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arXiv · 2608.15756

Bootstrapping bilinear relations of discrete Painlev\'e systems from 5d gauge theories

Minsung Kim·Xin Wang
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Abstract

We study tau-functions of elliptic and $ q $-Painlev\'e equations through their correspondence with 5d $ \mathcal{N}=1 $ supersymmetric field theories. Defining tau-functions associated with 5d field theories in terms of partition functions on the $ \Omega $-background, we propose a bootstrap method that determines their bilinear relations from the effective prepotential and the perturbative contributions, together with the global symmetry of the theory. Applying the method to 5d $ \mathrm{SU}(2) $ gauge theories coupled to $ N_f \leq 8 $ fundamental hypermultiplets, we obtain the bilinear relations of the elliptic and $ q $-Painlev\'e equations. This method can be applied to general 5d supersymmetric field theories beyond $ \mathrm{SU}(2) $ gauge theories.

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Minsung Kim, Xin Wang. 2026-08-16. Bootstrapping bilinear relations of discrete Painlev\'e systems from 5d gauge theories. https://arxiv.org/abs/2608.15756

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