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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.02107 |
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| _version_ | 1866908958943346688 |
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| author | Chandra, A. Ramesh Mukhi, Sunil Singh, Palash |
| author_facet | Chandra, A. Ramesh Mukhi, Sunil Singh, Palash |
| contents | We prove the equivalence of a class of generalised Schur partition functions $\mathcal Z_G(q;α)$ 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 $\mathcal Z_{USp(2N)}(q;α)$ 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 $\mathcal Z_{USp(2N)}(q;α,β)$ of the generalised Schur partition function. Finally, we relate the $α=-k$ specialisation to quantum monodromy traces ${\rm Tr}\,M^k$ and formulate a conjecture linking their $k$-dependence to MLDEs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02107 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalised 4d Partition Functions and Modular Differential Equations Chandra, A. Ramesh Mukhi, Sunil Singh, Palash High Energy Physics - Theory Mathematical Physics We prove the equivalence of a class of generalised Schur partition functions $\mathcal Z_G(q;α)$ 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 $\mathcal Z_{USp(2N)}(q;α)$ 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 $\mathcal Z_{USp(2N)}(q;α,β)$ of the generalised Schur partition function. Finally, we relate the $α=-k$ specialisation to quantum monodromy traces ${\rm Tr}\,M^k$ and formulate a conjecture linking their $k$-dependence to MLDEs. |
| title | Generalised 4d Partition Functions and Modular Differential Equations |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2512.02107 |