QCD meets gravity 2019 @ IPAM, UCLA Differential equations for - PowerPoint PPT Presentation

QCD meets gravity 2019 @ IPAM, UCLA ——————— Differential equations for one-loop string integrals ——————— Oliver Schlotterer (Uppsala University) based on 1908.09848, 1908.10830 with C. Mafra and 1911.03476 with J. Gerken & A. Kleinschmidt 09.12.2019

1 Intro I – String perturbation theory String amplitudes ← → worldsheets as “fattened” Feynman diag’s loop order in perturbation theory = genus of the worldsheet α ′ → 0 or − → + + + point- particle limit or − → + + others α ′ → 0 � �� � � �� � � �� � open-string states: closed-string states: convenient organization of loop integrand “gravity = (gauge theory) 2 ” (BCJ) external gravitons non-abelian gauge bosons

2 Intro I – String perturbation theory String amplitudes ← → worldsheets as “fattened” Feynman diag’s loop order in perturbation theory = genus of the worldsheet α ′ → 0 or − → + + + point- particle limit or − → + + others α ′ → 0 � �� � � �� � � �� � open-string states: closed-string states: convenient organization of loop integrand “gravity = (gauge theory) 2 ” (BCJ) external gravitons non-abelian gauge bosons This talk: Study corrections to field theory ∼ inverse string tension α ′ = ⇒ rewarding laboratory for (elliptic) multiple zeta values & modular forms governed by differential equations similar to those of Feynman integrals

3 Intro I – String perturbation theory Map external states to punctures • on the worldsheet, e.g. • 1 open strings 1 4 conformal • 2 • 4 symmetry at tree level: 2 3 • 3 String amplitudes ( n points, g loop) ↔ integrals over moduli spaces M g ; n of n -punctured worldsheets of genus g (with / without boundary), � � � � 4 • 4 • 4 • • • 3 3 + + + . . . 1 • • • 1 • 1 3 • • • 2 2 2 M 0;4 M 1;4 M 2;4 M 3;4 α ′ -expansions ↔ generating series for (large classes of) periods of M g ; n .

4 Intro II – From worldsheet cartoons to integral bases Universal integrand: α ′ -dependent “Koba–Nielsen factor” = − α ′ 2 k i · k j “Koba–Nielsen” punctures z i =1 , 2 ,...,n n � � � KN τ g,n = exp s ij G g ( z i , z j , τ ) moduli τ @ genus g> 0 � �� � 1 ≤ i<j extra 1 2 for Green function, e.g. − log | z ij | 2 at tree level open strings At tree level: additionally, Parke–Taylor factors ∈ integrand 1 PT(1 , 2 , 3 , . . . , n ) = , z ij = z i − z j z 12 z 23 . . . z n − 1 ,n z n, 1 Closed-string tree amplitudes ↔ basis of integrals over spheres ( σ, ρ ∈ S n ) � d 2 z 1 . . . d 2 z n W tree ( σ (1 , 2 , . . . , n ) | ρ (1 , 2 , . . . , n )) = • vol SL 2 ( C ) • C • • × KN 0 ,n PT( σ (1 , 2 , . . . , n )) PT( ρ (1 , 2 , . . . , n ))

5 Intro II – From worldsheet cartoons to integral bases Goals of this talk: • propose genus-one analogues of Parke–Taylor factors ϕ (1 , 2 , . . . , n | τ ) ↔ function on torus with poles ( z 12 z 23 . . . z n − 1 ,n ) − 1 [Mafra, OS 1908.09848, 1908.10830] • conjectural basis of torus integrals in one-loop string amplitudes n d 2 z j � � W τ ( σ (1 , 2 , . . . , n ) | (1 , 2 , . . . , n )) = • • Im τ torus j =1 • × KN τ • 1 ,n ϕ ( σ (1 , 2 , . . . , n ) | τ ) ϕ ( ρ (1 , 2 , . . . , n ) | τ ) universal to bosonic, heterotic & supersymmetric theories. [Gerken, Kleinschmidt, OS 1911.03476] • homogeneous first-order differential equation w.r.t. τ for W τ

6 Motivation I – Double copy • At tree level: Parke–Taylor basis revealed double copy • • � � � � � � open super disk- or = ⊗ superstring Yang Mills Z -integrals • � d z 1 . . . d z n Z tree ( σ (cycle) | ρ (1 , 2 , . . . , n )) = vol SL 2 ( R ) KN 0 ,n PT( ρ (1 , 2 , . . . , n )) σ {−∞ <z 1 <z 2 <...<z n < ∞} • double copy manifest by KLT-type formula (field-theory kernel S [ α | β ]) � A tree A tree SYM (1 , α, n, n − 1) S [ α | β ] Z tree ( σ | 1 , β, n − 1 , n ) open ( σ ) = α,β ∈ S n − 3 [Mafra, OS, Stieberger 1106.2645 & Br¨ odel, OS, Stieberger 1304.7267] • both PT( . . . ) and A tree SYM ( . . . ) fall into ( n − 3)! bases � � integration by parts = ⇒ PT(1 , 2 , . . . , n ) & perm(2 , 3 , . . . , n − 2) .

7 Motivation I – Double copy One-loop generalization of ( n − 3)! Parke–Taylors: − → conjectural ( n − 1)! basis of Kronecker–Eisenstein integrands � � integration by parts = ⇒ ϕ (1 , 2 , 3 , . . . , n | τ ) & perm(2 , 3 , . . . , n ) . • • • open strings: induces basis of cylinder- & M¨ obius-strip integrals • • � • Z τ ( σ (cycle) | ρ (1 , 2 , . . . , n )) = KN τ 1 ,n ϕ ( ρ (1 , 2 , . . . , n ) | τ ) σ (cycle) • this talk: conjecturally Eduardo’s & Piotr’s talk: monodromy rel’s among cycles ( n − 1)! (twisted) cocycles • long-term goal: one-loop double-copy construction for various theories: � one-loop bosonic / he- � above one-loop � � � � some field = ⊗ theory Z τ -integrals terotic / SUSY strings

8 Motivation II – All-order α ′ -expansion Expanding Z tree , W tree in s ij = − α ′ 2 k i · k j ⇒ multiple zeta values (MZVs) ∞ � k − n 1 k − n 2 . . . k − n r ζ n 1 ,n 2 ,...,n r = , n r ≥ 2 r 1 2 0 <k 1 <k 2 <...<k r @ uniform transcendentality: weight n 1 + n 2 + . . . + n r matches order in α ′ [Terasoma 9908045; Broedel, OS, Stieberger, Terasoma 1304.7304] Analogous α ′ -expansion at genus one → functions of τ (integrate later) • • � • • elliptic MZVs [Enriquez 1301.3042] Z τ ( ·|· ) ← → = ⇒ • [Br¨ odel, Mafra, Matthes, OS 1412.5535] • modular (graph) forms � [D’Hoker, • • W τ ( ·|· ) ← → = ⇒ Green, G¨ urdogan, Vanhove 1512.06779] • • [D’Hoker, Green 1603.00839]

9 Motivation II – All-order α ′ -expansion Generate one-loop α ′ -expansion from homogeneous differential eq. in τ : 2 πi ∂ � ∂τ Z τ ( ∗| 1 , ρ (2 , 3 , . . . , n )) = D τ ( ρ | α ) Z τ ( ∗| 1 , α (2 , 3 , . . . , n )) α ∈ S n − 1 [Mafra, OS 1908.09848, 1908.10830] Clue: matrix D τ ( ρ | α ) acting on integrands is linear in α ′ = ⇒ solution via path-ordered exponential has uniform transcendentality! � � τ � d τ ′ � 2 πi D τ ′ ( ρ | α ) Z τ ( ∗| 1 , ρ ) = Z i ∞ ( ∗| 1 , α ) exp i ∞ Z tree at α ∈ S n − 1 � �� � generates eMZVs ( n +2) points σ + = 0 • σ n • • z n • z n τ → i ∞ . . . . ∼ . . . = . . σ 2 • • z 1 • z 1 • z 2 z 2 σ 1 • • • σ − = ∞

10 Motivation II – All-order α ′ -expansion Generate one-loop α ′ -expansion from homogeneous differential eq. in τ : 2 πi ∂ � ∂τ Z τ ( ∗| 1 , ρ (2 , 3 , . . . , n )) = D τ ( ρ | α ) Z τ ( ∗| 1 , α (2 , 3 , . . . , n )) α ∈ S n − 1 [Mafra, OS 1908.09848, 1908.10830] Clue: matrix D τ ( ρ | α ) acting on integrands is linear in α ′ = ⇒ solution via path-ordered exponential has uniform transcendentality! � � τ � d τ ′ � 2 πi D τ ′ ( ρ | α ) Z τ ( ∗| 1 , ρ ) = Z i ∞ ( ∗| 1 , α ) exp i ∞ Z tree at α ∈ S n − 1 � �� � generates eMZVs ( n +2) points • resembles ǫ -form of diff. eq. for Feynman integrals in D = 4 − 2 ǫ dim [e.g. Henn 1304.1806; Adams, Weinzierl 1802.05020] • work in progress: similar expansion techniques for closed-string W τ ( ·|· )

11 Outline: some more details on the results I. The Kronecker–Eisenstein integrands II. Open-string differential equations III. Closed-string integrals and their differential equations IV. Summary & Outlook [Mafra, OS 1908.09848 & 1908.10830, Gerken, Kleinschmidt, OS 1911.03476]

12 Results I – The Kronecker–Eisenstein integrands Parke–Taylor factors are related by partial fraction ( z ij = z i − z j ) 1 1 1 = + = ⇒ KK relations among PT( . . . ) z 12 z 13 z 12 z 23 z 13 z 32 Naive genus-1 generalization of z − 1 ij : odd Jacobi theta function � � 1 quasi-periodic completion ∂ z log θ 1 ( z ij | τ ) = + , w.r.t. z → z +1 & z → z + τ z ij ... more specifically: ∞ � θ 1 ( z | τ ) = 2 e iπτ/ 4 sin( πz ) � 1 − e 2 πinτ �� 1 − e 2 πi ( nτ + z ) �� 1 − e 2 πi ( nτ − z ) � n =1 Problem: quasi-periodic completion spoils partial fraction: ∂ z log θ 1 ( z 12 | τ ) ∂ z log θ 1 ( z 13 | τ ) � = ∂ z log θ 1 ( z 12 | τ ) ∂ z log θ 1 ( z 23 | τ ) + ∂ z log θ 1 ( z 13 | τ ) ∂ z log θ 1 ( z 32 | τ )

13 Results I – The Kronecker–Eisenstein integrands Parke–Taylor factors are related by partial fraction ( z ij = z i − z j ) 1 1 1 = + = ⇒ KK relations among PT( . . . ) z 12 z 13 z 12 z 23 z 13 z 32 genus-1 generalization of z − 1 ij : doubly-periodic Kronecker–Eisenstein series � θ ′ 1 (0 | τ ) θ 1 ( z + η | τ ) � 2 πiη Im z Ω( z, η, τ ) = exp Im τ θ 1 ( z | τ ) θ 1 ( η | τ ) Partial fraction generalizes to Fay identity involving auxiliary var’s η Ω( z 12 , η 2 , τ ) Ω( z 13 , η 3 , τ ) = Ω( z 12 , η 2 + η 3 , τ ) Ω( z 23 , η 3 , τ ) + Ω( z 13 , η 2 + η 3 , τ ) Ω( z 32 , η 2 , τ ) Kronecker–Eisenstein integrand at n points: n − 1 auxiliary var’s η 2 , η 3 , . . . , η n n � ϕ τ η (1 , 2 , . . . , n ) = Ω( z j − 1 ,j , η j + η j +1 + . . . + η n , τ ) � j =2