Superstring amplitudes with the pure spinor formalism Carlos R. - - PowerPoint PPT Presentation

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Superstring amplitudes with the pure spinor formalism Carlos R. Mafra STAG Research Centre and Mathematical Sciences, University of Southampton, UK C.R. Mafra (Southampton) String amplitudes 11.06.2019 1 / 17 Motivation Calculate the L -loop

SLIDE 1

Superstring amplitudes with the pure spinor formalism

Carlos R. Mafra

STAG Research Centre and Mathematical Sciences, University of Southampton, UK

C.R. Mafra (Southampton) String amplitudes 11.06.2019 1 / 17

SLIDE 2

Motivation

Calculate the L-loop N-point superstring amplitude Find hidden structures in superstring and field-theory amplitudes Simplify the calculations and the results

C.R. Mafra (Southampton) String amplitudes 11.06.2019 2 / 17

SLIDE 3

Motivation

Calculate the L-loop N-point superstring amplitude Find hidden structures in superstring and field-theory amplitudes Simplify the calculations and the results

C.R. Mafra (Southampton) String amplitudes 11.06.2019 2 / 17

SLIDE 4

Motivation

Calculate the L-loop N-point superstring amplitude Find hidden structures in superstring and field-theory amplitudes Simplify the calculations and the results

C.R. Mafra (Southampton) String amplitudes 11.06.2019 2 / 17

SLIDE 5

D=10 SYM theory

Describes gluon and gluino with action

F 2 + χαγm

αβDmχβ

Covariant description using 10D superfields (Witten’86) Aα(x, θ), Am(x, θ), W α(x, θ), Fmn(x, θ) Linearized equations of motion DαAβ + DβAα = γm

αβAm,

DαAm = (γmW )α + ∂mAα DαW β = 1 4(γmn)

β α Fmn,

DαFmn = 2∂[m(γn]W )α where Dα is covariant derivative, {Dα, Dβ} = γm

αβ∂m.

Have standard θ expansions, e.g Am(x, θ) = am − (ξγmθ) − 1 8(θγmγpqθ)Fpq + . . .

C.R. Mafra (Southampton) String amplitudes 11.06.2019 3 / 17

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Pure Spinor Formalism (Berkovits’ 00)

S = 1 2πα′

d2z

∂xm∂xm + α′pα∂θα − α′wα∂λα

Pure spinor λα and its conjugate momenta wα: λαγm

αβλβ = 0

Manifest spacetime SUSY, pure spinor superspace (PSS): (λα, θα) Covariant BRST Quantization: Q = λαDα Free CFT up to PS constraint with “simple” OPEs, e.g. pα(zi)θβ(zj) → δβ

α

zij , zij ≡ zi − zj

C.R. Mafra (Southampton) String amplitudes 11.06.2019 4 / 17

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Pure Spinor Formalism

Massless N-point tree-level prescription (Berkovits ‘00): Atree = V1V2V3

U4. . .
Un

Vertex operators V = λαAα(x, θ), QV = 0 U = ∂θαAα + AmΠm + dαW α + 1 2NmnFmn, QU = ∂V Use OPEs to integrate non-zero modes Surviving zero modes give rise to pure spinor superspace expressions λαλβλγfαβγ(x, θ) . . . non-vanishing only for (λγmθ)(λγnθ)(λγpθ)(θγmnpθ) = 1

C.R. Mafra (Southampton) String amplitudes 11.06.2019 5 / 17

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Pure Spinor Superspace Cohomology

BRST charge Q = λαDα and SYM equations of motion are closely related DαAβ + DβAα = γm

αβAm,

DαAm = (γmW )α + ∂mAα DαW β = 1 4(γmn)

β α Fmn,

DαFmn = 2∂[m(γn]W )α Amplitudes are expressions in the BRST cohomology of pure spinor superspace Study PSS cohomology and use it as tool to simplify and antecipate amplitude calculations (CM ‘10)

C.R. Mafra (Southampton) String amplitudes 11.06.2019 6 / 17

SLIDE 9

Multiparticle SYM superfields (CM, Schlotterer ‘14, ‘15)

KB ∈ {AB

α, Am B , W α B , F mn B }

Recursive definition of multiparticle superfields inspired by OPE computations and satisfy generalized EOMs W α

12 = 1

4(γmnW 2)αF 1

mn + W α 2 (k2 · A1) − (1 ↔ 2)

DαW β

12 = 1

4(γmn)αβF 12

mn + (k1 · k2)(A1 αW β 2 − A2 αW β 1 )

Can be defined to satisfy generalized Jacobi identities (Elliot Bridges) 0 = KAℓ(B) + KBℓ(A) ℓ(A) is the left-to-right Dynkin bracket, e.g ℓ(123) = [[1, 2], 3] = 123 − 213 − 312 + 321

C.R. Mafra (Southampton) String amplitudes 11.06.2019 7 / 17

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Towards FT tree amplitudes

String tree-level amplitudes written in terms of multiparticle vertices VB = λαAB

α having BRST variations

QV1 = 0 QV12 = s12V1V2 QV123 = (k1 · k2)(V1V23 + V13V2) + (k12 · k3)V12V3 String amplitudes reduce to FT amplitudes in the α′ → 0 limit Guess FT amplitudes from BRST invariance Write down terms VAVBVC in the BRST cohomology with the correct poles QASYM(1, 2, . . . , n) = 0, ASYM(1, 2, . . . , n) = Q(something)

C.R. Mafra (Southampton) String amplitudes 11.06.2019 8 / 17

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Tree-level FT amplitudes

A(1, 2, 3, 4) = 2 1 s12 3 4 + 3 2 s23 4 1 = V12V3V4 s12 + V23V4V1 s23 In the BRST cohomology with correct kinematic poles Rewrite using Berends-Giele current M12 ≡ V12/s12, M1 ≡ V1 A(1, 2, 3, 4) = (M12M3 + M1M23)M4 Generalizes to N-pts ASYM(1, 2, . . . , n) =

XY =12...n−1

MXMY Mn

C.R. Mafra (Southampton) String amplitudes 11.06.2019 9 / 17

SLIDE 12

String tree-level amplitude

String tree amplitude can be rewritten in terms of (N − 3)! SYM amplitudes (CM, Schlotterer, Stieberger) A =

i<j

|zij|−sij N−2

k−1

smk zmk ASYM(1, 2, . . . , N)+P(2, . . . , N−2)

Recursive techniques to compute α′ expansion of associated integrals:

Drinfeld method (Drummond, Ragoucy 2013; Broedel et al 2013)

Berends-Giele method (CM, Schlotterer 2016)

C.R. Mafra (Southampton) String amplitudes 11.06.2019 10 / 17

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1-loop string amplitudes

String 1-loop amplitudes from OPEs (CM, Schlotterer ‘18) Building blocks VA, TA,B,C, T m

A,B,C,D, T mn ...

etc with BRST algebra QT1,2,3 = 0 QT12,3,4 = s12(V1T2,3,4 − V2T1,3,4) QT12,34,5 = s12(V1T2,34,5 − V2T1,34,5) + s34(V3T12,4,5 − V4T12,3,5) QT m

1,2,3,4 = km 1 V1T2,3,4 + (1 ↔ 2, 3, 4)

QT m

12,3,4,5 = s12(V1T m 2,3,4,5 − V2T m 1,3,4,5) + km 12V12T3,4,5

+

3 V3T12,4,5 + (3 ↔ 4, 5)

Guess FT integrands from BRST invariance (CM, Schlotterer 2014)

A(1, 2, 3, . . . , n) =

dDℓ

(2π)D A(1, 2, 3, . . . , n|ℓ) QA(1, 2, 3, . . . , n|ℓ) = 0

C.R. Mafra (Southampton) String amplitudes 11.06.2019 11 / 17

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FT 1-loop integrand

Amplitude with cubic graphs (Bern, Carrasco, Johansson ‘10) Propagators with Mandelstams and loop momentum ℓm The 4-point 1-loop FT integrand is in the BRST cohomology A(1, 2, 3, 4|ℓ) = V1T2,3,4 ℓ2(ℓ − k1)2(ℓ − k12)2(ℓ − k123)2 .

C.R. Mafra (Southampton) String amplitudes 11.06.2019 12 / 17

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5-point 1-loop integrand

ℓ 1 5 2 4 (2ℓm − k1

m)V1T m 2,3,4,5 +

V1T23,4,5 + (2, 3|2, 3, 4, 5)
3

Overall 5pt integrand is BRST closed!

C.R. Mafra (Southampton) String amplitudes 11.06.2019 13 / 17

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5-point 2-loop integrand

5pt closed-string amplitude computed with the (non-minimal) pure spinor formalism led to kinematic building blocks: TA,B|C,D and T m

A,B,C|D,E (Gomez, CM, Schlotterer ‘15)

Minimal pure spinor representation with BRST algebra (CM, Schlotterer ‘15) QT1,2|3,4 = 0 QT12,3|4,5 = s12(V1T2,3|4,5 − V2T1,3|4,5) QT m

1,2,3|4,5 = km 1 V1T2,3|4,5 + km 2 V2T1,3|4,5 + km 3 V3T1,2|4,5

Essentially the same algebraic structure as before!

C.R. Mafra (Southampton) String amplitudes 11.06.2019 14 / 17

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2-loop 5pt topologies

BCJ identities: master diagram (a) (Carrasco, Johansson ‘11) PSS representation 2N(a)

1,2,3|4,5(ℓ) ≡ (ℓm+ℓm−k123 m )T m 1,2,3|4,5+(T12,3|4,5+T13,2|4,5+T23,1|4,5)

C.R. Mafra (Southampton) String amplitudes 11.06.2019 15 / 17

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2-loop 5-pt integrand

BRST-invariant 2-loop 5-pt integrand (CM, Schlotterer ‘15) BCJ identities satisfied by construction, gravity amplitudes for free The solution for numerators look intuitive, hope for N-pt solution 4-point 3-loop FT integrand under construction

C.R. Mafra (Southampton) String amplitudes 11.06.2019 16 / 17

SLIDE 19

Conclusions

Many simplying features not discussed today for lack of space and time! The multiloop amplitudes have a nice underlying BRST algebra structure Compact expressions in 10D PS superspace; the benefits of 10D SYM Patterns are building up, closed formula solution? Can we find generating series of FT loop amplitudes? How much of the recent simplicity in 4D scattering amplitudes can be explained from a 10D perspective?

C.R. Mafra (Southampton) String amplitudes 11.06.2019 17 / 17