Generalised 4d Partition Functions and Modular Differential Equations

TL;DR

This paper establishes a link between 4d superconformal gauge theory partition functions and 2d rational conformal field theory modular forms, deriving differential equations and exploring extensions and specializations.

Contribution

It proves the equivalence of certain 4d gauge theory partition functions to vector-valued modular forms and derives associated modular differential equations, including extensions and special cases.

Findings

Partition functions satisfy order-(N+1) MLDEs with vanishing Wronskian index.

Connections are made between gauge theory partition functions and RCFT characters.

Proposed extensions relate parameters to quantum monodromy traces and MLDEs.

Abstract

We prove the equivalence of a class of generalised Schur partition functions $\mathcal Z_G(q;\alpha)$ of 4d $\mathcal N=2$ superconformal gauge theories to contour integral representations of vector-valued modular forms of the type that arise in 2d rational conformal field theories (RCFT). Concretely, we consider the $USp(2N)$ theory with $2N+2$ fundamental hypermultiplets and analytically prove that $\mathcal Z_{USp(2N)}(q;\alpha)$ satisfies an order-$(N+1)$ modular linear differential equation (MLDE) with vanishing Wronskian index, explaining how the parameter $\alpha$ of the former determines the parameters of the latter. Several connections are made to characters of RCFTs including unitary ones. We then propose a two-parameter extension $\mathcal Z_{USp(2N)}(q;\alpha,\beta)$ of the generalised Schur partition function. Finally, we relate the $\alpha=-k$ specialisation to quantum monodromy traces ${\rm Tr}\,M^k$ and formulate a conjecture linking their $k$-dependence to MLDEs.