Supersymmetric gauge theory, (2,0) theory and twisted 5d Super-Yang-Mills

TL;DR

This work proposes that the nonperturbative dynamics of class S theories, obtained from 6d ${\rm N}=(2,0)$ compactifications on punctured Riemann surfaces, are captured by a twisted 5d SYM theory on ${\rm R}^{1,2}\times C$. By deriving generalized Hitchin equations and a spectral-curve framework, it connects holomorphic data to chiral operators at punctures via 3d ${\rm T}_\rho[G]$ theories, and identifies mesons and baryons with puncture data and Higgs/Coulomb moduli, respectively. A simple, nonperturbative superpotential vev formula is obtained, enabling checks against many ${\cal N}=1$ and ${\cal N}=2$ examples, including SQCD-like theories and $T_N$-based quivers. The analysis shows how spectral curves encode Seiberg–Witten-like couplings and domain-wall tensions, and clarifies the role of the Higgs branch and the adjoint scalar ${\sigma}$ in the twisted 5d SYM. Together, these results provide a unified, Lagrangian-like description of class ${\rm S}$ holomorphic dynamics and illuminate the physical meaning of Hitchin-type curves in this broader setting.

Abstract

Twisted compactification of the 6d N=(2,0) theories on a punctured Riemann surface give a large class of 4d N=1 and N=2 gauge theories, called class S. We argue that nonperturbative dynamics of class S theories are described by 5d maximal Super-Yang-Mills (SYM) twisted on the Riemann surface. In a sense, twisted 5d SYM might be regarded as a "Lagrangian" for class S theories on R^{1,2} times S^1. First, we show that twisted 5d SYM gives generalized Hitchin's equations which was proposed recently. Then, we discuss how to identify chiral operators with quantities in twisted 5d SYM. Mesons, or holomorphic moment maps, are identified with operators at punctures which are realized as 3d superconformal theories T_rho[G] coupled to twisted 5d SYM. "Baryons" are identified qualitatively through a study of 4d N=2 Higgs branches. We also derive a simple formula for dynamical superpotential vev which is relevant for BPS domain wall tensions. With these tools, we examine many examples of 4d N=1 theories with several phases such as confining, Higgs, and Coulomb phases, and show perfect agreements between field theories and twisted 5d SYM. Spectral curve is an essential tool to solve generalized Hitchin's equations, and our results clarify the physical information encoded in the curve.

Figures (4)

• Figure 1: (a):The brane realization of the $T[\mathop{\rm SU}(N)]$ theory in type IIB string theory for $N=3$. Horizontal lines are D3 branes, vertical solid lines are NS5 branes, and vertical dashed lines are D5 branes. The NS5 branes and the D5 branes are extended to different directions in ten dimensions, but we do not explicitly show that in the figure. The $\mathrm{U}(k)$ vector multiplets in \ref{['eq:TSUNquiver']} are coming from D3 branes suspended between adjacent NS5 branes. (b):A 4d quiver superconformal gauge theory in type IIA brane construction. Horizontal lines are D4 branes, vertical solid lines are NS5 branes, and vertical dashed lines are D6 branes. There are two maximal punctures realized at the two sets of $N$ D6 branes at the ends of D4 branes, and several other simple punctures. (c):D4 branes ending on D6 branes give a maximal puncture of figure (b). By $S^1$ compactification of the $x^3$ direction and taking a $T$-dual, we get D3 branes ending on D5 branes in type IIB string theory. The $S$-dual of it gives the $T[\mathop{\rm SU}(N)]$ theory at the end of the D3 branes, which is realized by D3 branes suspended between NS5 branes.
• Figure 2: Generalized quiver gauge theory. Trivalent vertices are copies of the $T_N$ theory, circles are ${\cal N}=2$$\mathop{\rm SU}(N)$ vector multiplets, and boxes are flavor $\mathop{\rm SU}(N)$ symmetries. In this example, there is $g=1$ loop, and there are $n=8$ flavor groups. The $g$ and $n$ correspond to the genus of the Riemann surface and the number of punctures, respectively, in the corresponding twisted 5d SYM.
• Figure 3: $S$-duality in ${\cal N}=2$ theory. Both the left and right figures are just a Riemann sphere with four punctures $A,B,C$ and $D$, but different degeneration limits give different field theory realizations.
• Figure 4: Decomposition of the Riemann surface into pieces. By gluing them, we can get a 4d field theory. Although each piece does not have a direct field theory interpretation, it is convenient to consider this decomposition to understand the ${\cal N}=1$ dualities.