Dimitrios Tsimpis June 21, 2007 Arnold Sommerfeld Center ...€¦ · PLB 178 (1986) Q.H. Park, D. Zanon, PRD 35 (1987) Higher-order corrections Dimitrios Tsimpis. Introduction Various

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


Outline

1 IntroductionMotivationOverview

2 Various ApproachesString PerturbationSupersymmetryField Theory

3 M-theoryIntroduction11D superspaceSpinorial Cohomology

4 R4, purifiedIntroductionThe ingredientsThe four-point Lagrangian

5 Conclusions



Outline





5 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



Motivation

Important consequences

Qualitative modifications to vacua

Implications for no-go theorems

Testing dualities beyond leading-order

Black-hole precision measurements



Overview

Various approaches

Corrections not fully under control

Much recent progress



Overview

Various approaches

Corrections not fully under control

Much recent progress



Outline





5 Conclusions



Overview

Stringtheory

Super-symmetry

Fieldtheory

D = 10 Sugra

D = 11 Sugra

String perturbation

Conformalinvariance

Scatteringamplitudes



Overview

Stringtheory

Super-symmetry

Fieldtheory

D = 10 Sugra

D = 11 Sugra

Supersymmetry

Componentapproach

Harmonicsuperspace

Action principle

Spinorialcohomology



Overview

Stringtheory

Super-symmetry

Fieldtheory

D = 10 Sugra

D = 11 Sugra

Field theory

Higher-loopcounterterms

New techniques



Outline





5 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)




RNS

β = 0 implies target EOMs

WS n-loop maps to (α′)n−1

Up to four loops (superstring)


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)




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



Scattering Amplitudes

RNS/GSTime-honored technique

Expansions in α′, gs

General N-point expressions

Hard to extract LEEA


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




RNS/GSTime-honored technique

Expansions in α′, gs

General N-point expressions

Hard to extract LEEA


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




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



Outline





5 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)



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



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



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



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



Outline





5 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



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)



Outline





5 Conclusions



M-theory

Higher-order corrections

LEEA, series in lPlanck

No (fully-satisfactory) microscopic formulation

Susy is restrictive



M-theory




Susy is restrictive



M-theory




Susy is restrictive



Outline





5 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



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



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



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



Outline





5 Conclusions



Spinorial Cohomology

The Xs are elements of super-cohomology

Xf ;ab, Xf ;abcde ∈ H2res(M)

Similarly for form-formulation



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



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



Spinorial vs Pure-Spinor Cohomology

In the linearized approximation: Hsc −→ Hps

P.S. Howe, unpublished




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 • • • • •



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 • • • • • • • • •



Seven-form formulation

Supergravity at O(l6Planck )

Start with dG7 + 12G4 ∧ G4 = l3PlanckX8

Unique deformation at O(l6Planck )

Difficult to compute




Seven-form formulation

Supergravity at O(l6Planck )

Start with dG7 + 12G4 ∧ G4 = l3PlanckX8

Unique deformation at O(l6Planck )

Difficult to compute




Outline





5 Conclusions



Pure-spinor superstring

BRST operator

Q =∮

dzλαdα

whereλγaλ = 0

Massless vertex operators

QU = 0, QV = α′∂U



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



Outline





5 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



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



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



The Kawai-Lewellen-Tye Relations

∫d2z =

∫dz ⊗

∫dz




∫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 )




∫d2z =

∫dz ⊗

∫dz

Four-point Lagrangian factorizes

Lcl4 = Lop

4 ⊗ Lop4



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



Outline





5 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



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




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 >




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 >



Future Directions

Physical applications

Higher points

Factorization



Thank You


Documents

Dimitrios Tsimpis June 21, 2007 Arnold Sommerfeld Center ...€¦ · PLB 178 (1986) Q.H. Park, D. Zanon, PRD 35 (1987) Higher-order corrections Dimitrios Tsimpis. Introduction Various