Higher genus modular graph functions, string invariants, and their exact asymptotics

Abstract

The concept and the construction of modular graph functions are generalized from genus-one to higher genus surfaces. The integrand of the four-graviton superstring amplitude at genus-two provides a generating function for a special class of such functions. A general method is developed for analyzing the behavior of modular graph functions under non-separating degenerations in terms of a natural real parameter $t$. For arbitrary genus, the Arakelov Green function and the Kawazumi-Zhang invariant degenerate to a Laurent polynomial in $t$ of degree $(1,1)$ in the limit $t\to \infty$. For genus two, each coefficient of the low energy expansion of the string amplitude degenerates to a Laurent polynomial of degree $(w,w)$ in $t$, where $w+2$ is the degree of homogeneity in the kinematic invariants. These results are exact to all orders in $t$, up to exponentially suppressed corrections. The non-separating degeneration of a general class of modular graph functions at arbitrary genus is sketched and similarly results in a Laurent polynomial in $t$ of bounded degree. The coefficients in the Laurent polynomial are generalized modular graph functions for a punctured Riemann surface of lower genus.

63 pages, 8 figures; minor typos corrected in new version

References in corpus (4)

• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops
• Johnson's homomorphisms and the Arakelov-Green function
• Tetrahedral modular graph functions
• Asymptotics of the $D^8{R}^4$ genus-two string invariant

Cited by in corpus (40)

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• Two-Loop Scattering Amplitudes from Ambitwistor Strings: from Genus Two to the Nodal Riemann Sphere
• Tree-level amplitudes from the pure spinor superstring
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• Exploring transcendentality in superstring amplitudes
• All-order differential equations for one-loop closed-string integrals and modular graph forms
• One-loop open-string integrals from differential equations: all-order alpha'-expansions at n points
• Generating series of all modular graph forms from iterated Eisenstein integrals
• Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings
• Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality
• Modular graph functions and odd cuspidal functions -- Fourier and Poincaré series
• All loop structures in Supergravity Amplitudes on $Ad{S}_5\times S^5$ from CFT
• On integrands and loop momentum in string and field theory
• Canonical Differential Equations Beyond Genus One
• The SAGEX Review on Scattering Amplitudes, Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their $\mathcal{N}=4$ supersymmetric Yang-Mills duals
• All-order alpha'-expansion of one-loop open-string integrals
• Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems
• Modular graph forms from equivariant iterated Eisenstein integrals
• From Little String Free Energies Towards Modular Graph Functions
• Non-planar corrections in orbifold/orientifold $\mathcal{N}=2$ superconformal theories from localization
• On Genus-one String Amplitudes on $Ad{S}_5\times S^5$
• Elliptic modular graph forms I: Identities and generating series
• Two-loop superstring five-point amplitudes III, construction via the RNS formulation: even spin structures
• Poisson equations for elliptic modular graph functions
• Basis Decompositions and a Mathematica Package for Modular Graph Forms
• Relations between elliptic modular graphs
• Diagrammatic Expansion of Non-Perturbative Little String Free Energies
• Asymptotics of the $D^8{R}^4$ genus-two string invariant
• On the multiloop soft theorem of the dilaton in the bosonic string
• Cyclic products of Szegö kernels and spin structure sums I: hyper-elliptic formulation
• Symmetric Orbifold Theories from Little String Residues
• Supergravity limit of genus two modular graph functions in the worldline formalism
• String theory integrands and supergravity divergences
• Identities among higher genus modular graph tensors
• Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors
• Poisson equation for genus two string invariants: a conjecture
• Frobenius' theta function and Arakelov invariants in genus three
• Integrating simple genus two string invariants over moduli space
• A construction of single-valued elliptic polylogarithms
• One-loop matrix elements of effective superstring interactions: ${\alpha }^{\prime }$-expanding loop integrands