Exact Partition Functions for Gauge Theories on $\mathbb{R}^3_\lambda$

TL;DR

This paper derives exact partition functions for a family of gauge theories on a noncommutative space, revealing their factorization into simpler components and connections to integrable systems.

Contribution

It provides explicit formulas for partition functions of gauge theories on $ three_l$ and links them to integrable hierarchies, advancing understanding of noncommutative gauge models.

Findings

Partition functions factorize into products over fuzzy spheres

Exact determinant expressions for certain sub-family partition functions

Connection to integrable 2-D Toda lattice hierarchy

Abstract

The noncommutative space $\mathbb{R}^3_\lambda$, a deformation of $\mathbb{R}^3$, supports a $3$-parameter family of gauge theory models with gauge-invariant harmonic term, stable vacuum and which are perturbatively finite to all orders. Properties of this family are discussed. The partition function factorizes as an infinite product of reduced partition functions, each one corresponding to the reduced gauge theory on one of the fuzzy spheres entering the decomposition of $\mathbb{R}^3_\lambda$. For a particular sub-family of gauge theories, each reduced partition function is exactly expressible as a ratio of determinants. A relation with integrable 2-D Toda lattice hierarchy is indicated.