On Three Dimensional Quiver Gauge Theories and Integrability

TL;DR

This work unifies multiple UV descriptions of the vacuum geometry for 3d N=4 quiver gauge theories, showing how mirror symmetry and circle compactification yield interrelated Bethe-like equations and algebraic Lagrangians. By translating 3d quiver data into Bethe equations for XXZ spin chains and connecting 4d N=4 on a segment to tRS-type integrable systems, the authors establish a robust bispectral duality web linking spin chains, many-body systems, and gauge-theory boundaries. They develop explicit dictionaries between mass/FI deformations, conjugate momenta, and Lagrangian intersections, and use brane constructions to illuminate the dualities and Higgsing processes. The resulting framework unifies (and generalizes) several known dualities, provides a fertile ground for exploring limits to XXX/ Gaudin models and CM/rational variants, and suggests promising directions toward elliptic systems and broader gauge-theory/integrable-system correspondences.

Abstract

In this work we compare different descriptions of the space of vacua of certain three dimensional N=4 superconformal field theories, compactified on a circle and mass-deformed to N=2 in a canonical way. The original N=4 theories are known to admit two distinct mirror descriptions as linear quiver gauge theories, and many more descriptions which involve the compactification on a segment of four-dimensional N=4 super Yang-Mills theory. Each description gives a distinct presentation of the moduli space of vacua. Our main result is to establish the precise dictionary between these presentations. We also study the relationship between this gauge theory problem and integrable systems. The space of vacua in the linear quiver gauge theory description is related by Nekrasov-Shatashvili duality to the eigenvalues of quantum integrable spin chain Hamiltonians. The space of vacua in the four-dimensional gauge theory description is related to the solution of certain integrable classical many-body problems. Thus we obtain numerous dualities between these integrable models.

Figures (13)

• Figure 1: $A_L$ linear quiver with labels $(N_1,M_1),\dots, (N_L,M_L)$.
• Figure 2: Brane realization of $A_4$ quiver with $(N_i,M_i)$ labels $(4,6),(3,4),(2,3),(2,2)$. Here and further on in the paper red ovals denote D5 branes, horizontal blue lines show D3 branes and vertical black lines designate NS5 branes.
• Figure 3: The transformation of a portion of a linear quiver induced by Higgsing along the D5 branes at positions $2$ and $3$.
• Figure 4: The sequence of Hanany-Witten moves on $T[U(5)]$, which Higgses the quiver gauge theory as in figure \ref{['fig:TSUNHWmoves1']}.
• Figure 5: Transformation of the $T[U(N)]$ quiver corresponding to the brane manipulations in Fig. \ref{['fig:TSUNHWmoves']}.