The superstring n-point 1-loop amplitude Carlos R. Mafra (With - PowerPoint PPT Presentation

The superstring n-point 1-loop amplitude Carlos R. Mafra (With Oliver Schlotterer, arXiv:1812. { 10969,10970,10971 } ) Supported by a Royal Society University Fellowship STAG Research Centre and Mathematical Sciences, University of Southampton, UK C.R. Mafra (Southampton) March 2019 1 / 36

Motivation Compute the n-point open superstring correlator at one loop using worldsheet methods C.R. Mafra (Southampton) March 2019 2 / 36

Essential requirements Correlator K n ( ℓ ) defined by: � � � d D ℓ |I n ( ℓ ) | �K n ( ℓ ) � A n = d τ dz 2 dz 3 . . . dz n C top D top top such that: BRST invariant (ie susy and gauge invariant) 1 Q K n ( ℓ ) = 0 monodromy invariant 2 D K n ( ℓ ) = 0 C.R. Mafra (Southampton) March 2019 3 / 36

Summary of results Correlators built from: kinematic factors in pure spinor superspace 1 worldsheet functions at genus one surface 2 Outcome: a beautiful Lie-polynomial structure n − 4 1 � �� � V A 1 T m 1 ... m r A 2 ,..., A r +4 Z m 1 ... m r � K n ( ℓ ) = A 1 ,..., A r +4 + 12 . . . n | A 1 , . . . , A r +4 r ! r =0 + corrections Duality between BRST and monodromy operators (BRST invariants vs generalized elliptic integrands) Q ↔ D C.R. Mafra (Southampton) March 2019 4 / 36

Examples 4 points (Berkovits 2004) K 4 ( ℓ ) = V 1 T 2 , 3 , 4 Z 1 , 2 , 3 , 4 kinematic factor is BRST invariant V 1 T 2 , 3 , 4 ≡ 1 ( λγ m W 2 )( λγ m W 3 ) F 4 � � 3( λ A 1 ) mn + cyc (2 , 3 , 4) QV 1 T 2 , 3 , 4 = 0 worldsheet functions are monodromy invariant Z 1 , 2 , 3 , 4 ≡ 1 C.R. Mafra (Southampton) March 2019 5 / 36

5 point correlator K 5 ( ℓ ) = V 1 T m 2 , 3 , 4 , 5 Z m 1 , 2 , 3 , 4 , 5 + V 12 T 3 , 4 , 5 Z 12 , 3 , 4 , 5 + (2 ↔ 3 , 4 , 5) + V 1 T 23 , 4 , 5 Z 1 , 23 , 4 , 5 + (2 , 3 | 2 , 3 , 4 , 5) kinematic factors V A T B , C , D and V A T m B , C , D , E in pure spinor superspace with covariant BRST variations one-loop worldsheet functions Z A , B , C , D and Z m A , B , C , D , E from Kronecker–Einsestein series and loop momentum with covariant monodromy variations C.R. Mafra (Southampton) March 2019 6 / 36

5pt BRST & monodromy invariance There is a strong interplay between kinematics and worldsheet functions: The 5-pt correlator is BRST invariant due to a total derivative: � �� k m 2 Z m � Q K 5 ( ℓ ) = − V 1 V 2 T 3 , 4 , 5 1 , 2 , 3 , 4 , 5 + s 21 Z 21 , 3 , 4 , 5 + (1 ↔ 3 , 4 , 5) + (2 ↔ 3 , 4 , 5) ∼ = 0 The 5-pt correlator is single valued due to BRST cohomology ids (BRST exact terms) � �� k m 1 V 1 T m � D K 5 ( ℓ ) = Ω 1 2 , 3 , 4 , 5 + V 12 T 3 , 4 , 5 + 2 ↔ 3 , 4 , 5 � �� k m 2 V 1 T m � V 1 T 23 , 4 , 5 + 3 ↔ 4 , 5 + Ω 2 2 , 3 , 4 , 5 + V 21 T 3 , 4 , 5 + + (2 ↔ 3 , 4 , 5) ∼ = 0 C.R. Mafra (Southampton) March 2019 7 / 36

Examples 6 point correlator K 6 ( ℓ ) = 1 2 V 1 T mn 2 , 3 , 4 , 5 , 6 Z mn 1 , 2 , 3 , 4 , 5 , 6 V 12 T m 3 , 4 , 5 , 6 Z m � � + 12 , 3 , 4 , 5 , 6 + (2 ↔ 3 , 4 , 5 , 6) V 1 T m 23 , 4 , 5 , 6 Z m � � + 1 , 23 , 4 , 5 , 6 + (2 , 3 | 2 , 3 , 4 , 5 , 6) � � + V 123 T 4 , 5 , 6 Z 123 , 4 , 5 , 6 + V 132 T 4 , 5 , 6 Z 132 , 4 , 5 , 6 + (2 , 3 | 2 , 3 , 4 , 5 , 6) � ( V 12 T 34 , 5 , 6 Z 12 , 34 , 5 , 6 + cyc (2 , 3 , 4)) + (2 , 3 , 4 | 2 , 3 , 4 , 5 , 6) � + � � + ( V 1 T 2 , 34 , 56 Z 1 , 2 , 34 , 56 + cyc (3 , 4 , 5)) + (2 ↔ 3 , 4 , 5 , 6) � � + V 1 T 234 , 5 , 6 Z 1 , 234 , 5 , 6 + V 1 T 243 , 5 , 6 Z 1 , 243 , 5 , 6 + (2 , 3 , 4 | 2 , 3 , 4 , 5 , 6) Nice combinatorics of Stirling set and cycle numbers: 2 1 � �� � V A 1 T m 1 ... m r A 2 ,..., A r +4 Z m 1 ... m r K 6 ( ℓ ) = � 12 . . . 6 | A 1 , . . . , A r +4 A 2 ,..., A r +4 + r ! r =0 C.R. Mafra (Southampton) March 2019 8 / 36

6pt anomaly cancellation (Green, Schwarz 84) 6pt correlator is not BRST invariant by itself However BRST variation is a total derivative on moduli space Q K 6 ( ℓ ) = − 1 ∂ ∂τ log I 6 ( ℓ ) ∼ 2 V 1 Y 2 , 3 , 4 , 5 , 6 Z mm 1 , 2 , 3 , 4 , 5 , 6 = − 2 π i V 1 Y 2 , 3 , 4 , 5 , 6 = 0 , where Y 2 , 3 , 4 , 5 , 6 is the anomaly kinematic factor (CM, Berkovits 2006) Y 2 , 3 , 4 , 5 , 6 ≡ 1 2( λγ m W 2 )( λγ n W 3 )( λγ p W 4 )( W 5 γ mnp W 6 ) To show this need identities for τ derivatives of the Kronecker-Eisenstein series, several BRST variations etc So anomaly cancels after summing over one-loop topologies for SO (32) (Green, Schwarz 84) C.R. Mafra (Southampton) March 2019 9 / 36

Derivations C.R. Mafra (Southampton) March 2019 10 / 36

Pure spinor amplitude prescription at one loop � � � � � dz U 2 · · · ( µ, b )( PCs ) V 1 (0) dz U n A 1 = moduli vertex operators using SYM superfields A α ( x , θ ), A m ( x , θ ), W α ( x , θ ) and F mn ( x , θ ) V = λ α A α ( x , θ ) , U = ∂θ α A α + A m Π m + d α W α + 1 2 N mn F mn CFT calculation: zero modes and OPEs OPEs among vertices organized using multiparticle superfields with covariant BRST variations (CM, Schlotterer ‘14) b ghost and PCOs complications bypassed by completing the known parts of the correlators from OPEs to BRST-invariant and single-valued answers C.R. Mafra (Southampton) March 2019 11 / 36

SYM description in 10D Single-particle ( i is particle label) (Witten’86) K i ∈ { A i α , A m i , W α i , F mn } i Multiparticle ( B is a “word” with particle labels) K B ∈ { A B α , A m B , F mn B , W α B } Inspired by OPE computations and defined recursively, eg W α 1 = W α 1 12 = 1 2 ( k 2 · A 1 ) − (1 ↔ 2) W α 4( γ mn W 2 ) α F 1 mn + W α 12 + 1 123 = − ( k 12 · A 3 ) W α W α 4( γ rs W 3 ) α F 12 rs − (12 ↔ 3) + 1 2( k 1 · k 2 ) 2 ( A 1 · A 3 ) − (1 ↔ 2) W α � � C.R. Mafra (Southampton) March 2019 12 / 36

Generalized SYM equations of motion Superfields in K B satisfy generalized SYM EOMs, eg 1 = 1 D α W β 4( γ mn ) αβ F 1 mn 12 = 1 D α W β 4( γ mn ) αβ F 12 mn + ( k 1 · k 2 )( A 1 α W β 2 − A 2 α W β 1 ) 123 = 1 D α W β 4( γ mn ) αβ F 123 mn + ( k 1 · k 2 ) A 1 α W β 23 + A 13 α W β � � 2 − (1 ↔ 2) + ( k 12 · k 3 ) A 12 α W β � � 3 − (12 ↔ 3) , In general: P = 1 D α W β � α W β α W β 4( γ mn ) αβ F P � A XR jS − A jR � ( k X · k j ) mn + , XS P = XjY Y = R ✁ S Similar EOMs for A B α , A m B , F mn B C.R. Mafra (Southampton) March 2019 13 / 36

Generalized Jacobi symmetries The superfields K B satisfy generalized Jacobi symmetries 0 = K 12 + K 21 , 0 = K 123 + K 231 + K 312 , (Jacobi identity) 0 = K 1234 − K 1243 + K 3412 − K 3421 0 = K A ℓ ( B ) + K B ℓ ( A ) ℓ ( A ) is the Dynkin operator (left-to-right nested brackets) These are the same symmetries obeyed by nested commutators K 1234 ... p ≡ K ℓ ( P ) = K [ ... [[[1 , 2] , 3] , 4] ,..., p ] BCJ identities/numerators are natural in this framework BRST operator is λ α D α so multiparticle superfields lead to (a rich) BRST algebra, cohomology identities etc C.R. Mafra (Southampton) March 2019 14 / 36

Zero-mode prescription and building blocks An analysis of the PS prescription leads to a zero-mode contribution of d α d β N mn from the vertices (Berkovits ‘04) Four points K 4 = � V 1 U 2 U 3 U 4 � ddN = � V 1 T 2 , 3 , 4 � where T 2 , 3 , 4 = 1 3( λγ m W 2 )( λγ m W 3 ) F mn + cyc (2 , 3 , 4) 4 Higher points: multiparticle version (CM, Schlotterer ‘12) T A , B , C = 1 3( λγ m W A )( λγ m W B ) F mn + cyc ( A , B , C ) C at 5pts V 12 T 3 , 4 , 5 , V 1 T 23 , 4 , 5 + perm Also tensorial generalization ( V A T mn ... B , C , D , E ,... ) C.R. Mafra (Southampton) March 2019 15 / 36

One-loop superstring correlators C.R. Mafra (Southampton) March 2019 16 / 36

Recalling: Lie polynomials A Lie polynomial is an expression written in terms of nested commutators Ree theorem If Z P satisfies shuffle symmetries Z A ✁ B = 0 and t p i are non-commutative indeterminates then Z p 1 p 2 p 3 ... t p 1 t p 2 t p 3 · · · � P is a Lie polynomial Example: Z 12 satisfies shuffle if it is antisymmetric, so Z 12 t 1 t 2 + Z 21 t 2 t 1 = Z 12 [ t 1 , t 2 ] is a Lie polynomial C.R. Mafra (Southampton) March 2019 17 / 36

Lessons from tree-level (CM, Schlotterer, Stieberger 2011) n-point disk correlator can be rewritten in a suggestive way: K tree � Z tree Z tree � �� � = V n + perm (23 . . . n − 2) . 1 A V 1 A n − 1 , B V n − 1 , B n AB =23 ... n − 2 Worldsheet functions satisfy shuffle symmetries 1 1 Z tree Z tree 123 ... p ≡ − → A ✁ B = 0 z 12 z 23 . . . z p − 1 , p associated kinematics satisfy generalized Jacobi symmetries 2 V P ≡ λ α A P − → V A ℓ ( B ) + V B ℓ ( A ) = 0 α This has the same structure of a Lie polynomial! � Z tree V P P P C.R. Mafra (Southampton) March 2019 18 / 36

Ansatz for one-loop correlators Tree-level reinterpretation key to unlock the one-loop correlators 1 Assume Lie-polynomial structure for one-loop correlators: � K n → Z A , B , C , D V A T B , C , D + · · · 2 kinematic factors V A T B , C , D satisfying generalized Jacobi symmetries 3 one-loop worldsheet functions Z A , B , C ,... satisfying shuffle symmetries Singular behaviour of Z A , B ,... as vertices collide is known from OPEs Unlike at tree-level, OPEs don’t determine the complete functions as regular pieces are not fixed by singularities The shuffle-symmetry requirement was very helpful in fixing the functions C.R. Mafra (Southampton) March 2019 19 / 36