Walls, Lines, and Spectral Dualities in 3d Gauge Theories

TL;DR

This work develops a unified framework linking 3d ${\mathcal N}=2$ gauge theories to periods on a spectral curve ${\mathcal V}$, where half-BPS defects and domain walls are encoded as monodromies of a twisted superpotential ${\widetilde{\mathcal W}}$ and as period integrals of $\lambda=\sum_i \log y_i\, d\log x_i$. It introduces a 3d–4d coupled setup with $SL(2,\mathbb{Z})$ transformation walls realized by 2d $(0,2)$ theories, and provides brane constructions that realize these defects and their dualities. A central proposal is a 3d spectral duality, relating a theory ${\mathcal T}_{\text{here}}$ to a basic Nekrasov–Shatashvili sector ${\mathcal T}_{\text{NS}}^{\text{magnon}}$ via $S$-duality, with the spectral curve acting as a bridge to integrable systems such as XXZ, sinh-Gordon, and Ruijsenaars–Schneider models. The paper also extends these ideas to knot theory through (super) $A$-polynomial curves, their periods, and the spectrum of domain walls and line operators, including the interpretation of incompressible surfaces as boundary data for line operators. Together, these results reveal deep connections between 3d gauge dynamics, integrable systems, brane realizations, and topological invariants of knots and 3-manifolds.

Abstract

In this paper we analyze various half-BPS defects in a general three dimensional N=2 supersymmetric gauge theory T. They correspond to closed paths in SUSY parameter space and their tension is computed by evaluating period integrals along these paths. In addition to such defects, we also study wall defects that interpolate between T and its SL(2,Z) transform by coupling the 3d theory to a 4d theory with S-duality wall. We propose a novel spectral duality between 3d gauge theories and integrable systems. This duality complements a similar duality discovered by Nekrasov and Shatashvili. As another application, for 3d N=2 theories associated with knots and 3-manifolds we compute periods of (super) A-polynomial curves and relate the results with the spectrum of domain walls and line operators.

Figures (19)

• Figure 1: Different types of paths on the parameter space of 3d ${\mathcal{N}}=2$ theory and its circle compactification. Note, that closed cycles that go around asymptotic regions of the curve ${\mathcal{V}}$ disappear in the 3d / tropical limit.
• Figure 2: The defect $L_\gamma$ separating two domains $D_1$ and $D_2$ of the effective 2d theory.
• Figure 3: Period corresponding to domain wall interpolating between vacuum $i$ and vacuum $j$ of the theory at a given value of the SUSY parameter $x$.
• Figure 4: In all the figures $\times_{q}$ stands for the fibration on $S^{1}$ with equivariant parameter $q$. Figure $(a)$ shows the space on which we define $Z_{{\cal T}}^{T^{3}}(q,k)$. The line labelled $W$ stands for the twist operator insertion $e^{2\pi ikJ}$. Unwrapping the torus produces a cylindrical slice with vacuum $|i\rangle$ on one side and vacuum $|j\rangle$ on the other. The path integral on this geometry is saturated by the BPS kink configuration. This is shown in figure $(b)$. Figure $(c)$ isolates the support of the defect and illustrates that it is a two-torus with nome $q$.
• Figure 5: The first figure shows the theory ${\cal T}^{(4d)}[\tau]$ on the left half-space and the dual theory ${\cal T}^{(4d)}[\varphi(\tau)]$ on the right half-space. In the second figure we have dualized the theory on the right back to ${\cal T}^{(4d)}[\tau]$ while introducing new degrees of freedom ${\cal B}_{\varphi}$ on the interface.