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arXiv · 2401.17198

Euler transformation for multiple $q$-hypergeometric series from wall-crossing formula of $K$-theoretic vortex partition function

Yutaka Yoshida
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Abstract

We show that transformation formulas of multiple $q$-hypergeometric series agree with wall-crossing formulas of $K$-theoretic vortex partition functions obtained by Hwang, Yi and the author \cite{Hwang:2017kmk}. For the vortex partition function in 3d $\mathcal{N}=2$ gauge theory, we show that the wall-crossing formula agrees with the Kajihara transformation \cite{kajihara2004euler}. For the vortex partition function in 3d $\mathcal{N}=4$ gauge theory, we show that the wall-crossing formula agrees with the transformation formula by Halln\"as, Langmann, Noumi and Rosengren \cite{Halln_s_2022}. Since the $K$-theoretic vortex partition functions are related with indices such as the $\chi_t$-genus of the handsaw quiver variety, we discuss geometric interpretation of Euler transformations in terms of wall-crossing formulas of handsaw quiver variety.

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Yutaka Yoshida. 2024-01-30. Euler transformation for multiple $q$-hypergeometric series from wall-crossing formula of $K$-theoretic vortex partition function. https://arxiv.org/abs/2401.17198

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