M5-branes, toric diagrams and gauge theory duality

TL;DR

This work uncovers a precise duality between 5D N=1 linear quiver gauge theories with gauge groups SU(N)^{M-1} and SU(M)^{N-1} compactified on S^1. The authors derive an explicit map between the two theories’ UV parameters by parallel analyses with Seiberg-Witten curves (via M-theory) and Nekrasov partition functions (via topological strings and toric geometry), showing that the two descriptions describe the same low-energy Coulomb dynamics. The duality is geometrically realized as a rotation of the M5-brane and a corresponding reflection on toric diagrams, and it persists in the refined Omega-background. Through the AGTW correspondence, the results imply q-deformed Liouville/Toda relations in 2D CFTs, including a q-deformed DOZZ structure for U(1) quivers, thereby linking 5D gauge dynamics to 2D conformal data and suggesting broad extensions to higher-rank dual pairs and 4D limits.

Abstract

In this article we explore the duality between the low energy effective theory of five-dimensional N=1 SU(N)^{M-1} and SU(M)^{N-1} linear quiver gauge theories compactified on S^1. The theories we study are the five-dimensional uplifts of four-dimensional superconformal linear quivers. We study this duality by comparing the Seiberg-Witten curves and the Nekrasov partition functions of the two dual theories. The Seiberg-Witten curves are obtained by minimizing the worldvolume of an M5-brane with nontrivial geometry. Nekrasov partition functions are computed using topological string theory. The result of our study is a map between the gauge theory parameters, i.e., Coulomb moduli, masses and UV coupling constants, of the two dual theories. Apart from the obvious physical interest, this duality also leads to compelling mathematical identities. Through the AGTW conjecture these five-dimentional gauge theories are related to q-deformed Liouville and Toda SCFTs in two-dimensions. The duality we study implies the relations between Liouville and Toda correlation functions through the map we derive.

Figures (18)

• Figure 1: The circle $SU(N)_{(i)}$ corresponds to the $i$-th gauge group, the segments between two circles are bifundamental hypermultiplets, and the flavor symmetries are illustrated by the two blue boxes $SU(N)_{(0)}$ and $SU(N)_{(M)}$ at the ends of the quiver.
• Figure 2: In the definition of the framing factor we have $m=\det \left(\vec{v}_{\text{in}} \cdot \vec{v}_{\text{out}}\right)$. We graphically clarify its definition and give two examples.
• Figure 3: Brane configuration for $SU(2)$ gauge theory.
• Figure 4: In this figure the configuration of the M5-brane that leads to 5D $SU(2)$ four flavor is depicted.
• Figure 5: Brane setup for $SU(N)^{M-1}$ gauge theory, with vertical lines being D4-branes and horizontal ones being NS5-branes. Without the loss of generality, we assume that $|\tilde{m}_1| \ge |\tilde{m}_2| \ge \cdots \ge |\tilde{m}_N|$ and $|\tilde{m}_{N+1}| \ge |\tilde{m}_{N+2}| \ge \cdots \ge |\tilde{m}_{2N}|$.