A Four-dimensional Gauge Theory Perspective on Quantum K-theory

TL;DR

This paper develops a four-dimensional gauge theory framework to understand quantum K-theory, extending previous 2D and 3D models, and demonstrates how 4D invariants relate to quantum K-theoretic invariants.

Contribution

It introduces a 4D gauge theory model that generalizes lower-dimensional setups and connects 4D invariants to quantum K-theory.

Findings

Computed the 4D partition function on D^2×T^2.

Showed the partition function satisfies a difference equation.

Indicated 4D invariants reduce to 3D quantum K-theory invariants.

Abstract

The two-dimensional gauged linear sigma model has provided a physical model for the quantum cohomology of a K\"ahler manifold, $X$. A three-dimensional version of such construction has recently been shown to shed light on models of quantum K-theory of $X$. We consider an $\mathcal{N}=1$ four-dimensional version consisting of a $U(1)$ vector multiplet and chiral multiplets, generalizing the two-dimensional $\mathcal{N}=(2,2)$ setup. We compute the four-dimensional partition function on $D^2\times \mathbb{T}^2$ and demonstrates that it satisfies a difference equation which reduces to the deformed quantum K-theoretic one in the appropriate limit. We also demonstrate, though indirectly, that 4d invariants reduce to 3d quantum K-theory invariants in the same limit.