Ergin Sezgin and Per Sundell- Geometry and Observables in Vasiliev’s Higher Spin Gravity

8/3/2019 Ergin Sezgin and Per Sundell- Geometry and Observables in Vasilievs Higher Spin Gravity

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arXiv:1103

.2360v1[hep-th]

11Mar2011

MIFPA-11-8

Geometry and Observables in Vasilievs Higher Spin Gravity

Ergin Sezgin 1 and Per Sundell 2,3

1 George and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,

Texas A& M University, College Station, TX 77843, USA

email: [email protected]

2 Service de Mecanique et Gravitation, Universite de Mons

20, Place du Parc, B7000 Mons, Belgium

email: [email protected]

ABSTRACT

We provide global formulations of Vasilievs four-dimensional minimal bosonic higher

spin gravities by identifying structure groups, soldering one-forms and classical observables.

In the unbroken phase, we examine how decorated Wilson loops collapse to zero-form chargesand exploit them to enlarge the Vasiliev system with new interactions. We propose a metric

phase whose characteristic observables are minimal areas of higher spin metrics and on-shell

closed abelian forms of positive even degrees. We show that the four-form is an on-shell

tree-amplitude of the generalized Hamiltonian action recently proposed by Boulanger and

one of the authors. In the metric phase, we also introduce tensorial coset coordinates and

demonstrate how single derivatives with respect to coordinates of higher spins factorize into

multiple derivatives with respect to the coordinates of spin one.

3Ulysse Incentive Grant for Mobility in Scientific Research, F.R.S.-FNRS
http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1http://arxiv.org/abs/1103.2360v1


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Contents

1 Introduction 3

2 Local Formulation 4

2.1 Vasilievs Master Field Formalism . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Minimal Bosonic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Manifest Lorentz Covariance and Type A and B Models . . . . . . . . . . . 10

3 Global Formulation 12

3.1 Structure Groups and Observables . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Initial Data, Transition Functions and Boundary Values . . . . . . . . . . . 13

4 Unbroken Phase: Wilson Loops and Zero-Form Charges 14

5 Geometrical Phase Based on hs+(4) 175.1 -Even Structure Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.2 Abelian p-Form Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.3 Soldering Mechanism and Higher Tensorial Coordinates . . . . . . . . . . . 21

5.4 Generalized Metrics and Minimal Areas . . . . . . . . . . . . . . . . . . . . 23

6 Deformation of Bulk Action and Tree Amplitudes 24

7 Conclusions 27

A Lorentz Covariantization 28

B Additional Zero-Form Invariants in the Duality Extended Model 29

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1 Introduction

Vasilievs higher spin gravities are fully nonlinear extensions of ordinary gravities by higher

spin gauge fields as well as specific lower-spin fields [1, 2] (for reviews, see [3, 4]). Their

classical equations of motion admit anti-de Sitter or de Sitter spacetimes as exact solutions

with unbroken higher spin symmetries. The corresponding fluctuation fields are towers of

symmetric tensor gauge fields, known as Fronsdal tensors, forming higher spin multiplets.

In this paper, we shall study four dimensional minimal bosonic models consisting pertur-

batively of a physical scalar, a metric and a tower of real Fronsdal tensors of even ranks

greater than or equal to four each occurring once [1, 3, 5, 6]1

So far Vasilievs theory has been studied primarily at the level of the field equations and

in terms of locally defined quantities in four dimensional spacetimes. It is clearly desirable

to develop an action principle, and a globally defined framework for classical and quan-tum observables. A topological action that preserves all locally defined gauge symmetries

was proposed already in [7] and recently revisited in [8, 9]. The action constructed in [8]

takes into account on-shell duality extensions and additional couplings that are higher than

first order in the Lagrange multiplies, known as generalized Poisson structures, and the

possibility to formulate a topological sigma model of the generalized Hamiltonian type.

In this paper, we shall add new terms to this action, that define nonvanishing semi-

classical amplitudes on-shell, and break the full higher spin algebra off-shell down to a

subalgebra that defines a structure group of a particular broken geometric phase of thetheory. The choice of structure group is not unique and one may have several global de-

scriptions of one and the same locally defined Vasiliev system (or any unfolded system

for that matter) with physically distinct properties as we shall illustrate below. Here we

shall propose a structure group that one may think of as a particular infinite-dimensional

generalization of the four-dimensional Lorentz algebra.

We shall treat classical observables given by abelian p-form charges and minimal p-

volumes, which are invariant under generalized local Lorentz transformations off-shell and

under the remaining gauge transformations on-shell and up to total derivatives (and henceunder diffeomorphisms which are incorporated into the Cartan gauge algebra)2.

The classical observables are defined intrinsically in a coordinate-free fashion at the fully

1The characterization of these models as minimal, essentially refers to the fact that their physical spectrum

cannot be truncated further; for a more precise definition, see [5].2For a discussion of actions and charges in unfolded dynamics which does not, however, incorporate the

effects of structure groups, see [10].

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nonlinear level. Their evaluation on exact solutions may require, however, the specification

of local coordinates. The unfolded dynamics approach indeed facilitates the introduction

of tensorial coordinates for an infinite-dimensional base manifold. As we shall illustrate

below, the dependence of all perturbatively defined Weyl tensor components on the higher

tensorial coordinates is completely constrained. For example, the derivative of the physical

scalar with respect to the n-th rank symmetric traceless tensorial coordinate is expressed

in terms of nth order derivative with respect to the four-dimensional spacetime coordinate.

The framework that we shall set up may also facilitate the nonlinear completions of earlier

attempts at understanding higher spin geometry at the level of free fields [11, 12].

The paper is organized as follows: we begin in the next Section by outlining certain key

features of Vasilievs formalism. Using these we then move on to define global formulations

in Section 3, stressing the role of the structure group. In Section 4 we examine the unbroken

phase, where the structure group is the full higher spin algebra, and in Section 5 we examine

a broken phase of the theory, where the structure group is a certain natural extension of

sl(2,C). In Section 6 we discuss the perturbation of the generalized Hamiltonian action of [8]

by terms that are nonvanishing on-shell. In Section 7 we summarize our results and comment

on select open problems. Appendix A contains details of the Lorentz covariantization and

Appendix B contains a nonvanishing amplitude in twistor space that depends on a physical

two-form arising due to duality extension.

2 Local Formulation

The geometric formulation of higher spin gravity that to be presented in the next Section

rests on Vasilievs unfolded master field formalism (see [3, 4] for reviews). In this Section

we shall therefore first present the general ideas of Vasilievs formalism after which we focus

on the manifestly Lorentz-covariant formulation of the minimal bosonic models, referring

to [5] for more details.

2.1 Vasilievs Master Field Formalism

In a fairly general setting, the Vasiliev system of equations for four dimensional higher spin

gravities can be written as a quasi-free differential algebra consisting of a set of differential

forms living on a noncommutative base manifold B of Poisson type. These forms take

their values in an associative -product algebra A which encodes the detailed information

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about the symmetry algebra and spectrum.3 The space (B) of differential forms on B is a

noncommutative and associative algebra equipped with a canonical product, including the

graded anti-0commutative wedge product, that we also denote by . In other words, the

master fields are elements of the tensor-product algebra

A = (B) A . (2.1)The quasi-free differential algebra4 consists of a master one-form A, a master zero-form and a set of globally well-defined two-forms Jr (r=1,2,...,N) obeying [1]

F + Fr() Jr = 0 , (2.2)D = 0 , (2.3)

d

Jr = 0 , (2.4)

where Jr is subject to the additional algebraic constraintf Jr = Jr (f) , (2.5)

for any element f in the full algebra A. The curvature and covariant derivative are definedby

F = d A + A A , D = d + A,

. (2.6)

where we have used the notation

f,

g

=

f

g (1)

fg

g (

f) and is an automorphism

of A obeying d = d. Furthermore, Fr are -functions representing interaction freedomFr() = f1, r + f3, r () + , (2.7)

where f2n+1, r() (n = 0, 1, 2, . . . ) are complex valued and on-shell closed functionals of ,i.e. d f2n+1,r = 0 , f2n+1,r, f

= 0 f A . (2.8)

The definitions given above suffice to show that the equations of motion are formally

Cartan integrable, i.e. compatible with

d2 0 on universal base manifolds, or equivalently,

with the Bianchi identities DF 0 and D2 F , . It follows that the equations ofmotion remain invariant under the Cartan higher spin gauge transformations

A = D , = ,

, (2.9)

3The algebra A can contain Grassmann odd generators, which lead to higher spin supergravities, though

here we shall soon focus on the bosonic models.4The terminology of quasi-free differential algebra refers to the fact that Jr are constrained by both

differential and algebraic constraints.

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with locally defined zero-form parameters . These transformations close to form the fullhigher spin algebra.

So far in the literature, f2n+1,r have been taken to be constants. Here we propose

to extend the Vasiliev system by introducing allowing f2n+1,r to depend on the master

zero-form . We shall come back to this point at the end of Section 2.3 and Section 4.It remains to specify the local geometry of B and the attendant properties (Jr, ), for

which it is essential that

the symplectic leaves of B form a noncommutative twistor space Z of topology C2

with globally defined coordinates Z = (z, z) where = 1, . . . , 4 label an Sp(4;R)-

quartet and , = 1, 2 label two SL(2;C)-doublets [1];

the algebra A consists of the tensor product of the algebra [0](Y) of zero-forms on a

copy Y of Z with coordinates Y = (y, y) and an internal associative algebra Aint

[13, 14], viz.

A = [0](Y) Aint . (2.10)

In what follows we assume that B has the bundle structure5

Z B M , (2.11)

where M is a commuting manifold (or supermanifold in the case of supersymmetric models)

and the projection amounts to restriction to some given base point in Z, say Z = 0. Theexterior derivative on B is thus given

d = dXMM + dz + dz , (2.12)where =

z , =

z

and XM coordinatize M. Without loss of generality, the twistor

coordinates can be taken to obey the following canonical oscillator algebra

[y, y ] = 2i , [z, z ] = 2i , (2.13)

[y, y ] = 2i , [z, z ] = 2i . (2.14)

and as the Y and Z spaces are two mutually commuting copies one has

[y, z ] = 0 , [y, z ] = 0 . (2.15)

5We note that for certain exact solutions [16] the topology of the fiber space Zmay wary as goes from one

point to another on the base manifold in which case the bundle picture is replaced by a more sophisticated

geometrical structure.

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One may also assume the reality conditions

(y) = y , (z) = z , d = d . (2.16)It follows that A admits two basic types of automorphisms (, ) that act on Aint in a modeldependent fashion, namely as inner automorphisms (int, int) generated by idempotent

elements (int, int) in Aint, and on B Y such that d(, ) = (, )d and(XM; y, y; z, z) = (XM; y, y; z, z) , (2.17)

(XM; y, y; z, z) = (XM; y, y; z, z) . (2.18)

Their actions on [0](Y Z) is thus inner and generated by the adjoint action by the

Kleinian operators

= cos( N) , = cos(N) , (2.19)where the ( N, N) are Weyl-ordered chiral number operators defined by

N = 12

a ,a+ , a ,a+ = , N = ( N) . (2.20)Using the realization (a+ ,a ) = 12 (y + z, iy + iz) yields

N =

i

2y z , N =

i

2y z . (2.21)

The Kleinians can be expressed in various ordering schemes (for details, see [15]); for the

perturbative weak-field expansion it is convenient to normal order (a+ ,a ) which yields [1] = [exp(iyz)]Normal , = exp(iyz)Normal , (2.22)

whereas for the purpose of performing chiral supertraces (see below) it is more convenient

to work with Weyl order which yields6

=

(2)22(y)2(z)

Weyl,

=

(2)22(y)2(z)

Weyl. (2.23)

It follows that there exists a doublet of globally defined two-forms Jr = (J1,J1) given byJ1 = i

4dz2 int , J1 = i

4dz2 int , (2.24)

6For the purpose of finding exact solutions it is more useful to factorize = y z [16] where y and

z , respectively, are Kleinians for the chiral oscillators y and z; one can then proceed by working in Weyl

order where y = 22(y) and z = 2

2(z) or normal orders where y and z are Gaussian.

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provided that the elements f A obey(f) = f . (2.25)

As a result, the source term Fr() Jr cannot be redefined away, and it yields the correctlinearized source terms for the two-form curvatures on M upon reducing to Z = 0 providedthat f1, r(0) does not vanish [1].

Vasilievs equations can be conveniently rewritten in terms of the variables

S = z 2i A , S = z 2i A , (2.26)obeying (S) = (S) and (S) = (S). The internal equations take the form ofgeneralized versions of Wigners deformed oscillator algebra [1]7:

S[ S] = i(1 F1 int) , S[ S] = i(1 F1 int) , (2.27)S S = S S , (2.28)S + (S) = 0 , S + (S) = 0 . (2.29)The Vasiliev equations can thus be written on two equivalent forms, namely

as in Eqs. (2.2)(2.4), which one may refer to as an associative quasi-free differential

algebra on an extended correspondence space8, namely the product space B Y; or

as in Eqs. (2.27)(2.29), (2.48) and (2.49), which one may refer to as a graded

commutative quasi-free differential algebra on a commutative base manifold M with

fiber generated by functions on Y Z, generalized curvature constraints (2.48) and

(2.49) and algebraic zero-form constraints (2.27)(2.29);

The correspondence space formulation in the spirit of string field theory, while the quasi-free

differential algebra formulation relies on gauging a higher-spin Lie algebra on an ordinary

commutative base manifold.7Starting from Wigners algebra [Si, S

,j ] = ji + (1)N

ji (i = 1, 2); R;

ji a rank one projector;

Si = Si(aj, aj); and [ai, a

j] = ji , Vasilievs generalization consists of complexifying the oscillators and

taking the anyonic parameter to be a function of the oscillators.8This terminology refers to the fact that one has a parent theory formulated on a higher-dimensional

space that contains projections to theories formulated on lower-dimensional submanifolds. These latter two

theories thus correspond to each other via the parent theory and one may refer to the space in which the latter

is defined as the correspondence space. In this sense and taking into account dynamical symmetry breaking

mechanisms, one may think of Vasilievs formulation of higher spin gravity as the natural framework for

understanding the correspondence between the formulations of gauge theory in four-dimensional spacetime

and in Penroses twistor space.

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2.2 Minimal Bosonic Models

Clearly, the simplest case is to take

Aint = 1 , int = int = 1 . (2.30)

The master one-form is then given by

A = dXM AM(X,Y,Z) + dz A(X,Y,Z) + dz A(X,Y,Z) , (2.31)and the master zero-form = (X,Y,Z). Bosonic models are obtained by imposing theconditions

( A) = A , () = . (2.32)Minimal bosonic models whose weak-field expansions are in terms of symmetric tensors of

even ranks require the stronger conditions

( A) = A , () = () , (2.33)where the anti-automorphism is defined by

(XM; y, y; z, z) = (XM; iy, iy; iz, iz) , (2.34)

and (

Jr) =

Jr. Models with Lorentzian spacetime signature and negative cosmological

constant require [1, 5, 6]

9

( A) = A , () = () , (2.35)(J1) = J1 , (F1()) = F1() , (2.36)

where C. Field redefinitions

F() = f1 + f3 () + , (2.37)where f2n+1() obeydf2n+1 = 0 , (f2n+1) = f2n+1 , f1(0) = 0 , (2.38)lead to the identifications

F1() F1(F()) . (2.39)9For other signatures, see [17].

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In terms of gauging an algebra, one thus has a gauge field on M valued in the higher

spin Lie algebra

hs(4) =

P(Y, Z) : (

P) = (

P) =

P

, ad P1(

P2) =

P1,

P2

, (2.40)

coupled to algebraically constrained zero-forms and S = (S, S) that take their values,respectively, in the twisted-adjoint representation

T[hs(4)] = T(Y, Z) : (T) = (T) = (T) , (P)(T) = [P , T] , (2.41)and the quasi-adjoint representation

T[hs(4)] = T(Y, Z) : (T) = iT , (T) = T , (P)(T) = [Q, T] ,(2.42)

in accordance with (2.33), (2.36) and (2.26).

2.3 Manifest Lorentz Covariance and Type A and B Models

As first observed in [3], and studied further in [5], manifest local Lorentz symmetry is

achieved by first introducing a connection and = () transforming canonically

as

= d 2(

) , (2.43)

under local Lorentz transformations, and then shifting the connection on M as follows [3, 5]

AM = WM + 14i

MM + MM , (2.44)

M = M(0) + M(S) , M = M(0) + M(S) , (2.45)M(0) = y( y) z( z) , M(0) = y( y) z( z) , (2.46)M(S) = S( S) , M(S) = S( S) . (2.47)The constraints on M now takes the form (see Appendix A)

W + W2 + r(0) + r(S) = 0 , (2.48) + [W , ] = 0 , S + [W , S] = 0 , S + [W , S] = 0 , (2.49)

where we have defined

r(0) = 14i

rM(0) + rM(0) , r(S) = 14i rM(S) + rM(S) , (2.50)

r = d , r = (r) = d

, (2.51)

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and the Lorentz covariant derivative

= d + ((0)) , (0) = 14i

M(0) + M(0) , (2.52)

acts in accordance with the following canonical representation of the Lorentz algebra:

W = (0) ,W

, (0) = d(0) + (0),(0)

, (2.53)

= (0) ,

, S = S (0) , S

, (2.54)

where ((0) ) and the gauge parameter(0) = 14i

M(0) + M(0)

. (2.55)

The Lorentz-covariantized Cartan gauge transformations take the form

W = + [W , ] , (0) = 0 , (2.56) = [, ] , S = [, S] . (2.57)

The introduction of the canonical Lorentz connection leads to an over parameterization

of connection on M, thus inducing a shift symmetry with unconstrained parameters

(Mab, M

) acting such that

A = 0 , = 0 , S = 0 , (2.58)which ensures shift symmetry of the master equations and implies

W = 14i

M + M , (M , M ) = (M , M ) . (2.59)

Provided that M has nontrivial components along y( y) , the shift symmetry can beused to impose the condition that W has vanishing components along y( y) .

Assigning the scalar field intrinsic spacetime parity 1, and imposing parity symmetry,

drastically reduces the interaction freedom leading to the condition [6]

F1 = F1 , (2.60)

which can be redefined away using (2.37) leading to the minimal bosonic

Type A model : F1 = , (2.61)Type B model : F1 = i . (2.62)

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3 Global Formulation

In this section we set up the global formulation of Vasilievs minimal bosonic models based

the notions of structure groups, intrinsically defined observables and other geometric data;

for related discussions, see [18, 19, 8].

3.1 Structure Groups and Observables

In order to provide a formulation of Vasilievs theory that is defined globally on M a

structure group needs to be defined. By its definition, the structure group is generated by

a structure algebra given by a subalgebra

t

hs(4) sl(2,C) . (3.1)

Decomposing the manifold M into coordinate charts MI labelled by I, i.e.

M =I

MI , (3.2)

the classical moduli space then consists of gauge orbits of locally defined field configurationsWI,(0)I , I, SI, (3.3)glued together by transition functions

GI

I = exp(

tI

I ) with

tI

I

t defined on MI MI.

The classical moduli space thus encodes a principalt-bundle with connection = t(W + (0)) , (3.4)where t denotes the projection to

t, and an associated t-bundle with section (E, , S)where E = (1 t)(W + (0)) . (3.5)Viewing the classical moduli space as an infinite-dimensional manifold, it is equipped by a

space of functions generated by classical observables, i..e functionals O

WI,

(0)I ,

I,

SI,

whose invariance properties under gauge transformations with locally defined parametersI +(0) are: IO 0 off-shell for locally defined

t-valued parameters I = t(I +(0)I ) (definedindependently for each I); and

{ I}

O = 0 on-shell for parameters I = (1 t)(I + (0) ) that form sections of at-bundle associated to the principal bundle.12


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Taken together, these two conditions imply that the observables are left invariant on-shell

by the diffeomorphisms of M.

Each structure group corresponds to a specific phase of the theory with its own classical

moduli space and associated set of classical observables. The minimal bosonic higher spin

gravity has four natural phases:

an unbroken phase with structure algebra hs(4) sl(2,C) and classical observablesgiven by decorated Wilson lines and zero-form charges, which we shall treat in more

detail in Section 4;

a broken phase with -even higher-spin structure algebra hs+(4) sl(2,C) where

hs+(4) =

1

2(1 + )

hs(4) , (3.6)

and characteristic classical observables given by minimal areas and charges of on-shell

closed abelian forms, which will be treated in Section 5;

a broken phase with chiral higher-spin structure algebra10 hl(2,C) sl(2,C) wherehl(2,C) = P(y, z) + P(y, z) hs(4) , (3.7)

that is, all polynomials in hs+(4) that are purely holomorphic or anti-holomorphic.The geometrical significance of this choice of structure group and its consequences for

deforming the theory off-shell remains to be investigated;

the broken Lorentz invariant phase with structure algebra given by the canonical

sl(2,C) which is implicitly assumed in most of the existing literature on higher spin

gravity.

3.2 Initial Data, Transition Functions and Boundary Values

Two locally defined configurations (3.3) belong to the same gauge orbit if they are related

by locally defined gauge transformations GI = exptI where tI is a t-valued otherwiseunrestricted function on MI. The local configurations can thus be written as [20, 21]

11

WI = (LI)1 dLI 14i

MM + MM , (3.8)

I = (LI)1 I(Y, Z) (LI) , SI = (LI)1 SI(Y, Z) LI , (3.9)10This choice was pointed out to us by M. Vasiliev.

11For a more general discussion of gauge functions and initial data in Cartan integrable systems, see [8].

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where LI are gauge functions valued in the coset generated by hs(4)/t and related acrosschart boundaries via the transition functions, viz.

LI =

GI

I

LI . (3.10)

Assuming that LIpI

= 1 for base points pI MI, the reduced master fields

I = IpI

, SI(Y, Z) = SI pI

, (3.11)

obey the deformed oscillator algebra (2.27)(2.29) to be solved subject to initial data

CI(Y) = IZ=0

, (3.12)

and boundary conditions on

SI that we refer to as S-moduli and that are related to

projectors in the -product algebra; for example, see [17]. It suffices to provide these dataat single chart, say at the point pI0 MI0 , since the other values can then be obtained by

combining gauge functions and transition functions. The classical observables depend on

the gauge functions via their values at M modulo boundary gauge transformations and

the homotopy classes GII = GI GII (GI)1 (3.13)Thus, a globally defined solution is characterized by (see also [8])

the initial data CI0(Y) and the SI0-moduli in twistor space ;

the homotopy classes [GII ]; the values of the gauge functions LI at M.

In summary, the unfolded equations of motion on MI can be integrated using gauge func-

tions and initial data given by the values of the zero-forms at a single point in MI, which

implies that no new strictly local degrees of freedom are introduced if new dimensions are

added to M. When combined with the soldering mechanism this can be used to formulate

higher spin gravity in extended spacetimes with higher tensorial coordinates as we shall

exemplify in Section 5.

4 Unbroken Phase: Wilson Loops and Zero-Form Charges

In the unbroken phase, the principal bundle has the connection

= W + (0) , (4.1)14


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and all Cartan gauge symmetries as well as local Lorentz symmetries remain locally defined.

The canonical Lorentz connection can be shifted away by choosing the gauge

= = 0

O =

W . (4.2)

As a result, the Vasiliev equations consist of the deformed oscillator algebra (2.27)(2.29)

and the differential constraints

dW + W W = 0 , d + [W , ] = 0 , dS + [W , S] = 0 , (4.3)where d = dXMM. To define classical observables as functionals of the master fields that

do not depend on the ordering prescription, a trace operation on the oscillator algebra is

needed. We shall use

Tr f(y,z,dz; y, z, dz) = Y

d2

y2

d2

y2

Z

f(y,z,dz; y, z, dz) , (4.4)with normalization and integration domain chosen such that

Tr f(y,z, y, z) := Tr d2z2

d2z

2f(y,z, y, z) =

Y

d2y

2

d2y

2

Z

d2z

2

d2z

2f(y, z; y, z) ,

(4.5)

where (y, z) and (y, z) are treated as real and independent variables, and we are using the

convention that the integration operationZ() projects onto the top form degree in Z. As

discussed in Appendix B, the key formal properties of this trace operation are:

independence of the choice of ordering scheme used to represent operators by symbols;

cyclic invariance modulo boundary terms in the twistor spaces;

-invariance, i.e. Tr[(f)] = Tr[f] , (4.6)idem , and for this reason we refer to Tr as the chiral trace.

The insertion of inner Kleinians into the chiral trace operation yields graded cyclic chiral

trace operations Tr [ ()], Tr () and Tr () referred to as chiral supertraces ofthe oscillator algebra. The last one reduces to a cyclic trace operation in the bosonic model

due to (2.32). The Kleinian produces phase factors that localize the chiral integral toy = z = 0 in the Weyl ordering scheme as can be seen from (2.23), and idem .

Natural classical observables in the unbroken phase are Wilson loops along paths

M with impurities in the form of adjoint vertex operators inserted at points xi for

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i = 1, . . . , N . A basis for these observables consists of ( = 0, 1)

W;{xi}

ki, ki; i, i

= Tr (

) P Ni=1

Vki,kii,i

(, , S)xi

U[W] , (4.7)where

the vertex operator

Vk,k,

(, , S) = exp i(S + S) ()k ()k , (4.8)with k, k,n, n N and we have defined the adjoint elements

= , = , (4.9)obeying the following relations

d + [W , ] = 0 , d + [W , ] = 0 , (4.10){S, } = [S, ] = 0 , {S, } = [S, ] = 0 , (4.11) = , [, ] = 0 , = ; (4.12)

the path-ordered exponential U[W] is defined byP

V U[W] := PVl

GIlIl1pl expl

WIl

, (4.13)

where has been cut into oriented links l with end-points pl and pl+1 passing through

charts Il, viz.

= l l , l = {pl+1} {pl} , (4.14)

where the sign indicates the orientation, and insertions of transition functions GIlIl1in passages from l1 to l assure gauge invariance.

From (3.8) and since = = 0 it follows that if there is no decoration along a link l then

Pexp

l

WIl = (

LIl)

1pl

(

LIl)

pl+1

. (4.15)

Thus, if V = 1 then

P

U[W] = 1 . (4.16)Formally, the flatness conditions (4.3) and (4.10) imply that the Wilson loops remain in-

variant under smooth deformations of and xi. If it is possible to move all impurities to a

single point on , say x, then

W;x

k, k; ,

= Tr () PVk,k,

(, , S)x

U[W] , (4.17)16


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which can b e collapsed further using (4.16). Thus, provided no singularities appear, the

decorated Wilson loops are equivalent to the zero-form charges

I

k, k; ,

=

Tr

(

) Vk,k

,(

,

,

S)

. (4.18)

Expanding in (, ) yields expressions of the form

Tr () S(1 Sn) S( 1 Sn) ()k ()k . (4.19)If m + m > 0 then (4.11) and (A.5) imply that the argument of the trace becomes a

commutator that vanishes formally by cyclicity, whereas a rigorous calculation requires

taking into account the fall-off behavior of the deformed oscillators at the infinity of twistor

space. Setting m + m = 0 yields the Lorentz-scalar zero-form charges [21, 17]12

I k, k = I k, k; , (,)=(0,0)

= Tr () ()k ()k , (4.20)where one may take = 0 ifk + k > 0. These charges are on-shell closed and can therefore

be used in (2.7) as

f2n+1,r = f2n+1,r

I(k, k)

. (4.21)

More explicitly, we can take F1() to have an expansion of the formF1 = b1

+ b3

(

)

+

=0,1b3

Tr

(

)

(

) (4.22)

+=0,1

b3Tr () () 1 + O(5) , (4.23)

with multi-traces arising from O(5) and onwards. An interaction ofO(2n+1) contains 2nintegrals over Y Z if it contains at least one untraced , i.e. if it is non-central in A, and2n + 1 integrals over Y Z if it is proportional to 1, i.e. if it is central. Consequently the

new non-central interaction terms are as nonlocal in twistor space as the original interaction

terms that involve no traces.

5 Geometrical Phase Based on hs+(4)In this section we exhibit various observables and the soldering mechanism of the geometrical

formulation of Vasilievs theory based on the structure algebra hs+(4) given in (3.6).12For the perturbative regularization of these charges, see [19].

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5.1 -Even Structure Group

Taking the structure algebra to be the direct sum of the canonical Lorentz algebra and the

-even subalgebra

hs+(4) of the full Cartan gauge algebra

hs(4) leads to connection

and

soldering one-form E given by = W+ + (0) , E = W , W = 12

(1 )W . (5.1)Letting denote the canonical representation oft given by the adjoint representation of12 (1 + )

hs(4) and the canonical representation of sl(2,C) defined in (2.54) and (2.43),and introducing the covariant derivative

(

) = d +

(

) = + ad W+ , (5.2)

where the Lorentz covariant derivative is defined in (2.52), one has the Bianchi identities

2 (R) , R 0 , (5.3)R = W+ + W+ W+ + r(0) , (5.4)

where r(0) is defined in (2.50). Splitting also and S into even and odd parts, = 1

2(1 ) , S = 12 (1 )S , S = 12 (1 )S , (5.5)

the master field equations in X-space can be written asR + E E+ r(S)+ = 0 , E+ r(S) = 0 , (5.6) + E,

= 0 , S + E, S

= 0 , (5.7)

where we have defined the projections

r(S) = 12

(1 )r(S) , (5.8)of the quantity r

(S) given in (2.50) using 12 (1 )M(S) = S

+ S

+ S

S

. The Cartan

integrability is a consequence of

r(S) = [E,r(S)] . (5.9)The locally defined gauge symmetries are

the shift symmetry (2.58); and

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the generalized Lorentz transformations () with parameter = + +(0) , = 12 (1 ) , (5.10)

where the parameter (0) of the manifest local Lorentz symmetry acts in accordancewith (2.54) and (2.43), and + generates the transformations

+W+ = + , +(0) = 0 , + E = [+, E] , (5.11)+ = [+, ] , + S = [+, S ] . (5.12)

The broken gauge symmetries, referred to as generalized local translations, are

= [

E,

] ,

(0) = 0 ,

E =

, (5.13)

= {, } , S = [, S ] , (5.14)where parameter is given by the -odd projection

= . (5.15)We recall that (E, ) is a section of the hs+(4)-bundle. Thus, on the overlap MI MIthe local representatives (EI, I) and (EI , I) are related by a transition function GIIgenerated by

hs+(4). Moreover, the Lie derivative LV along a globally defined vector field

V is equivalent on-shell to the composite parameters

+V = iVW+ , (0)V = iV(0) , V = iV E , V = V , (5.16)5.2 Abelian p-Form Charges

The definition of a soldering one-form facilitates the construction of intrinsically defined

classical observables given by the charges of on-shell closed abelian p-form, viz.

Q[, H] =

H(

E,

,

S,

S, r , r) , (5.17)

where are closed p-cycles in X-space and H are globally defined differential forms that

are cohomologically nontrivial on-shell, i.e.

dH = 0 , H = dK on-shell , (5.18)

(H, K) = 0 off-shell . (5.19)

Thus, the charges Q are invariant

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off-shell under generalized Lorentz gauge transformations (these are manifest symme-

tries of the charge densities);

on-shell under diffeomorphisms in X-space (modulo the equations of motion, the Lie

derivative LVH = d(iVH) where iVH is globally defined implying that d(iVH) =0);

on-shell under deformations of (which is to say that Q are generalized conserved

charges).

The systematic search for abelian charges is tantamount to looking for cohomology groups

in de Rham chain complexes consisting of globally defined differential forms. In particular,

focusing on single chiral traces, one has complexes labelled by

(q ;p) N7 , q = (k, k; m, m; m, m) N6 , (5.20)

consisting of elements of the form

M(q , p; K) = Tr M(q ;p) K , K 1,,, , (5.21)where M(q ;p) are -monomials of degree q in (, (); S, (S); S, (S)) and of totalhomogeneous degree p in (E,r(S), (r(S))) where r(S) is defined in (2.50). In other words,

M(q ;p) belongs to the linear space of multi-linear functions with the above degree assign-

ments; these are generated by all possible permutations modulo the graded cyclicity and

-invariance of T r ( ) K.In the simplest sector, q =

0 , one has Tr Ep K 0 for odd p and all K as well as

for even p and K 1,, using the parity arguments described above. Thus, for = 0the cohomology in the q =

0 sector consists ofTr Ep with p = 2, 4, . . . and their

hermitian conjugates. For finite their r(S)-dressed versions read (p = 2, 4, 6, . . . ) :q =

0 : H[p] =

p/2r=0

p

r

Tr SE(p2r), (

r(S)+)r

, (5.22)

where S( X1, . . . , XN) = 1N! SNX(1) X(N) denotes the totally symmetric -monomialof degree N, that is

q =0 : H[p] = Tr (E E+ r(S)+)(p/2) , (5.23)

whose conservation on-shell can be checked directly using the fact that (E E+r(S)+) = 0on-shell. Considering chain complexes with nontrivial q , we have examined various low-

lying levels without finding any nontrivial cohomologies.

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5.3 Soldering Mechanism and Higher Tensorial Coordinates

The nontriviality of the abelian p-form charges in (5.17) requires to be a nontrivial

closed p-cycle and the soldering one-form

E to have at least rank p. However, the study of

minimal areas and dynamical p-branes requires a soldering mechanism whereby the coseths(4)/hs+(4) in the fiber, or a subspace thereof, becomes identified with the tangent spaceof M. The one-form E then gives rise to a frame field EMA, also referred to as generalizedvielbein, defined by EM = EMA PA , (PA) = PA , (5.24)where PA are the coset generators. The local translations with gauge parameters = A PAare then identified via (5.16) as infinitesimal diffeomorphisms with globally defined vector

field VM =

EMA

A combined with local generalized Lorentz transformations.

The soldering can be examined perturbatively by projecting the quasi-free differential

algebra on M Z to a free differential algebra on a submanifold M M {Z = 0} with

local coordinates XM. To this end, one first solves the constraints on Z given initial data

in the form of the reduced master fields

W(X, Y) = dXMWMZ=0

, (X, Y) = Z=0

, (5.25)

belonging to the reduced higher spin algebra hs(4) and its twisted-adjoint representation

T[hs(4)] given by the Z = 0 projections of (2.40) and (2.41), respectively. Decomposing

underhs

+(4

), the remaining constraints on M {Z = 0} read

+ [, ] + {E, } + P = 0 , (5.26)

+ + E E+ J+ + r+ = 0 , E+ [, E] + J + r = 0 , (5.27)

where J and P are quadratic and linear in W and of O() and O(2), respectively, and r

is linear in (r, r) and ofO(2). Projecting down to M using

E = EZ=0

M

= dXMEAMPA , (PA) = PA , (5.28)

where PA is a basis for the -odd elements ofhs(4) and EAM is invertible, yields

A(, ) + {PA, } + PA = 0 , (5.29)

RAB + PA PB + J+AB + r

+AB = 0 , TAB + J

AB + r

AB = 0 , (5.30)

where we have defined covariant derivative (, ) = + ad, curvature R = +

and generalized torsion T = E+ {, E} and expanded in the frame field as follows:

(, ) = EAA(, ) , R =1

2EAEBRAB , T =

1

2EAEBTAB . (5.31)

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The resulting generalized -cohomology can be analyzed by decomposing PA = {PA}=0

into levels of increasing tensorial rank, viz.

PA =

Pa(2+1),b(2k)

k=0=

M{a1b1 Ma2kb2k Pa2k+1 Pa2+1}

k=0, (5.32)

where (Mab, Pa) are the generators of so(3, 2) and Pa(2+1),b(2k) is a Lorentz tensor of

type (2 + 1, 2k). The invertibility can be achieved by working in a triangular gauge

for EMA which requires the notion of a level expansion of the coordinates as well, say

XM =

XM=0

where XM =

X(2+1),(2k)k=0

, such that it can be assumed that

EMA

(XM )=0= 0 , > , > 0 . (5.33)

The gauge function L = exp(iXMAMPA) leads to a higher spin generalization of AdS4

with frame field and connection given by non-Gaussian integrals for (XM ) = 0 that can

be expanded in terms of Gaussian integrals at (XM ) = 0.

The zeroth level of the zero-form constraint reads

a(, ) + {Pa, } + Pa = 0 , (5.34)

where the translation generator is given by the twistor relation

Pa =1

4(a)

yy , (5.35)

which implies that (n+2s),(n) is identifiable as the nth order symmetrized covariant vec-torial derivative of the primary spin-s Weyl tensor (2s) [5]. On the other hand, the th

level of the zero-form constraint implies

a(2+1)(, ) + {Pa(2+1), } = 0 , (5.36)

where the higher translation generator is now given by the enveloping formula

Pa(2+1) = P{a1 Pa2+1} . (5.37)

As a result, the tensorial derivatives a(2+1)(, ) factorize into multiple vectorial deriva-tives; for example, the tensorial derivative of the physical scalar = |Y=0 factorizes into

a(2+1)(, ) {a1(, ) a2+1}(, ) . (5.38)

More generally, the derivatives A(, )C(2s) are given by vectorial derivatives of primary

Weyl tensors (2s) of spins s depending on s and symmetry property of the tensorial

coordinate A.

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Thus, for fixed , the second order operators A(2+1)B(2+1)A(2+1)(, )B(2+1)(, )

are constrained: for = 0 the resulting second-order equations yield the well-known physical

spectrum upon imposing suitable boundary conditions; for > 0 the resulting higher-

derivative equations hold locally in a trivial fashion in view of the = 0 equations and

the constraints. Not surprisingly, the nontrivial effects from the extension thus reside in

boundary conditions; for example, a closed cycle of dimension p with p = 6, 8, . . . may

activate the corresponding abelian p-form charge.

5.4 Generalized Metrics and Minimal Areas

Given a topologically nontrivial p-cycle [] in X-space, one can construct classical observ-

ables as minimal areas with respect to rank-s metrics

ds = dXM1 dXMsGM1...Ms = (s)(, S, S; E , . . . , E) , (5.39)where (s)(, S, S; , . . . , ) denotes s-linear and totally symmetrict-invariant functions13.In other words, letting A[, G(s)] denote the area of the surface

[], the minimal area

Amin[, G(s)] = min[]

A[, G(s)] , (5.40)

is an intrinsically defined and t-invariant classical observable. A class of metrics are givenby the totally symmetric parts of single chiral traces of strings of -products ofs generalized

vielbeins E = dXM EM with insertions of s vertices Vki,kii,i(, , S) (i = 1, . . . , s) as givenin (4.8), viz.

GM1...Ms = T(M1,...,Ms) , (5.41)

where the rank-s tensor

TM1,...,Ms = TrK E(M1 Vk1,k11,1 EMs) Vks,kss,s , (5.42)

with

K

1,

,

,

. Given such a metric, one introduces local coordinates on , say

m, and computes the induced rank-s metric

(fG)m1...ms = m1XM1 msX

MsGM1...Ms . (5.43)

Letting m1...mp denote the totally anti-symmetric tensor density on , one can then take

A[, G(s)] =

dp

I

k, (fG(s))l1/k

, pk = sl , (5.44)

13For an analog in three dimensional higher spin gravity, see [22].

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where I is a scalar density of degree k. For example, if s is even one may take

I = m1[p] ms [p] (fG)m1...ms (fG)m1...ms

p times

, k = s , l = p , (5.45)

whereas odd s require more involved invariants. In the case ofs = 2 the Riemann curvature

RM(2);N(2) (symmetric convention) has a perturbative expansion in terms of generalized

Riemann tensors, viz. R,(s1);(1),(s1) C(s),(s). The tensorial calculus pertinent to

examining the variational principles for s > 2 remains to be investigated further.

Clearly there are infinitely many choices of metrics and related minimal-area observables.

Moreover, one may couple p-branes to anti-symmetric forms and other poly-forms formed

by various Young projections of TM1,...,Ms . One may therefore speculate that for a given

p, there exists a preferred coupling to dynamical p-brane that requires the Vasiliev equa-

tions. Moreover, one may envisage an embedding scenario in which an infinite-dimensional

submanifold ofhs(4)/hs+(4) itself being a higher-spin coset can be viewed as the worldvol-

ume of an infinite-dimensional brane. We expect this approach to b e most p owerful when

both manifolds are supermanifolds, in which case a simple superembedding condition may

produce the higher spin field equations in both targetspace and worldvolume.

6 Deformation of Bulk Action and Tree Amplitudes

In this section we shall show how the zero-form charges and the abelian p-form chargesthat we have constructed can be given an interpretation as on-shell actions within the

off-shell formulation for Vasilievs higher spin gravity based on generalized Hamiltonian

actions [7, 8, 9]. Bulk actions for the minimal bosonic models can be obtained by consistent

truncation of enlarged models with external Kleinian operators (K, K) [8]. For example,

an action with linear source term in the Q-structure (leading to linear F1) and a quadratic

Poisson structure is given by

Sbulk = MTr 1 + KK

2 U D B + V (F + B J + U)K= K=0 , (6.1)

where M is odd-dimensional; J = J[2] + J[4] is closed and central term; (A, V) have oddform degree and (B, U) have even form degree. For example, if dimM = 5, then

B = B[0] + B[2] + B[4] + B[6] , A = A[1] + A[3] + A[5] + A[7] , (6.2)U = U[2] + U[4] + U[6] + U8] , V = V[1] + V[3] + V[5] + V[7] . (6.3)

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The variational principle implies the Cartan integrable bulk equations of motion

D B + V 0 , F + B J + U 0 , (6.4)DU V J 0 , DV + B, U 0 , (6.5)

andMTr[U B V A] 0. The latter are solved by imposing

U|M = 0 , V|M = 0 , (6.6)which one may view as part of the definition of a generalized Hamiltonian action princi-

ple. The on-shell Cartan gauge transformations are symmetries of the action as follows:

the gauge parameters associated with (A, B) generate transformations that leave the La-grangian invariant, while those of (

U ,

V) do so up to total derivatives and must hence form

a section. Taking into account the boundary conditions, it thus follows that (U , V) can beset equal to the zero section on-shell by fixing a gauge, leaving

F + B J = 0 , DB = 0 , (6.7)which is a duality extended version of Vasilievs original system of equations with linear

interaction freedom. Assuming that the gauge symmetries associated with the master fields

of form degrees p 2 remain unbroken, the duality extended and original systems share

the same initial and boundary data, and are hence equivalent on-shell (modulo transition

functions) as discussed in Section 3.2.

Since the bulk action vanishes on-shell, semi-classical amplitudes can only be generated

by deformations Stop, referred to as topological vertex operators, which are functionals

integrated over submanifolds of M where the Lagrange multipliers (U , V) vanish, with thefollowing properties:

Stop = 0 for unbroken gauge parameters associated with ( A, B); the general variation Stop 0 on the shell of Sbulk; and

Stop has a nontrivial value on-shell.

One simple class of deformations consists of the following generating functionals for twistor

space amplitudes in the unbroken phase (n = 0, 1, 2, . . . ; K {1,k, k, kk}):Stopn, K

= Tr 1 + KK2

K F Bn + nn + 1

B(n+1) JK=K=0

p

, (6.8)

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Stopn, K

= Tr 1 + KK2

K F2 Bn nn + 2

B(n+2) J2K=K=0

p

, (6.9)

where the point p M is chosen such that

U|p = 0 , V|p = 0 . (6.10)Their on-shell values are given by

Stopn, 1

n + 1Tr 1 + KK

2 K B(n+1) J

p

, (6.11)

Stopn, 2

n + 2Tr 1 + KK

2 K B(n+2) J2

p

. (6.12)

There are two truncations to duality extended minimal bosonic models given by [8]

A[1] = 1 + K K2 A[1] , A[3] = C[3] K+ K2

, (6.13)

B[0] = 1 + KK2 , B[2] = C[2]

K+ K

2

+1, (6.14)

where = 0, 1 (corresponding to k = +1, 1 in [8]). For suitable K, the deformations oftype Stop reproduce the zero-form invariants in (4.20), viz.

Stop2m, 1

m + 1

Tr

(

) (

(

))m+1

p

, (6.15)

Stop2m+1 22m + 1Tr ( ())m

p. (6.16)

The on-shell values of the deformations of type Stop, on the other hand, are given by

linear combinations of (4.20) and additional, separately on-shell closed, zero-form invariants

involving as well as C[2]; see Appendix B. In other words, the duality extension isnontrivial in the sense that it gives rise to classical observables that are functionals of

the initial data C(Y) = Z=0,p

for the Weyl zero-form but that cannot be given a local

description within the original unextended system.

Another class of deformations consists of the following generating functionals for 2-form

and 4-form amplitudes in the broken phase:

Stop[2] = R e

2

2

Tr R , (6.17)Stop[4] = R e

4

Tr 4R R + 4 (E E+ r(S)+) R + 12

(E E+ r(S)+)2 ,(6.18)

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where 2, 4 and 4 are complex constants and 2,4 are submanifolds of M where the

Lagrange multipliers vanish. Their values on-shell are given by

Stop[2] Re

2

2H[2]

, (6.19)

Stop[4] Re

(4 1

24)

4

H[4]

, (6.20)

where H[2] and H[4] are given in (5.23).

7 Conclusions

In this paper we have proposed geometric formulations of Vasilievs higher spin gravity

with generalized vielbeins and curvature zero-forms forming sections of bundles associated

to a principle bundle of a structure group. The choice of structure group is not unique,

The choice made here utilizes an automorphsim of the -product algebra and naturally

lends itself to the construction of intrinsically defined classical observables as traces. We

have constructed two types of classical observables: minimal area functionals and charges

of abelian p-forms that are closed on-shell. We have shown that the latter have a natural

interpretation as semi-classical amplitudes within the off-shell formulation proposed in [8].

The classical observables are manifestly invariant off-shell under the structure group gauge

transformations and invariant on-shell under the local higher-spin translations. The latter

symmetries arise on-shell via various mechanisms depending on the nature of the observablein question.

It remains to study in detail the precise relation between the topological vertex operator

(6.18) and the FradkinVasiliev cubic action [23] and to compare the on-shell amplitudes

(6.20) with those derived directly at the level of Weyl tensors by Giombi and Yin [24,

25]. The same analysis needs to be made in the case of the alternative structure groups.

In particular, there is an intriguing choice based on the holomorphic enveloping algebra

extension of the two copies of the Lorentz algebra acting on the two copies of the twistor

space.A study that may shed light on the first-quantized origin of Vasilievs theory, is to

compare the twistor space amplitudes arising from (6.8) and (6.9) with the topological

open string in twistor space [26].

Acknowledgment

We are grateful to N. Boulanger for collaborations at certain stages of this work. We

thank C. Iazeolla and M. Vasiliev for illuminating discussions. We both thank Scuola

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Normale Superiore in Pisa for hospitality during early stages of this work. E. S. thanks

the University of Mons and P. S. thanks the Mitchell Institute for Fundamental Physics

and Astronomy for hospitality. The research of E. S. is supported in part by NSF grants

PHY-0555575 and PHY-0906222.

A Lorentz Covariantization

The canonical Lorentz transformations and Lorentz covariant derivatives are defined by

= ,

M = M

2(M

) , (A.1)

= d , R

= d . (A.2)

The master field equations imply that the Cartan gauge transformations with parameters

= 14i

M + M , (A.3)

generate the canonical Lorentz transformations (2.54) and (2.43) of the full Vasiliev system.

To demonstrate this, one first notes that by their very definition, a Cartan gauge transfor-

mation acts on a general composite X = m,n YmZnXm,n where Xm,n are functionalsof the components of basic master fields, as X = m,n YmZnXm,n; in particular, onehas y = 0 = z. To proceed, one shows that

[M , ] = [M(0) , ] , [M , S] = [M(0) , S] 4i(S) , (A.4)where the last equation follows from

[M(S) , S] = 4i(S) . (A.5)It follows that the S-dependent terms drop out from and S, that hence assume theform given in (2.54) and (2.43). From (A.5) it also follows that A = A [(0) , A]idem

A. For

WM one has

WM = AM 14i(MM) + (MM) , (A.6)

where is given above and

AM = 14i

(M

)M + (M)M [(0) , AM] , (A.7)M = S( S) + S( S) = 2(M) [(0) , M ] , (A.8)

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implying that the terms containing M cancel such that WM transforms as in (2.54).

Furthermore, substituting the redefinition of AM into the constraints on d and dS,and using (A.4), one finds (2.49). One can the proceed amd substitute the redefinition into

the constraint on dA; using (2.49) and (A.5) one can then reduce the quartic -product ofSs down to a quadratic one after which (2.48) follows by the definition of W and R .Finally, we note that the full Lorentz generators M form the field-dependent algebra

[M , M ] = 2iM + 3 perms ;M + ;M , (A.9)which obeys the Jacobi identity. To this end, if g is a Lie algebra with generators X

represented in an associative -product algebra spanned by Vr by(X) :=

Vrr(X; ) , (A.10)

where r(X; ) are functions on a manifold with local coordinates i carrying a represen-

tation ofg given by Lie derivatives LX , viz.

LX(Y) = VrLXr(Y; ) , LXr(Y; ) = Xiir(Y; ) , (A.11)then it follows that the representation property

[(X), (Y)] := ([X, Y]) LX(Y) + LY(X) , (A.12)

is compatible with associativity and [LX , LY] = L[X,Y].

B Additional Zero-Form Invariants in the Duality Extended

Model

For suitable K, the on-shell values of the deformations Stop in (6.8) are given for = 0 byStop2m,

1

2m + 1Tr () ( ())m J[4] + (2m + 1)C[2] J[2]

p, (B.1)

Stop

2m+1

1

2(m + 1)Tr ( ())m () J[4] + 2(m + 1)(C[2]) J[2]p ,(B.2)

and for = 1 by

Stop2m 1

2m + 1Tr ( ())m J[4] + (2m + 1)C[2] J[2]

p, (B.3)

Stop2m+1, 1

2(m + 1)Tr () ( ())m () J[4] + 2(m + 1)(C[2]) J[2]

p,

(B.4)

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which we identify as linear combinations of the zero-form invariants (4.20) of the duality

unextended system and a new set of invariants involving C[2]. The latter are closed on-shellas can be seen using (2.3) and the additional duality extended equations of motion at the

point p, which read

DC[2] + C[3],

= 0 , DC[3] + J[4] + C[2] J[2] = 0 , (B.5)where the covariant derivatives and the projected J[4] are defined by

= 0 : DC[2] = dC[2] + A[1], C[2]

, DC[3] = dC[3] + A[1], C[3]

, (B.6)J[4] = i(c + c)d2zd2z , c C , (B.7) = 1 :

D

C[2] =

d

C[2] +

A[1],

C[2]

,

D

C[3] =

d

C[3] +

A[1],

C[3]

, (B.8)

J[4] = ( + )d2zd2z , , R . (B.9)

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Documents

Ergin Sezgin and Per Sundell- Geometry and Observables in Vasiliev’s Higher Spin Gravity