# Modular Forms and $SL(2, {\mathbb Z})$-covariance of type IIB superstring theory

## TL;DR

This paper investigates the modular properties of higher-derivative interactions in type IIB superstring theory, deriving differential equations for their coefficients based on $SL(2, Z)$-covariance and supersymmetry, revealing their structure and contributions to string amplitudes.

## Contribution

It introduces first-order differential equations and Laplace eigenvalue equations for modular form coefficients of BPS interactions, advancing understanding of their structure and non-perturbative effects.

## Key findings

- Derived $SL(2, Z)$-covariant Laplace eigenvalue equations for modular coefficients.
- Identified two independent modular forms for certain interactions, one with no tree-level contribution.
- Connected amplitude analysis with modular form equations, constraining coefficients.

## Abstract

The local higher-derivative interactions that enter into the low-energy expansion of the effective action of type IIB superstring theory with constant complex modulus generally violate the $U(1)$ R-symmetry of IIB supergravity by $q_U$ units. These interactions have coefficients that transform as non-holomorphic modular forms under $SL(2, {\mathbb Z})$ transformations with holomorphic and anti-holomorphic weights $(w,-w)$, where $q_U=-2w$. In this paper $SL(2, {\mathbb Z})$-covariance and supersymmetry are used to determine first-order differential equations on moduli space that relate the modular form coefficients of classes of BPS-protected maximal $U(1)$-violating interactions that arise at low orders in the low-energy expansion. These are the moduli-dependent coefficients of BPS interactions of the form $d^{2p} \mathcal{P}_n$ in linearised approximation, where $\mathcal{P}_n$ is the product of $n$ fields that has dimension $=8$ with $q_U=8-2n$, and $p=0$, $2$ or $3$. These first-order equations imply that the coefficients satisfy $SL(2, {\mathbb Z})$-covariant Laplace eigenvalue equations on moduli space with solutions that contain information concerning perturbative and non-perturbative contributions to superstring amplitudes. For $p=3$ and $n\ge 6$ there are two independent modular forms, one of which has a vanishing tree-level contribution. The analysis of super-amplitudes for $U(1)$-violating processes involving arbitrary numbers of external fluctuations of the complex modulus leads to a diagrammatic derivation of the first-order differential relations and Laplace equations satisfied by the coefficient modular forms. Combining this with a $SL(2, {\mathbb Z})$-covariant soft axio-dilaton limit that relates amplitudes with different values of $n$ determines most of the modular invariant coefficients, leaving a single undetermined constant.