Gravity, strings, modular and quasimodular forms

TL;DR

This paper explores the role of modular and quasimodular forms, especially Eisenstein series, in gravity and string theory, highlighting their appearance in solutions, spectrum calculations, and non-perturbative effects, and introduces new gravitational instanton solutions with modular properties.

Contribution

It presents a class of four-dimensional gravitational instanton solutions with modular properties obtained via (conformal) self-duality and Bianchi IX symmetry, expanding the understanding of modular structures in gravity.

Findings

Existence of gravitational instantons with modular properties.

Eisenstein series appear in spectrum and non-perturbative effects.

Discussion of modular structures in string theory expansions.

Abstract

Modular and quasimodular forms have played an important role in gravity and string theory. Eisenstein series have appeared systematically in the determination of spectrums and partition functions, in the description of non-perturbative effects, in higher-order corrections of scalar-field spaces, ... The latter often appear as gravitational instantons i.e. as special solutions of Einstein's equations. In the present lecture notes we present a class of such solutions in four dimensions, obtained by requiring (conformal) self-duality and Bianchi IX homogeneity. In this case, a vast range of configurations exist, which exhibit interesting modular properties. Examples of other Einstein spaces, without Bianchi IX symmetry, but with similar features are also given. Finally we discuss the emergence and the role of Eisenstein series in the framework of field and string theory perturbative expansions, and motivate the need for unravelling novel modular structures.