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Introduction Various Approaches M-theory R4, purified Conclusions
Higher-order corrections in String/M-theory
Dimitrios Tsimpis
June 21, 2007Arnold Sommerfeld Center, Munich
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Outline
1 IntroductionMotivationOverview
2 Various ApproachesString PerturbationSupersymmetryField Theory
3 M-theoryIntroduction11D superspaceSpinorial Cohomology
4 R4, purifiedIntroductionThe ingredientsThe four-point Lagrangian
5 Conclusions
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Outline
1 IntroductionMotivationOverview
2 Various ApproachesString PerturbationSupersymmetryField Theory
3 M-theoryIntroduction11D superspaceSpinorial Cohomology
4 R4, purifiedIntroductionThe ingredientsThe four-point Lagrangian
5 Conclusions
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Motivation
String/M-theory can be approximated by LEEA
String/M-theory predicts higher-derivative corrections
String-theory (10D) implies lString corrections
M-theory (11D) receives lPlanck corrections
κ-symmetric branes receive lString , lPlanck corrections
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Motivation
Important consequences
Qualitative modifications to vacua
Implications for no-go theorems
Testing dualities beyond leading-order
Black-hole precision measurements
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Overview
Various approaches
Corrections not fully under control
Much recent progress
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Overview
Various approaches
Corrections not fully under control
Much recent progress
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Outline
1 IntroductionMotivationOverview
2 Various ApproachesString PerturbationSupersymmetryField Theory
3 M-theoryIntroduction11D superspaceSpinorial Cohomology
4 R4, purifiedIntroductionThe ingredientsThe four-point Lagrangian
5 Conclusions
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Overview
Stringtheory
Super-symmetry
Fieldtheory
D = 10 Sugra
D = 11 Sugra
String perturbation
Conformalinvariance
Scatteringamplitudes
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Overview
Stringtheory
Super-symmetry
Fieldtheory
D = 10 Sugra
D = 11 Sugra
Supersymmetry
Componentapproach
Harmonicsuperspace
Action principle
Spinorialcohomology
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Overview
Stringtheory
Super-symmetry
Fieldtheory
D = 10 Sugra
D = 11 Sugra
Field theory
Higher-loopcounterterms
New techniques
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Outline
1 IntroductionMotivationOverview
2 Various ApproachesString PerturbationSupersymmetryField Theory
3 M-theoryIntroduction11D superspaceSpinorial Cohomology
4 R4, purifiedIntroductionThe ingredientsThe four-point Lagrangian
5 Conclusions
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Conformal invariance
RNS
β = 0 implies target EOMs
WS n-loop maps to (α′)n−1
Up to four loops (superstring)
Difficulty with RR fields
M.T. Grisaru, A. van de Ven,D. Zanon, PLB 173 (1986)
M.T. Grisaru, D. Zanon,PLB 177 (1986)
M.D. Freeman, C.N. Pope,M. Sohnius, K.S. Stelle,PLB 178 (1986)
Q.H. Park, D. Zanon,PRD 35 (1987)
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Conformal invariance
RNS
β = 0 implies target EOMs
WS n-loop maps to (α′)n−1
Up to four loops (superstring)
Difficulty with RR fields
M.T. Grisaru, A. van de Ven,D. Zanon, PLB 173 (1986)
M.T. Grisaru, D. Zanon,PLB 177 (1986)
M.D. Freeman, C.N. Pope,M. Sohnius, K.S. Stelle,PLB 178 (1986)
Q.H. Park, D. Zanon,PRD 35 (1987)
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Conformal invariance
Pure-spinor
δQ = 0 implies target EOMs
Up to one loop (superstring)
N. Berkovis, P.S. Howe,NPB 635 (2002)
O. Chandia, B.C. Vallilo,JHEP 0404
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Scattering Amplitudes
RNS/GSTime-honored technique
Expansions in α′, gs
General N-point expressions
Hard to extract LEEA
Difficulty with RR fields
M.B. Green, J. Schwarz,PLB 109 (1982)
D. Gross, E. Witten,NPB 277 (1986)
N. Sakai, Y. Tanii,NPB 287 (1987)
D. Gross, J. Sloan,NPB 291 (1987)
K. Peeters, P. Vanhove,A. Westerberg,CQG 18 (2001); 19 (2002)
K. Peeters, A. Westerberg,CQG 21 (2004)
D. Oprisa, S. Stieberger,hep-th/0509042
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Scattering Amplitudes
RNS/GSTime-honored technique
Expansions in α′, gs
General N-point expressions
Hard to extract LEEA
Difficulty with RR fields
M.B. Green, J. Schwarz,PLB 109 (1982)
D. Gross, E. Witten,NPB 277 (1986)
N. Sakai, Y. Tanii,NPB 287 (1987)
D. Gross, J. Sloan,NPB 291 (1987)
K. Peeters, P. Vanhove,A. Westerberg,CQG 18 (2001); 19 (2002)
K. Peeters, A. Westerberg,CQG 21 (2004)
D. Oprisa, S. Stieberger,hep-th/0509042
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Scattering Amplitudes
Pure-spinor
Bypasses difficulties withRNS/GS
Easier to treat RR fields
N. Berkovits, JHEP 0004;0409; 0601
N. Berkovits B.C. Vallilo,JHEP 0007
N. Berkovits, CRP 6 (2005)
C.R. Mafra, JHEP 0601
G. Policastro, DT,CQG 23 (2006)
N. Berkovits, C.R. Mafra,JHEP 0611
C. Stahn, 0704.0015
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Outline
1 IntroductionMotivationOverview
2 Various ApproachesString PerturbationSupersymmetryField Theory
3 M-theoryIntroduction11D superspaceSpinorial Cohomology
4 R4, purifiedIntroductionThe ingredientsThe four-point Lagrangian
5 Conclusions
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Component Approach
Component susy
Most direct
Successful in N = 1, D = 10
Structure of invariants
E. Bergshoeff, M. de Roo,NPB 328 (1989)
M. de Roo, H. Suelmann,A. Wiedemann,NPB 405 (1993)
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Harmonic Superspace
G-analyticity
Append group to spacetime
Extended susy
Fewer-θ integrals
Linearized approximation
A. Galperin, E. Ivanov,S. Kalitsyn, V. Ogievetsky,E. Sokatchev, CQG 1 (1984)
G. Hartwell, P.S. Howe,IJMP A10 (1995)
J. Drummond, P. Heslop,P.S. Howe, S. Kerstan,JHEP 0308
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Harmonic Superspace
G-analyticity
Append group to spacetime
Extended susy
Fewer-θ integrals
Linearized approximation
A. Galperin, E. Ivanov,S. Kalitsyn, V. Ogievetsky,E. Sokatchev, CQG 1 (1984)
G. Hartwell, P.S. Howe,IJMP A10 (1995)
J. Drummond, P. Heslop,P.S. Howe, S. Kerstan,JHEP 0308
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Action Principle
Super D-form
Useful in presence of CS
κ-symmetric branes
M-theory
R. D’Auria, P. Fre,P. Townsend,P. van Nieuwenhuizen,AP 155 (1984)
S.J. Gates, M. Grisaru,M. Knutt-Wehlau, W. Siegel,PLB 421 (1998)
I. Bandos, D. Sorokin,D. Volkov, PLB 352 (1995)
P.S. Howe, O. Raetzel,E. Sezgin, JHEP 9808
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Spinorial Cohomology
Super H-groups
General structure
Explicit computations
Related to pure-spinor
M. Cederwall,B. E. W. Nilsson, DT,JHEP 0106
M. Cederwall,B. E. W. Nilsson, DT,JHEP 0202
P.S. Howe, DT, JHEP 0309
DT, JHEP 0410
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Outline
1 IntroductionMotivationOverview
2 Various ApproachesString PerturbationSupersymmetryField Theory
3 M-theoryIntroduction11D superspaceSpinorial Cohomology
4 R4, purifiedIntroductionThe ingredientsThe four-point Lagrangian
5 Conclusions
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Counterterms
Higher-loop counterterms
Earliest approach
Generally applicable
Superparticle (D = 10, 11)
S. Deser, J.H. Kay, K.S. Stelle,PRL 38 (1977)
R.E. Kallosh, PLB 99 (1981)
D.C. Dunbar, B. Julia,D. Seminara and M. Trigiante,JHEP 0001
M. B. Green, M. Gutperle,P. Vanhove, PLB 409 (1997)
M. B. Green, H. Kwon,P. Vanhove, PRD 61 (2000)
K. Peeters, J. Plefka, S. Stern,JHEP 0508
M. B. Green, J. G. Russo,P. Vanhove,PRL 98 (2007); JHEP 0702
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Mixed Techniques
Mixed techniques
Unitarity cuts, KLT, twistors
N = 8 supergravity
Z. Bern, J.J. Carrasco, L.J. Dixon,H. Johansson, D.A. Kosower,R. Roiban, hep-th/0702112
Z. Bern, L. J. Dixon, R. Roiban,PLB 644 (2007)
N.E.J. Bjerrum-Bohr, D.C. Dunbar,H. Ita, W.B. Perkins, K. Risager,JHEP 0612
Z. Bern, N.E.J. Bjerrum-Bohr,D.C. Dunbar, JHEP 0505
N.E.J. Bjerrum-Bohr, D.C. Dunbar,H. Ita, PLB 621 (2005)
Z. Bern, L.J. Dixon, D.C. Dunbar,M. Perelstein, J.S. Rozowsky,NPB 530 (1998)
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Outline
1 IntroductionMotivationOverview
2 Various ApproachesString PerturbationSupersymmetryField Theory
3 M-theoryIntroduction11D superspaceSpinorial Cohomology
4 R4, purifiedIntroductionThe ingredientsThe four-point Lagrangian
5 Conclusions
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
M-theory
Higher-order corrections
LEEA, series in lPlanck
No (fully-satisfactory) microscopic formulation
Susy is restrictive
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
M-theory
Higher-order corrections
LEEA, series in lPlanck
No (fully-satisfactory) microscopic formulation
Susy is restrictive
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
M-theory
Higher-order corrections
LEEA, series in lPlanck
No (fully-satisfactory) microscopic formulation
Susy is restrictive
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Outline
1 IntroductionMotivationOverview
2 Various ApproachesString PerturbationSupersymmetryField Theory
3 M-theoryIntroduction11D superspaceSpinorial Cohomology
4 R4, purifiedIntroductionThe ingredientsThe four-point Lagrangian
5 Conclusions
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
11D superspace
Superobjects
vielbein: EMA
connection: ΩMAB
torsion, curvature: T A = DEA, RAB = dΩA
B + ΩACΩC
B
Bianchi identities: DT A = EBRBA, DRA
B = 0
CJS supergravity
T fαβ = γf
αβ
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
11D superspace
Superobjects
vielbein: EMA
connection: ΩMAB
torsion, curvature: T A = DEA, RAB = dΩA
B + ΩACΩC
B
Bianchi identities: DT A = EBRBA, DRA
B = 0
Deformed supergravity
T fαβ = γf
αβ + γabαβX f
;ab + γabcdeαβ X f
;abcde
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Deformed Supergravity
Super-Bianchi identities
Rab − 12ηabR = 1
12(G2ab − 1
8ηabG2) + fab(X f ;a1a2 , X f ;a1...a5)
M. Cederwall, U. Gran, B.E.W. Nilsson, DT, JHEP 0505
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Deformed Supergravity
Super-Bianchi identities
Rab − 12ηabR = 1
12(G2ab − 1
8ηabG2) + fab(X f ;a1a2 , X f ;a1...a5)
M. Cederwall, U. Gran, B.E.W. Nilsson, DT, JHEP 0505
The Xs are black boxes
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Outline
1 IntroductionMotivationOverview
2 Various ApproachesString PerturbationSupersymmetryField Theory
3 M-theoryIntroduction11D superspaceSpinorial Cohomology
4 R4, purifiedIntroductionThe ingredientsThe four-point Lagrangian
5 Conclusions
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Spinorial Cohomology
The Xs are elements of super-cohomology
Xf ;ab, Xf ;abcde ∈ H2res(M)
Similarly for form-formulation
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Four-form formulation
11D supergravity at O(l3Planck )
Start with dG4 = 0
Deformations ∈ H0,4res (M)
Unique deformation at O(l3Planck )G4 −→ G4 + l3Planckp1(M)
DT, JHEP 0410
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
11D Supergravity at O(l3Planck)
G2A ∝ Ga1a2
ijGb1b2ij |(02000)
G2B ∝ εa1...a5
ijklmnGijklGb1b2mn|(01002)
IA ∝ Ga1a2ij(γb1b2
Tij)α|(02001)
(1) (1)(1)
(1)
(1)AI
2B
T
2AG G R
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Spinorial vs Pure-Spinor Cohomology
In the linearized approximation: Hsc −→ Hps
P.S. Howe, unpublished
P.S. Howe, DT, JHEP 0309
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
The D = 10 SYM cohomology complex
M. Cederwall, B. E. W. Nilsson, DT,JHEP 0202
n = 0 n = 1 n = 2 n = 3 n = 4
2× dim = 0 (00000)
1 • •2 • (10000) •3 • (00001) • •4 • • • • •5 • • (00010) • •6 • • (10000) • •7 • • • • •8 • • • (00000) •9 • • • • •
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
The D = 11 supergravity cohomology complex
n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8
2×dim = −6 (00000)
−5 • •
−4 • (10000) •
−3 • • • •
−2 • •(01000)(10000)
• •
−1 • • (00001) • • •
0 • • •
(00000)(00100)(20000)
• • •
1 • • •(00001)(10001)
• • • •
2 • • • • • • • • •
3 • • • •(00001)(10001)
• • • •
4 • • • •
(00000)(00100)(20000)
• • • •
5 • • • • • (00001) • • •
6 • • • • •(01000)(10000)
• • •
7 • • • • • • • • •
8 • • • • • • (10000) • •
9 • • • • • • • • •
10 • • • • • • • (00000) •
11 • • • • • • • • •
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Seven-form formulation
Supergravity at O(l6Planck )
Start with dG7 + 12G4 ∧ G4 = l3PlanckX8
Unique deformation at O(l6Planck )
Difficult to compute
P.S. Howe, DT, JHEP 0309
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Seven-form formulation
Supergravity at O(l6Planck )
Start with dG7 + 12G4 ∧ G4 = l3PlanckX8
Unique deformation at O(l6Planck )
Difficult to compute
P.S. Howe, DT, JHEP 0309
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Outline
1 IntroductionMotivationOverview
2 Various ApproachesString PerturbationSupersymmetryField Theory
3 M-theoryIntroduction11D superspaceSpinorial Cohomology
4 R4, purifiedIntroductionThe ingredientsThe four-point Lagrangian
5 Conclusions
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Pure-spinor superstring
BRST operator
Q =∮
dzλαdα
whereλγaλ = 0
Massless vertex operators
QU = 0, QV = α′∂U
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Tree Amplitudes
N-point amplitude
A = 〈U1(z1)U2(z2)U3(z3)∫
dz4V4(z4) . . .∫
dzNVN(zN)〉Integrate out nonzero modes:A =
∫dz4 . . .
∫dzN〈λαλβλγ fαβγ(zr , kr , θ)〉
Zero-mode integration
A = T αβγ∫
dz4 . . .∫
dzN fαβγ(zr , kr , θ)
where:T αβγ(γ iθ)α(γ jθ)β(γkθ)γ(θγijkθ) = 1
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Outline
1 IntroductionMotivationOverview
2 Various ApproachesString PerturbationSupersymmetryField Theory
3 M-theoryIntroduction11D superspaceSpinorial Cohomology
4 R4, purifiedIntroductionThe ingredientsThe four-point Lagrangian
5 Conclusions
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Vertices
Integrated vertex
U = λαAα(x , θ)
10D SYM in Superspace
Field-strength: F = DA
Bianchi identities: DF = 0
Fαβ = 0 =⇒ ordinary SYM
B.E.W. Nilsson, Göteborg-ITP-81-6
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
The expansion of the vertices
Wess-Zumino gauge
θαAα = 0
so that:θαDα = θα ∂
∂θα
Integrate on-shell conditions
Am = [cosh√O]m
qaq + [O−12 sinh
√O]m
q(θγqξ)
where:[O]m
q := 12(θγm
qpθ)∂p; aq := Aq|, ξα := W α|
Solvability in linearized approximation is generic
DT, JHEP 0411
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
The expansion of the vertices
Wess-Zumino gauge
θαAα = 0
so that:θαDα = θα ∂
∂θα
Integrate on-shell conditions
Am = [cosh√O]m
qaq + [O−12 sinh
√O]m
q(θγqξ)
where:[O]m
q := 12(θγm
qpθ)∂p; aq := Aq|, ξα := W α|
Solvability in linearized approximation is generic
DT, JHEP 0411
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
The Kawai-Lewellen-Tye Relations
∫d2z =
∫dz ⊗
∫dz
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
The Kawai-Lewellen-Tye Relations
∫d2z =
∫dz ⊗
∫dz
Four-point amplitude factorizes
Acl4 = −g2 sin(πα′k2 · k3)Aop
4 (α′s2 , α′t
2 ) ⊗ Aop4 (α′t
2 , α′u2 )
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
The Kawai-Lewellen-Tye Relations
∫d2z =
∫dz ⊗
∫dz
Four-point Lagrangian factorizes
Lcl4 = Lop
4 ⊗ Lop4
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Pole Subtraction
Unitarity
Poles in N-pt amplitudes come from 1PRD’s with N ′ vertices,where N ′ < N
Before taking the 1PRD’s into account:
L ∼ f (s, t , u) . . . where:f (s, t , u) = − 8π
α′3stu − 2πζ(3) + O(α′2)
Taking 1PRD’s into account:
f (s, t , u) −→ G(s, t , u) := f (s, t , u)+ 8π
α′3stu
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Outline
1 IntroductionMotivationOverview
2 Various ApproachesString PerturbationSupersymmetryField Theory
3 M-theoryIntroduction11D superspaceSpinorial Cohomology
4 R4, purifiedIntroductionThe ingredientsThe four-point Lagrangian
5 Conclusions
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Four-point Lagrangian: schematically
Four-point Lagrangian
L4pt ∝ (α′)3G
R4 + (∂F )2R2 + (∂F )4
where:Rmn
pq := Rmnpq + 2κe−
κD√2∇[mHn]
pq −√
2κδ[m[p∇n]∇q]D
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Four-point Lagrangian: the gore and the glory
NS-NS
LNS =(α′)3
4!G t8t8R4
where:
Rmnpq := Rmn
pq + 2κe−κD√
2∇[mHn]pq −
√2κδ[m
[p∇n]∇q]D
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Four-point Lagrangian: the gore and the glory
RR-NS
L(∂F )2R2 = −26κ(α′)3G (A1 +12
A2 +14
A3)
where:
A1 := R in
jn′Ripjp′ < γn∂p 6Fγ(n′
∂p′) 6F Tr >
A2 := Rmnin′Rpqip′
(< γmnp∂q 6Fγ(n′
∂p′) 6F Tr > +F ↔ F Tr)
A3 := Rmnm′n′Rpqp′q′ < γ[mnp∂q] 6Fγm′n′p′∂q′ 6F Tr >
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Four-point Lagrangian: the gore and the glory
RR-RR
L(∂F )4 =329
(α′)3κ2G (B1 − 5B2 + B3 + 4B4 − B5)
where:
B1 :=< ∂m∂p 6Fγq∂m∂p 6F Trγn 6Fγq 6F Trγn >
B2 :=< ∂m∂p 6Fγq 6F Trγn∂m∂p 6Fγq 6F Trγn >
B3 :=< ∂m∂p 6Fγq 6F Trγn 6Fγq∂m∂p 6F Trγn >
B4 :=< ∂m∂p 6Fγq 6F Trγn >< ∂m∂p 6Fγq 6F Trγn >
B5 :=< 6Fγq 6F Trγn >< ∂m∂p 6Fγq∂m∂p 6F Trγn >
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Future Directions
Physical applications
Higher points
Factorization
Higher-order corrections Dimitrios Tsimpis
Introduction Various Approaches M-theory R4, purified Conclusions
Thank You
Higher-order corrections Dimitrios Tsimpis
Recommended