Modular graph forms from equivariant iterated Eisenstein integrals

Abstract

The low-energy expansion of closed-string scattering amplitudes at genus one introduces infinite families of non-holomorphic modular forms called modular graph forms. Their differential and number-theoretic properties motivated Brown's alternative construction of non-holomorphic modular forms in the recent mathematics literature from so-called equivariant iterated Eisenstein integrals. In this work, we provide the first validations beyond depth one of Brown's conjecture that equivariant iterated Eisenstein integrals contain modular graph forms. Apart from a variety of examples at depth two and three, we spell out the systematics of the dictionary and make certain elements of Brown's construction fully explicit to all orders.

45 pages; submission includes ancillary data files; v2: typos corrected / minor improvements, matches published version

References in corpus (8)

• Multi-Regge kinematics and the moduli space of Riemann spheres with marked points
• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops
• All-order differential equations for one-loop closed-string integrals and modular graph forms
• Generating series of all modular graph forms from iterated Eisenstein integrals
• 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
• Poincaré series for modular graph forms at depth two. II. Iterated integrals of cusp forms
• Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems
• Poisson equation for genus two string invariants: a conjecture

Cited by in corpus (9)

• Tree-level amplitudes from the pure spinor superstring
• Bootstrapping spinning two body problem in dynamical Chern-Simons gravity using worldline QFT
• Basis decompositions of genus-one string integrals
• Canonical differential equations and intersection matrices
• Advanced tools for basis decompositions of genus-one string integrals
• Integral of depth zero to three basis of Modular Graph Functions
• Type II superstring amplitude at one-loop and transcendentality
• From Modular Graph Forms to Iterated Integrals
• Modular Features of Superstring Scattering Amplitudes: Generalised Eisenstein Series and Theta Lifts