Generalised 4d partition functions and modular differential equations

A

bstract

We prove the equivalence of a class of generalised Schur partition functions 𝒵 _G ( 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 (2 N ) theory with 2 N + 2 fundamental hyper-multiplets and analytically prove that 𝒵 _{USp (2 N )} ( 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 𝒵 _{USp (2 N )} ( q ; α, β ) of the generalised Schur partition function. Finally, we relate the α = − k specialisation to quantum monodromy traces Tr M ^k and formulate a conjecture linking their k -dependence to MLDEs.