Asymptotics of higher genus string integrands Boris Pioline String - PowerPoint PPT Presentation

Asymptotics of higher genus string integrands Boris Pioline “String theory from a worldsheet perspective”, GGI, FLorence, 16/4/2019 based on 1712.06135, 1806.02691, 1810.11343 in collaboration with Eric d’Hoker and Michael B. Green B. Pioline (LPTHE) Higher genus string integrands GGI, 16/4/2019 1 / 31

Modular graph functions beyond genus one I In perturbative closed string theory, scattering amplitudes of N external states at h loop involve an integral over the moduli space M h , N of genus h curves Σ with N punctures z 1 , . . . z N . At fixed complex structure, this reduces to an integral over N copies of Σ . At genus one, the low energy expansion of the resulting integrand produces an infinite family of real-analytic modular forms labelled by certain graphs, known as modular graph functions. They exhibit remarkable asymptotics near the cusp ( finite Laurent polynomial plus exponential corrections ) and transcendentality properties. My goal will be to describe some hints of a similar structure at higher genus, primarily for N = 4 , g = 2. B. Pioline (LPTHE) Higher genus string integrands GGI, 16/4/2019 2 / 31

String integrand I Recall that the genus h contribution to N -point scattering amplitude is of the form � A h ( { k i , ǫ i } ) = I h ( { k i , ǫ i } ) M h , N where I h ( { k i , ǫ i } ) ∼ � � N i = 1 V k i ,ǫ i ( z i ) � 3 h − 3 | ( b , µ ) | 2 � is a correlator j = 1 in the worldsheet conformal field theory. The moduli space M h , N is fibered over M h ≡ M h , 0 , with fiber Σ 1 × · · · × Σ N . The vertex operators V k i ,ǫ i are ( 1 , 1 ) -forms over Σ i , while the ghost part produces a volume form on the base M h . I will focus on 1 ≤ h ≤ 3, where M h is isomorphic to a fundamental domain F h for the action of Sp ( h , Z ) on the Siegel upper half-plane H h (away from some divisors). B. Pioline (LPTHE) Higher genus string integrands GGI, 16/4/2019 3 / 31

String integrand II In the RNS formalism, I h originates from an integrand I h over the moduli space of super Riemann curves M h , N , after summing over spin structures and integrating over fermionic moduli. There is no canonical projection M h , N → M h , n , leading to total derivative ambiguities [Donagi Witten’13] . State of the art: h = 2 , N = 4 [D’Hoker Phong ’05] In Berkovits’ pure spinor formalism, b is a composite field involving powers of ( λ ¯ λ ) − 1 , leading to potential divergences from the region λ ¯ λ → 0. State of the art: idem, with partial results for h = 2 , N = 5 [Gomez Mafra Schlotterer ’15] and h = 3 , N = 4 [Gomez Mafra ’13] B. Pioline (LPTHE) Higher genus string integrands GGI, 16/4/2019 4 / 31

Modular graph functions beyond genus one I Although we do not know the integrand I h in general, for type II strings on T d we expect a result of the form � I h = δ ( k i ) Z KN ( { k i } ) Z T d P h ( { k i , ǫ i , z i } ) µ h where Z KN ( { k i , z i } ) = exp ( � i < j s ij G ( z i , z j )) is the Koba-Nielsen factor, with s ij = − α ′ 2 k i · k j the Mandelstam variables and G ( z i , z j ) the scalar Green function. Z T d is the genus- h Siegel theta series for a lattice of signature ( d , d ) P h ( { k i , ǫ i , z i } ) is a (1,1)-form in each position z i , and a Lorentz invariant homogeneous polynomial in k i , ǫ i µ h is the Siegel volume form over M h ≃ F h B. Pioline (LPTHE) Higher genus string integrands GGI, 16/4/2019 5 / 31

Canonical forms on compact curves I Let A I , B I ( I = 1 . . . h ) be a canonical basis of H 1 (Σ) , such that A I ∩ A J = B I ∩ B J = 0 , A I ∩ B J = δ IJ . Choose ω I a basis of � holomorphic differentials on Σ such that A I ω J = δ IJ . The period � matrix τ IJ = B I ω J satisfies τ > 0, hence τ ∈ H h . Under Sp ( h , Z ) change of basis, τ �→ ( a τ + b )( c τ + d ) − 1 . The I ≤ J | d τ IJ | 2 � Siegel volume form µ h = ( det Im τ ) h + 1 is invariant. An example of (1,1) form on Σ i is the canonical Kähler form 2 h Im τ IJ ω I ( z i )¯ i κ ( z i ) = ω J ( z i ) . Examples of ( 1 , 1 ) forms on Σ i × Σ j are κ ( z i ) κ ( z j ) and | ν ( z i , z j ) | 2 with ν ( z i , z j ) = Im τ IJ ω I ( z i ) ω J ( z j ) . Another example on h copies of Σ is | ∆( z 1 , . . . z h ) | 2 where ∆( z 1 , . . . z h ) = ǫ I 1 ... I h ω I 1 ( z 1 ) . . . ω I h ( z h ) is a (1,0)-form in each variable. B. Pioline (LPTHE) Higher genus string integrands GGI, 16/4/2019 6 / 31

Canonical forms on compact curves II The Arakelov Green function G ( z , w ) is a symmetric function on Σ × Σ defined by ∂ z ¯ ∂ z G ( z , w ) = − πδ ( 2 ) ( z , w ) + π κ ( z ) , � G ( z , w ) κ ( w ) = 0 , Σ � The r.h.s. integrates to zero thanks to Σ κ = 1, allowing G to be globally well-defined. � w In genus one, setting v = z ω = α + βτ , ′ 2 + 2 π � � ( Im v ) 2 = τ 2 e 2 π i ( m β − n α ) � ϑ 1 ( v ,τ ) � G ( z , w ) = − log � � η π | m + n τ | 2 τ 2 � ( m , n ) ∈ Z 2 B. Pioline (LPTHE) Higher genus string integrands GGI, 16/4/2019 7 / 31

Genus-one modular graph functions I After Taylor expanding in powers of s ij , and integrating over the positions z i of the vertex operators, the integrand is a linear combination of genus one modular graph functions, of the type κ ( z ) = d z d ¯ � z � � B Γ ( τ ) = G ( z i , z j ) κ ( z i ) 2 i Im τ Σ N / Σ ( i , j ) ∈ Γ i where ( i , j ) runs over the edges of the graph Γ (possibly dressed with derivatives wrt z i ) [d’Hoker Green Gurdogan Vanhove ’15] By construction, B Γ are real analytic modular functions. e.g. � • • G ( v ) k κ ( v ) D k ( τ ) = Σ gives E 2 for k = 2, E 3 + ζ ( 3 ) for k = 3. B. Pioline (LPTHE) Higher genus string integrands GGI, 16/4/2019 8 / 31

Genus-one modular graph functions II Modular graph functions satisfy an intricate set of algebraic and differential equations. Their expansion near the cusp τ → i ∞ has the remarkable form 1 − w a n ( πτ 2 ) n + O ( e − 2 πτ 2 ) � B Γ ( τ ) = n = w where w is the number of edges (i.e. degree in G ). The coefficients a n are single valued multizeta values of transcendentality degree w − n . d’Hoker Green Gurdogan Vanhove Zerbini Kaidi Basu Kleinschmidt Verschinin Gerken Schlotterer Duke . . . B. Pioline (LPTHE) Higher genus string integrands GGI, 16/4/2019 9 / 31

Genus-one modular graph functions III After integrating over τ ∈ F 1 , the coefficient of the effective 3 R 4 at one-loop (where σ n = s n + t n + u n ) is given interaction σ p 2 σ q by a regularized theta lifting, � E ( 1 , d ) � B Γ ( τ ) Z T d ( G ij , B ij ; τ ) µ 1 ( p , q ) = α Γ F 1 ( L ) Γ O ( d , d ) where G ij , B ij parametrize the Narain moduli space O ( d ) × O ( d ) and F 1 ( L ) = F 1 ∩ { τ 2 < L } is the truncated fundamental domain. The powerlike terms in B Γ ( τ ) are responsible for infrared divergences as L → ∞ . The full amplitude including non-analytic terms is finite when D > 4. More on this later. B. Pioline (LPTHE) Higher genus string integrands GGI, 16/4/2019 10 / 31

Higher genus modular graph functions I For h = 2, N = 4, the precise integrand is d’Hoker Phong ’05 k i ) R 4 |Y| 2 Z KN Z T d det Im τ µ 2 � I 2 = δ ( where Y = ( t − u )∆( z 1 , z 2 ) ∆( z 3 , z 4 ) + perm Z KN = e s [ G ( z 1 , z 2 )+ G ( z 3 , z 4 )]+ t [ G ( z 1 , z 4 )+ G ( z 2 , z 3 )]+ u [ G ( z 1 , z 3 )+ G ( z 2 , z 4 )] Σ 4 | ∆( 1 , 2 ) ∆( 3 , 4 ) | 2 ∝ 1 / det Im τ . At leading order, Z KN ≃ 1 and � The coefficient of the ∇ 4 R 4 interaction at genus 2 is then ( 1 , 0 ) = π � E ( d , 2 ) Z T d µ 2 ∝ E O ( d , d ) d − 3 2 2 Λ 2 F 2 ( L ) B. Pioline (LPTHE) Higher genus string integrands GGI, 16/4/2019 11 / 31

Higher genus modular graph functions II At next-to-leading order, one power of G ( z 1 , z 2 ) comes down from Z KN . The integral over z 3 , z 4 is easy. The coefficient of the ∇ 6 R 4 interaction at genus-two is [d’Hoker Green’13] � E ( d , 2 ) ( 0 , 1 ) = π ϕ KZ Z T d µ 2 F 2 ( L ) where ϕ KZ is the Kawazumi-Zhang invariant, defined by ϕ KZ = − 1 � | ν ( z 1 , z 2 ) | 2 G ( z 1 , z 2 ) 4 Σ × Σ where ν ( z 1 , z 2 ) = Im τ IJ ω I ( z 1 ) ω J ( z 2 ) . This is the simplest genus-two modular graph function, associated • • 1 2 (but the measure need to be specified) to the graph B. Pioline (LPTHE) Higher genus string integrands GGI, 16/4/2019 12 / 31

Higher genus modular graph functions III Using standard theory of complex structure deformations, one may show that ϕ KZ is an eigenmode of the Laplacian on M 2 , ϕ KZ = − 2 π det Im τ δ ( 2 ) ( v ) � � ∆ Sp ( 4 ) − 5 � which makes it east to compute F 2 ϕ KZ µ 2 [d’Hoker Green BP Russo’14] Integrating against Z T d , it follows that [BP Russo’15] � 2 � 2 � � E ( d , 2 ) E ( d , 1 ) E O ( d , d ) � � ∆ O ( d , d ) − ( d + 2 )( 5 − d ) ( 0 , 1 ) = − ∝ d − 2 ( 0 , 0 ) 2 Λ 1 hence E ( d , 2 ) ( 0 , 1 ) is not an Eisenstein series. Combined with information about asymptotics (see below), this shows that ϕ KZ can be obtained as a Borcherds’ type theta lift of 1 /η 6 from SL ( 2 ) to SO ( 3 , 2 ) = Sp ( 4 ) . This the weak Jacobi form θ 2 gives access to the full Fourier expansion [BP’15] . B. Pioline (LPTHE) Higher genus string integrands GGI, 16/4/2019 13 / 31