Generalised 4d partition functions and modular differential equations

Abstract

We prove the equivalence of a class of generalised Schur partition functions 𝒵G(q; α) of 4d 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 hyper-multiplets and analytically prove that 𝒵USp(2N)(q; α) satisfies an order-(N + 1) modular linear differential equation (MLDE) with vanishing Wronskian index, explaining how the parameter α 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 𝒵USp(2N)(q; α, β) of the generalised Schur partition function. Finally, we relate the α = −k specialisation to quantum monodromy traces Tr Mk and formulate a conjecture linking their k-dependence to MLDEs.