Noncommutative Jacobi identity, and gauge theory

TL;DR

This work proves a noncommutative analogue of the Jacobi triple product identity within a four-dimensional gauge-theory framework, motivated by the BPS/CFT correspondence and gauge origami. It develops a noncommutative algebraic setting using an infinite matrix ${\bf Y}$ and a shift ${\bf S}$ to formulate ${\mathcal{Z}}({\bf Y})$ and its transpose, along with a noncommutative Hirota bilinear identity, and provides a boson-fermion correspondence in this setting. The paper then derives important applications, including a $W_{1+\infty}$-Jacobi identity for Toeplitz matrices and a Theta-transform of $q$-characters via $Y$-observables, culminating in a factorized transform ${\mathcal{D}}_i = {\mathcal{D}}_i^{+} y_i(x) {\mathcal{D}}_i^{-}$ with explicit infinite-product realizations. Finally, it offers a gauge-theory interpretation, linking to $SL_N$-opers on elliptic curves, Taub-Nut blowups, and instanton counting, and outlines future work to connect these identities to Lax operators, isomonodromic systems, and Seiberg-Witten geometry for quiver theories.

Abstract

We prove the noncommutative analogue of Jacobi triple product identity. As an application we organizing the q-characters of circular quiver gauge theories into an infinite product. We conjecture the gauge origami theory interpretation of the Jacobi identity.