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<p>We prove the equivalence of a class of generalised Schur partition functions 𝒵<sub><i>G</i></sub>(<i>q</i>; <i>α</i>) 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 <i>USp</i>(2<i>N</i>) theory with 2<i>N</i> + 2 fundamental hyper-multiplets and analytically prove that 𝒵<sub><i>USp</i>(2<i>N</i>)</sub>(<i>q</i>; <i>α</i>) satisfies an order-(<i>N</i> + 1) modular linear differential equation (MLDE) with vanishing Wronskian index, explaining how the parameter <i>α</i> 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<sub><i>USp</i>(2<i>N</i>)</sub>(<i>q</i>; <i>α, β</i>) of the generalised Schur partition function. Finally, we relate the <i>α</i> = −<i>k</i> specialisation to quantum monodromy traces Tr <i>M</i><sup><i>k</i></sup> and formulate a conjecture linking their <i>k</i>-dependence to MLDEs.</p>

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Generalised 4d partition functions and modular differential equations

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  • 发表日期：

    2026年6月19日

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  • A. Ramesh Chandra,
  • Sunil Mukhi,
  • Palash Singh

摘要

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.