ℓ2)(r − ℓ)2 − 4k2ℓ4,
b(r) = c(r) = −12
√(r − 3ℓ)(r + ℓ),
f(r) =
√r2 − ℓ2
2(3.115)
together with
F1(r) =kℓ2(r − 3ℓ)(r + ℓ)
(1 + k2ℓ2)(r − ℓ)2 − 4k2ℓ4. (3.116)
Here, we used a radial coordinate r (r ≥ 3ℓ) defined by dt = (r − ℓ)dr/√
(r − 3ℓ)(r + ℓ).
3.4 Conclusion
We have derived G2T equations in Abelian heterotic supergravity theory. When a G2 mani-fold is locally given by R+ × S3 × S3, the G2T equations are reduced to ordinary differentialequations(3.42)–(3.49). A numerical analysis of these equations shows that a global solutiondoesn’t exist for the general six radial functions ai(t), bi(t) (i = 1, · · · , 3) and thus we study thereduced case, a1(t) = a2(t) and b1(t) = b2(t). The Abelian heterotic solutions (g,H, ϕ, F ) are ob-tained from (3.70)–(3.73) by using Ricci-flat G2 holonomy metrics. To construct regular solutionsof the reduced G2T equations, we have investigated S3-bolt solutions and T 1,1-bolt solutions bynumerical analysis. These solutions are shown graphically in figures 1 and 2. The formulas (3.70)–(3.73) generate only S3-bolt solutions. A problem of finding analytic expressions for T 1,1-boltsolutions remains as a future work. We have also derived Spin(7) with torsion equations basedon 3-Sasakian manifolds. The explicit ALC solution to Abelian heterotic supergravity theory hasbeen obtained from these equations.
46
Chapter 4
Supersymmetric heterotic solutionswith SU(3)-structure
This chapter is based on Ref. [1]. We consider a geometry of a NS sector in E8 ×E8 heteroticstring theory. This theory has Bianchi identity,
dH = α′(trF ∧ F − trR− ∧R−), (4.1)
and this constraints the back ground geometry. Here R− is a curvature of Hull connection ∇−,which is given by
∇− = ∇− 12H , (4.2)
where ∇ is a Levi-Civita connection. On the other hand, Bismut connection ∇+ appears in theKilling spinor equations arising from supersymmetry variations of the gravitino and it is given by
∇+ = ∇ +12H . (4.3)
We assume that ten-dimensional space-time takes the form R1,3 × M6, where M6 is a six-dimensional space with Killing spinors obeying the Strominger system. Thus we should considera geometry on M6. If a field strength is identified with a curvature of a Hull connection, F = R−,a part of the gauge symmetry E8×E8 is embedded into the background geometry, which is knownas the standard embedding especially in the case H = 0 [42]. In H = 0 case, the holonomy of∇± is in SU(3) and thus a part of the gauge symmetry is SU(3) by the standard embedding.Therefore E8 × E8 breaks to E6 × E8 and this is known as a hallmark of Calabi–Yau compacti-fication. In H = 0 case, the holonomy of ∇+ belongs to SU(3), whereas the holonomy of ∇− isin SO(6) generally. Hence E8 × E8 may break to SO(10) × E8 by the standard embedding andthis is desirable in a phenomenological view.
In this chapter, we construct a supersymmetric heterotic supergravity solution such that aholonomy of Bismut connection ∇+ is in SU(3) but a holonomy of Hull connection ∇− is not, bysuperposing two hyper-Kahlers with torsion (HKT) geometries. As already pointed out in Ref.[40], one can obtain HKT geometries by conformally transforming hyper-Kahler geometries. We
47
choose the Gibbons–Hawking space as the starting point and apply a conformal transformation toobtain a HKT geometry. Then, we superpose these two HKT geometries and require a resultingmetric for some conditions. Consequently, we find that the holonomy of Hull connection on thesuperposed solution remains to be SO(4). We also show that by T dualizing this solution theholonomy of Hull connection turns into SO(5) or SO(6).
We take a two-dimensional periodic array of the “intersecting HKT” solutions to get a com-pact six-dimensional solution. We find that the fundamental parallelogram of the two-dimensionalperiodic array is separated into distinct smooth regions bordered by codimension-1 singularityhypersurfaces, hence the name “supersymmetric domain wall.”
4.1 HKT geometry as a conformal transform
We start with a four-dimensional hyper-Kahler with torsion (HKT) metric gHKT obtained asa conformal transform of a hyper-Kahler metric, where for the latter we specifically consider theGibbons–Hawking (GH) metric gGH ,
gHKT = Φ gGH . (4.4)
The GH metric is given by [41]
gGH =1φ
(dτ −
3∑i=1
ψidxi
)2
+ φ
3∑i=1
(dxi)2 , (4.5)
where φ and ψ = (ψ1, ψ2, ψ3) are scalar functions of the coordinates (x1, x2, x3) of R3 obeyingthe relation
gradφ = rotψ. (4.6)
Φ is a scalar field of which the properties will be described shortly. We define the orthonormalbasis
E0 =
√Φφ
(dτ −
3∑i=1
ψidxi
), Ei =
√Φ φdxi (i = 1, 2, 3), (4.7)
so that the hypercomplex structure is given by the three complex structures Ja (a = 1, 2, 3)satisfying the quaternionic identities,
Ja(Eµ) = ηaµνE
ν , (4.8)
where ηaµν are the ’t Hooft matrices. The corresponding fundamental 2-forms are
Ωa = −ηaµνE
µ ∧ Eν . (4.9)
The HKT structure is defined by the 3-form torsion T satisfying [43, 44]
T = J1dΩ1 = J2dΩ2 = J3dΩ3 . (4.10)
48
In the present case, we have
T = −E0 log ΦE123 + E1 log ΦE023 + E2 log ΦE031 + E3 log ΦE012 (4.11)
in terms of dual vector fields Eµ to the 1-forms (4.7),
E0 =
√φ
Φ∂
∂τ, Ei =
1√Φ φ
(∂
∂xi+ ψi
∂
∂τ
)(4.12)
and
Eµνλ = Eµ ∧ Eν ∧ Eλ. (4.13)
The exterior derivative is calculated as
dT = − 1Φ2φ
3∑µ=0
V 2µ Φ
E0123 (4.14)
with the vector fields Vµ =√
Φ φEµ. Therefore, if Φ is chosen to be a harmonic function withrespect to the GH metric (4.5), then the torsion T becomes a closed 3-form.
Using this T , we introduce the two types of connections ∇±,
∇±XY = ∇XY ± 1
2
3∑µ=0
T (X,Y,Eµ)Eµ, (4.15)
where ∇ is a Levi-Civita connection. The corresponding connection 1-forms ω±µν are defined by
∇±Eµ
Eν = ω±λν(Eµ)Eλ, (4.16)
and the curvature 2-forms are written as
R±µν = dω±µ
ν + ω±µλ ∧ ω±λ
ν . (4.17)
The torsion curvature R+ µν satisfies the SU(2) holonomy condition
R+01 + R+
23 = 0, R+02 + R+
31 = 0, R+03 + R+
12 = 0. (4.18)
On the other hand, if the torsion T is a closed 3-form, that is, Φ is a harmonic function, thenthe curvature R−µ
ν becomes an Anti–Self–Dual 2-form, which may be regarded as a Yang–Millsinstanton with the gauge group SU(2) × SU(2) = SO(4).
49
4.2 Intersecting HKT metrics
In the previous section we have seen that the HKT metrics obtained by a conformal transfor-mation have ω+ µ
ν in SU(2) but ω−µν in SO(4) strictly larger than SU(2) as long as the original
GH space is not a flat Euclidean space. In this section we construct their six-dimensional analogsby superposing two such HKT metrics embedded in different four-dimensional subspaces. Thisconstruction is motivated by that used in constructing intersecting brane solutions [45, 46] 1;namely, we assume the form of the metric as
g = ΦΦφφ((dx1)2 +(dx2)2)+Φφ(dx3)2 +Φφ
(dx4 −ψdx3)2 +Φφ(dx5)2 +Φφ
(dx6− ψdx5)2 . (4.19)
The HKT metric that we have considered in the previous section is characterized by a triplet(Φ, φ, ψ) on R3 = (x1, x2, x3) obeying (4.6). So at first it might seem that (Φ, φ) or (Φ, φ) couldto be functions of (x1, x2, x3) or (x1, x2, x5), and dx4 − ψdx3 or dx6 − ψdx5 could be replacedwith a more general form dx4 −
∑i=1,2,3 ψidxi or dx6 −
∑i=1,2,5 ψidxi, respectively. However, it
turns out that such a more general ansatz does not lead to a metric with SU(3) holonomy evenin the case Φ = Φ = 1. Thus we are led to consider the metric of the form (4.19), assuming thefollowing:
• (Φ, φ) and (Φ, φ) are harmonic functions on the two-dimensional flat space R2 = (x1, x2).
• ψ = (0, 0, ψ) and ψ = (0, 0, ψ), of which the components are harmonic functions on R2
satisfying the Cauchy–Riemann conditions
∂φ
∂x2= − ∂ψ
∂x1,
∂φ
∂x1=
∂ψ
∂x2,
∂φ
∂x2= − ∂ψ
∂x1,
∂φ
∂x1=
∂ψ
∂x2. (4.20)
Under these assumptions, we will show that a six-dimensional space M6 with the metric (4.19)has the following KT structure:
(a) a closed Bismut torsion [see, Eq. (4.26)],
(b) an exact Lee form [see, Eq. (4.27)],
(c) a Bismut connection ∇+ with SU(3) holonomy [see, Eq. (4.34)].
We first introduce an orthonormal basis
e1 =√
ΦΦφφdx1 , e2 =√
ΦΦφφdx2 , e3 =√
Φφdx3 , e4 =
√Φφ
(dx4 − ψdx3) ,
e5 =√
Φφdx5 , e6 =
√Φφ
(dx6 − ψdx5) (4.21)
1The term “intersecting” in the (commonly used) name is misleading since they are smeared and hence do nothave intersections with larger codimensions. See, e.g., Ref. [47] for recent developments in constructing localizedintersecting brane solutions in supergravity.
50
and their exterior derivatives dea are as follows:
de1 = − 1ΦΦφφ
∂2
√ΦΦφφe12 , de2 =
1ΦΦφφ
∂1
√ΦΦφφe12 ,
de3 =1
2Φφ
√ΦΦφφ
[(φ∂1Φ + Φ∂1φ)e13 + (φ∂2Φ + Φ∂2φ)e23] ,
de4 =1
2Φ√
ΦΦφφ
(∂1Φ − Φ
φ∂1φ
)e14 +
(∂2Φ − Φ
φ∂2φ
)e24
− 1
φ
√ΦΦφφ
(∂1ψe13 + ∂2ψe23) ,
de5 =1
2Φφ
√ΦΦφφ
[(φ∂1Φ + Φ∂1φ)e15 + (φ∂2Φ + Φ∂2φ)e25] ,
de6 =1
2Φ√
ΦΦφφ
(∂1Φ − Φ
φ∂1φ
)e16 +
(∂2Φ − Φ
φ∂2φ
)e26
− 1
φ
√ΦΦφφ
(∂1ψe15 + ∂2ψe25) .
(4.22)
The space M6 has a natural complex structure J defined by
J(e1) = ϵ1e2 , J(e3) = ϵ2e
4 , J(e5) = ϵ3e6 (4.23)
together with the conditions |ϵi| = 1(i = 1, 2, 3) and ϵ1ϵ2 = ϵ1ϵ3 = −1. Indeed, we can directlycheck that a Nijenhuis tensor associated with J vanishes under these conditions (see AppendixA). Then, the metric (4.19) becomes Hermitian with respect to the complex structure J , and therelated fundamental 2-form κ takes the form
κ = ϵ1e1 ∧ e2 + ϵ2e
3 ∧ e4 + ϵ3e5 ∧ e6 . (4.24)
The Bismut torsion T is uniquely determined by
∇+Xg = 0 , ∇+
Xκ = 0 . (4.25)
Explicitly, we have
T = −J dκ
=1
Φ√
ΦΦφφ(∂1Φe234 − ∂2Φe134) +
1
Φ√
ΦΦφφ(∂1Φe256 − ∂2Φe156). (4.26)
It should be noticed that in our case the Bismut torsion is a closed 3-form, dT = 0. We shallrefer to ∇+ and ∇− as the Bismut connection and Hull connection, respectively, according toRef. [10]. The Lee form θ is a 1-form defined by Θ = −Jδκ [11], which becomes a exact 1-form,
Θ = 2dϕ , ϕ = log√
ΦΦ. (4.27)
51
The space M6 admits a SU(3)-structure, which is classified by
d(e−2ϕΥ) = 0 , (4.28)d(e−2ϕ ∗ κ) = 0 , (4.29)T = −e2ϕ ∗ d(e−2ϕκ) , (4.30)
where Υ is the complex (3, 0)-form associated with κ and it is explicitly given by
Υ =√
ϵ1(e1 + i e2) ∧√
ϵ2(e3 − i e4) ∧√
ϵ3(e5 − i e6) . (4.31)
The triplet (κ, Υ, T ) satisfies these equations (see Appendix C). We will identify the Bismuttorsion with 3-form flux, T = H, and the function ϕ with a dilaton. The equations (4.28), (4.29),and (4.30) are defining equations of the SU(3)-structure admitted by a Calabi–Yau with torsionmanifold.
We should study the holonomy of the Bismut connection. The spin connection with the Bismuttorsion ω+
µν are as follows:
ω+12 =
1
2ΦΦφφ
√ΦΦφφ
( ∂2(ΦΦφφ)e1 − ∂1(ΦΦφφ)e2 ) ,
ω+13 = − 1
2√
ΦΦφφ
(1
Φφ∂1(Φφ)e3 +
(1φ
∂2 log φ − 1Φ
∂2 log Φ)
e4
),
ω+14 = − 1
2√
ΦΦφφ
((1φ
∂2φ +1Φ
∂2Φ)
e3 +(
1Φ
∂1Φ − 1φ
∂1φ
)e4
),
ω+15 = − 1
2√
ΦΦφφ
(1
Φφ∂1(Φφ)e5 +
(1φ
∂2φ − 1Φ
∂2Φ)
e6
),
ω+16 = − 1
2√
ΦΦφφ
((1φ
∂2φ +1Φ
∂2Φ)
e5 +(
1Φ
∂1Φ − 1φ
∂1φ
)e6
),
ω+23 =
1
2√
ΦΦφφ
(− 1
Φφ∂2(Φφ)e3 +
(1φ
∂1φ − 1Φ
∂1Φ)
e4
),
ω+24 =
1
2√
ΦΦφφ
((1φ
∂1φ +1Φ
∂1Φ)
e3 −(
1Φ
∂2Φ − 1φ
∂2φ
)e4
),
ω+25 =
1
2√
ΦΦφφ
(− 1
Φφ∂2(Φφ)e5 +
(1φ
∂1φ − 1Φ
∂1Φ)
e6
),
52
ω+26 =
1
2√
ΦΦφφ
((1φ
∂1φ +1Φ
∂1Φ)
e5 −(
1Φ
∂2Φ − 1φ
∂2φ
)e6
),
ω+34 =
1
2√
ΦΦφφ
((1φ
∂2φ +1Φ
∂2Φ)
e1 −(
1φ
∂1φ +1Φ
∂1Φ)
e2
),
ω+35 = 0 , ω+
36 = 0 , ω+45 = 0 , ω+
46 = 0 ,
ω+56 =
1
2√
ΦΦφφ
((1φ
∂2φ +1Φ
∂2Φ)
e1 −(
1φ
∂1φ +1Φ
∂1Φ)
e2
). (4.32)
By the definition of spin connection with torsion, ω+µν for each µ, ν are clearly skew symmetric
for exchanging µ with ν and thus non-trivial ω+µν are above all. The spin connection 1-forms ω+
µν
clearly satisfy the condition
6∑µ,ν=1
ω+µνκµν = 2(ϵ1ω+
12 + ϵ2ω+34 + ϵ3ω
+56) = 0 , (4.33)
which is equivalent to the condition (see Sec. 2.1)
ϵ1R+12 + ϵ2R+
34 + ϵ3R+56 = 0 . (4.34)
Therefore the Ricci form of the Bismut connection vanishes and this is implies that the holonomyof ∇+ is contained in SU(3) [11]. In addition, M6 admits two independent Killing spinors obeying∇+
Xε = 0 in type II theory.Thus, the triplet (g,H, ϕ) gives rise to a supersymmetric solution tothe type II supergravity theory.
4.3 Embedding into heterotic string theory and T-duality
We study supersymmetric solutions describing heterotic flux compactification. The bosonicpart of the string frame action, up to the first order in the α’ expansion, is given by
S =1
2κ2
∫d10x
√−ge−2ϕ
(R + 4(∇ϕ)2 − 1
12HMNP HMNP − α′(trFMNFMN − trR−
MNR−MN ))
.
(4.35)It is assumed that ten-dimensional spacetimes take the form R1,3 × M6, where M6 is a six-dimensional space admitting a Killing spinor ε,
∇+µ ε = 0,
(γµ∂µϕ − 1
12Hµνργ
µνρ
)ε = 0, Fµνγ
µνε = 0. (4.36)
This system together with the anomaly cancellation condition
dH = α′ (trF ∧ F − trR− ∧R−)(4.37)
is known as the Strominger system [48].
53
Now, we turn to the heterotic solution obeying the Strominger system. If the curvature R− inthe anomaly condition (4.37) is given by the Hull connection ∇−, we can choose a non-Abeliangauge field as F = R− since the 3-form flux (4.26) is closed by the identification T = H. This isa form of the usual standard embedding. Combining the well-known identity
R+abµν −R−
µνab =12(dT )abµν = 0 (4.38)
with the holonomy condition (4.34), we can see that the gauge field F is an instanton satisfyingthe third equation in (4.36).
Apparently, F seems to take values in SO(6) ⊂ E8, which would describe a symmetry breakingfrom E8 to SO(10). However, for generic choices of the harmonic functions φ, Φ, φ and Φ, it isnot ensured that the metric (4.19) can remain non-negative, and the dilaton (4.27) can remainreal valued. Therefore, to get a meaningful solution, we are forced to impose
φ = φ = Φ = Φ . (4.39)
Under the condition (4.39), we study the holonomy of the Bismut connection and the Hullconnection. In case of the Bismut connection, ϵ4 = −1, these connections are given by
ω+12 =
2φ3
( ∂2φe1 − ∂1φe2 ) , ω+34 =
1φ3
( ∂2φe1 − ∂1φe2 ) ,
ω+56 =
1φ3
( ∂2φe1 − ∂1φe2 ) , ω+35 = 0 , ω+
36 = 0 , ω+45 = 0 , ω+
46 = 0 ,
ω+13 = − 1
φ3∂1φe3 , ω+
14 = − 1φ3
∂2φe3 , ω+15 = − 1
φ3∂1φe5 ,
ω+16 = − 1
φ3∂2φe5 , ω+
23 = − 1φ3
∂2φe3 , ω+24 =
1φ3
∂1φe3 ,
ω+25 = − 1
φ3∂2φe5 , ω+
26 =1φ3
∂1φe5 (4.40)
and these teach us the relations
ω+13 = −ω+
24 , ω+14 = ω+
23 , ω+15 = −ω+
26 , ω+16 = ω+
25 , ω+12 = ω+
34 + ω+56 . (4.41)
The relation ω+12 = ω+
34 + ω+56 give a relation R+
12 = R+34 + R+
56 and thus the Ricci form of theBismut connection vanishes. The Bismut curvature 2-form R+ = dω+ + ω+ ∧ ω+ is
R+ =12R+
abMab
= R+13(M
13 − M24) + R+14(M
14 + M23) + R+15(M
15 − M26) + R+16(M
16 + M25)+R+
35(M35 + M46) + R+
36(M36 − M45) (4.42)
54
and commutation relations of their generators are following:
[M12 + M34 , M12 + M56] = 0 , [M12 + M34 , M13 − M24] = −2(M14 + M23) ,
[M12 + M34 , M14 + M23] = 2(M13 + M24) , [M12 + M34 , M15 − M26] = −(M16 + M25) ,
[M12 + M34 , M16 + M25] = (M15 − M26) , [M12 + M56 , M13 − M24] = −(M14 + M23) ,
[M12 + M56 , M14 + M23] = (M13 − M24) , [M12 + M56 , M15 − M26] = −2(M16 + M25) ,
[M12 + M56 , M16 + M25] = 2(M15 − M26) , [M13 − M24 , M14 + M23] = −(M12 + M34)[M13 − M24 , M15 − M26] = −(M35 + M46) , [M13 + M24 , M16 + M25] = −(M36 − M45)[M14 + M23 , M15 − M26] = (M36 − M45) , [M14 + M23 , M16 + M25] = −(M35 + M46)[M15 − M26 , M16 + M25] = −2(M12 + M56) , (4.43)
where so(6) generators Mab (a, b = 1, · · · , 6) satisfy the commutation relation
[Mab,M cd] = δbcMad − δacM bd − δbdMac + δadM bc . (4.44)
Thus eight independent generators M12 +M34, M12 +M56, M13−M24, M14 +M23, M15−M26,M16 + M25, M35 + M46, and M36 −M45 generate su(3) Lie algebra. Therefore the holonomy ofthe Bismut connection is contained in SU(3). On the other hand, the case of the Hull connection,ϵ4 = 1, these connections are given by
ω−12 =
2φ3
( ∂2φe1 − ∂1φe2 ) , ω−13 = − 1
φ3( ∂1φe3 + ∂2φe4 ) ,
ω−15 = − 1
φ3( ∂1φe5 + ∂2φe6 ) , ω−
23 =1φ3
(−∂2φe3 + ∂1φe4 ) ,
ω−25 =
1φ3
(−∂2φe5 + ∂1φe6 ) ,
ω−14 = ω−
16 = ω−24 = ω−
26 = ω−34 = ω−
35 = ω−36 = ω−
45 = ω−46 = ω−
56 = 0 . (4.45)
The Hull curvature 2-form R− = dω− + ω− ∧ ω− is
R− =12R−
abMab = R−
12M12 + R−
13M13 + R−
15M15 + R−
23M23 + R−
25M25 + R−
35M35 (4.46)
and commutation relations of their generators are the following:
[M12 , M13] = −M23 , [M12 , M15] = −M25 , [M12 , M23] = M13 , [M12 , M25] = M15
[M13 , M15] = −M35 , [M13 , M23] = −M12 , [M13 , M25] = 0 , [M15 , M23] = 0[M23 , M25] = −M35 , [M35 , M13] = −M15 , [M35 , M15] = M13 , [M35 , M23] = −M25
[M35 , M25] = −M23 .
Thus six independent generators M12, M13, M15, M23, M25, and M35 generate the Lie algebraso(4).Consequently, with the condition (4.39), the holonomy of ∇+ remains SU(3), whereas theinstanton F reduces to a proper Lie subalgebra SO(4) of SO(6).
55
To recover the SO(6) instanton, we apply a T-duality transformation. From (4.19), (4.26),and (4.27) with φ = φ = Φ = Φ, we have the following metric with the holonomy of ∇+ which iscontained in SU(3), 3-form flux, and dilaton:
g = φ4((dx1)2 + (dx2)2) + φ2( (dx3)2 + (dx5)2 ) + (dx4 − ψdx3)2 + (dx6 − ψdx5)2 ,(4.47)
H = − 1φ3
(∂2φe134 − ∂1φe234 + ∂2φe156 − ∂1φe256) , (4.48)
ϕ =12
log φ2 . (4.49)
The metric (4.47) has isometries U(1)4 generated by Killing vector fields ∂a (a = 3, 4, 5, 6).Therefore, we can T-dualize the type II solution (g,H, ϕ) along directions of these isometries. Itis easy to see that the solution is inert under the T-duality along x4 and x6; the T-dualities alongthe remaining directions give nontrivial deformations of the solutions, preserving one-quarter ofsupersymmetries. 2
We first T-dualize the solution along x3. The resulting solution (g, H, ϕ) is given by
g = φ4( (dx1)2 + (dx2)2 ) +1
φ2 + ψ2(dx3 + ψdx4)2
+φ2
φ2 + ψ2(dx4)2 + φ2(dx5)2 + (dx6 − ψdx5)2 , (4.50)
H =1
φ3(φ2 + ψ2)( ( (φ2 − ψ2)∂2φ + 2φψ∂1φ )e134 − ( (φ2 − ψ2)∂1φ − 2φψ∂2φ )e234 )
− 1φ3
(∂2φe156 − ∂1φe256) , (4.51)
ϕ =12
log(
φ2
φ2 + ψ2
). (4.52)
Here, the orthonormal basis is defined by
e1 = φ2dx1 , e2 = φ2dx2 , e3 =1√
φ2 + ψ2
(dx3 + ψdx4
), e4 =
φ√φ2 + ψ2
dx4 ,
e5 = φdx5 , e6 = dx6 − ψdx5 (4.53)
and their exterior derivatives are given by
de1 = −2φ∂φ
∂x2dx12 = − 2
φ3
∂φ
∂x2e12 , de2 =
2φ3
∂φ
∂x1e12 ,
de3 =1
φ2(φ2 + ψ2)(−(φ∂1φ − ψ∂2φ)e13 − (φ∂2φ + ψ∂1φ)e23 ) − 1
φ3∂2φe14 +
1φ3
∂1φe24 ,
de4 =ψ
φ3(φ2 + ψ2)( (ψ∂1φ + φ∂2φ)e14 + (ψ∂2φ − φ∂1φ)e24 ) ,
de5 =1φ3
(∂1φe15 + ∂2φe25) , de6 =1φ3
(∂2φe15 − ∂1φe25) . (4.54)2See, e.g., Ref. [49] for the classification of supersymmetric solutions to heterotic supergravity.
56
Then, we have a deformed complex structure J ,
J e1 = ϵ1e2 , J e3 = ϵ2e4 , J e5 = ϵ3e6 (4.55)
with |ϵi| = 1(i = 1, 2, 3) and ϵ1ϵ2 = ϵ1ϵ3 = −1. Indeed, a Nijenhuis tensor related with J vanishesunder the conditions (see Appendix D). The associated fundamental 2-form κ takes the sameform as (4.24),
κ = ϵ1e1 ∧ e2 + ϵ2e
3 ∧ e4 + ϵ3e5 ∧ e6 (4.56)
and the associated complex (3, 0)-form Υ is given by
Υ = iφ + iψ√φ2 + ψ2
( e135 + e245 + e236 − e146 + i(e235 − e145 − e136 − e246) ) . (4.57)
A Bismut torsion T is defined by T = −J∗dκ. Then, we find that the torsion T is a closed 3-formand T = H. An Lee form Θ is defined by Θ = −J∗δκ and it takes the form, Θ = 2dϕ, whichis clearly an exact 1-form. A SU(3) structure admitted on the space M6 is defined by the sameform equations as (4.28), (4.29), and (4.30) and (κ, T , Υ) satisfies these equations (see AppendixD). This suggests that the space (M6, g, κ, Υ) is a Calabi–Yau with torsion manifold. We shouldconfirm that the Ricci form of the Bismut connection ∇+ vanishes. The Bismut connection1-forms ω+
µν are presented as follows:
ω+12 = ω+
12 , ω+56 = ω+
56 , ω+34 =
ψ
2φd log (φ2 + ψ2) ,
ω+35 = 0 , ω+
36 = 0 , ω+45 = 0 , ω+
46 = 0 ,
ω+13 =
1φ2(φ2 + ψ2)
((φ∂1φ − ψ∂2φ)e3 +
ψ
φ(ψ∂2φ − φ∂1φ)e4
),
ω+14 =
1φ2(φ2 + ψ2)
((φ∂2φ + ψ∂1φ)e3 − ψ
φ(ψ∂1φ + φ∂2φ)e4
),
ω+23 =
1φ2(φ2 + ψ2)
((φ∂2φ + ψ∂1φ)e3 − ψ
φ(ψ∂1φ + φ∂2φ)e4
),
ω+24 = − 1
φ2(φ2 + ψ2)
((φ∂1φ − ψ∂2φ)e3 +
ψ
φ(ψ∂2φ − φ∂1φ)e4
),
ω+15 = − 1
φ3∂1φe5 , ω+
16 = − 1φ3
∂2φe5 , ω+25 = − 1
φ3∂2φe5 , ω+
26 =1φ3
∂1φe5 . (4.58)
We notice that following four relations hold as same as the Bismut connection 1-forms for beforethe T-duality transformation along ∂3,
ω+13 = −ω+
24 , ω+14 = ω+
23 , ω+15 = −ω+
26 , ω+16 = ω+
25 , (4.59)
57
but ω+12 = ω+
34 + ω+56. Here we can rewrite ω+
34 as follows,
ω+34 =
ψ
φ(φ2 + ψ2)
(ψ
φ∂2 log φ − ∂1 log φ
)e1 − ψ
φ(φ2 + ψ2)
(ψ
φ∂1 log φ + ∂2 log φ
)e2
=1φ2
(∂2 log φe1 − ∂1 log φe2)
− 1φ(φ2 + ψ2)
((φ∂2 log φ + ψ∂1 log φ)e1 − (φ∂1 log φ − ψ∂2 log φ)e2)
= ω+34 − Λ , (4.60)
where the 1-form Λ is defined by
Λ =1
φ2(φ2 + ψ2)((φ∂2φ + ψ∂1φ)e1 − (φ∂1φ − ψ∂2φ)e2) . (4.61)
Then, we obtain∑µ<ν
ω+µν κµν = ϵ1ω
+12 + ϵ2ω
+34 + ϵ3ω
+56 = −ω+
12 + ω+34 − Λ + ω+
56 = −Λ (4.62)
Therefore we don’t see whether the Ricci form of the Bismut connection ∇+ vanishes or not inthe view of the connection. We need to investigate it at the level of the Bismut curvature. Thecondition ρ(X,Y ) = 0 is equivalent to
∑µ,ν R+
µν κµν = 0, where R+ is curvature 2-form relatedwith the Bismut connection ω+.
∑µ,ν R+
µν κµν is given by
6∑µ,ν=1
R+µν κµν = ϵ1R+
12 + ϵ2R+34 + ϵ3R+
56 = −R+12 + R+
34 + R+56 , (4.63)
where each terms are
R+12 = dω+
12 +∑
ρ
ω+1ρ ∧ ω+
ρ2 = − 2φ4
[∂2
1(log φ) + ∂22(log φ)
]e12 = R+
12
R+34 = dω+
34 − dΛ = R+34 − dΛ = − 1
φ4(∂2
1 log φ + ∂22 log φ)e12 − dΛ ,
R+56 = − 1
φ4(∂2
1 log φ + ∂22 log φ)e12 = R+
56 , (4.64)
We see that the 1-from Λ is closed-form as follows,
dΛ = − 2φ4(φ2 + ψ2)
(φ∂1φ − ψ∂2φ)e1 + (φ∂2φ + ψ∂1φ)e2 ∧ (φ∂2φ + ψ∂1φ)e1 − (φ∂1φ − ψ∂2φ)e2
− 1φ6(φ2 + ψ2)
∂1(φ2)e1 + ∂2(φ2)e2 ∧ (φ∂2φ + ψ∂1φ)e1 − (φ∂1φ − ψ∂2φ)e2
− 2φ4(φ2 + ψ2)
(∂2φ)2 + (∂1φ)2e12
− 2φ6(φ2 + ψ2)
( (φ∂2φ + ψ∂1φ)(∂2φ)e12 + (φ∂1φ − ψ∂2φ)(∂1φ)e12 )
= 0 .
58
Hence we have6∑
µ,ν=1
R+µν κµν = −R+
12 + R+34 + R+
56 = 0 . (4.65)
Thus the Ricci form of the Bismut connection ∇+ vanishes. The Bismut curvature 2-form R+
takes the same form as (4.42).Therefore the Bismut connection ∇+ has an SU(3) holonomy.
We consider the holonomy of the Hull connection ∇−. The Hull connection 1-forms ω−µν are
presented as follows:
ω−12 = ω−
12 , ω−34 =
1φ2(φ2 + ψ2)
( (φ∂2φ + ψ∂1φ)e1 − (φ∂1φ − ψ∂2φ)e2 ) ,
ω−13 =
1φ2(φ2 + ψ2)
( (φ∂1φ − ψ∂2φ)e3 + (φ∂2φ + ψ∂1φ)e4 ) ,
ω−14 = − ψ
φ3(φ2 + ψ2)( (φ∂1φ − ψ∂2φ)e3 + (ψ∂1φ + φ∂2φ)e4 ) ,
ω−23 =
1φ2(φ2 + ψ2)
( (φ∂2φ + ψ∂1φ)e3 − (φ∂1φ − ψ∂2φ)e4 ) ,
ω−24 =
ψ
φ3(φ2 + ψ2)( (φ∂2φ + ψ∂1φ)e3 − (ψ∂2φ − φ∂1φ)e4 ) ,
ω−15 = − 1
φ3(∂1φe5 + ∂2φe6) , ω−
25 = − 1φ3
(∂2φe5 − ∂1φe6) ,
ω−16 = ω−
26 = ω−35 = ω−
36 = ω−45 = ω−
46 = ω−56 = 0 . (4.66)
We introduce so(6) generators Mab (a, b = 1, · · · , 6) satisfying (4.44). Then, the curvature 2-formR− (see Appendix D) is given by
R− = R−12M
12+R−13M
13+R−14M
14+R−15M
15+R−23M
23+R−24M
24+R−25M
25+R−35M
35+R−45M
45
(4.67)and commutation relations to the generators Mab are the following:
[M12 , M13] = −M23 , [M12 , M14] = −M24 , [M12 , M15] = −M25 , [M12 , M23] = M13
[M12 , M24] = M14 , [M12 , M25] = M15 , [M12 , M34] = 0 , [M12 , M35] = 0[M12 , M45] = 0 , [M13 , M14] = −M34 , [M13 , M15] = −M35 , [M13 , M23] = −M12
[M13 , M24] = 0 , [M13 , M25] = 0 [M13 , M34] = M14 , [M13 , M35] = M15
[M13 , M45] = 0 , [M14 , M15] = −M45 , [M14 , M23] = 0 , [M14 , M24] = −M12
[M14 , M25] = 0 , [M14 , M34] = −M13 , [M14 , M35] = 0 , [M14 , M45] = M15
[M15 , M23] = 0 , [M15 , M24] = 0 , [M15 , M34] = 0 , [M15 , M35] = −M13
[M15 , M45] = −M14 , [M23 , M24] = −M34 , [M23 , M25] = −M35 , [M23 , M34] = M24
[M23 , M35] = M25 , [M23 , M45] = 0 , [M24 , M25] = −M45 , [M24 , M34] = −M23
[M24 , M35] = 0 , [M24 , M45] = M25 , [M34 , M35] = −M45 , [M34 , M45] = M35
[M35 , M45] = −M34 . (4.68)
59
These ten independent generators generate the Lie algebra so(5). In this case, it turns out thatthe Hull connection ∇− is in SO(5), which is still smaller than SO(6).
Thus, we further T-dualize the solution (g, H, ϕ) once more along x5 and finally obtain (g, H, ϕ):
g = φ4( (dx1)2 + (dx2)2 ) +1
φ2 + ψ2(dx3 + ψdx4)2
+1
φ2 + ψ2(dx5 + ψdx6)2 +
φ2
φ2 + ψ2( (dx4)2 + (dx6)2 ) , (4.69)
H =1
φ3(φ2 + ψ2)( ((φ2 − ψ2)∂2φ + 2φψ∂1φ)e134 − ((φ2 − ψ2)∂1φ − 2φψ∂2φ)e234 )
+1
φ3(φ2 + ψ2)( ((φ2 − ψ2)∂2φ + 2φψ∂1φ)e156 − ((φ2 − ψ2)∂1φ − 2φψ∂2φ)e256 ),
(4.70)
ϕ =12
log(
φ2
(φ2 + ψ2)2
). (4.71)
The orthonormal basis is defined by
e1 = e1 , e2 = e2 , e3 = e3 , e4 = e4 ,
e5 =1√
φ2 + ψ2(dx5 + ψdx6) , e6 =
φ√φ2 + ψ2
dx6 (4.72)
and exterior derivatives of them are as follows:
de1 = − 2φ3
∂φ
∂x2e12 , de2 =
2φ3
∂φ
∂x1e12 ,
de3 = − 1φ2(φ2 + ψ2)
( (φ∂1φ − ψ∂2φ)e13 − (φ∂2φ + ψ∂1φ)e23 ) − 1φ3
∂2φe14 +1φ3
∂1φe24 ,
de4 =ψ
φ3(φ2 + ψ2)( (ψ∂1φ + φ∂2φ)e14 + (ψ∂2φ − φ∂1φ)e24 ) ,
de5 =1
φ2(φ2 + ψ2)(−(φ∂1φ − ψ∂2φ)e15 − (φ∂2φ + ψ∂1φ)e25 ) − 1
φ3∂1B2e
16 − 1φ3
∂2B2e26 ,
de6 =ψ
φ3(φ2 + ψ2)( (ψ∂1φ + φ∂2φ)e16 + (ψ∂2φ − φ∂1φ)e26 ) . (4.73)
In this basis the complex structure J is given by
J e1 = ϵ1e2 , J e3 = ϵ2e
4 , J e4 = ϵ3e2 (4.74)
with |ϵi| = 1(i = 1, 2, 3) and ϵ1ϵ2 = ϵ1ϵ3 = −1. Indeed, a Nijenhuis tensor associated with Jvanishes under the conditions (see Appendix E). The fundamental 2-form κ takes the form
κ = ϵ1e1 ∧ e2 + ϵ2e
3 ∧ e4 + ϵ3e5 ∧ e6 (4.75)
60
and the associated complex (3, 0)-form Υ is given by
Υ = iφ + iψ
φ2 + ψ2( e135 + e245 + e236 − e146 + i(e235 − e145 − e136 − e246) ) . (4.76)
A Bismut torsion T is defined by T = −J∗dκ, which is a closed 3-form and satisfies T = H. An Leeform Θ is defined by Θ = −J∗δκ and it takes the form, Θ = 2dϕ. A SU(3) structure admittedon the space M6 is classified by the same form equations as (4.28), (4.29), and (4.30). Then,(κ, T , Υ) satisfies these equations (see Appendix E). This implies that the space (M6, g, κ, Υ) is aCalabi–Yau with torsion manifold. We should verify that the Ricci form of the Bismut connection∇+ vanishes. The Bismut connection 1-forms ω+
µν are presented as follows:
ω+12 = ω+
12 , ω+34 = ω+
34 , ω+56 =
ψ
2φd log (φ2 + ψ2)
ω+13 = ω+
13 , ω+14 = ω+
14 , ω+23 = ω+
23 , ω+24 = ω+
24 ,
ω+15 =
1φ2(φ2 + ψ2)
((φ∂1φ − ψ∂2φ)e5 +
ψ
φ(ψ∂2φ − φ∂1φ)e6
)
ω+16 =
1φ2(φ2 + ψ2)
((φ∂2φ + ψ∂1φ)e5 − ψ
φ(ψ∂1φ + φ∂2φ)e6
)
ω+25 =
1φ2(φ2 + ψ2)
((φ∂2φ + ψ∂1φ)e5 − ψ
φ(ψ∂1φ + φ∂2φ)e6
)
ω+26 = − 1
φ2(φ2 + ψ2)
((φ∂1φ − ψ∂2φ)e5 +
ψ
φ(ψ∂2φ − φ∂1φ)e6
)ω+
35 = 0 , ω+36 = 0 , ω+
45 = 0 , ω+46 = 0
and we notice that following four relations hold as same as the Bismut connection for before theT-duality transformation,
ω+13 = −ω+
24 , ω+14 = ω+
23 , ω+15 = −ω+
26 , ω+16 = ω+
25 , (4.77)
but ω+12 = ω+
34 + ω+56. As it was already mentioned, ω+
34 is given by ω+34 = ω+
34 − Λ, where Λ isdefined by (4.61). Also, ω+
56 is rewritten as follows,
ω+56 =
1φ2
(∂2 log φe1 − ∂1 log φe2)
− 1φ2(φ2 + ψ2)
( (φ∂2φ + ψ∂1φ)e1 − (φ∂1φ − ψ∂2φ)e2 )
= ω+56 − Λ . (4.78)
61
The relation ω+12 = ω+
34 + ω+56 is not valid,∑
µ<ν
ω+µν κµν = ϵ1ω
+12 + ϵ2ω
+34 + ϵ3ω
+56 = −ωTB
12 + ωTB34 − Λ + ωTB
56 − Λ = −2Λ .
Therefore we don’t see whether the Ricci form ρ of the Bismut connection ∇+ vanishes in theview of the connection. We need to investigate it at the level of the Bismut curvature. Thecondition ρ = 0 is equivalent to
∑µ,ν R+
µν κµν = 0, where R+ is curvature 2-form related with theBismut connection ω+. Hence we confirm that
∑µ,ν R+
µν κµν vanishes.∑
µ,ν R+µν κµν is given by
6∑µ,ν=1
R+µν κµν = ϵ1R+
12 + ϵ2R+34 + ϵ3R+
56 = −R+12 + R+
34 + R+56 , (4.79)
where each terms are following:
R+12 = − 2
φ4( ∂2
1 log φ + ∂22 log φ )e12 = R+
12 ,
R+34 = R+
34 − dΛ = − 1φ4
(∂21 log φ + ∂2
2 log φ)e12 − dΛ ,
R+56 = dω+
56 − dΛ = R+56 − dΛ = − 1
φ4(∂2
1 log φ + ∂22 log φ)e12 − dΛ . (4.80)
As was already mentioned, the 1-form Λ is closed-form. Thus we have
6∑µ,ν=1
R+µν κµν = −R+
12 + R+34 + R+
56 = 0 (4.81)
and thus the Ricci form of the Bismut connection ∇+ vanishes. The curvature 2-form R+ alsotakes the same form as (4.42). Therefore the Bismut holonomy is contained in SU(3).
We should investigate the holonomy of the Hull connection ∇−. The Hull connection 1-formsω−
µν are presented as follows:
ω−12 = ω−
12 , ω−34 = ω−
34 , ω−35 = ω−
36 = ω−45 = ω−
46 = 0 ,
ω−56 =
1φ2(φ2 + ψ2)
( (φ∂2φ + ψ∂1φ)e1 − (φ∂1φ − ψ∂2φ)e2 ) ,
ω−13 = ω−
13 , ω−14 = ω−
14 , ω−23 = ω−
23 , ω−24 = ω−
24 ,
ω−15 =
1φ2(φ2 + ψ2)
( (φ∂1φ − ψ∂2φ)e5 + (φ∂2φ + ψ∂1φ)e6 ) ,
ω−16 = − ψ
φ3(φ2 + ψ2)( (φ∂1φ − ψ∂2φ)e5 + (ψ∂1φ + φ∂2φ)e6 ) ,
ω−25 =
1φ2(φ2 + ψ2)
( (φ∂2φ + ψ∂1φ)e5 − (φ∂1φ − ψ∂2φ)e6 ) ,
62
ω−26 =
ψ
φ3(φ2 + ψ2)( (φ∂2φ + ψ∂1φ)e5 − (ψ∂2φ − φ∂1φ)e6 ) . (4.82)
We introduce so(6) generators Mab (a, b = 1, · · · , 6) satisfying (4.44). Then, the Hull curvature2-form R− (see Appendix E) is given by
R− = R−12M
12 + R−13M
13 + R−14M
14 + R−15M
15 + R−16M
16 + R−23M
23 + R−24M
24
+R−25M
25 + R−26M
26 + R−35M
35 + ω−36M
36 + R−45M
45 + ω−46M
46 (4.83)
and commutation relations to the generators Mab are the following:
[M12 , M13] = −M23 , [M12 , M14] = −M24 , [M12 , M15] = −M25 , [M12 , M16] = −M26
[M12 , M23] = M13 , [M12 , M24] = M14 , [M12 , M25] = M15 , [M12 , M26] = M16
[M12 , M34] = 0 , [M12 , M35] = 0 , [M12 , M45] = 0 , [M12 , M46] = 0 , [M12 , M56] = 0[M13 , M14] = −M34 , [M13 , M15] = −M35 , [M13 , M16] = −M36 [M13 , M23] = −M12
[M13 , M24] = 0 , [M13 , M25] = 0 , [M13 , M26] = 0 , [M13 , M34] = M14
[M13 , M35] = M15 , [M13 , M36] = M16 , [M13 , M45] = 0 , [M13 , M45] = 0[M13 , M56] = 0 , [M14 , M15] = −M45 , [M14 , M16] = −M46 , [M14 , M23] = 0[M14 , M24] = −M12 , [M14 , M25] = 0 , [M14 , M26] = 0 , [M14 , M34] = −M13
[M14 , M35] = 0 , [M14 , M36] = 0 , [M14 , M45] = M15 , [M14 , M46] = M16
[M14 , M56] = 0 , [M15 , M16] = −M56 , [M15 , M23] = 0 , [M15 , M24] = 0[M15 , M25] = −M12 , [M15 , M26] = 0 , [M15 , M34] = 0 , [M15 , M35] = −M13
[M15 , M36] = 0 , [M15 , M45] = −M14 , [M15 , M46] = 0 , [M15 , M56] = 0[M16 , M23] = 0 , [M16 , M24] = 0 , [M16 , M25] = 0 , [M16 , M26] = −M12
[M16 , M34] = 0 , [M16 , M35] = 0 , [M16 , M36] = −M13 , [M16 , M45] = 0[M16 , M46] = −M14 , [M16 , M56] = −M15 , [M23 , M24] = −M34 , [M23 , M25] = −M35
[M23 , M26] = −M36 , [M23 , M34] = M24 , [M23 , M35] = M25 , [M23 , M36] = M36
[M23 , M45] = 0 , [M23 , M46] = 0 , [M23 , M56] = 0 , [M24 , M25] = −M45
[M24 , M26] = −M46 , [M24 , M34] = −M23 , [M24 , M35] = 0 , [M24 , M36] = 0[M24 , M45] = M25 , [M24 , M46] = M26 , [M24 , M56] = 0 , [M25 , M26] = −M56
[M25 , M34] = 0 , [M25 , M35] = −M23 , [M25 , M36] = 0 , [M25 , M45] = −M24
[M25 , M46] = 0 , [M25 , M56] = 0 , [M26 , M34] = 0 , [M26 , M35] = 0[M26 , M36] = −M23 , [M26 , M45] = 0 , [M26 , M46] = −M24 , [M26 , M56] = −M25
[M34 , M35] = −M45 , [M34 , M36] = −M46 , [M34 , M45] = M35 , [M34 , M46] = M36
[M34 , M56] = 0 . (4.84)
These fifteen generators Mab (a < b , a, b = 1, · · · 6) generate the Lie algebra so(6) and thus theholonomy of the Hull connection ∇− is SO(6) as desired.
63
4.4 SUSY domain wall metric
The last topic concerns the construction of type II/heterotic supersymmetric solutions on acompact six-dimensional space with the Hull connection not being in SU(3). Since the triplesobtained in the previous section depend only on x1 and x2, we can compactify the x3, x4, x5 andx6 spaces on T 4 by simply identifying periodically, whereas we consider a periodic array of copiesof the solution along the x1 and x2 directions.
Let us consider a periodic array of (g,H, ϕ) [Eqs. (4.47), (4.48), and (4.49)], (g, H, ϕ) [eqs.(4.50), (4.51), and (4.52)], or (g, H, ϕ) [Eqs. (4.69), (4.70), and (4.71)], which are characterizedby a pair of harmonic functions φ and ψ. In two dimensions both the real and imaginary partsof any holomorphic function are harmonic. Thus, we can take φ to be, say, the real part of anydoubly periodic, holomorphic function. In this case, ψ may be taken to be the imaginary part ofthe same doubly periodic function.
Figure 4.1: The real (left plot) and imaginary (right plot) parts of the ℘ function. The funda-mental parallelogram can be taken to be −1
2 ≤ x1
l ≤ 12 and −1
2 ≤ x2
l ≤ 12 .
Since the only nonsingular holomorphic function on T 2 is a constant function, we need to allowsome pole singularities in the fundamental parallelogram of the periodic array, which may be seento be in accordance with the no-go theorems against smooth flux compactifications [50, 51]. Thedoubly periodic meromorphic functions are known as elliptic functions. It is well known that, fora given periodicity, the field of elliptic functions is generated by Weierstrass’ ℘ function and itsderivative ℘′. In the following we consider, as a typical example, the compactification of (g,H, ϕ),(g, H, ϕ) and (g, H, ϕ) on a square torus of side l by taking
φ(x1, x2) = Re ℘(z), (4.85)ψ(x1, x2) = Im ℘(z), (4.86)
where ℘(z) is of modulus τ = i or τ = eπi5 and z = l−1(x1 + ix2). Our solutions are determined
entirely by Weierstrass’s ℘ function without any reference to α′ because of the choice F = R−
that causes the rhs of (4.37) to be closed. Note that they solve the heterotic equations of motionup to O(α′).
64
Figure 4.2: The zero loci of the real and imaginary parts of the ℘ function for the modulus τ = i
(left plot) and τ = eπi5 (right plot). The shaded region is the fundamental parallelogram.
The real and imaginary parts of ℘(z) are shown in Fig. 4.1. We see that φ may take negativeas well as positive values, but note that the metric (4.47), (4.50) or (4.50) depends on φ throughφ2 as we designed, so the solution is only singular where φ vanishes (as well as φ diverges). Alsonegative ψ causes no problem as long as φ is nonzero.
For any case of (g,H, ϕ), (g, H, ϕ), or (g, H, ϕ), some of the components of the metric vanishwhere φ = 0, and hence the solution is singular. Also, the “string coupling” (= exponentialof the dilaton) vanishes there. The φ = 0 curves are shown in Fig. 4.2 for the cases τ = i
and τ = eπi5 . For both cases, we see that the fundamental parallelogram (shown by the shaded
region) is separated into two distinct smooth regions bordered by the codimension-1 singularityhypersurfaces. The two singularity hypersurfaces intersect at x1 = x2 = 0, where the ℘ functionhas a unique double pole; its real and imaginary parts rapidly fluctuate at x1 = x2 = 0. Moredetails about the solution will be reported elsewhere.
4.5 Conclusions
In this chapter, we have shown that two HKT metrics given by (Φ, φ,ψ) and (Φ, φ, ψ) can besuperposed and lifted to a six-dimensional smeared intersecting solution of type II supergravity ifthe functions Φ, φ, Φ, and φ are restricted to harmonic functions on the two-dimensional flat spaceR2 = (x1, x2), together with ψ = (0, 0, ψ) and ψ = (0, 0, ψ) satisfying the Cauchy–Riemannconditions. The simplest geometry that we have considered has an SO(4) ∇− connection thatleads to the SO(10) unbroken gauge symmetry if it is embedded to heterotic string theory as aninternal space. By T-duality transformations we have obtained one having an SO(5) or SO(6) ∇−
holonomy. We have also compactified this six-dimensional KT space by taking a periodic array tofind a supersymmetric domain wall solution of heterotic supergravity in which the fundamental
65
parallelogram of the two-dimensional periodic array is separated into distinct smooth regionsbordered by codimension-1 singularity hypersurfaces.
66
Chapter 5
Summary
In this thesis, we studied supersymmetric solutions in 6-, 7-, 8-, dimensional supergravitytheories. In 7 dimensions, we introduced the class of G2-structure associated with the Abelianheterotic supergravity theory. The class has a closed 3-form torsion T7 and an exact Lee form Θ7.Hence we identified the flux H7 with T7 and the dilaton ϕ7 with Θ7, T7 = H7 and 2dϕ7 = Θ7.The class was determined by the choice of (g7, Ω, F7), where g7 was the metric associated with thefundamental 3-form Ω and F7 was the field strength of U(1) gauge field.In type II supergravitytheory, a 3-form flux of the B-field is of closed form; however, the flux H7 and the field strength F7
should satisfy the Bianchi identity, dH7 = F7 ∧F7. In addition, F7 should satisfy the generalizedself-dual equation, ∗(Ω ∧ F7) = F7, which arises from G2 irreducible representation. Thus F7
was determined algebraically. Then, the defining equations of the class on the cohomogeneity onemanifold of the form R+×S3×S3 became first order ordinary differential equations. We obtainedthe formulae which give the S3-bolt solutions from regular and Ricci-flat G2 holonomy metrics,and T 1,1-bolt solutions were also obtained numerically. In 8 dimensions , similarly to the case of7 dimensions, we introduced the class of Spin(7)-structure associated with the Abelian heteroticsupergravity theory. We identified the flux H8 with the 3-form torsion T8 and the dilaton ϕ8
with the exact Lee form Θ8. The class was determined by the choice of (g8, Ψ, F8), where g8 is ametric associated with the fundamental 4-form Ψ and F8 is the field strength of U(1) gauge field.The field strength F8 was required to satisfy the Bianchi identity and the generalized self-dualequation. We assumed the manifold R+ × M3−Sasaki, where M3−Sasaki denotes a manifold witha 3-Sasakian structure, and then defining equations reduced to first order ordinary differentialequations. Regular solutions with F8 = 0 were obtained from regular and Ricci-flat Spin(7)holonomy metrics. It is of great interest to study general solutions, which may shed new light onG2 or Spin(7) with torsion geometry.
The role of Green–Schwarz mechanism [52] is twofold. First, to cancel the gauge and gravi-tational anomalies of ten-dimensional type I and heterotic superstring theories, the gauge groupmust be restricted to SO(32) or E8 × E8. Second, the mechanism requires the 2-form B to varyunder both the gauge and local Lorentz transformations so that the invariant 3-form field H mustbe of the form
H = dB − α′ (ω3Y − ω−3L
), (5.1)
67
where ω3Y is the Chern–Simons 3-form associated with the Yang–Mills connection and ω−3L is
also a Chern–Simons 3-form. The equation (5.1) leads to the Bianchi identity
dH = α′ (trF ∧ F − trR− ∧R−). (5.2)
The background geometry is restricted in such a way that the curvature 2-form R− satisfies thisidentity. The second term in the Bianchi identity (5.2) comes from Chern–Simons ω−
3L and R−
arises from the Hull connection ∇− (4.2). Note that the connection (4.2) is different from the onethat appears in the supersymmetry variation of the gravitino, δψM ∝ ∇+
Mε, where ∇+ is calledthe ”Bismut connection” (4.3). The relevance of the difference between the two connections waspointed out by Bergshoeff and de Roo [53], and later emphasized by e.g. [10, 51]. If a field strengthis identified with the curvature of a Hull connection, F = R−, a part of the gauge symmetryE8 × E8 is embedded into the background geometry, which is known as standard embedding,especially in the case where H = 0 [42]. In the H = 0 case, the holonomy of ∇± is in SU(3), andthus, a part of the gauge symmetry is SU(3) by the standard embedding. Therefore, E8 × E8
breaks into E6 × E8, and this is known as a hallmark of Calabi–Yau compactification. In theH = 0 case, the holonomy of ∇+ belongs to SU(3), whereas the holonomy of ∇− is in SO(6)generally. Hence E8 × E8 may break into SO(10) × E8 by the standard embedding, and this isdesirable in a phenomenological view. This is striking contrast to the H = 0, Calabi–Yau case. Ifone doesn’t require the identification F = R− on the at most-SO(6) Hull connection, then s/heshould use the nonstandard embedding that needs complicated mathematical machinery [7, 42].
In dimension 6, we constructed an intersecting metric g6 by superposing two Gibbons–Hawkingmetrics with the conformal factors, i.e., HKT metrics. The metric g6, the fundamental 2-form κ,and the complex (3, 0)-form Υ satisfy the defining equations of the SU(3)-structure associatedwith the NS–NS sector in type II supergravity, where the Lee form Θ6 is an exact 1-form andthe Bismut torsion T6 is closed 3-form. Also, the Ricci form vanishes, which gives a necessarycondition for the holonomy of the Bismut connection to be contained in SU(3). We identifiedthe flux H6 with the torsion T6 and the dilaton ϕ6 with Θ6. Then, we confirmed that twoindependent Killing spinors existed that satisfy supersymmetric equations in type II supergravitytheory. Therefore, the manifold (M6, g6, κ, Υ) is a Calabi–Yau with torsion manifold, and thus,(g6,H6, ϕ6) are supersymmetric solutions in the theory [11]. The solution has 2-dimensionalharmonic functions φ, φ, Φ, and Φ; however, the condition φ = φ = Φ = Φ is required such thatthe metric is non-negative and the dilaton takes real value. Thus, the solution is characterizedby a pair of harmonic functions (φ, ψ), which are related by Cauchy–Riemann equations. If thecurvature 2-form of the Hull connection is identified with the gauge field strength in E8 × E8
heterotic supergravity theory, then (g6, ϕ6,H6,F6) is the solution of the heterotic supergravitytheory. Then, the gauge group embedded into E8 is expected to be at most SO(6), and thus,SO(10) ⊂ E8 will be obtained. Hence, the supersymmetric solutions should have a holonomygroup SO(6) of the Hull connection. However, the holonomy group of the obtained manifold is anSO(4). The metric g6 has four Killing vectors ∂3, ∂4, ∂5, and ∂6. To recover the SO(6) holonomy,we T-dualized along ∂3 and obtained the supersymmetric solution (g6, ϕ, H) having a holonomygroup SO(5) of the Hull connection. Since this solution had a Killing vector ∂5, we further T-dualized along ∂5. Finally, we obtained the supersymmetric solution (g6, ϕ, H) with the SO(6)holonomy of the Hull connection. To obtain the compact 6-dimensional space, we chose φ + iψ
68
as the Weierstrass’s ℘ function. Then, the solution had codimension-1 singularity hypersurfacesand thus it was interpreted as a ”domain wall”. It would be interesting to solve the gauginoDirac equation on this background and compare the spectrum with the corresponding E8-typesupersymmetric nonlinear sigma model [54], similarly to what has been done in the SU(3) ∇−
case [55].
69
Appendix A
Nijenhuis tensor associated with J
In this section we study the Nijenhuis tensor associated with almost complex structure J . Wefirst introduce an orthonormal basis
e1 =√
ΦΦφφdx1 , e2 =√
ΦΦφφdx2 ,
e3 =√
Φφdx3 , e4 =
√Φφ
(dx4 − ψdx3) ,
e5 =√
Φφdx5 , e6 =
√Φφ
(dx6 − ψdx5) (A.1)
and their exterior derivatives deµ (µ = 1, · · · , 6) are given by
de1 = − 1ΦΦφφ
∂2
√ΦΦφφe12 ,
de2 =1
ΦΦφφ∂1
√ΦΦφφe12 ,
de3 =1
2Φφ
√ΦΦφφ
[(φ∂1Φ + Φ∂1φ)e13 + (φ∂2Φ + Φ∂2φ)e23] ,
de4 =1
2Φ√
ΦΦφφ
(∂1Φ − Φ
φ∂1φ
)e14 +
(∂2Φ − Φ
φ∂2φ
)e24
− 1
φ
√ΦΦφφ
(∂1ψe13 + ∂2ψe23) ,
de5 =1
2Φφ
√ΦΦφφ
[(φ∂1Φ + Φ∂1φ)e15 + (φ∂2Φ + Φ∂2φ)e25] ,
de6 =1
2Φ√
ΦΦφφ
(∂1Φ − Φ
φ∂1φ
)e16 +
(∂2Φ − Φ
φ∂2φ
)e26
− 1
φ
√ΦΦφφ
(∂1ψe15 + ∂2ψe25) .
(A.2)
An almost complex structure J : TpM6 −→ TpM6 is defined by
Je1 = −ϵ1e2 , Je3 = −ϵ2e4 , Je5 = −ϵ3e6, (A.3)
70
where eµ is dual vector for eµ and ϵa = ±1 (a = 1, 2, 3). Then the almost complex structure actson 1-form eµ as follows:
J∗e1 = ϵ1e
2 , J∗e3 = ϵ2e
4 , J∗e5 = ϵ3e
6 . (A.4)
Associated symplectic form κ is defined by
κ := ϵ1e1 ∧ e2 + ϵ2e
3 ∧ e4 + ϵ3e5 ∧ e6 . (A.5)
The functions fµνρ are defined by
deµ =12fµ
νρeνρ (A.6)
and can be read from (4.22),
f112 = − 1
ΦΦφφ
∂
√ΦΦφφ
∂x2= −f1
21 , f212 =
1ΦΦφφ
∂
√ΦΦφφ
∂x1= −f2
21 ,
f313 =
1
2Φφ
√ΦΦφφ
(φ∂1Φ + Φ∂1φ) = −f331 , f3
23 =1
2Φφ
√ΦΦφφ
(φ∂2Φ + Φ∂2φ) = −f332 ,
f413 = − 1
φ
√ΦΦφφ
∂1ψ = −f431 , f4
14 =1
2Φ√
ΦΦφφ
(∂1Φ − Φ
φ∂1φ
)= −f4
41 ,
f423 = − 1
φ
√ΦΦφφ
∂2ψ = −f432 , f4
24 =1
2Φ√
ΦΦφφ
(∂2Φ − Φ
φ∂2φ
)= −f4
42 ,
f515 =
1
2Φφ
√ΦΦφφ
(φ∂1Φ + Φ∂1φ) = −f551 , f5
25 =1
2Φφ
√ΦΦφφ
(φ∂2Φ + Φ∂2φ) = −f552 ,
f615 = − 1
φ
√ΦΦφφ
∂1ψ = −f651 , f6
16 =1
2Φ√
ΦΦφφ
(∂1Φ − Φ
φ∂1φ
)= −f6
61 ,
f625 = − 1
φ
√ΦΦφφ
∂2ψ = −f652 , f6
26 =1
2Φ√
ΦΦφφ
(∂2Φ − Φ
φ∂2φ
)= −f6
62 . (A.7)
Nijenhuis tensor NJ related to an almost complex structure J is defined by
NJ(X,Y ) := [JX, JY ] − [X,Y ] − J [X,JY ] − J [JX, Y ] , (A.8)
where commutation relation between eµ is given by
[eµ, eν ] = −fρµνeρ . (A.9)
Each components of the Nijenhuis tensor are the following:
NJ(e1, e2) = [Je1, Je2] − [e1, e2] − J [e1, Je2] − J [Je1, e2] = [−ϵ1e2, ϵ1e1] − [e1, e2]= 0 ,
71
NJ(e1, e3) = [Je1, Je3] − [e1, e3] − J [e1, Je3] − J [Je1, e3]= [−ϵ1e2,−ϵ2e4] − [e1, e3] − J [e1,−ϵ2e4] − J [−ϵ1e2, e3]= −ϵ1ϵ2f
µ24eµ + fµ
13eµ − ϵ2fµ
14Jeµ − ϵ1fµ
23Jeµ
= −ϵ1ϵ2f4
24e4 + f313e3 + f4
13e4 − ϵ2f4
14Je4 − ϵ1f3
23Je3 − ϵ1f4
23Je4
= (f313 − f4
14 − ϵ1ϵ2f4
23)e3 + (ϵ1ϵ2f323 − ϵ1ϵ2f
424 + f4
13)e4
=1
φ
√ΦΦφφ
(∂1φ + ϵ1ϵ2∂2ψ) e3 +1
φ
√ΦΦφφ
(ϵ1ϵ2∂2φ − ∂1ψ) e4 .
If ϵ1ϵ2 = −1, NJ(e1, e3) = 0 by Cauchy-Riemann equation (4.20). Therefore if we require NJ = 0,relative sign of ϵ1 and ϵ2 is determined by the following relation,
ϵ1ϵ2 = −1 . (A.10)
The reminder of this sections, we impose ϵ1 and ϵ2 on ϵ1ϵ2 = −1.
NJ(e1, e4) = [Je1, Je4] − [e1, e4] − J [e1, Je4] − J [Je1, e4]= [−ϵ1e2, ϵ2e3] − [e1, e4] − J [e1, ϵ2e3] − J [−ϵ1e2, e4]= ϵ1ϵ2f
µ23eµ + fµ
14eµ + ϵ2fµ
13Jeµ − ϵ1fµ
24Jeµ
= ϵ1ϵ2f3
23e3 + ϵ1ϵ2f4
23e4 + f414e4 + ϵ2f
313Je3 + ϵ2f
413Je4 − ϵ1f
424Je4
= (ϵ1ϵ2f323 + f4
13 − ϵ1ϵ2f4
24)e3 + (ϵ1ϵ2f423 + f4
14 − f313)e4
= (−f323 + f4
13 + f424)e3 + (−f4
23 + f414 + f3
13)e4 .
= 0 ,
NJ(e1, e5) = [Je1, Je5] − [e1, e5] − J [e1, Je5] − J [Je1, e5]= [−ϵ1e2,−ϵ3e6] − [e1, e5] − J [e1,−ϵ3e6] − J [−ϵ1e2, e5]= −ϵ1ϵ3f
µ26eµ + fµ
15eµ − ϵ3fµ
16Jeµ − ϵ1fµ
25Jeµ
= −ϵ1ϵ3f6
26e6 + f515e5 + f6
15e6 − ϵ3f6
16Je6 − ϵ1f5
25Je5 − ϵ1f6
25Je6
= (f515 − f6
16 − ϵ1ϵ3f6
25)e5 + (f615 + ϵ1ϵ3f
525 − ϵ1ϵ3f
626)e6
=1
φ
√ΦΦφφ
(∂1φ + ϵ1ϵ3∂2ψ)e5 +1
φ
√ΦΦφφ
(−∂1ψ + ϵ1ϵ3∂2φ)e6 .
If ϵ1ϵ3 = −1, NJ(e1, e5) = 0 by Cauchy-Riemann equation (4.20). Therefore if we require NJ = 0,relative sign of ϵ1 and ϵ3 is determined by the following relation,
ϵ1ϵ3 = −1 . (A.11)
The rest of this sections, we impose ϵ1 and ϵ3 on ϵ1ϵ3 = −1.
NJ(e1, e6) = [Je1, Je6] − [e1, e6] − J [e1, Je6] − J [Je1, e6]= (ϵ1ϵ3f5
25 + f615 − ϵ1ϵ3f
626)e5 + (ϵ1ϵ3f6
25 + f616 − f5
15)e6 .
= 0 .
72
By similar calculation or Je1 = −ϵ1e2 , Je2 = ϵ1e1, we can see immediately
NJ(e2, eµ) = 0 ,
NJ(e3, e4) = [Je3, Je4] − [e3, e4] − J [e3, Je4] − J [Je3, e4]= [−ϵ2e4, ϵ2e3] − [e3, e4]= 0 ,
NJ(e3, e5) = [Je3, Je5] − [e3, e5] − J [e3, Je5] − J [Je3, e5]= [−ϵ2e4,−ϵ3e6] − [e3, e5] − J [e3,−ϵ3e6] − J [−ϵ2e4, e5]= −ϵ2ϵ3f
µ46eµ + fµ
35eµ − ϵ3fµ
36Jeµ − ϵ2fµ
45Jeµ
= 0 ,
NJ(e3, e6) = [Je3, Je6] − [e3, e6] − J [e3, Je6] − J [Je3, e6]= [−ϵ2e4, ϵ3e5] − [e3, e6] − J [e3, ϵ3e5] − J [−ϵ2e4, e6]= 0 ,
By similar calculation or Je3 = −ϵ2e4 , Je4 = ϵ2e3, we can see immediately
NJ(e4, eµ) = 0.
NJ(e5, e6) = [Je5, Je6] − [e5, e6] − J [e5, Je6] − J [Je5, e6]= [−ϵ3e6, ϵ3e5] − [e5, e6]= 0
Consequently under conditions ϵ1ϵ2 = −1 and ϵ1ϵ3 = −1 1 Nijenhuis tensor is vanish,
NJ = 0 . (A.12)
Then almost complex structure J is also complex structure.
1A condition for ϵ2ϵ3 can be obtained from ϵ1ϵ2 = −1 and ϵ1ϵ3 = −1, ϵ2ϵ3 = 1.
73
Appendix B
(1, 0)-form
We introduce complex 1-form ζa, ζ a by the following,
ζ1 =√
φφ(dx1 + idx2) , ζ2 =√
φdx3 − i1√φ
(dx4 − ψdx3) , ζ3 =√
φdx5 − i1√φ
(dx6 − ψdx5) ,
ζ 1 =√
φφ(dx1 − idx2) , ζ 2 =√
φdx3 + i1√φ
(dx4 − ψdx3) , ζ 3 =√
φdx5 + i1√φ
(dx6 − ψdx5) .
(B.1)
Then the dual vector fields are
ζ1 =1
2√
φφ
(∂
∂x1− i
∂
∂x2
), ζ1 =
1
2√
φφ
(∂
∂x1+ i
∂
∂x2
),
ζ2 =12
1√φ
∂
∂x3+ i
(√φ − i
ψ√φ
)∂
∂x4
, ζ2 =
12
1√φ
∂
∂x3− i
(√φ + i
ψ√φ
)∂
∂x4
,
ζ3 =12
1√φ
∂
∂x3+ i
√φ − i
ψ√φ
∂
∂x4
, ζ3 =12
1√φ
∂
∂x3− i
√φ + i
ψ√φ
∂
∂x4
.
(B.2)
Using J∗e1 = ϵ1e
2 , J∗e3 = ϵ2e
4 , J∗e5 = ϵ3e
6 and the definition of ea, we can obtain J∗dxµ andJ∗ζ
a as follows: J∗dxµ (µ = 1, · · · , 6) are
J∗dx1 = ϵ2dx2 , J∗dx2 = −ϵ2dx1 ,
J∗dx3 =ϵ2φ
(dx4 − ψdx3) , J∗dx4 = ϵ2
ψ
φdx4 − φ
(1 +
ψ2
φ2
)dx3
,
J∗dx5 =ϵ3
φ(dx6 − ψdx5) , J∗dx6 = ϵ3
ψ
φdx6 − φ
(1 +
ψ2
φ2
)dx5
(B.3)
and J∗ζa (a = 1, · · · , 6) are
J∗ζ1 = −iϵ1ζ
1 , J∗ζ2 = iϵ2ζ
2 , J∗ζ3 = iϵ3ζ
3 . (B.4)
74
When ϵ1 = −1 together with ϵ2 = 1 and ϵ3 = 1, ζ1, ζ2 and ζ3 are (1,0)-form [56].
Let us rewrite fundamental 2-form κ,
κ = ϵ1ΦΦφφdx12 + ϵ2Φdx34 + ϵ3Φdx56 . (B.5)
By using ζa and ζ a, dxµ (µ = 1, · · · , 6) are given by
dx1 =1
2√
φφ(ζ1 + ζ 1) , dx2 =
1
2i
√φφ
(ζ1 − ζ 1) ,
dx3 =1
2√
φ(ζ2 + ζ 2) , dx4 = −
√φ
2i(ζ2 − ζ 2) +
ψ
2√
φ(ζ2 + ζ 2) ,
dx5 =1
2√
φ(ζ3 + ζ 3) , dx6 = −
√φ
2i(ζ3 − ζ 3) +
ψ
2√
φ(ζ3 + ζ 3) .
(B.6)
Also dxµν are calculated as follows:
dx12 =i
2φφζ1ζ 1 , dx34 = − i
2ζ2ζ 2 , dx56 = − i
2ζ3ζ 3 ,
dx13 =1
4φ
√φ
(ζ1ζ2 + ζ1ζ 2 + ζ 1ζ2 + ζ 1ζ 2) ,
dx14 =1
4√
φ
[(i +
ψ
φ)ζ1ζ2 + (−i +
ψ
φ)ζ1ζ 2 + (i +
ψ
φ)ζ 1ζ2 + (−i +
ψ
φ)ζ 1ζ 2
],
dx15 =1
4φ√
φ(ζ1ζ3 + ζ1ζ 3 + ζ 1ζ3 + ζ 1ζ 3) ,
dx16 =1
4√
φ
[(i +
ψ
φ
)ζ1ζ3 +
(−i +
ψ
φ
)ζ1ζ 3 +
(i +
ψ
φ
)ζ 1ζ3 +
(−i +
ψ
φ
)ζ 1ζ 3
],
dx23 = − i
4φ
√φ
(ζ1ζ2 + ζ1ζ 2 − ζ 1ζ2 − ζ 1ζ 2) ,
dx24 =1
4√
φ
[(1 − i
ψ
φ
)ζ1ζ2 −
(1 + i
ψ
φ
)ζ1ζ 2 +
(−1 + i
ψ
φ
)ζ 1ζ2 +
(1 + i
ψ
φ
)ζ 1ζ 2
],
dx25 = − i
4φ√
φ(ζ1ζ3 + ζ1ζ 3 − ζ 1ζ3 − ζ 1ζ 3) ,
dx26 =1
4√
φ
[(1 − i
ψ
φ
)ζ1ζ3 −
(1 + i
ψ
φ
)ζ1ζ 3 +
(−1 + i
ψ
φ
)ζ 1ζ3 +
(1 + i
ψ
φ
)ζ 1ζ 3
]
75
and then dζa and dζa are given by
dζ1 = − 1
4√
φφ∂1 log φφ + i∂2 log φφζ1ζ 1 ,
dζ 1 =1
4√
φφ∂1 log φφ − i∂2 log φφζ1ζ 1 ,
dζ2 =1
4√
φφ
[(∂1 log φ − i∂2 log φ)ζ1ζ2 + 2(∂1 log φ − i∂2 log φ)ζ1ζ 2 − (∂1 log φ + i∂2 log φ)ζ 1ζ2
],
dζ 2 =1
4√
φφ
[(∂1 log φ + i∂2 log φ)ζ 1ζ 2 − 2(∂1 log φ + i∂2 log φ)ζ2ζ 1 − (∂1 log φ − i∂2 log φ)ζ1ζ 2
],
dζ3 =1
4√
φφ
[(∂1 log φ − i∂2 log φ)ζ1ζ3 + 2(∂1 log φ − i∂2 log φ)ζ1ζ 3 − (∂1 log φ + i∂2 log φ)ζ 1ζ3
],
dζ3 =1
4√
φφ
[(∂1 log φ + i∂2 log φ)ζ 1ζ 3 − 2(∂1 log φ + i∂2 log φ)ζ3ζ 1 − (∂1 log φ − i∂2 log φ)ζ1ζ 3
].
(B.7)
Therefore we have κ expressed as (1,1)-form,
κ =i
2ϵ1ΦΦζ1ζ 1 − ϵ2Φζ2ζ 2 − ϵ3Φζ3ζ 3 . (B.8)
We consider the Bismut torsion T = ϵ4J∗dκ. The exterior derivative of κ is given by
dκ = − i
4√
φφ
[ϵ2
(∂1Φ − i∂2Φ)ζ12ζ 2 − (∂1Φ + i∂2Φ)ζ2ζ 12
+ϵ3
(∂1Φ − i∂2Φ)ζ13ζ 3 − (∂1Φ + i∂2Φ)ζ3ζ 13
]and thus we have
J∗dκ = − i
4√
φφ
[ϵ2
−iϵ1(∂1Φ − i∂2Φ)ζ12ζ 2 − iϵ1(∂1Φ + i∂2Φ)ζ2ζ 12
+ϵ3
−iϵ1(∂1Φ − i∂2Φ)ζ13ζ 3 − iϵ1(∂1Φ + i∂2Φ)ζ3ζ 13
]=
1
4√
φφ
[(∂1Φ − i∂2Φ)ζ12ζ 2 + (∂1Φ + i∂2Φ)ζ2ζ 12
+
(∂1Φ − i∂2Φ)ζ13ζ 3 + (∂1Φ + i∂2Φ)ζ3ζ 13
]=
1
4√
φφ
[(∂1Φ)(ζ1 − ζ 1)ζ2ζ 2 − i(∂2Φ)(ζ1 + ζ 1)ζ2ζ 2 + (∂1Φ)(ζ1 − ζ 1)ζ3ζ 3 − i(∂2Φ)(ζ1 + ζ 1)ζ3ζ 3
]= (∂2Φ)dx134 − (∂1Φ)dx234 + (∂2Φ)dx156 − (∂1Φ)dx256 .
Indeed, this is coincident with the Bismut torsion (4.26) under ϵ4 = −1.
76
Appendix C
The SU(3)-structure on the superposed HKT metrics
The necessary and sufficient conditions for preservation of supersymmetry in type II super-gravity are that the manifold M6 has an SU(3) structure satisfying the differential conditions[9],
d(e−2ϕΥ) = 0 (C.1)d(e−2ϕ ∗ κ) = 0 (C.2)T = −e2ϕ ∗ d(e−2ϕκ) , (C.3)
where ϕ is dilaton and Υ is complex (3,0)-form. In this appendix, we check the manifold (M6, g, J)admits the SU(3)-structure, where the metric g is given by (4.19) and the complex structure Jis defined by (4.23).
Lee form Θ is defined by
Θ(X) = −12
6∑µ=1
T (JX, eµ, Jeµ) , (C.4)
where X is an vector field on TM6. From (4.23), an complex structure J : TpM6 −→ TpM6 isgiven by
Je1 = −ϵ1e2 , Je3 = −ϵ2e4 , Je5 = −ϵ3e6, (C.5)
where eµ is dual vector for eµ and ϵa = ±1 (a = 1, 2, 3). Then, Θ(X) is
Θ(X) = ϵ1T (JX, e1, e2) + ϵ2T (JX, e3, e4) + ϵ3T (JX, e5, e6) . (C.6)
Hence components of the Lee form θ in the orthonormal base eµ are the following,
Θ(e1) = −ϵ1ϵ2T (e2, e3, e4) − ϵ1ϵ3T (e2, e5, e6) ,
= − ϵ4
Φ√
ΦΦφφ∂1Φ − ϵ4
Φ√
ΦΦφφ∂1Φ ,
Θ(e2) = ϵ1ϵ2T (e1, e3, e4) + ϵ1ϵ3T (e1, e5, e6) ,
= − ϵ4
Φ√
ΦΦφφ∂2Φ − ϵ4
Φ√
ΦΦφφ∂2Φ ,
77
Θ(e3) = −ϵ1ϵ2T (e4, e1, e2) − ϵ2ϵ3T (e4, e5, e6) ,
= 0 ,
Θ(e4) = ϵ1ϵ2T (e3, e1, e2) + ϵ2ϵ3T (e3, e5, e6) ,
= 0 ,
Θ(e5) = −ϵ1ϵ3T (e6, e1, e2) − ϵ2ϵ3T (e6, e3, e4) ,
= 0 ,
Θ(e6) = ϵ1ϵ3T (e5, e1, e2) + ϵ2ϵ3T (e5, e3, e4) ,
= 0 .
Therefore the Lee form Θ is
Θ = − ϵ4√ΦΦφφ
[(1Φ
∂1Φ +1Φ
∂1Φ)
e1 +(
1Φ
∂2Φ +1Φ
∂2Φ)
e2
]. (C.7)
Then it should be noticed that Θ can be rewritten as an exact form (dΘ = 0);
Θ = −ϵ4
(d log Φ + d log Φ
)= −ϵ4d log ΦΦ . (C.8)
When we identify dilaton with Lee form,
dϕ = −12ϵ4Θ , (C.9)
the dilaton ϕ is related to Θ under ϵ4 = −1,
Θ = 2dϕ , ϕ = log√
ΦΦ . (C.10)
The Bismut torsion T is given byT = ϵ4J∗dκ (C.11)
and the torsion T satisfies (C.3) under the condition ϵ4 = −1 as follows: Firstly we consider(C.3). The exterior derivative of κ is
dκ =ϵ2
Φ√
ΦΦφφ(∂1Φe134 + ∂2Φe234) +
ϵ3
Φ√
ΦΦφφ(∂1Φe156 + ∂2Φe256)
and thus
d(e−2ϕκ) = d(1
ΦΦκ) ,
=1
ΦΦ√
ΦΦφφ
[−ϵ2(∂1 log Φ)e134 − ϵ2(∂2 log Φ)e234 − ϵ3(∂2 log Φ)e156 − ϵ3(∂2 log Φ)e256
].
78
The Hodge dual associated with (4.19) of e134, e234, e156 and e256 are given by
∗e134 = ιe4ιe3ιe1e123456 = e256 , ∗e234 = ιe4ιe3ιe2e
123456 = −e156 ,
∗e156 = ιe6ιe5ιe1e123456 = e234 , ∗e256 = ιe6ιe5ιe2e
123456 = −e134 , (C.12)
and thus the torsion T in (C.3) is the following,
T = − ϵ2√ΦΦφφ
[(∂2 log Φ)e134 − (∂2 log Φ)e234 + (∂2 log Φ)e156 − (∂1 log Φ)e256] (∵ ϵ2 = ϵ3)(C.13)
Secondly, we consider (C.11). J∗eabc are given by
J∗e134 = J∗e
1 ∧ J∗e3 ∧ J∗e
4 = −ϵ1e243 = ϵ1e
234 ,
J∗e234 = J∗e
2 ∧ J∗e3 ∧ J∗e
4 = ϵ1e143 = −ϵ1e
134 ,
J∗e156 = J∗e
1 ∧ J∗e5 ∧ J∗e
6 = −ϵ1e265 = ϵ1e
256 ,
J∗e256 = J∗e
2 ∧ J∗e5 ∧ J∗e
6 = ϵ1e165 = −ϵ1e
156 ,
and then the torsion T in (C.11) takes the form
T = − ϵ4√ΦΦφφ
− 1
Φ(∂2Φe134 − ∂1Φe234) − 1
Φ(∂2Φe156 − ∂1Φe256)
,
= −ϵ4−(∂2Φdx134 − ∂1Φdx234) − (∂2Φdx156 − ∂1Φdx256) . (C.14)
Thus, the Bismut torsion (C.11) coincides with (C.14) when ϵ4 = −1 and ϵ2 = 1. In addition, wecalculate dT ,
dT = ϵ1ϵ4ϵ2(∂22Φ + ∂2
1Φ)dx1234 + ϵ3(∂22Φ + ∂2
1Φ)dx1256 . (C.15)
Namely, the 3-form torsion T is closed form, dT = 0, if Φ and Φ are harmonic function on R2.
We consider (C.2). The Hodge dual associated with (4.19) of the fundamental 2-form κ isgiven by
∗κ = ϵ1e3456 + ϵ2e
1256 + ϵ3e1234 ,
= ϵ1ΦΦdx3456 + ϵ2ΦΦ2φφdx1256 + ϵ3Φ2Φφφdx1234 .
Therefore d ∗ κ is
d ∗ κ = ϵ1∂1(ΦΦ)dx13456 + ϵ1∂2(ΦΦ)dx23456 ,
=ϵ1
ΦΦ√
ΦΦφφ
∂1(ΦΦ)e13456 + ∂2(ΦΦ)e23456
,
79
and thus we have
d(e−2ϕ ∗ κ) = e−2ϕd ∗ κ − 2e−2ϕdϕ ∧ ∗κ ,
= e−2ϕ ϵ1
ΦΦ√
ΦΦφφ
∂1(ΦΦ)e13456 + ∂2(ΦΦ)e23456
−e−2ϕ 1
ΦΦ√
ΦΦφφ
∂1(ΦΦ)e1 + ∂2(ΦΦ)e2
∧ (ϵ1e3456 + ϵ2e
1256 + ϵ3e1234) ,
= 0 .
The rest task is to check (C.1). (3,0)-form Υ and fundamental 2-form κ have a volume matchingcondition [25],
Υ ∧ Υ = i43κ3 . (C.16)
Because Υ is (3,0)-form it takes the ensuing form under conditions ϵ1 = −1, ϵ2 = 1, ϵ3 = 1,
Υ = hζ123 , (C.17)
where ζ1, ζ2 and ζ3 are given by
ζ1 =√
φφ(dx1 + idx2) , ζ2 =√
φdx3 − i1√φ
(dx4 − ψdx3) , ζ3 =√
φdx5 − i1√φ
(dx6 − ψdx5).
(C.18)Then, eµ are rewritten as follows (see also Appendix 2):
e1 =1
2√
φφ
√ΦΦφφ(ζ1 + ζ 1) =
12
√ΦΦ(ζ1 + ζ 1) , e2 =
1
2i
√φφ
√ΦΦφφ(ζ1 − ζ 1) =
12i
√ΦΦ(ζ1 − ζ 1) ,
e3 =1
2√
φ
√Φφ(ζ2 + ζ 2) =
12
√Φ(ζ2 + ζ 2) , e4 = −
√φ
2i
√Φφ
(ζ2 − ζ 2) = − 12i
√Φ(ζ2 − ζ 2) ,
e5 =1
2√
φ
√Φφ(ζ3 + ζ 3) =
12
√Φ(ζ3 + ζ 3) , e6 = −
√φ
2i
√Φφ
(ζ3 − ζ 3) = − 12i
√Φ(ζ3 − ζ 3) .
Hence κ3 is calculated as follows: eab (a, b = 1, · · · , 6) are written by means of ζi and ζ i
e12 = − 12i
ΦΦζ1 ∧ ζ 1 =i
2ΦΦζ1 ∧ ζ 1 ,
e34 =12i
Φζ2 ∧ ζ 2 = − i
2Φζ2 ∧ ζ 2 ,
e56 =12i
Φζ3 ∧ ζ 3 = − i
2Φζ3 ∧ ζ 3 ,
80
and thus κ3 is given by
κ3 = 6ϵ1ϵ2ϵ3e123456 ,
=34iϵ1ϵ2ϵ3(ΦΦ)2ζ123 ∧ ζ 123 .
By (C.17) Υ ∧ Υ is also calculated as
Υ ∧ Υ = hhζ123 ∧ ζ 123 .
Therefore from (C.16) absolute value of the function h is given by
|h(z, z)|2 = −ϵ1ϵ2ϵ3(ΦΦ)2 (C.19)
and thus (3,0)-form Υ is written as
Υ =√
−ϵ1ϵ2ϵ3(ΦΦ)2fpζ123 , (C.20)
where fp is a phase function and |fp| = 1. Now, (3,0)-form Υ should satisfy (C.1) and we have
d(e−2ϕΥ) = d(√
ϵ1ϵ2ϵ3fpζ123
)(∵ e2ϕ = ΦΦ)
=√
ϵ1ϵ2ϵ3(dfp ∧ ζ123 + fpdζ1 ∧ ζ23 − fpζ1 ∧ dζ2 ∧ ζ3 + fpζ
12 ∧ dζ3)
=√
ϵ1
(∂1fp)1
2√
φφ(ζ1 + ζ 1) + (∂2fp)
1
2i
√φφ
(ζ1 − ζ 1) + (∂3fp)1
2√
φ(ζ2 + ζ 2)
+(∂4fp)−√
φ
2i(ζ2 − ζ 2) +
ψ
2√
φ(ζ2 + ζ 2)
+ (∂5fp)
1
2√
φ(ζ3 + ζ 3)
+(∂6fp)
−
√φ
2i(ζ3 − ζ 3) +
ψ
2√
φ(ζ3 + ζ 3)
ζ123 + 0
=√
ϵ1
2√
φφ(∂1fp + i∂2fp)ζ 1ζ123 +
√ϵ1
2√
φ[∂3fp + (ψ − iφ)∂4fp] ζ 2ζ123
+√
ϵ1
2√
φ
[∂5fp + (ψ − iφ)∂6fp
]ζ 3ζ123
To satisfy (C.1), the following equation required for a phase function fp:
∂1fp + i∂2fp = 0 , ∂3fp + (ψ − iφ)∂4fp = 0 , ∂5fp + (ψ − iφ)∂6fp = 0 . (C.21)
Decomposing fp into the real part and the imaginary part, fp = FR + iFI , the equations aboutfp are the following:
∂1FR − ∂2FI = 0 , ∂1FI + ∂2FR = 0 ,
∂3FR + ψ∂4FR + φ∂4FI = 0 , ∂3FI − φ∂4FR + ψ∂4FI = 0 ,
∂5FR + ψ∂6FR + φ∂6FI = 0 , ∂5FI − φ∂6FR + ψ∂6FI = 0 . (C.22)
81
Here φ, ψ, φ, ψ depend only x1, x2 and thus we obtain the following relation by (C.21),
∂3fp
∂4fp= −(ψ − iφ) independent of x3, x4 ,
∂5fp
∂6fp= −(ψ − iφ) independent of x5, x6 . (C.23)
Then, fp isfp(x1, x2, x3, x4, x5, x6) = f(x1, x2)ei(c3x3+c4x4)ei(c5x5+c6x6) , (C.24)
where c3, c4, c5, c6 are dependent on x1 and x2. Substituting this fp for (C.21), the ensuingequations is lead;
c3 + c4(ψ − iφ) = 0 , c5 + c6(ψ − iφ) = 0 . (C.25)
To be valid for any harmonic functions φ, ψ, φ, ψ, c3, c4, c5, c6 vanish,
c3 = c4 = c5 = c6 = 0 , (C.26)
Therefore the phase function fp isfp = fp(x1, x2) . (C.27)
Also fp is holomorphic function because it’s real part FR and it’s imaginary part FI satisfyCauchy–Riemann equations
∂1FR = ∂2FI , ∂1FI = −∂2FR . (C.28)
Hence fp is bounded on complex plane C, an entire function. Let us recall Liouville’s theorem,
theorem 1 (Liouville’s theorem). If and only if an entire function is bounded, the function is aconstant.
By this theorem and |fp| = 1,fp = 1 . (C.29)
After all, Υ is written as
Υ =√
ϵ1ΦΦζ123 , (∵ ϵ2ϵ3 = 1 from ϵ1ϵ2 = −1 and ϵ1ϵ3 = −1) (C.30)
or by using orthonormal base eµ,
Υ =√
ϵ1(e1 + i e2) ∧√
ϵ2(e3 − i e4) ∧√
ϵ3(e5 − i e6) . (C.31)
Clearly, the (3,0)-form Υ satisfy (C.1).
Consequently, the fundamental 2-form κ(4.24), the (3,0)-form Υ(4.31), the Lee form Θ(C.7),and the Bismut torsion T (4.26) satisfy (C.1), (C.2), and (C.3). Namely, the triplet (g, T, ϕ),
g6 = ΦΦφφ(dx1)2 + (dx2)2 + Φφ(dx3)2 +Φφ
(dx4 − ψdx3)2 + Φφ(dx5)2 +Φφ
(dx6 − ψdx5)2 ,
(C.32)T = −(∂2Φdx134 − ∂1Φdx234) − (∂2Φdx156 − ∂1Φdx256) , (C.33)
ϕ = log√
ΦΦ + const. , (C.34)
constructs the geometry admitting the SU(3)-structure classified by (C.1), (C.2) and (C.3). Ob-viously, this result is valid for the condition φ = φ = Φ = Φ.
82
C.1 Curvature
When Φ = Φ = φ = φ, the components of the Bismut curvature are listed as follows:
R+1212 = − 2
φ4(∂1
2 log |φ| + ∂22 log |φ|) ,
R+3412 = R+
5612 = − 1φ4
(∂12 log |φ| + ∂2
2 log |φ|) ,
R+1313 = −R+
2413 = − 1φ4
(∂2
1 log |φ| − (∂1 log |φ|)2 + 3(∂2 log |φ|)2)
,
R+1323 = −R+
2423 = − 1φ4
(∂1∂2 log |φ| − 4(∂1 log |φ|)(∂2 log |φ|)) ,
R+1413 = R+
2313 = − 1φ4
(∂1∂2 log |φ| − 4(∂1 log |φ|)(∂2 log |φ|)) ,
R+1423 = R+
2323 = − 1φ4
(∂2
2 log |φ| − (∂2 log |φ|)2 + 3(∂1 log |φ|)2)
,
R+1515 = −R+
2615 = − 1φ4
(∂1
2 log |φ| − (∂1 log |φ|)2 + 3(∂2 log |φ|)2)
,
R+1525 = −R+
2625 = − 1φ4
(∂1∂2 log |φ| − 4(∂1 log |φ|)(∂2 log |φ|)) ,
R+1615 = R+
2515 = − 1φ4
( ∂1∂2 log |φ| − 4(∂1 log |φ|)(∂2 log |φ|) ) ,
R+1625 = R+
2525 = − 1φ4
( ∂22 log |φ| − (∂2 log |φ|)2 + 3(∂1 log |φ|)2 ) ,
R+3535 = R+
4635 = − 1φ4
( (∂1 log |φ|)2 + (∂2 log |φ|)2 ) ,
R+1234 = R+
1256 = R+1314 = R+
1324 = R+1414 = R+
1424 = R+1516 = R+
1526 = R+1616 = R+
1626 = 0 ,
R+2314 = R+
2324 = R+2414 = R+
2424 = R+2516 = R+
2526 = R+2616 = R+
2626 = 0 ,
R+3434 = R+
3536 = R+3545 = R+
3546 = R+3635 = R+
3645 = R+4535 = R+
4536 = R+5656 = 0 .
Also, the components of the Hull curvature are listed as follows:
R−1212 = − 2
φ4(∂1
2 log |φ| + ∂22 log |φ|) , R−
1234 = R−1256 =
1φ4
( (∂1 log |φ|)2 + (∂2 log |φ|)2 ) ,
R−1313 = − 1
φ4( ∂2
1 log |φ| − (∂1 log |φ|)2 + 3(∂2 log |φ|)2 ) ,
R−1314 = − 1
φ4( ∂1∂2 log |φ| − 4(∂1 log |φ|)(∂2 log |φ|) ) ,
83
R−1323 = − 1
φ4( ∂1∂2 log |φ| − 4(∂1 log |φ|)(∂2 log |φ|) ) ,
R−1324 = − 1
φ4( ∂2
2 log |φ| − 2(∂2 log |φ|)2 + 2(∂1 log |φ|)2 ) ,
R−1515 = − 1
φ4( ∂1
2 log |φ| − (∂1 log |φ|)2 + 3(∂2 log |φ|)2 ) ,
R−1516 = − 1
φ4( ∂1∂2 log |φ| − 4(∂1 log |φ|)(∂2 log |φ|) ) ,
R−1525 = − 1
φ4( ∂1∂2 log |φ| − 4(∂1 log |φ|)(∂2 log |φ|) ) ,
R−1526 = − 1
φ4( (∂2
2 log |φ|) − 2(∂2 log |φ|)2 + 2(∂1 log |φ|)2 ) ,
R−2313 =
1φ4
(−∂1∂2 log |φ| + 4(∂1 log |φ|)(∂2 log |φ|) ) ,
R−2314 =
1φ4
( ∂12 log |φ| − 2(∂1 log |φ|)2 + 2(∂2 log |φ|)2 ) ,
R−2323 =
1φ4
(−∂22 log |φ| + (∂2 log |φ|)2 − 3(∂1 log |φ|)2 ) ,
R−2324 =
1φ4
( ∂2∂1 log |φ| − 4(∂1 log |φ|)(∂2 log |φ|) ) ,
R−2515 =
1φ4
(−∂1∂2 log |φ| + 4(∂1 log |φ|)(∂2 log |φ|) ) ,
R−2516 =
1φ4
( ∂12 log |φ| − 2(∂1 log |φ|)2 + 2(∂2 log |φ|)2 ) ,
R−2525 =
1φ4
(−∂22 log |φ| + (∂2 log |φ|)2 − 3(∂1 log |φ|)2 ) ,
R−2526 =
1φ4
( ∂1∂2 log |φ| − 4(∂1 log |φ|)(∂2 log |φ|) ) ,
R−3535 = − 1
φ4( (∂1 log |φ|)2 + (∂2 log |φ|)2 ) ,
R−3546 = − 1
φ4( (∂2 log |φ|)2 + (∂1 log |φ|)2 ) ,
R−1413 = R−
1414 = R−1423 = R−
1424 = R−1615 = R−
1616 = R−1625 = R−
1626 = 0 ,
R−2413 = R−
2414 = R−2423 = R−
2424 = R−2615 = R−
2616 = R−2625 = R−
2626 = 0 ,
R−3412 = R−
3434 = R−3536 = R−
3545 = R−3635 = R−
3645 = R−4535 = R−
4536 = R−4635 = R−
5612 = R−5656 = 0 .
84
Appendix D
SU(3) structure of the T-dualized solution along ∂3
We T-dualize the solutions (4.47), (4.48), and (4.49) along ∂3 and then they take the followingform:
g6 =1
(φ2 + ψ2)(dx3 − B1dx4)2 + φ4(dx1)2 + (dx2)2 +
φ2
φ2 + ψ2(dx4)2 (D.1)
+φ2(dx5)2 + (dx6 − ψdx5)2 , (D.2)
B = − ψ
φ2 + ψ2dx4 ∧ dx3 + B2dx56 , (D.3)
e2ϕ =1
(φ2 + ψ2)e2ϕ . (D.4)
Let us introduce an new orthonormal frame;
e1 = φ2dx1 , e2 = φ2dx2 , e3 =1√
φ2 + ψ2(dx3 − B1dx4) , e4 =
φ√φ2 + ψ2
dx4 ,
e5 = φdx5 , e6 = dx6 − ψdx5 (D.5)
and exterior derivatives of them are given by
de1 = −2φ∂φ
∂x2dx12 = − 2
φ3
∂φ
∂x2e12 , de2 =
2φ3
∂φ
∂x1e12 ,
de3 =1
φ2(φ2 + ψ2)(−(φ∂1φ − ψ∂2φ)e13 − (φ∂2φ + ψ∂1φ)e23 ) − 1
φ3∂1B1e
14 − 1φ3
∂2B1e24 ,
de4 =ψ
φ3(φ2 + ψ2)( (ψ∂1φ + φ∂2φ)e14 + (ψ∂2φ − φ∂1φ)e24 ) ,
de5 =1φ3
(∂1φe15 + ∂2φe25) ,
de6 =1φ3
(∂2φe15 − ∂1φe25) . (D.6)
85
Thus we can read the structure constants fµνρ which are defined by deµ = 1
2 fµνρeνρ as follows;
f112 = − 2
φ3∂2φ , f2
12 =2φ3
∂1φ ,
f313 = − 1
φ2(φ2 + ψ2)(φ∂1φ − ψ∂2φ) , f3
23 = − 1φ2(φ2 + ψ2)
(φ∂2φ + ψ∂1φ) ,
f314 = − 1
φ3∂1B1 , f3
24 = − 1φ3
∂2B1 ,
f414 =
ψ
φ3(φ2 + ψ2)(ψ∂1φ + φ∂2φ) , f4
24 =ψ
φ3(φ2 + ψ2)(ψ∂2φ − φ∂1φ) ,
f515 =
1φ3
∂1φ , f525 =
1φ3
∂2φ , f615 =
1φ3
∂2φ , f625 = − 1
φ3∂1φ . (D.7)
A flux of the B-field is defined byH = −dB (D.8)
and thus the flux H is given by
H = − 1φ3(φ2 + ψ2)
( ((ψ2 − φ2)∂2φ − 2φψ∂1φ)e134 + ((φ2 − ψ2)∂1φ − 2φψ∂2φ)e234 )
− 1φ3
(∂1B2)e156 − 1φ3
(∂2B2)e256 . (D.9)
We should investigate that the triplet (g6, H, ϕ) satisfies supersymmetry preserving conditions:
d(e−2ϕ ∗ κ) = 0 , (D.10)d(e−2ϕΥ) = 0 , (D.11)∗H = e2ϕd(e−2ϕκ) , (D.12)
where J , κ, Υ(3,0) represent an almost complex structure, a fundamental 2-form, and a (3,0)-formrespectively.
We assume that a fundamental 2-form takes the form
κ = K(x1, x2)(ε1e12 + ε2e
34 + ε3e56) . (D.13)
Let us determine a function K(x1, x2) such that κ satisfies (D.10). We define a volume form tothe metric (D.2) by
vol. = ∗1 , vol. = e123456 . (D.14)
Then Hodge dual of e12, e34 and e56 are given by
∗e12 = ιe2ιe1vol. = e3456 , ∗e34 = e1256 , ∗e56 = e1234 .
We have∗κ = K(ε1e
3456 + ε2e1256 + ε3e
1234) (D.15)
86
ande−2ϕ ∗ κ =
Kh
φ2(ε1e
3456 + ε2e1256 + ε3e
1234) ,
whereh ≡ φ2 + ψ2 . (D.16)
Therefore the left hand side of (D.10) takes the form,
d(e−2ϕ ∗ κ) =K
φ2dh ∧ (ε1e
3456 + ε2e1256 + ε3e
1234) − hK
φ4d(φφ) ∧ (ε1e
3456 + ε2e1256 + ε3e
1234)
+h
φ2dK ∧ (ε1e
3456 + ε2e1256 + ε3e
1234) +hK
φ2(ε1de3456 + ε2de1256 + ε3de1234)
(D.17)
and each terms of (D.17) are the following:
1st term = ε12Kφ2
φ6( (φ∂1φ − ψ∂2φ)e13456 + (φ∂2φ + ψ∂1φ)e23456 ) ,
2nd term = −ε1hK
φ6( ∂1φ
2e13456 + ∂2φ2e23456 ) ,
3rd term = ε1h
φ6( φ2(∂1K)e13456 + φ2(∂2K)e23456 ) ,
4th term = ε1hK
φ6( φ4(f3
13 + f414 + f5
15)e13456 + φ4(f323 + f4
24 + f525)e23456 ) .
Then we have
d(e−2ϕ ∗ κ) =ε1
φ6( h(φ2∂1K + φ4K(f3
13 + f414 + f5
15)) − hK∂1φ2 + 2Kφ2(φ∂1φ − ψ∂2φ) )e13456
+ε1
φ6( h(φ2∂2K + φ4K(f3
23 + f424 + f5
25)) − hK∂2φ2 + 2Kφ2(φ∂2φ + ψ∂1φ) )e23456 ,
where
hφ4(f313 + f4
14 + f515) = −φ2(φ∂1φ − ψ∂2φ) + ψ2φ∂1φ + φ2ψ∂2φ + hφ∂1φ
and
hφ4(f323 + f4
24 + f525) = −φ2(φ∂2φ + ψ∂1φ) + ψ2φ∂2φ − φ2ψ∂1φ + hφ∂2φ .
Therefore we have
d(e−2ϕ ∗ κ) =ε1
φ6hφ2( (∂1K)e13456 + (∂2K)e23456 )
and thus we obtain the following equations for the function K(x1, x2) by (D.10)
∂1K = 0 , ∂2K = 0 .
87
Then, the function K(x1, x2) = 1. Consequently, the fundamental 2-form κ takes the form
κ = ε1e12 + ε2e
34 + ε3e56 . (D.18)
From (D.18), an almost complex structure J is defined by
J e1 = −ε1e2 , J e3 = −ε2e4 , J e5 = −ε3e6 (D.19)
and J∗eµ (µ = 1, · · · 6) are given by
J∗e1 = ε1e2 , J∗e3 = ε2e4 , J∗e5 = ε3e6 . (D.20)
A Nijenhuis tensor N associated with the almost complex structure J is defined by
NJ(X,Y ) = [JX , JY ] − [X , Y ] − J [JX , Y ] − J [X , JY ] , (D.21)
where a commutation relation between vector fields are given by [eµ , eν ] = −fρµν eρ. We require
ε1ε2 = −1 and then we have
NJ(e1, e3) =1φ3
(−(∂1φ + ∂2B1)e3 + (∂2φ − ∂1B1)e4 ) ,
NJ(e2, e3) =1φ3
( (−∂2φ + ∂1B1)e3 − (∂1φ + ∂2B1)e4 ) ,
NJ(e1, e4) =1φ3
(∂2φ − ∂1B1)e3 +1φ3
(∂1φ + ∂2B1)e4 ,
NJ(e2, e4) = − 1φ3
(∂1φ + ∂2B1)e3 −1φ3
(−∂2φ + ∂1B1)e4 . (D.22)
If B1 = −ψ, then from Cauchy–Riemann equations (4.20) the components of the Nijenhuis tensorare the following,
NJ(e1, e3) = 0 , NJ(e2, e3) = 0 , NJ(e1, e4) = 0 , NJ(e2, e4) = 0 . (D.23)
Other components of the Nijenhuis tensor NJ(eµ, eν) are also clearly zero under the conditionε1ε3 = −1. Hence the almost complex structure J is a complex structure when B1 = −ψ,ε1ε2 = ε1ε3 = −1.
A Bismut torsion associated with J is defined by
T := ε4J∗dκ . (D.24)
The exterior derivative of κ is given by
dκ =ε2
hφ3( (ψ2 − φ2)∂1φ + 2φψ∂2φ )e134 +
ε2
hφ3( (ψ2 − φ2)∂2φ − 2φψ∂1φ )e234
+ε3
φ3∂1φe156 +
ε3
φ3∂2φe256 (D.25)
88
and we have
J∗dκ =ε2
hφ3( (ψ2 − φ2)∂1φ + 2φψ∂2φ )J∗e
1 ∧ J∗e3 ∧ J∗e
4
+ε2
hφ3( (ψ2 − φ2)∂2φ − 2φψ∂1φ )J∗e
2 ∧ J∗e3 ∧ J∗e
4
+ε3
φ3∂1φJ∗e
1 ∧ J∗e5 ∧ J∗e
6 +ε3
φ3∂2φJe2 ∧ J e5 ∧ J e6
= − 1hφ3
( (ψ2 − φ2)∂1φ + 2φψ∂2φ )e234 +1
hφ3( (ψ2 − φ2)∂2φ − 2φψ∂1φ )e134
− 1φ3
∂1φe256 +1φ3
∂2φe156 .
Hence the Bismut torsion takes the following form,
T =ε4
hφ3( ( (φ2 − ψ2)∂1φ − 2φψ∂2φ )e234 + ( (ψ2 − φ2)∂2φ − 2φψ∂1φ )e134 − h(∂1φe256 − ∂2φe156) ) .
On the other hand, deformed flux H with the conditions B1 = −ψ and B2 = −ψ is given by
H =−1hφ3
( ( (ψ2 − φ2)∂2φ − 2φψ∂1φ )e134 + ( (φ2 − ψ2)∂1φ − 2φψ∂2φ )e234 + h(∂2φe156 − ∂1φe256) )
and thus we see that H = T when ε4 = −1.
Let us check whether the torsion T with ε4 = −1 satisfies the equation dT κ = 0 or not.According to Ref. [57], an exterior derivative with a torsion for deformed fundamental 2-form κis given by
dT κ = dκ −6∑
µ=1
(ιeµ T ) ∧ (ιeµ κ) . (D.26)
The first term of dT κ has been given by (D.25). On the second term, (ιeµ T ) ∧ (ιeµ κ) for eachvalues of µ = 1, · · · , 6 are given by
(ιe1 T ) ∧ (ιe1 κ) =ε1ε4
hφ3( ( (ψ2 − φ2)∂2φ − 2φψ∂1φ )e234 + h∂2φe256 ) ,
(ιe2 T ) ∧ (ιe2 κ) =ε1ε4
hφ3( ( (ψ2 − φ2)∂1φ + 2φψ∂2φ )e134 + h∂1φe156 ) ,
(ιe3 T ) ∧ (ιe3 κ) = 0 , (ιe4 T ) ∧ (ιe4 κ) = 0 , (ιe5 T ) ∧ (ιe5 κ) = 0 , (ιe6 T ) ∧ (ιe6 κ) = 0
and thus we have
6∑µ=1
(ιeµ T ) ∧ (ιeµ κ) =ε1ε4
hφ3
(( (ψ2 − φ2)∂1φ + 2φψ∂2φ )e134
+( (ψ2 − φ2)∂2φ − 2φψ∂1φ )e234 + h(∂1φe156 + ∂2φe256))
.
89
Therefore dT κ is
dT κ =ε2
hφ3( ( (ψ2 − φ2)∂1φ + 2φψ∂2φ )e134 + ( (ψ2 − φ2)∂2φ − 2φψ∂1φ )e234 )
+ε3
φ3(∂1φe156 + ∂2φe256) − ε1ε4
hφ3
(( (ψ2 − φ2)∂1φ + 2φψ∂2φ )e134
+( (ψ2 − φ2)∂2φ − 2φψ∂1φ )e234 + h(∂1φe156 + ∂2φe256))
and thus we see that dT κ = 0 when ε1ε2 = −1, ε1ε3 = −1 and ε4 = −1.
A Lee form Θ is defined byΘ = ε4J∗δκ , (D.27)
where the operator δ is a co-derivative on p-form that is defined by δ = (−1)np−p2+p ∗ d∗. Theexterior derivatives of ∗κ is given by
d ∗ κ =2ε1ψ
φ3(φ2 + ψ2)( (ψ∂1φ + φ∂2φ)e13456 + (−φ∂1φ + ψ∂2φ)e23456 ) (D.28)
and thus the Lee form takes the following form
Θ = −ε4d
(log
φ2
φ2 + ψ2
). (D.29)
If we identify a dilaton with the Lee form by dϕ = −12ε4Θ, the dilaton ϕ is given by
ϕ =12
logφ2
φ2 + ψ2(D.30)
and this is consistent with (4.52).
We consider (D.12). The Hodge dual of H is given by
∗H = − 1φ3(φ2 + ψ2)
( ( (ψ2 − φ2)∂2φ − 2φψ∂1φ )e256 − ( (φ2 − ψ2)∂1φ − 2φψ∂2φ )e156 )
− 1φ3
∂1B2e234 +
1φ3
∂2B2e134 . (D.31)
The right hand side of (D.12) becomes
e2ϕd(e−2ϕκ) = − ε2
hφ3∂1φe134 − ε2
hφ3∂2φe234 +
ε3
hφ3( (φ2 − ψ2)∂1φ − 2φψ∂2φ )e156
+ε3
hφ3( (φ2 − ψ2)∂2φ + 2φψ∂1φ )e256
and thus we have
∗H − e2ϕd(e−2ϕκ) =1φ3
(∂2B2 + ε2∂1φ)e134 − 1φ3
(∂1B2 − ε2∂2φ)e234
+1
hφ3( (φ2 − ψ2)∂1φ − 2φψ∂2φ − ε3( (φ2 − ψ2)∂1φ − 2φψ∂2φ ) )e156
− 1hφ3
( (ψ2 − φ2)∂2φ − 2φφ∂1φ + ε3( (φ2 − ψ2)∂2φ + 2φψ∂1φ ) )e256 .
90
If ε2 = ε3 = 1 and B1 = B2 = −ψ, we have ∗H = e2ϕd(e−2ϕκ). Namely, the triplet (g, H, ϕ)satisfies (D.12) when ε2 = ε3 = 1 and B1 = B2 = −ψ.
We consider a (3,0)-form Υ which satisfies (D.11).We assume that Υ takes the form
Υ =√
ε1ε2ε3W (x1, x2)(e1 + i e2) ∧ (e3 − i e4) ∧ (e5 − i e6)=
√ε1ε2ε3W (x1, x2)( e135 + e245 + e236 − e146 + i(e235 − e145 − e136 − e246) ) , (D.32)
where the function W is a complex function. A (3, 0)-form Υ has the relation between thefundamental 2-form κ by the volume matching condition,
Υ ∧ ¯Υ =43i κ3 . (D.33)
Calculating (D.33) directly, we obtain the conditions
ε1ε2ε3 = −1 , |W |2 = 1 (D.34)
and thus the complex function W is a phase function,
W (x1, x2) = fp(x1, x2) , (D.35)
where |fp| = 1. We determine fp so as to satisfy (D.11). e−2ϕΥ is given by
e−2ϕΥ = ihfp
φ2( e135 + e245 + e236 − e146 + i(e235 − e145 − e136 − e246) ) .
The exterior derivative of e−2ϕΥ is calculated as follows,
d(e−2ϕΥ) =i
φ4( ih∂1fp + ifp(φ∂1φ − ψ∂2φ)
−h∂2fp − fp(φ∂2φ + ψ∂1φ) )(e1235 − ie1245 − ie1236 − e1246) . (D.36)
From (D.11), we obtain
ih∂1fp + ifp(φ∂1φ − ψ∂2φ) − h∂2fp − fp(φ∂2φ + ψ∂1φ) = 0 (D.37)
and this is rewritten as follows,
∂1fpI + fpI∂1 log(φ2 + ψ2)12 + ∂2fpR + fpR∂2 log(φ2 + ψ2)
12 = 0 (D.38)
∂1fpR + fpR∂1 log(φ2 + ψ2)12 − ∂2fpI + fpI∂2 log(φ2 + ψ2)
12 = 0 , (D.39)
where fpR denotes a real part of fp and fpI denotes an imaginary part of fp. We see that thefunctions,
fpR =φ√
φ2 + ψ2, fpI =
ψ√φ2 + ψ2
, (D.40)
are the solutions of the equations (D.38) and (D.39) and they satisfy |fp| = 1. Therefore whenB1 = B2 = −ψ, the deformed (3,0)-form Υ takes the following form,
Υ = iφ + iψ√φ2 + ψ2
( e135 + e245 + e236 − e146 + i(e235 − e145 − e136 − e246) ) . (D.41)
91
D.1 Bismut curvature and Hull curvature
Curvature 2-forms of connection ∇± are defined by
R±ab = dω±
ab + ω±ab ∧ ω±
ab =12R±
abµν eµν , (D.42)
where R±abµν (a, b, µ, ν = 1, · · · , 6) denotes curvature components. In this section, we use the
following notations
Φ1 = ∂1 log |φ| , Φ2 = ∂2 log |φ| , Ψ1 = ∂21 log |φ| , Ψ2 = ∂2
2 log |φ| , Ξ = ∂2∂1 log |φ| . (D.43)
The components of Bismut curvature are listed as follows:
R+1212 = − 2
φ4(Ψ1 + Ψ2) , R+
1234 = 0 , R+1256 = 0 ,
R+1313 =
−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φψ2(−Φ21 + Φ2
2 + Ψ1) + φ3(−3Φ21 + 3Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R+1314 = − ψ
φ4(φ2 + ψ2)2(−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φψ2(−Φ2
1 + Φ22 + Ψ1)
+φ3(−3Φ21 + 3Φ2
2 + Ψ1)) ,
R+1323 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−3Φ21 + Φ2
2 − Ψ2) − φ2ψ(5Φ21 − 3Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R+1324 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(5Φ21 − 3Φ2
2 + Ψ2) + ψ3(3Φ21 − Φ2
2 + Ψ2))φ4(φ2 + ψ2)2
,
R+1413 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−Φ21 + 3Φ2
2 + Ψ1) + φ2ψ(−3Φ21 + 5Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R+1414 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(3Φ21 − 5Φ2
2 − Ψ1) + ψ3(Φ21 − 3Φ2
2 − Ψ1))φ4 (φ2 + ψ2)2
,
R+1423 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ21 − 3Φ2
2 + Ψ2) + φψ2(Φ21 − Φ2
2 + Ψ2)φ3 (φ2 + ψ2)2
,
R+1424 = −ψ(φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ2
1 − 3Φ22 + Ψ2) + φψ2(Φ2
1 − Φ22 + Ψ2))
φ4(φ2 + ψ2)2,
R+1515 =
Φ21 − 3Φ2
2 − Ψ1
φ4, R+
1516 = 0 , R+1525 = −Ξ − 4Φ1Φ2
φ4, R+
1526 = 0 ,
R+1615 = −Ξ − 4Φ1Φ2
φ4, R+
1616 = 0 , R+1625 =
−3Φ21 + Φ2
2 − Ψ2
φ4, R+
1626 = 0 ,
R+2313 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−Φ21 + 3Φ2
2 + Ψ1) + φ2ψ(−3Φ21 + 5Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
92
R+2314 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(3Φ21 − 5Φ2
2 − Ψ1) + ψ3(Φ21 − 3Φ2
2 + Ψ1))φ4(φ2 + ψ2)2
,
R+2323 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ21 − 3Φ2
2 + Ψ2) + φψ2(Φ21 − Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R+2324 = − ψ
φ4(φ2 + ψ2)2(φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φψ2(Φ2
1 − Φ22 + Ψ2)
+φ3(3Φ21 − 3Φ2
2 + Ψ2)) ,
R+2413 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ21 − 3Φ2
2 − Ψ1) + φψ2(Φ21 − Φ2
2 − Ψ2)φ3 (φ2 + ψ2)2
,
R+2414 =
ψ(−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φ3(−3Φ21 + 3Φ2
2 + Ψ1) + φψ2(−Φ21 + Φ2
2 + Ψ1))φ4(φ2 + ψ2)2
,
R+2423 =
6φ3Φ1Φ2 − 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + ψ3(3Φ21 − Φ2
2 + Ψ2) + φ2ψ(5Φ21 − 3Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R+2424 = −ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(5Φ2
1 − 3Φ22 + Ψ2) + ψ3(3Φ2
1 − Φ22 + Ψ2))
φ4 (φ2 + ψ2)2,
R+2515 = −Ξ − 4Φ1Φ2
φ4, R+
2516 = 0 , R+2525 =
−3Φ21 + Φ2
2 − Ψ2
φ4, R+
2526 = 0 ,
R+2615 =
−Φ21 + 3Φ2
2 + Ψ1
φ4, R+
2616 = 0 , R+2625 =
Ξ − 4Φ1Φ2
φ4, R+
2626 = 0 ,
R+3412 =
φ2(Φ21 + Φ2
2) − ψ2(Ψ1 + Ψ2)φ4(φ2 + ψ2)
, R+3434 = 0 , R+
3456 = 0 ,
R+3535 =
Φ21 + Φ2
2
φ4 + φ2ψ2, R+
3536 = 0 , R+3545 = − ψ(Φ2
1 + Φ22)
φ3(φ2 + ψ2), R+
3546 = 0 ,
R+3635 = − ψ(Φ2
1 + Φ22)
φ3 (φ2 + ψ2), ˆ3636 = 0 , ˆ3645 =
ψ2(Φ21 + Φ2
2)φ4(φ2 + ψ2)
, R+3646 = 0 ,
R+4535 =
ψ(Φ21 + Φ2
2)φ3 (φ2 + ψ2)
, R+4536 = 0 , R+
4545 = −ψ2(Φ21 + Φ2
2)φ4(φ2 + ψ2)
, R+4546 = 0 ,
R+4635 =
(Φ21 + Φ2
2)φ2 (φ2 + ψ2)
, R+4636 = 0 , R+
4645 = − ψ(Φ21 + Φ2
2)φ3(φ2 + ψ2)
, R+4646 = 0 ,
R+5612 = −Ψ1 + Ψ2
φ4, R+
5634 = 0 , ˆ5656 = 0 .
One can notice the relations
−R+12 + R+
34 + R+56 = 0 , R+
13 = −R+24 , R+
14 = R+23 , R+
15 = −R+26 , R+
16 = R+25 . (D.44)
93
Also, the components of Hull curvature are listed as follows:
R−1212 = − 2
φ4(Ψ1 + Ψ2) , R−
1234 =(Φ1 + Φ2)
φ4, R+
1256 =(Φ1 + Φ2)
φ4,
R−1313 =
−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φψ2(−Φ21 + Φ2
2 + Ψ1) + φ3(−3Φ21 + 3Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R−1314 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−Φ21 + 3Φ2
2 + Ψ1) + φ2ψ(−3Φ21 + 5Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R−1323 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) − ψ3(2Φ21 − 2Φ2
2 + Ψ2) − φ2ψ(4Φ21 − 4Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R−1324 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(4Φ21 − 2Φ2
2 + Ψ2) + φψ2(2Φ21 + Ψ2)
φ3(φ2 + ψ2)2,
R−1413 = − ψ
φ4(φ2 + ψ2)2(−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φψ2(−Φ2
1 + Φ22 + Ψ1)
+φ3(−3Φ21 + 3Φ2
2 + Ψ1)) ,
R−1414 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(3Φ21 − 5Φ2
2 − Ψ1) + ψ3(Φ21 − 3Φ2
2 − Ψ1))φ4(φ2 + ψ2)2
,
R−1423 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(4Φ21 − 4Φ2
2 − Ψ2) + ψ3(2Φ21 − 2Φ2
2 + Ψ2))φ4(φ2 + ψ2)2
,
R−1424 = −ψ(φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(4Φ2
1 − 2Φ22 + Ψ2) + φψ2(2Φ2
1 + Ψ2))φ4(φ2 + ψ2)2
,
R−1515 =
Φ21 − 3Φ2
2 − Ψ1
φ4, R−
1516 = −Ξ − 4Φ1Φ2
φ4,
R−1525 = −Ξ − 4Φ1Φ2
φ4, R−
1526 = −2Φ21 − 2Φ2
2 + Ψ2
φ4,
R−1615 = R−
1616 = R−1625 = R−
1626 = 0 ,
R−2313 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−2Φ21 + 2Φ2
2 + Ψ1) + φ2ψ(−4Φ21 + 4Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R−2314 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(2Φ21 − 4Φ2
2 − Ψ1) − φψ2(2Φ22 + Ψ1)
φ3(φ2 + ψ2)2,
R−2323 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ21 − 3Φ2
2 + Ψ2) + φψ2(Φ21 − Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R−2324 =
6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + ψ3(3Φ21 − Φ2
2 + Ψ2) + φ2ψ(5Φ21 − 3Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
94
R−2413 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(4Φ21 − 4Φ2
2 − Ψ1) + ψ3(2Φ21 − 2Φ2
2 − Ψ1))φ4(φ2 + ψ2)2
,
R−2414 =
ψ(−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φ3(−2Φ21 + 4Φ2
2 + Ψ1) + φψ2(2Φ22 + Ψ1))
φ4(φ2 + ψ2)2,
R−2423 = − ψ
φ4(φ2 + ψ2)2(φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φψ2(Φ2
1 − Φ22 + Ψ2)
+φ3(3Φ21 − 3Φ2
2 + Ψ2)) ,
R−2424 = −ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(5Φ2
1 − 3Φ22 + Ψ2) + ψ3(3Φ2
1 − Φ22 + Ψ2))
φ4 (φ2 + ψ2)2,
R−2515 = −Ξ − 4Φ1Φ2
φ4, R−
2516 =−2Φ2
1 + 2Φ22 + Ψ1
φ4,
R−2525 =
−3Φ21 + Φ2
2 − Ψ2
φ4, R+
2526 =Ξ − 4Φ1Φ2
φ4,
R−2615 = R−
2616 = R−2625 = R−
2626 = 0 , R−3412 = R−
3434 = R−3456 = 0 ,
R−3535 =
Φ21 + Φ2
2
φ4 + φ2ψ2, R−
3536 = − ψ(Φ21 + Φ2
2)φ3(φ2 + ψ2)
,
R−3545 =
ψ(Φ21 + Φ2
2)φ3(φ2 + ψ2)
, R−3546 =
(Φ21 + Φ2
2)φ2(φ2 + ψ2)
,
R−3635 = R−
3636 = R−3645 = R−
3646 = 0 ,
R−4535 = − Φ2
1 + Φ22
φ4 + φ2ψ2, R−
4536 =ψ2(Φ2
1 + Φ22)
φ4(φ2 + ψ2),
R−4545 = −ψ2(Φ2
1 + Φ22)
φ4(φ2 + ψ2), R−
4546 = − (Φ21 + Φ2
2)ψφ3(φ2 + ψ2)
,
R−4635 = R−
4636 = R−4645 = R−
4646 = 0 , R−5612 = R−
5634 = R−5656 = 0 .
It should be notice that well-known identity R+abµν − R−
µνab = dTabµν = 0 is valid.
95
Appendix E
SU(3) structure of the T-dualized solution along ∂5
We T-dualized the solutions (4.50), (4.51), and (4.52) along ∂5 and then take the followingform:
g6 =1
φ2 + ψ2(dx5 − B2dx6) + φ4((dx1)2 + (dx2)2) +
1φ2 + ψ2
(dx3 + ψdx4)2
+φ2
φ2 + ψ2( (dx4)2 + (dx6)2 ) , (E.1)
(E.2)
B = − ψ
φ2 + ψ2dx6 ∧ dx5 +
ψ
φ2 + ψ2dx3 ∧ dx4 , (E.3)
e2ϕ =1
φ2 + ψ2e2ϕ =
1(φ2 + ψ2)2
φ2 . (E.4)
The orthonormal basis is defined by Let us introduce an new orthonormal basis;
e1 = φ2dx1 , e2 = φ2dx2 , e3 =1√
φ2 + ψ2(dx3 + ψdx4) , e4 =
φ√φ2 + ψ2
dx4
e5 =1√
φ2 + ψ2(dx5 − B2dx6) , e6 =
φ√φ2 + ψ2
dx6 (E.5)
and exterior derivatives of them are the following:
de1 = − 2φ3
∂φ
∂x2e12 , de2 =
2φ3
∂φ
∂x1e12 ,
de3 = − 1φ2(φ2 + ψ2)
(φ∂1φ − ψ∂2φ)e13 +1
φ2(φ2 + ψ2)(φ∂2φ + ψ∂1φ)e23 − 1
φ3∂2φe14 +
1φ3
∂1φe24 ,
de4 =ψ
φ3(φ2 + ψ2)( (ψ∂1φ + φ∂2φ)e14 + (ψ∂2φ − φ∂1φ)e24 ) ,
de5 =1
φ2(φ2 + ψ2)(−(φ∂1φ − ψ∂2φ)e15 − (φ∂2φ + ψ∂1φ)e25 ) − 1
φ3∂1B2e
16 − 1φ3
∂2B2e26 ,
de6 =ψ
φ3(φ2 + ψ2)( (ψ∂1φ + φ∂2φ)e16 + (ψ∂2φ − φ∂1φ)e26 ) . (E.6)
96
Thus we obtain the function fµνρ that are defined by deµ = 1
2 fµνρe
νρ as follows;
f112 = − 2
φ3∂2φ , f2
12 =2φ3
∂1φ ,
f313 = − 1
φ2(φ2 + ψ2)(φ∂1φ − ψ∂2φ) , f3
23 = − 1φ2(φ2 + ψ2)
(φ∂2φ + ψ∂1φ) ,
f314 = − 1
φ3∂2φ , f3
24 =1φ3
∂1φ ,
f414 =
ψ
φ3(φ2 + ψ2)(ψ∂1φ + φ∂2φ) , f4
24 =ψ
φ3(φ2 + ψ2)(ψ∂2φ − φ∂1φ) ,
f515 = − 1
φ2(φ2 + ψ2)(φ∂1φ − ψ∂2φ) , ˜f5
25 = − 1φ2(φ2 + ψ2)
(φ∂2φ + ψ∂1φ) ,
f516 = − 1
φ3∂1B2 , f5
26 = − 1φ3
∂2B2 ,
f616 =
ψ
φ3(φ2 + ψ2)(ψ∂1φ + φ∂2φ) , f6
26 =ψ
φ3(φ2 + ψ2)(ψ∂2φ − φ∂1φ) . (E.7)
A flux of the B-field B is derived byH = −dB (E.8)
and thus the flux H is explicitly given by
H =1
φ3(φ2 + ψ2)(((φ2 − ψ2)∂2φ + 2φψ∂1φ
)e134 −
((φ2 − ψ2)∂1φ − 2φψ∂2φ
)e234
−((ψ2 − φ2)∂2φ − 2φψ∂1φ
)e156 −
((φ2 − ψ2)∂1φ − 2φψ∂2φ
)e256) . (E.9)
We should investigate that the triplet (g, H, ϕ) satisfies supersymmetry preserving conditions:
d(e−2ϕ ∗ κ) = 0 , (E.10)d(e−2ϕΥ) = 0 , (E.11)∗H = e2ϕd(e−2ϕκ) , (E.12)
where J , κ, Υ(3,0) represent an almost complex structure, a fundamental 2-form, and a (3,0)-formrespectively.We assume a fundamental 2-form as follows,
κ = K(x1, x2)(α1e12 + α2e
34 + α3e56) . (E.13)
Let us determine a function K(x1, x2) such that κ satisfies (E.10). We define a volume form tothe metric (E.2) by
Vol. = ∗1 , Vol. = e123456 . (E.14)
Then Hodge dual of e12, e34 and e56 are given by
∗e12 = ιe2ιe1Vol. = e3456 , ∗e34 = e1256 , ∗e56 = e1234 .
97
We have∗κ = α1e
3456 + α2e1256 + α3e
1234 (E.15)
and
e−2ϕ ∗ κ =h2
φ2(α1e
3456 + α2e1256 + α3e
1234) ,
whereh = φ2 + ψ2 (E.16)
and
dh =2φ2
((φ∂1φ − ψ∂2φ)e1 + (φ∂2φ + ψ∂1φ)e2) . (E.17)
The left hand side of (E.10) takes the form
d(e−2ϕ ∗ κ) =2hK
φ2dh ∧ (α1e
3456 + α2e1256 + α3e
1234) − h2K
φ4d(φ2) ∧ (α1e
3456 + α2e1256 + α3e
1234)
+h2
φ2dK ∧ (α1e
3456 + α2e1256 + α3e
1234) +h2K
φ2(α1de3456 + α2de1256 + α3de1234)
(E.18)
and each terms of (E.18) are the following:
1st term = α14hKφ2
φ6((φ∂1φ − ψ∂2φ)e13456 + (φ∂2φ + ψ∂1φ)e23456)
2nd term = −α1h2K
φ6(∂1(φ2)e13456 + ∂2(φ2)e23456)
3rd term = α1h2
φ6(φ2(∂1K)e13456 + φ2(∂2K)e23456)
4th term = α1h2K
φ6(φ4(f3
13 + f414 + f5
15 + f616)e13456 + φ4(f3
23 + f424 + f5
25 + f626)e23456) .
Then, we have
d(e−2ϕ ∗ κ) =α1
φ6( h2(φ2(∂1K) + φ4K(f3
13 + f414 + f5
15 + f616)) − h2K∂1(φ2)
+2Kφ2(h(φ∂1φ − ψ∂2φ) + h(φ∂1φ − ψ∂2φ))]e13456
+α1
φ6[h2(φ2(∂2K) + φ4K(f3
23 + f424 + f5
25 + f626)) − h2K∂2(φ2)
+2Kφ2(h(φ∂2φ + ψ∂1φ) + h(φ∂2φ + ψ∂1φ)) )e23456 ,
where
h2φ4(f313 + f4
14 + f515 + f6
16) = hφ( (ψ2 − φ2)∂1φ + 2φψ∂2φ ) − hφ2(φ∂1φ − ψ∂2φ)+hφψ2∂1φ + hφ2ψ∂2φ ,
98
h2φ4(f323 + f4
24 + f525 + f6
26) = hφ( (ψ2 − φ2)∂2φ − 2φψ∂1φ ) − hφ2(φ∂2φ + ψ∂1φ)+hφψ2∂2φ − hφ2ψ∂1φ .
Thus left hand side of (E.18) is
d(e−2ϕ ∗ κ) =α1
φ6hφ2( (∂1K)e13456 + (∂2K)e23456 )
and thus we obtain the following equations for the function K(x1, x2) by (E.10)
∂1K = 0 , ∂2K = 0 .
Then, the function K(x1, x2) = 1. Consequently, the fundamental 2-form κ takes the form
κ = α1e12 + α2e
34 + α3e56 . (E.19)
From (E.19), an almost complex structure J is defined by
J e1 = −α1e2 , J e3 = −α2e , J e5 = −α3e6 (E.20)
and J∗eµ (µ = 1, · · · 6) are given by
J∗e1 = α1e2 , J∗e3 = α2e4 , J∗e5 = α3e6 . (E.21)
A Nijenhuis tensor N associated with the almost complex structure J is defined by
NJ(X,Y ) = [JX , JY ] − [X , Y ] − J [JX , Y ] − J [X , JY ] , (E.22)
where a commutation relation between vector fields are given by [eµ , eν ] = −fρµν eρ. We require
α1α2 = −1 and α1α3 = −1 and then we have
NJ(e1, e5) =1φ3
(−(∂1φ + ∂2B2)e5 + (∂2φ − ∂1B2)e6 ) ,
NJ(e2, e5) =1φ3
( (−∂2φ + ∂1B2)e5 − (∂1φ + ∂2B2)e6 ) ,
NJ(e1, e6) =1φ3
( (∂2φ − ∂1B2)e5 + (∂1φ + ∂2B2)e6 ) ,
NJ(e2, e6) = − 1φ3
( (∂1φ + ∂2B2)e5 − (−∂2φ + ∂1B2)e6 ) , (E.23)
If B2 = −ψ, then from Cauchy–Riemann equations (4.20) the components of the Nijenhuis tensorare the following,
NJ(e1, e5) = 0 , NJ(e2, e5) = 0 , NJ(e1, e6) = 0 , NJ(e2, e6) = 0 . (E.24)
The rest of NJ(eµ, eν) are clearly zero by using Cauchy–Riemann conditions. Thus an almostcomplex structure J is a complex structure when B2 = −ψ.
99
A Bismut torsion associated with J is defined by
T := α4J∗dκ . (E.25)
The exterior derivative of κ is given by
dκ =α2
φ3(φ2 + ψ2)( ((ψ2 − φ2)∂1φ + 2φψ∂2φ)e134 + ((ψ2 − φ2)∂2φ − 2φψ∂1φ)e234 )
+α3
φ3(φ2 + ψ2)( ((ψ2 − φ2)∂1φ + 2φψ∂2φ)e156 + ((ψ2 − φ2)∂2φ − 2φψ∂1φ)e256 )
(E.26)
and thus we have
J∗dκ = − 1φ3(φ2 + ψ2)
( ((ψ2 − φ2)∂1φ + 2φψ∂2φ)e234 − ((ψ2 − φ2)∂2φ − 2φψ∂1φ)e134
+((ψ2 − φ2)∂1φ + 2φψ∂2φ)e256 − ((ψ2 − φ2)∂2φ − 2φψ∂1φ)e156 ) .
Therefore the Bismut torsion T takes the following form,
T = − α4
φ3(φ2 + ψ2)(((ψ2 − φ2)∂1φ + 2φψ∂2φ)e234 − ((ψ2 − φ2)∂2φ − 2φψ∂1φ)e134
+((ψ2 − φ2)∂1φ + 2φψ∂2φ)e256 − ((ψ2 − φ2)∂2φ − 2φψ∂1φ)e156)
(E.27)
This is obviously coincident with (E.9) under the condition α4 = −1,
H = T for α4 = −1 . (E.28)
Let us check whether the torsion T with α4 = −1 satisfies the equation dT κ = 0 or not.According to Ref. [57], an exterior derivative with a torsion for the fundamental 2-form κ,
dT κ = dκ −6∑
µ=1
(ιeµ T ) ∧ (ιeµ κ) . (E.29)
The first term of dT κ has been given by (E.26). On the second term, (ιeµ T ) ∧ (ιeµ κ) for eachvalues of µ = 1, · · · , 6 are given by
(ιe1 T ) ∧ (ιe1 κ) =α1α4
φ3(φ2 + ψ2)((ψ2 − φ2)∂2φ − 2φψ∂1φ)(e234 + e256) ,
(ιe2 T ) ∧ (ιe2 κ) =α1α4
φ3(φ2 + ψ2)((ψ2 − φ2)∂1φ + 2φψ∂2φ)(e134 + e156) ,
(ιe3 T ) ∧ (ιe3 κ) = 0 , (ιe4 T ) ∧ (ιe4 κ) = 0 , (ιe5 T ) ∧ (ιe5 κ) = 0 , (ιe6 T ) ∧ (ιe6 κ) = 0
and thus we have6∑
µ=1
(ιeµ T ) ∧ (ιeµ κ) =α1α4
φ3(φ2 + ψ2)( ((ψ2 − φ2)∂1φ + 2φψ∂2φ)(e134 + e156)
+((ψ2 − φ2)∂2φ − 2φψ∂1φ)(e234 + e256) ) .
100
Therefore dT κ is given by
dT κ =α2
φ3(φ2 + ψ2)( ((ψ2 − φ2)∂1φ + 2φψ∂2φ)e134 + ((ψ2 − φ2)∂2φ − 2φψ∂1φ)e234 )
+α3
φ3(φ2 + ψ2)( ((ψ2 − φ2)∂1φ + 2φψ∂2φ)e156 + ((ψ2 − φ2)∂2φ − 2φψ∂1φ)e256 )
− α1α4
φ3(φ2 + ψ2)( ((ψ2 − φ2)∂1φ + 2φψ∂2φ)(e134 + e156)
+((ψ2 − φ2)∂2φ − 2φψ∂1φ(e234 + e256) )
and thus we see that dT κ = 0 when α1α2 = −1, α1α3 = −1 and α4 = −1.
A Lee form Θ is defined byΘ = α4J∗δκ , (E.30)
where the operator δ is a co-derivative on p-form that is defined by δ = (−1)np−p2+p ∗ d∗. Theexterior derivatives of ∗κ is given by
d∗κ =2α1
φ3(φ2 + ψ2)( (ψ2−φ2)∂1φ+2φψ∂2φ )e13456+
2α1
φ3(φ2 + ψ2)(−2φψ∂1φ+(ψ2−φ2)∂2φ )e23456
(E.31)and thus the Lee form takes the following form
Θ = −α4d
(log
φ2
(φ2 + ψ2)2
). (E.32)
If we identify a dilaton with the Lee form by dϕ = −12α4Θ, the dilaton ϕ is given by
ϕ =12
logφ2
(φ2 + ψ2)2(E.33)
and this is consistent with (4.71).
We consider (E.12).The Hodge dual of ∗H is given by
∗H =1
φ3(φ2 + ψ2)(((φ2 − ψ2)∂2φ + 2φψ∂1φ
)e256 +
((φ2 − ψ2)∂1φ − 2φψ∂2φ
)e156 )
+1
φ3(φ2 + ψ2)(−
((ψ2 − φ2)∂2φ − 2φψ∂1φ
)e234 +
((φ2 − ψ2)∂1φ − 2φψ∂2φ
)e134 )
(E.34)
The right hand side of (E.12) becomes
e2ϕd(e−2ϕκ) = dκ − Θ ∧ κ
= − α2
φ3(φ2 + ψ2)( ((ψ2 − φ2)∂1φ + 2φψ∂2φ)e134 + ((ψ2 − φ2)∂2φ − 2φψ∂1φ)e234 )
− α3
φ3(φ2 + ψ2)( ((ψ2 − φ2)∂1φ + 2φψ∂2φ)e156 + ((ψ2 − φ2)∂2φ − 2φψ∂1φ)e256 ) .
101
If α2 = α3 = 1, we obtain∗H − e2ϕd(e−2ϕκ) = 0 . (E.35)
Namely, the triplet (g, H, ϕ) satisfies (E.12) under the condition α2 = α3 = 1.
We consider a (3, 0)-form Υ which satisfies (E.11).We assume that Υ takes the form
Υ =√
α1α2α3W (x1, x2)(e1 + ie2) ∧ (e3 − ie4) ∧ (e5 − ie6)=
√α1α2α3W (x1, x2)( e135 + e245 + e236 − e146 + i(e235 − e145 − e136 − e246) ) ,(E.36)
where the function W is a complex function. A (3, 0)-form Υ has the relation between thefundamental 2-form κ by the volume matching condition,
Υ ∧ ¯Υ =43i κ3 . (E.37)
Calculating (E.37) directly, we obtain the conditions
α1α2α3 = −1 , |W |2 = 1 (E.38)
and thus the complex function W is a phase function,
W (x1, x2) = fp(x1, x2) , (E.39)
where |fp| = 1. We determine fp so as to satisfy (E.11). e−2ϕΥ is given by
e−2ϕΥ = ih2fp
φ2( e135 + e245 + e236 − e146 + i(e235 − e145 − e136 − e246) ) .
The exterior derivative of e−2ϕΥ is calculated as follows,
d(e−2ϕΥ) =ih
φ5(−(hφ∂2fp + 2φfp(ψ∂1φ + φ∂2φ))
+i(hφ∂1fp + 2φfp(φ∂1φ − ψ∂2φ)) )(e1235 − ie1236 − ie1245 − e1246) . (E.40)
From (E.11) we have
φ(φ2 + ψ2)∂1fp + 2φfp(φ∂1φ − ψ∂2φ) + i( φ(φ2 + ψ2)∂2fp + 2φfp(ψ∂1φ + φ∂2φ) ) = 0 (E.41)
and this is rewritten as follows:
∂1fpR − ∂2fpI + fpR∂1 log(φ2 + ψ2) − fpI∂2 log(φ2 + ψ2) = 0 , (E.42)∂1fpI + ∂2fpI + fpI∂1 log(φ2 + ψ2) + fpR∂2 log(φ2 + ψ2) = 0 , (E.43)
where fpR denotes a real part of fp and fpI denotes an imaginary part of fp. We see that thefunctions,
fpR =φ
φ2 + ψ2, fpI =
ψ
φ2 + ψ2(E.44)
are the solutions of the equations (E.42) and (E.43) and they satisfy |fp| = 1. Therefore whenB1 = B2 = −ψ, the deformed (3,0)-form Υ takes the following form
Υ = iφ + iψ
φ2 + ψ2( e135 + e245 + e236 − e146 + i(e235 − e145 − e136 − e246) ) . (E.45)
102
E.1 Bismut curvature and Hull curvature
Curvature 2-forms of connection ∇± are defined by
R±ab = dω±
ab + ω±ab ∧ ω±
ab =12R±
abµν eµν , (E.46)
where R±abµν (a, b, µ, ν = 1, · · · , 6) denotes curvature components. In what follows in this section,
we use the notations
Φ1 = ∂1 log |φ| , Φ2 = ∂2 log |φ| , Ψ1 = ∂21 log |φ| , Ψ2 = ∂2
2 log |φ| , Ξ = ∂2∂1 log |φ| . (E.47)
The components of Bismut curvature is listed as follows:
R+1212 = − 2
φ4(Ψ1 + Ψ2) , R+
1234 = 0 , R+1256 = 0 ,
R+1313 =
−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φψ2(−Φ21 + Φ2
2 + Ψ1) + φ3(−3Φ21 + 3Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R+1314 = − ψ
φ4(φ2 + ψ2)2(−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φψ2(−Φ2
1 + Φ22 + Ψ1)
+φ3(−3Φ21 + 3Φ2
2 + Ψ1)) ,
R+1323 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−3Φ21 + Φ2
2 − Ψ2) − φ2ψ(5Φ21 − 3Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R+1324 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(5Φ21 − 3Φ2
2 + Ψ2) + ψ3(3Φ21 − Φ2
2 + Ψ2))φ4(φ2 + ψ2)2
,
R+1413 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−Φ21 + 3Φ2
2 + Ψ1) + φ2ψ(−3Φ21 + 5Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R+1414 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(3Φ21 − 5Φ2
2 − Ψ1) + ψ3(Φ21 − 3Φ2
2 − Ψ1))φ4 (φ2 + ψ2)2
,
R+1423 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ21 − 3Φ2
2 + Ψ2) + φψ2(Φ21 − Φ2
2 + Ψ2)φ3 (φ2 + ψ2)2
,
R+1424 = −ψ(φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ2
1 − 3Φ22 + Ψ2) + φψ2(Φ2
1 − Φ22 + Ψ2))
φ4(φ2 + ψ2)2,
R+1515 =
−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φψ2(−Φ21 + Φ2
2 + Ψ1) + φ3(−3Φ21 + 3Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R+1516 = − ψ
φ4(φ2 + ψ2)2(−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φψ2(−Φ2
1 + Φ22 + Ψ1)
+φ3(−3Φ21 + 3Φ2
2 + Ψ1)) ,
103
R+1525 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−3Φ21 + Φ2
2 − Ψ2) − φ2ψ(5Φ21 − 3Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R+1526 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(5Φ21 − 3Φ2
2 + Ψ2) + ψ3(3Φ21 − Φ2
2 + Ψ2))φ4(φ2 + ψ2)2
,
R+1615 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−Φ21 + 3Φ2
2 + Ψ1) + φ2ψ(−3Φ21 + 5Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R+1616 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(3Φ21 − 5Φ2
2 − Ψ1) + ψ3(Φ21 − 3Φ2
2 − Ψ1))φ4 (φ2 + ψ2)2
,
R+1625 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ21 − 3Φ2
2 + Ψ2) + φψ2(Φ21 − Φ2
2 + Ψ2)φ3 (φ2 + ψ2)2
,
R+1626 = −ψ(φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ2
1 − 3Φ22 + Ψ2) + φψ2(Φ2
1 − Φ22 + Ψ2))
φ4(φ2 + ψ2)2,
R+2313 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−Φ21 + 3Φ2
2 + Ψ1) + φ2ψ(−3Φ21 + 5Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R+2314 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(3Φ21 − 5Φ2
2 − Ψ1) + ψ3(Φ21 − 3Φ2
2 + Ψ1))φ4(φ2 + ψ2)2
,
R+2323 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ21 − 3Φ2
2 + Ψ2) + φψ2(Φ21 − Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R+2324 = − ψ
φ4(φ2 + ψ2)2(φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φψ2(Φ2
1 − Φ22 + Ψ2)
+φ3(3Φ21 − 3Φ2
2 + Ψ2)) ,
R+2413 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ21 − 3Φ2
2 − Ψ1) + φψ2(Φ21 − Φ2
2 − Ψ2)φ3 (φ2 + ψ2)2
,
R+2414 =
ψ(−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φ3(−3Φ21 + 3Φ2
2 + Ψ1) + φψ2(−Φ21 + Φ2
2 + Ψ1))φ4(φ2 + ψ2)2
,
R+2423 =
6φ3Φ1Φ2 − 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + ψ3(3Φ21 − Φ2
2 + Ψ2) + φ2ψ(5Φ21 − 3Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R+2424 = −ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(5Φ2
1 − 3Φ22 + Ψ2) + ψ3(3Φ2
1 − Φ22 + Ψ2))
φ4 (φ2 + ψ2)2,
R+2515 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−Φ21 + 3Φ2
2 + Ψ1) + φ2ψ(−3Φ21 + 5Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R+2516 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(3Φ21 − 5Φ2
2 − Ψ1) + ψ3(Φ21 − 3Φ2
2 + Ψ1))φ4(φ2 + ψ2)2
,
R+2525 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ21 − 3Φ2
2 + Ψ2) + φψ2(Φ21 − Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
104
R+2526 = − ψ
φ4(φ2 + ψ2)2(φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φψ2(Φ2
1 − Φ22 + Ψ2)
+φ3(3Φ21 − 3Φ2
2 + Ψ2)) ,
R+2615 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ21 − 3Φ2
2 − Ψ1) + φψ2(Φ21 − Φ2
2 − Ψ2)φ3 (φ2 + ψ2)2
,
R+2616 =
ψ(−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φ3(−3Φ21 + 3Φ2
2 + Ψ1) + φψ2(−Φ21 + Φ2
2 + Ψ1))φ4(φ2 + ψ2)2
,
R+2625 =
6φ3Φ1Φ2 − 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + ψ3(3Φ21 − Φ2
2 + Ψ2) + φ2ψ(5Φ21 − 3Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R+2626 = −ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(5Φ2
1 − 3Φ22 + Ψ2) + ψ3(3Φ2
1 − Φ22 + Ψ2))
φ4 (φ2 + ψ2)2,
R+3412 =
φ2(Φ21 + Φ2
2) − ψ2(Ψ1 + Ψ2)φ4(φ2 + ψ2)
, R+3434 = 0 , R+
3456 = 0 ,
R+3535 = − Φ2
1 + Φ22
φ2(φ2 + ψ2), R+
3536 =(Φ2
1 + Φ22)ψ
φ3(φ2 + ψ2),
R+3545 =
ψ(Φ21 + Φ2
2)φ3(φ2 + ψ2)
, R+3546 = −(Φ2
1 + Φ22)ψ
2
φ4(φ2 + ψ2),
R+3635 = ˜3636 = ˜3645 = R+
3646 = 0 , R+4535 = R+
4536 = R+4545 = R+
4546 = 0 ,
R+4635 = − Φ2
1 + Φ22
φ2(φ2 + ψ2), R+
4636 =(Φ2
1 + Φ22)ψ
φ3(φ2 + ψ2),
R+4645 =
ψ(Φ21 + Φ2
2)φ3(φ2 + ψ2)
, R+4646 = −(Φ2
1 + Φ22)ψ
2
φ4(φ2 + ψ2),
R+5612 =
φ2(Φ21 + Φ2
2) − ψ2(Ψ1 + Ψ2)φ4(φ2 + ψ2)
, R+5634 = 0 , ˜5656 = 0 .
Clearly, we can find the relations
−R+12 + R+
34 + R+56 = 0 , R+
13 = −R+24 , R+
14 = R+23 , R+
15 = −R+26 , R+
16 = R+25 . (E.48)
Also, the components of Hull curvature are listed as follows:
R−1212 = − 2
φ4(Ψ1 + Ψ2) , R−
1234 =(Φ1 + Φ2)
φ4, R+
1256 =(Φ1 + Φ2)
φ4,
R−1313 =
−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φψ2(−Φ21 + Φ2
2 + Ψ1) + φ3(−3Φ21 + 3Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R−1314 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−Φ21 + 3Φ2
2 + Ψ1) + φ2ψ(−3Φ21 + 5Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
105
R−1323 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) − ψ3(2Φ21 − 2Φ2
2 + Ψ2) − φ2ψ(4Φ21 − 4Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R−1324 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(4Φ21 − 2Φ2
2 + Ψ2) + φψ2(2Φ21 + Ψ2)
φ3(φ2 + ψ2)2,
R−1413 = − ψ
φ4(φ2 + ψ2)2(−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φψ2(−Φ2
1 + Φ22 + Ψ1)
+φ3(−3Φ21 + 3Φ2
2 + Ψ1)) ,
R−1414 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(3Φ21 − 5Φ2
2 − Ψ1) + ψ3(Φ21 − 3Φ2
2 − Ψ1))φ4(φ2 + ψ2)2
,
R−1423 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(4Φ21 − 4Φ2
2 − Ψ2) + ψ3(2Φ21 − 2Φ2
2 + Ψ2))φ4(φ2 + ψ2)2
,
R−1424 = −ψ(φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(4Φ2
1 − 2Φ22 + Ψ2) + φψ2(2Φ2
1 + Ψ2))φ4(φ2 + ψ2)2
,
R−1515 =
−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φψ2(−Φ21 + Φ2
2 + Ψ1) + φ3(−3Φ21 + 3Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R−1516 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−Φ21 + 3Φ2
2 + Ψ1) + φ2ψ(−3Φ21 + 5Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R−1525 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) − ψ3(2Φ21 − 2Φ2
2 + Ψ2) − φ2ψ(4Φ21 − 4Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R−1526 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(4Φ21 − 2Φ2
2 + Ψ2) + φψ2(2Φ21 + Ψ2)
φ3(φ2 + ψ2)2,
R−1615 = − ψ
φ4(φ2 + ψ2)2(−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φψ2(−Φ2
1 + Φ22 + Ψ1)
+φ3(−3Φ21 + 3Φ2
2 + Ψ1)) ,
R−1616 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(3Φ21 − 5Φ2
2 − Ψ1) + ψ3(Φ21 − 3Φ2
2 − Ψ1))φ4(φ2 + ψ2)2
,
R−1625 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(4Φ21 − 4Φ2
2 − Ψ2) + ψ3(2Φ21 − 2Φ2
2 + Ψ2))φ4(φ2 + ψ2)2
,
R−1626 = −ψ(φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(4Φ2
1 − 2Φ22 + Ψ2) + φψ2(2Φ2
1 + Ψ2))φ4(φ2 + ψ2)2
,
R−2313 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−2Φ21 + 2Φ2
2 + Ψ1) + φ2ψ(−4Φ21 + 4Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R−2314 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(2Φ21 − 4Φ2
2 − Ψ1) − φψ2(2Φ22 + Ψ1)
φ3(φ2 + ψ2)2,
106
R−2323 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ21 − 3Φ2
2 + Ψ2) + φψ2(Φ21 − Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R−2324 =
6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + ψ3(3Φ21 − Φ2
2 + Ψ2) + φ2ψ(5Φ21 − 3Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R−2413 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(4Φ21 − 4Φ2
2 − Ψ1) + ψ3(2Φ21 − 2Φ2
2 − Ψ1))φ4(φ2 + ψ2)2
,
R−2414 =
ψ(−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φ3(−2Φ21 + 4Φ2
2 + Ψ1) + φψ2(2Φ22 + Ψ1))
φ4(φ2 + ψ2)2,
R+2423 = − ψ
φ4(φ2 + ψ2)2(φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φψ2(Φ2
1 − Φ22 + Ψ2)
+φ3(3Φ21 − 3Φ2
2 + Ψ2)) ,
R−2424 = −ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(5Φ2
1 − 3Φ22 + Ψ2) + ψ3(3Φ2
1 − Φ22 + Ψ2))
φ4 (φ2 + ψ2)2,
R−2515 =
−6φ3Φ1Φ2 − 2φψ2Φ1Φ2 + Ξφ(φ2 + ψ2) + ψ3(−2Φ21 + 2Φ2
2 + Ψ1) + φ2ψ(−4Φ21 + 4Φ2
2 + Ψ1)φ3(φ2 + ψ2)2
,
R−2516 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(2Φ21 − 4Φ2
2 − Ψ1) − φψ2(2Φ22 + Ψ1)
φ3(φ2 + ψ2)2,
R+2525 =
φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φ3(3Φ21 − 3Φ2
2 + Ψ2) + φψ2(Φ21 − Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R+2526 =
6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + ψ3(3Φ21 − Φ2
2 + Ψ2) + φ2ψ(5Φ21 − 3Φ2
2 + Ψ2)φ3(φ2 + ψ2)2
,
R−2615 =
ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(4Φ21 − 4Φ2
2 − Ψ1) + ψ3(2Φ21 − 2Φ2
2 − Ψ1))φ4(φ2 + ψ2)2
,
R−2616 =
ψ(−φ2ψ(Ξ − 8Φ1Φ2) − (Ξ − 4Φ1Φ2)ψ3 + φ3(−2Φ21 + 4Φ2
2 + Ψ1) + φψ2(2Φ22 + Ψ1))
φ4(φ2 + ψ2)2,
R−2625 = − ψ
φ4(φ2 + ψ2)2(φ2ψ(Ξ − 8Φ1Φ2) + (Ξ − 4Φ1Φ2)ψ3 + φψ2(Φ2
1 − Φ22 + Ψ2)
+φ3(3Φ21 − 3Φ2
2 + Ψ2)) ,
R−2626 = −ψ(6φ3Φ1Φ2 + 2φψ2Φ1Φ2 − Ξφ(φ2 + ψ2) + φ2ψ(5Φ2
1 − 3Φ22 + Ψ2) + ψ3(3Φ2
1 − Φ22 + Ψ2))
φ4 (φ2 + ψ2)2,
R−3412 = R−
3434 = R−3456 = 0 ,
R−3535 = − Φ2
1 + Φ22
φ4 + φ2ψ2, R−
3536 = R−3545 = 0 , R−
3546 = − (Φ21 + Φ2
2)ψφ3(φ2 + ψ2)
,
107
R−3635 = − (Φ2
1 + Φ22)ψ
φ3(φ2 + ψ2), R−
3636 = R−3645 = 0 , R−
3646 =(Φ2
1 + Φ22)ψ
φ3(φ2 + ψ2),
R−4535 =
(Φ21 + Φ2
2)ψφ3(φ2 + ψ2)
, R−4536 = R−
4545 = 0 , R−4546 =
(Φ21 + Φ2
2)ψφ3(φ2 + ψ2)
,
R−4635 = −(Φ2
1 + Φ22)ψ
2
φ4(φ2 + ψ2), R−
4636 = R−4645 = 0 , R−
4646 = −(Φ21 + Φ2
2)ψ2
φ4(φ2 + ψ2),
R−5612 = R−
5634 = R−5656 = 0 .
These components of the Bismut and the Hull curvature satisfy the identity R+abµν − R−
µνab =dTabµν = 0.
108
Appendix F
General Wahlquist metrics in all dimensions
As was discussed previous chapters, totally skew symmetric torsion tensors appear in variousclass of G-structures naturally. In case of related physics, the torsion tensors play an importantrole. Apart from G-structure, totally skew symmetric torsion tensors also contribute to gener-alizing Killing-Yano symmetry. Killing-Yano symmetry is significant for integrability on higher-dimensional rotating black hole spacetimes with spherical horizon topology [58, 59, 60, 61]. Inthis appendix, we give an outline of the general Wahlquist solution. This appendix is based onRef. [3].
The Wahlquist metric [62, 63, 64] is a stationary, axially symmetric perfect fluid solution withthe state equation ρ + 3p = const. The Wahlquist space time has some geometric properties ofthe Kerr spacetime, which is obtained by limiting case of the Wahlquist spacetime [62, 65]. Thekerr metric is the only vacuum solution admitting rank-2 Killing-Yano tensor [66]. It is knownthat the rank-2 Killing-Yano tensor generates two Killing vector fields and Killing-Stakel tensorin the Kerr spacetime. This implies that existence of the rank-2 Killing-Yano tensor characterizegeometric properties of the Kerr spacetime. However, Killing-Yano symmetry of the Wahlquistspacetime is still unclear. In the following of this appendix, we firstly find a rank-2 generalizedclosed conformal Killing-Yano (GCCKY) tensor with torsion [57]. Secondly, we attempt to con-struct a new family of rotating perfect fluid solutions which generalize the Wahlquist metric tohigher dimension.
F.1 Killing-Yano symmetry of the Wahlquist spacetime
The Wahlquist metric in 4 dimensions [62, 63, 64] can be written in a local coordinate system(z, w, τ, σ) as
g = (v1 + v2)(
dz2
U+
dw2
V
)+
U
v1 + v2(dτ + v2dσ)2 − V
v1 + v2(dτ − v1dσ)2 , (F.1)
109
where
U = Q0 + a1sinh(2βz)
2β− ν0
cosh(2βz) − 12β2
− µ0
β2
(cosh(2βz) − 1
2β2− z sinh(2βz)
2β
),
V = Q0 + a2sin(2βw)
2β+ ν0
1 − cos(2βw)2β2
+µ0
β2
(1 − cos(2βw)
2β2− w sin(2βw)
2β
), (F.2)
andv1 =
cosh(2βz) − 12β2
, v2 =1 − cos(2βw)
2β2. (F.3)
The metric contains six real constants Q0, a1, a2, ν0, µ0 and β. As was shown in [62], one cantake the limit β → 0. In the limit, the metric reduces to the Kerr-NUT-(A)dS metric [67].
The Wahlquist metric provides the stress-energy tensor for perfect fluids of the energy densityρ, pressure p and 4-velocity u with uµuµ = −1, which is written as
Tµν = (ρ + p)uµuν + pgµν . (F.4)
The 4-velocity is given by
uµ ∂
∂xµ=
1√−gττ
∂
∂τ, (F.5)
where gττ = (U − V )/(v1 + v2). If u lies on the 2-plane spanned by two Killing vector fields ∂τ
and ∂σ, then u can be written as u = N(∂τ + Ω∂σ), where N and Ω are functions in general.In particular, when Ω is constant, the perfect fluid is said to be rigidly rotating. Namely, theWahlquist solution represents rigidly rotating perfect fluids. The energy density and pressure aregiven by
ρ = −µ0 − 3β2gττ , p = µ0 + β2gττ . (F.6)
Thus, the equation of state is ρ + 3p = 2µ0.
F.1.1 Killing-Yano symmetry
From now on, we demonstrate that the Wahlquist metric (F.1) admits a rank-2 GCCKYtensor. For the purpose, we introduce the coordinates x and y defined by
x2 = v1 , y2 = v2 , (F.7)
and hence
dz2 =dx2
β2x2 + 1, dw2 =
dy2
1 − β2y2. (F.8)
The metric is then written as
g =x2 + y2
U(1 + β2x2)dx2 +
x2 + y2
V (1 − β2y2)dy2 +
U
x2 + y2(dτ + y2dσ)2 − V
x2 + y2(dτ − x2dσ)2
110
with the functions
U = Q0 + a1x√
1 + β2x2 − ν0x2 − µ0
β2
(x2 − xArcsinh(βx)
√1 + β2x2
β
),
V = Q0 + a2y√
1 − β2y2 + ν0y2 +
µ0
β2
(y2 − yArcsin(βy)
√1 − β2y2
β
).
Furthermore, taking the Wick rotation y →√−1y (with a2 → −
√−1a2 to keep the metric
function V real) and changing the sign σ → −σ, we obtain the Euclideanised expression for theWahlquist metric
gE =f1(x2 − y2)
Ξ1dx2 +
f2(y2 − x2)Ξ2
dy2 +Ξ1
x2 − y2(dτ + y2dσ)2 +
Ξ2
y2 − x2(dτ + x2dσ)2 , (F.9)
where
f1 =1√
1 + β2x2, f2 =
1√1 + β2y2
,
Ξ1 = Q0 + a1x√
1 + β2x2 − ν0x2 − µ0
β2
(x2 − xArcsinh(βx)
√1 + β2x2
β
),
Ξ2 = Q0 + a2y√
1 + β2y2 − ν0y2 − µ0
β2
(y2 − yArcsinh(βy)
√1 + β2y2
β
).
The form of the metric (F.9) is symmetric with respect to the coordinates (x, y) and precisely fitsinto type A of the classification in [68], that is, the Wahlquist spacetime admits a rank-2 GCCKYtensor. In fact, if we introduce an orthonormal frame
e1 = f1
√x2 − y2
Ξ1dx , e2 = f2
√y2 − x2
Ξ2dy ,
e1 =
√Ξ1
x2 − y2(dτ + y2dσ) , e2 =
√Ξ2
y2 − x2(dτ + x2dσ) ,
the rank-2 GCCKY tensor is given by
h = x e1 ∧ e1 + y e2 ∧ e2 (F.10)
with the skew-symmetric torsion
T =2x(f1 − f2)
f1f2(x2 − y2)
√Ξ2
y2 − x2e1 ∧ e1 ∧ e2 +
2y(f2 − f1)f1f2(y2 − x2)
√Ξ1
x2 − y2e2 ∧ e2 ∧ e1 . (F.11)
The torsion vanishes when we take the limit β → 0. This suggests that the torsion is related tothe perfect fluid. However, the physical meaning of the torsion is unclear.
111
F.2 Higher-dimensional Wahlquist spacetimes
We have seen that the Wahlquist metric (F.9) admits a rank-2 GCCKY tensor and its Euclideanform precisely fits into type A of the classification [68]. It seems to be reasonable to consider ahigher-dimensional generalisation of the Wahlquist metric. In this section, we attempt to solveEinstein equations for perfect fluids in higher dimensions by employing type A metrics obtainedin [68] as an ansatz.
Hereafter, we slightly change our notation. To treat higher-dimensional metrics in both evenand odd dimensions, we introduce ε, where ε = 0 for even and ε = 1 for odd dimensions. Thedimension is denoted by D = 2n + ε. The Latin indices a, b, · · · run from 1 to D and the Greeceindices µ, ν, · · · run from 1 to n. In the notation, we consider the following form of metricsadmitting a rank-2 GCCKY tensor in D dimensions:
g(D) =n∑
µ=1
f2µ
Pµdx2
µ +n∑
µ=1
Pµ
(n−1∑k=0
A(k)µ dψk
)2
+ εS
(n∑
k=0
A(k)dψk
)2
, (F.12)
where
Pµ =Ξµ
Uµ, Uµ =
∏ν =µ
(x2µ − x2
ν) , S =k2
A(n), fµ =
1√1 + β2x2
µ
, (F.13)
and A(k)µ (k = 0, · · · , n− 1) and A(k) (k = 0, · · · , n) are the k-th elementary symmetric functions
in (x21, x
22, · · ·x2
n) defined by
n−1∑k=0
A(k)µ tk =
∏ν =µ
(1 + tx2ν) ,
n∑k=0
A(k)tk =n∏
ν=1
(1 + tx2ν). (F.14)
The metric contains n unknown functions Ξµ(xµ) which depend only on single valuables xµ, anda constant k2.
F.2.1 Even dimensions
We determine the functions Ξµ from the Einstein equation for perfect fluids in even dimensions.The metric ansatz in 2n dimensions is given by
g(2n) =n∑
µ=1
f2µ
Pµdx2
µ +n∑
µ=1
Pµ
(n−1∑k=0
A(k)µ dψk
)2
. (F.15)
For the metric, the off-diagonal components of the Ricci curvature are
Rµµ = Rµν = Rµν = 0 , Rµν = β2(D − 2)√
Pµ
√Pν . (F.16)
The diagonal components are
Rµµ = Iµ(PT ) + β2
(Iµ(P (2)
T ) +32xµ∂µPT + PT
), Rµµ = Rµµ + β2(D − 2)Pµ , (F.17)
112
where
PT =n∑
µ=1
Pµ , P(2)T =
n∑µ=1
x2µPµ (F.18)
and Iµ are differential operators given by
Iµ = −12
∂2
∂x2µ
+1
x2µ − x2
ν
(xµ
∂
∂xµ− xν
∂
∂xν
). (F.19)
It should be emphasized that our metric ansatz is now expressed with Euclidean signature, sothat we have to consider the Euclideanised Einstein equation for perfect fluids,
Rab −12Rgab = −(ρ + p)uaub + pgab , (F.20)
where uaua = 1. Eliminating the scalar curvature, we obtain the equation in a more convenientform
Rab = −(ρ + p)uaub +ρ − p
D − 2gab . (F.21)
Moreover, to solve the equation, we assume that perfect fluids are rigidly rotating, namely, thevelocity u is written as u = N∂ψ0 where N is the normalization function. Since we have N =1/
√PT from uaua = 1, the velocity is given in the canonical frame as
u =1√PT
n∑µ=1
√Pµeµ . (F.22)
Under the assumption, together with (F.16) and (F.17), the Einstein equation to solve reducesto
ρ − p
D − 2= R11 = R22 = · · · = Rnn , (F.23)
ρ + p
D − 2= −β2PT . (F.24)
To solve the equations (F.23), we need to notice that the µµ-components of the Ricci curvature,Rµµ, can be written in a simple form. Calculating Rµµ in terms of Ξµ and its derivatives, weobtain
Rµµ = − 12xµ
Gµ
Uµ−
∑ν =µ
2xµ
x2µ − x2
ν
(Fµ
Uµ+
Fν
Uν
) − 2β2PT , (F.25)
where
Gµ = xµ(1 + β2x2µ)Ξ′′
µ + β2x2µΞ′
µ − 4β2xµΞµ , (F.26)
Fµ = xµ(1 + β2x2µ)Ξ′
µ − (1 + 2β2x2µ)Ξµ . (F.27)
113
Noticing that Gµ = F ′µ and that
∂FT
∂xµ=
F ′µ
Uµ−
∑ν =µ
2xµ
x2µ − x2
ν
(Fµ
Uµ+
Fν
Uν
), (F.28)
where
FT =n∑
ρ=1
Fρ
Uρ, (F.29)
we obtain the following expressions for µµ-components of the Ricci curvature:
Rµµ = − 12xµ
∂FT
∂xµ− 2β2PT . (F.30)
From (F.30), Rµµ − Rνν = 0 implies that[1xµ
∂
∂xµ− 1
xν
∂
∂xν
]FT = 0 . (F.31)
This can be solved by FT = FT (ξ), where FT (ξ) is an arbitrary function of ξ =∑n
µ=1 x2µ.
Substituting it into (F.29) and differentiating by ∂x1∂x2 · · · ∂xn the both sides of the equationmultiplied by the factor
∏µ=ν(x
2µ − x2
ν), we arrive at the condition F(n)T (ξ) = 0, which implies
that FT (ξ) is an n-th order polynomial in ξ. Furthermore, going back to (F.29) again andcomparing the coefficients of the equation, we find that FT must be a linear function. Namely,to be consistent with (F.29), the function must be chosen as FT (ξ) = C1ξ + C2 where C1 and C2
are constants. Then, using the identities
n∑µ=1
x2jµ
Uµ= 0 (j = 0, · · · , n − 2) ,
n∑µ=1
x2(n−1)µ
Uµ= 1 ,
n∑µ=1
x2nµ
Uµ=
n∑µ=1
x2µ , (F.32)
we obtain
Fµ =n∑
k=0
c2kx2kµ , (F.33)
where c2k (k = 0, 1, · · · , n) are constants with C1 = c2n and C2 = c2(n−1). In the end, using(F.27) and (F.33), the present problem of solving the Einstein equation (F.20) has been reducedto that of solving first-order ordinary differential equations for Ξµ:
Ξ′µ −
1 + 2β2x2µ
xµ(1 + β2x2µ)
Ξµ −∑n
k=0 c2kx2kµ
xµ(1 + β2x2µ)
= 0 (F.34)
The general solution is
Ξµ =n∑
k=0
c2kφ2k(xµ) + aµxµ
√1 + β2x2
µ , (F.35)
114
where aµ are integral constants, φ2k(x) for k ≥ 1 are given by
φ2k(x) = x√
1 + β2x2
∫ x
0
t2(k−1)dt
(1 + β2t2)3/2(F.36)
and φ0(x) ≡ −1. The solution in 2n dimensions contains parameters c2k (k = 0, · · · , n), aµ
(µ = 1, · · · , n) and β. Note that for low k, we have
φ2(x) = x2 , φ4(x) = − x
β2
(x − Arcsinh(βx)
√1 + β2x2
β
),
φ6(x) =3x
2β4
(x +
β2
3x3 − Arcsinh(βx)
√1 + β2x2
β
),
φ8(x) = −15x
8β6
(x +
β2
3x3 − 2β4
15x5 − Arcsinh(βx)
√1 + β2x2
β
). (F.37)
In 4 dimensions, for µ = 1, 2, we obtain
Ξµ = −c0 + c2x2µ + aµxµ
√1 + β2x2
µ − c4xµ
β2
xµ −Arcsinh(βxµ)
√1 + β2x2
µ
β
. (F.38)
The form coincides with the Wahlquist solution.
In the limit β → 0, we have φ2k → x2k/(2k − 1). That is,
Ξµ =n∑
k=0
c2kx2kµ + aµxµ , (F.39)
where c2k = c2k/(2k−1). This takes the same form as Kerr-NUT-(A)dS metrics in 2n dimensionsfound by Chen-Lu-Pope [61].
Finally, let us comment about the equation of state. From (F.23), (F.24), (F.30) and (F.33),we have
2ρ
D − 2= −c2n − 3β2PT ,
2p
D − 2= c2n + β2PT . (F.40)
Hence, the equation of state is ρ + 3p = (D − 2)c2n.
F.2.2 Odd dimensions
Let us consider odd dimensions D = 2n + 1. The metric ansatz in 2n + 1 dimensions is givenby
g(2n+1) =n∑
µ=1
f2µ
Pµdx2
µ +n∑
µ=1
Pµ
(n−1∑k=0
A(k)µ dψk
)2
+ S
(n∑
k=0
A(k)dψk
)2
(F.41)
115
with unknown functions Ξµ. The off-diagonal components of the Ricci curvature are
Rµν = Rµν = Rµµ = Rµ0 = 0 , Rµν = β2(D − 2)√
Pµ
√Pν , Rµ0 = β2(D − 2)
√Pµ
√S . (F.42)
The diagonal components are
Rµµ = Iµ(PT ) + β2
(Iµ(P (2)
T ) +32xµ∂µPT + PT
)− 1
2xµ
∂
∂xµ
(PT + β2P
(2)T
),
Rµµ = Rµµ + β2(D − 2)Pµ , (F.43)
R00 = −n∑
µ=1
1xµ
∂
∂xµ
(PT + β2P
(2)T
)+ β2PT + β2(D − 2)S ,
where
PT =n∑
µ=1
Pµ = PT + S , P(2)T =
n∑µ=1
x2µPµ = P
(2)T (F.44)
and
Pµ =Ξµ
Uµ, Ξµ = Ξµ − (−1)n k2
x2µ
. (F.45)
We assume that the velocity u lies in the plane of the Killing vectors
u =1√PT
n∑µ=1
√Pµeµ +
√Se0
. (F.46)
The equation reduces to
ρ − p
D − 2= R11 = R22 = · · · = Rnn , (F.47)
ρ + p
D − 2= −β2PT , (F.48)
and for all µ,R00 = Rµµ + β2(D − 2)S . (F.49)
Similarly to even dimensions, we find from the direct calculation that the µµ- and 00-components of the Ricci curvature can be written in the simple form
Rµµ = − 12xµ
∂FT
∂xµ− 2β2PT , (F.50)
R00 = −n∑
µ=1
Fµ
x2µUµ
− 2β2PT + β2(D − 2)S , (F.51)
where
FT =n∑
µ=1
Fµ
Uµ(F.52)
116
andFµ = xµ(1 + β2x2
µ)Ξ′µ − β2x2
µΞµ . (F.53)
As was discussed in even dimensions (cf. (F.33)), the equation (F.47) requires that Fµ are givenas
Fµ =n∑
k=0
c2kx2kµ . (F.54)
Indeed, by virtue of (F.50) and (F.51), we easily see that (F.54) together with c0 = 0 solves (F.47)and (F.49). From the equality of (F.53) and (F.54), we obtain the first-order ordinary differentialequations
Ξ′µ − β2xµ
1 + β2x2µ
Ξµ −∑n
k=1 c2kx2k−1µ
1 + β2x2µ
= 0 . (F.55)
The general solution is
Ξµ =n∑
k=1
c2kφ2k(xµ) + aµ
√1 + β2x2
µ , (F.56)
where aµ are integral constants and
φ2k(x) =√
1 + β2x2
∫ x
0
t2k−1dt
(1 + β2t2)3/2. (F.57)
Note that for low k, φ2k (k = 1, 2, · · · ) are written as
φ2(x) = − 1β2
(1 −
√1 + β2x2
), φ4(x) =
2β4
(1 +
β2
2x2 −
√1 + β2x2
),
φ6(x) = − 83β6
(1 +
β2
2x2 − β4
8x4 −
√1 + β2x2
),
φ8(x) =165β8
(1 +
β2
2x2 − β4
8x4 +
β6
16x6 −
√1 + β2x2
). (F.58)
Thus, we obtain the (2n + 1)-dimensional solution
Ξµ =n∑
k=1
c2kφ2k(xµ) + aµ
√1 + β2x2
µ − (−1)nk2
x2µ
. (F.59)
The (2n + 1)-dimensional solution contains parameters c2k (k = 1, · · · , n), aµ (µ = 1, · · · , n), k2
and β. In the limit β → 0, we have φ2k → x2k/2k, which reproduces Kerr-NUT-(A)dS metrics in2n + 1 dimensions [61]. We have
2ρ
D − 2= −c2n − 3β2PT ,
2p
D − 2= c2n + β2PT . (F.60)
Hence, the equation of state is ρ + 3p = (D − 2)c2n like the even dimensional case.
117
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