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

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 Hallnäs, Langmann, Noumi and Rosengren \cite{Halln_s_2022}. Since the $K$-theoretic vortex partition functions are related with indices such as the $χ_t$-genus of the handsaw quiver variety, we discuss geometric interpretation of Euler transformations in terms of wall-crossing formulas of handsaw quiver variety.

Figures (2)

• Figure 1: The quiver diagram representing 1d $\mathcal{N}=2$ SQM associated with the ADHM-like description of the $k$-vortex moduli for the 3d $\mathcal{N}=2$ gauge theory in Table \ref{['table:charge1']}. The circle with '$k$' denotes the 1d $\mathcal{N}=2$$U(k)$ vector multiplet. An arrow with solid line denotes a 1d $\mathcal{N}=2$ chiral multiplet. The dashed line denotes the 1d $\mathcal{N}=2$ fermi multiplet. '$I$' and '$J$' denote the scalars in each chiral multiplet. '$\psi$' denotes the fermion in the fermi multiplet. The representation of the multiplets is summarized in Table \ref{['table:HScharge']}
• Figure 2: The quiver diagram representing the 1d $\mathcal{N}=4$ SQM associated with the $k$-vortex moduli for the 3d $\mathcal{N}=4$ gauge theory. The circle with '$k$' denotes the 1d $\mathcal{N}=4$$U(k)$ vector multiplet. An arrow with solid line denotes a 1d $\mathcal{N}=4$ chiral multiplet. '$B$', '$I$' and '$J$' denote the scalars in each 1d $\mathcal{N}=4$ chiral multiplet. The representation of the 1d multiplets are the same as the 1d $\mathcal{N}=2$ case and summarized in Table \ref{['table:chargeN4']}