On 't Hooft Defects, Monopole Bubbling and Supersymmetric Quantum Mechanics

TL;DR

This paper analyzes the expectation values of 't Hooft operators in supersymmetric gauge theory, revealing their relation to moduli spaces, supersymmetric quantum mechanics, and brane constructions, with explicit computations and confirmations.

Contribution

It provides a new understanding of monopole bubbling contributions as equivariant integrals and Witten indices, and details the quiver quantum mechanics and brane setups for all N.

Findings

Monopole bubbling contributions are described as equivariant integrals over Kronheimer-Nakajima moduli spaces.

The contributions can be computed as Witten indices of supersymmetric quiver quantum mechanics.

Explicit calculations for SU(2) confirm agreement with previous formulas.

Abstract

We revisit the localization computation of the expectation values of 't Hooft operators in $\mathcal{N}=2^*$ SU(N) theory on $\mathbb{R}^3 \times S^1$. We show that the part of the answer arising from "monopole bubbling" on $\mathbb{R}^3$ can be understood as an equivariant integral over a Kronheimer-Nakajima moduli space of instantons on an orbifold of $\mathbb{C}^2$. It can also be described as a Witten index of a certain supersymmetric quiver quantum mechanics with $\mathcal{N}=(4,4)$ supersymmetry. The map between the defect data and the quiver quantum mechanics is worked out for all values of N. For the SU(2) theory, we compute several examples of these line defect expectation values using the Witten index formula and confirm that the expressions agree with the formula derived by Okuda, Ito and Taki. In addition, we present a Type IIB construction -- involving D1-D3-NS5-branes -- for monopole bubbling in $\mathcal{N}=2^*$ SU(N) SYM and demonstrate how the quiver quantum mechanics arises in this brane picture.