5D SYM and 2D q-Deformed YM

TL;DR

The paper tests an AGT-like duality by deriving the 4D ${\cal N}=2$ gauge theory on ${S}^3\times{S}^1$ from a 5D ${\cal N}=2$ SYM on ${S}^3\times\Sigma$ and showing its partition function coincides with the 2D ${\cal q}$-deformed YM theory on $\Sigma$ via localization. The localization analysis shows the vector multiplet drives the nontrivial $q$-deformed measure, while the hypermultiplet contributes only a constant, yielding a finite-dimensional integral with a $\chi(\Sigma)$-powered $\sin$-measure that encodes the $q$-deformation. The deformation parameter is fixed as $q=\exp(-{g^2}/{2\pi l})$, consistent with the Douglas–Lambert relation between instantons in 5D and KK modes of the 6D theory, thereby connecting the 4D index to the 6D origin. Overall, the work substantiates the proposed AGT-like correspondence and clarifies how 5D localization reproduces 2D $q$YM, reinforcing the dimensional uplift from the 6D ${\cal N}=(2,0)$ framework. The results pave the way for exploring more general deformations (e.g., $(q,t)$-deformations) and extensions to higher-rank gauge groups.

Abstract

We study the AGT-like conjectured relation of a four-dimensional gauge theory on the direct product of a three-sphere and a circle to two-dimensional q-deformed Yang-Mills theory on a Riemann surface by using a five-dimensional N=2 supersymmetric Yang-Mills theory on the direct product of the three-sphere and the Riemann surface, following the conjectured relation of the six-dimensional N=(2,0) theory on the circle to the five-dimensional Yang-Mills theory. Our results are in perfect agreement with both of the conjectures.