Review: Modular Graph Functions and the string effective action

Transcription

1 Review: Modular Graph Functions and the string effective action Jan E. Gerken Max-Planck-Institute for Gravitational Physics Albert-Einstein Institute Potsdam Golm Germany International School of Subnuclear Physics 2017 Jan Gerken (AEI) Modular Graph Functions ISSP / 15

2 Introduction Outline 1 Introduction 2 Moduli space 3 Feynman diagrams on the worldsheet 4 α -corrections to the string effective action 5 Conclusion Jan Gerken (AEI) Modular Graph Functions ISSP / 15

3 Introduction Introduction Modular graph functions = Tool to calculate higher order corrections in α to string amplitudes Motivation Effective action is physical prediciton of theory Coefficients in α exhibit interesting systematics Using U-duality: coefficients in α might be useful to constrain non-perturbative effects Jan Gerken (AEI) Modular Graph Functions ISSP / 15

4 Introduction Structure Three main points 1. String amplitudes are integrals over moduli space 2. Modular graph functions are Feynman diagrams on the worldsheet 3. Use this to calculate higher order corrections to string amplitudes Jan Gerken (AEI) Modular Graph Functions ISSP / 15

5 Moduli space Outline 1 Introduction 2 Moduli space 3 Feynman diagrams on the worldsheet 4 α -corrections to the string effective action 5 Conclusion Jan Gerken (AEI) Modular Graph Functions ISSP / 15

6 Moduli space String amplitudes are integrals over moduli space String perturbation theory is an expansion in the genus of the worldsheet. Specialize to a torus (genus 1) Use diffeomorphism Weyl gauge freedom to fix the worldsheet-metric up to a complex number τ (modular parameter). Therefore: DXDg e is[x,g] DXdτ e is[x,g(τ)]. The space in which τ lives is called moduli space M. This means that string amplitudes are integrals over τ of some τ-dependent functions. Jan Gerken (AEI) Modular Graph Functions ISSP / 15

7 Moduli space The integrand is modular invariant Modular transformations τ aτ + b cτ + d, with ( ) a b SL(2, Z) c d Modular transformations are just (large) diffeomorphisms, so identify tori which are related by this transformation. Hence, only integrate over the fundamental domain F, the subset of M which generates the hole space. Since the amplitude should be invariant under a change of choice of fundamental domain, the integrand of the integral over moduli space has to be modular invariant. Jan Gerken (AEI) Modular Graph Functions ISSP / 15

8 Feynman diagrams on the worldsheet Outline 1 Introduction 2 Moduli space 3 Feynman diagrams on the worldsheet 4 α -corrections to the string effective action 5 Conclusion Jan Gerken (AEI) Modular Graph Functions ISSP / 15

9 Feynman diagrams on the worldsheet General form of the string amplitude [9910 Green,Vanhove] [0801 Green,Russo,Vanhove] Four graviton scattering amplitude at genus one in type IIB string theory A 1 (ɛ i, k i ) R 4 (ɛ i, k i ) dτ B(s, t, u τ) F R 4 is a contraction of 4 powers of the Riemann tensor B is modular invariant This is very hard to evaluate, hence expand B in powers of s ij = α k i k j 2 The coefficients are Feynman diagrams on the worldsheet with 4 vertices These Feynman diagrams are called Modular Graph Functions Jan Gerken (AEI) Modular Graph Functions ISSP / 15

10 α -corrections to the string effective action Outline 1 Introduction 2 Moduli space 3 Feynman diagrams on the worldsheet 4 α -corrections to the string effective action 5 Conclusion Jan Gerken (AEI) Modular Graph Functions ISSP / 15

11 α -corrections to the string effective action α -corrections to the four-graviton amplitude [1502 D Hoker,Green,Vanhove] Due to their modular invariance, there are many non-trivial relations between the modular graph functions These can be proven by making use of graphical simplification techniques Or by using algebraic and group theoretic methods that make use of modular invariance Also the integrals over F can be evaluated α -expansion of the four graviton, genus one amplitude ( A 1 (ɛ i, k i ) R 4 Ξ (0,0) + Ξ (1,0) σ 2 + Ξ (0,1) σ 3 ) +Ξ (2,0) σ2 2 + Ξ (1,1) σ 2 σ 3 + Ξ (0,2) σ with σ n = s n + t n + u n where s = α k 1 k 2 2 etc. Jan Gerken (AEI) Modular Graph Functions ISSP / 15

12 α -corrections to the string effective action α -corrections to the string effective action [1502 D Hoker,Green,Vanhove] α -expansion of the four graviton, genus one amplitude ( A 1 (ɛ i, k i ) R 4 Ξ (0,0) + Ξ (1,0) σ 2 + Ξ (0,1) σ 3 ) +Ξ (2,0) σ2 2 + Ξ (1,1) σ 2 σ 3 + Ξ (0,2) σ A Ξ (p,q) σ p 2 σq 3-term in the amplitude corresponds to a Ξ (p,q) D 2(2p+3q) R 4 -contribution in the effective action At the lowest orders we have (ζ = Riemann zeta function) Ξ (0,0) = π Ξ (1,0) = 0 3 Ξ (2,0) = 0 Ξ (1,1) = π ζ(5) Only odd Riemann zeta values appear! Ξ (0,1) = π 9 ζ(3) Jan Gerken (AEI) Modular Graph Functions ISSP / 15

13 Conclusion Outline 1 Introduction 2 Moduli space 3 Feynman diagrams on the worldsheet 4 α -corrections to the string effective action 5 Conclusion Jan Gerken (AEI) Modular Graph Functions ISSP / 15

14 Conclusion Conclusion Modular graph functions are Feynman diagrams on the worldsheet Their rich algebraice structure implies many simplifying relations between them and allows the application of powerful tools (e.g. rep. theory) They can be used to calculate higher-order corrections to the genus-one effective action which leads to a deeper understanding of the structure of the theory and reveals hidden simplicities Jan Gerken (AEI) Modular Graph Functions ISSP / 15

15 Conclusion Thank you! Jan Gerken (AEI) Modular Graph Functions ISSP / 15

16 The Propagator on the Worldsheet τ 2 G(z = α + τβ τ) = π mτ + n 2 e2πi(nα mβ) (m,n) (0,0) Z 2 Jan Gerken (AEI) Modular Graph Functions ISSP / 15