5D partition functions, q-Virasoro systems and integrable spin-chains

TL;DR

The work develops a unifying framework connecting 5d ${\cal N}=1$ gauge theories on $S^5$ and $S^4\times S^1$ to 3d theories via degeneration of partition functions, structured by 5d holomorphic blocks that glue according to $SL(3,\mathbb{Z})$-like data. It embeds these blocks into a $q$-Virasoro/CFT picture, with two correlator pairings (${\cal S}$ and ${\rm id}$) that reproduce the $S^5$ and $S^4\times S^1$ results and explain degenerations to 3d partitions as residues at pinched poles. The paper then defines reflection coefficients from $q$-deformed 3-point functions and recasts them in terms of Harish-Chandra $c$-functions, showing that affinization recovers exact non-perturbative expressions and ties to integrable-system S-matrices, especially the XXZ/XYZ spin-chain limits. This establishes a deep correspondence between higher-dimensional gauge theory partition functions, $q$-Virasoro CFT, and 2d integrable systems, with implications for non-perturbative structure, dualities, and exact S-matrix data. The results offer a concrete route to explore non-perturbative completions via affinization and to leverage spin-chain intuition for understanding 5d/3d correspondences.

Abstract

We analyze N = 1 theories on S5 and S4 x S1, showing how their partition functions can be written in terms of a set of fundamental 5d holomorphic blocks. We demonstrate that, when the 5d mass parameters are analytically continued to suitable values, the S5 and S4 x S1 partition functions degenerate to those for S3 and S2 x S1. We explain this mechanism via the recently proposed correspondence between 5d partition functions and correlators with underlying q-Virasoro symmetry. From the q-Virasoro 3-point functions, we axiomatically derive a set of associated reflection coefficients, and show they can be geometrically interpreted in terms of Harish-Chandra c-functions for quantum symmetric spaces. We then link these particular c-functions to the types appearing in the Jost functions encoding the asymptotics of the scattering in integrable spin chains, obtained taking different limits of the XYZ model to XXZ-type.

Figures (2)

• Figure 1: Integration path for the fusion of two non-degenerate states.
• Figure 2: Pinching of the integration contour as the 3-point functions are continued to the degenerate values.