Generalised 4d Partition Functions and Modular Differential Equations

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 $\mathrm{T}\mathrm{r}{M}^k$ and formulate a conjecture linking their $k$-dependence to MLDEs.