# Wall-crossing and operator ordering for 't Hooft operators in N=2 gauge theories

## TL;DR

This paper investigates the algebraic and wall-crossing properties of 't Hooft line operators in 4d N=2 gauge theories, proposing a Moyal product framework for their expectation values and confirming it through explicit Witten index calculations.

## Contribution

It introduces a novel approach linking operator ordering, wall-crossing, and Moyal products for 't Hooft operators in N=2 theories, supported by explicit computations.

## Key findings

- Witten indices can be derived from Moyal products of expectation values.
- Wall-crossing occurs when the operator ordering changes.
- Explicit calculations confirm the conjectured relations in all chambers.

## Abstract

We study half-BPS 't Hooft line operators in 4d $\mathcal{N}=2$ $U(N)$ gauge theories on $S^1\times \mathbb{R}^3$ with an $\Omega$-deformation. The recently proposed brane construction of 't Hooft operators shows that non-perturbative contributions to their correlator are identified with the Witten indices of quiver supersymmetric quantum mechanics. For the products of minimal 't Hooft operators, a chamber in the space of Fayet-Iliopoulos parameters in the quantum mechanics corresponds to an ordering of the operators inserted along a line. These considerations lead us to conjecture that the Witten indices can be read off from the Moyal products of the expectation values of the minimal 't Hooft operators, and also that wall-crossing occurs in the quantum mechanics only when the ordering of the operators changes. We confirm the conjectures by explicitly computing the Witten indices for the products of two and three minimal 't Hooft operators in all possible chambers.