Mirrors of 3d Sicilian theories

TL;DR

This work shows that 3d $ obreak ext{N}=4$ mirrors of 4d Sicilian theories—arising from 6d $ obreak ext{N}=(2,0)$ compactifications on punctured Riemann surfaces—are always Lagrangian star-shaped quivers, with a central $ ext{SU}(N)$ (or $ ext{SO}(2N)$) node and tails corresponding to punctures. It develops two complementary routes: a brane-web derivation that identifies the $T_N$ mirror and extends to general Sicilian theories, and a field-theoretic construction via $ obreak ext{N}=4$ SYM on a graph, whose S-duality acts component-wise to implement the mirror map. The analysis extends to $D_N$-type Sicilian theories, where positive/negative punctures and twist lines produce different central diag groups and adjoint/fundamental matter in the mirror, consistently matching Hitchin moduli considerations. Together, these results illuminate how 3d IR fixed points encode the topology of the defining Riemann surface and demonstrate the robustness of mirror symmetry under diverse constructions and boundary conditions.

Abstract

We consider the compactification of the 6d N=(2,0) theories, or equivalently of M-theory 5-branes, on a punctured Riemann surface times a circle. This gives rise to what we call 3d N=4 Sicilian theories, and we find that their mirror theories are star-shaped quiver gauge theories. We also discuss an alternative construction of these 3d theories through 4d N=4 SYM on a graph, which allows us to obtain the 3d mirror via 4d S-duality.

Figures (20)

• Figure 1: The mirror of the 3d $T_N$ theory. A circle with $n$ inside stands for a $\mathrm{U}(n)$ gauge group, a doubled circle an $\mathrm{SU}(n)$, and a line between two circles signifies a bifundamental hypermultiplet transforming under the two gauge groups connected by the line.
• Figure 2: Relations of various constructions we will use in the paper. The horizontal arrow designates Kaluza-Klein reduction on $S^1$. The solid vertical arrow is a reduction on an $S^1$ inside the Riemann surface. The dotted vertical arrow signifies taking the long-distance limit compared to the size of the graph.
• Figure 3: Left: Table of directions spanned by the objects forming the web. Right: The web of $N$ D5-branes, NS5-branes and $(1,1)$ 5-branes; here $N=3$. In the figure the D5's are semi-infinite, while NS5's and $(1,1)$ 5-branes terminate each on a 7-brane $\otimes$ of the same type. The Coulomb branch of the 5d low energy theory is not sensitive to this difference.
• Figure 4: a) Quiver diagram resulting from a configuration of D3-branes suspended between NS5-branes. b) Quiver diagram of the $T[\mathrm{SU}(N)]$ theory. Circles are $\mathrm{U}(r_a)$ gauge groups, the square is an $\mathrm{SU}(N)$ global symmetry group and lines are bifundamental hypermultiplets.
• Figure 5: Left: $(p,q)$-web realizing the $T_N$ theory, with aligned 7-branes. Right: Quiver diagram of the mirror of $T_N$, with gauge groups $\mathrm{U}(r)$. The group at the center is taken to be $\mathrm{SU}$, to remove the decoupled overall $\mathrm{U}(1)$. Here $N = 3$.