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arXiv · 2512.02107

Generalised 4d Partition Functions and Modular Differential Equations

A. Ramesh Chandra·Sunil Mukhi·Palash Singh
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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.

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A. Ramesh Chandra, Sunil Mukhi, Palash Singh. 2025-12-01. Generalised 4d Partition Functions and Modular Differential Equations. https://arxiv.org/abs/2512.02107

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