Coulomb branch Hilbert series and Three Dimensional Sicilian Theories

TL;DR

This work develops a comprehensive framework to compute the Coulomb branch Hilbert series of mirrors to three-dimensional Sicilian theories by gluing Hall-Littlewood building blocks T_{\boldsymbol{\rho}}(G). The authors formulate and apply the monopole and Hall-Littlewood formulas, including background fluxes and gluing, to produce explicit series for A- and D-type Sicilian theories across genera. They verify genus-0 results against four-dimensional Higgs branch Hall-Littlewood indices and extend to higher genus, providing nontrivial predictions for non-Lagrangian cases and Lagrangian checks in special instances (e.g., A1, T_N, and tri-vertex theories). The results clarify why Hall-Littlewood polynomials appear in both 3d Coulomb-branch Hilbert series and 4d Schur/Hilbert-series limits, and demonstrate the power of a gluing approach to assemble complex moduli spaces from simple, well-understood building blocks. Overall, the paper offers a robust, scalable method to access Higgs/Coulomb branch data of non-Lagrangian Sicilian theories via 3d mirrors, with broad implications for dualities and moduli-space geometry.

Abstract

We evaluate the Coulomb branch Hilbert series of mirrors of three dimensional Sicilian theories, which arise from compactifying the $6d$ $(2,0)$ theory with symmetry $G$ on a circle times a Riemann surface with punctures. We obtain our result by gluing together the Hilbert series for building blocks $T_{\mathbfρ}(G)$, where $\mathbfρ$ is a certain partition related to the dual group of $G$, which we evaluated in a previous paper. The result is expressed in terms of a class of symmetric functions, the Hall-Littlewood polynomials. As expected from mirror symmetry, our results agree at genus zero with the superconformal index prediction for the Higgs branch Hilbert series of the Sicilian theories and extend it to higher genus. In the $A_1$ case at genus zero, we also evaluate the Coulomb branch Hilbert series of the Sicilian theory itself, showing that it only depends on the number of external legs.

Figures (7)

• Figure 1: Gluing $T_{\bm{\rho}_1}(G), \ldots, T_{\bm{\rho}_n}(G)$ via the common centerless flavor symmetry $G/Z(G)$. This is a mirror theory of the theory on M5-brane compactified on a circle times a Riemann sphere with punctures $\bm{\rho}_1, \ldots, \bm{\rho}_n$.
• Figure 2: The mirror theory of a tri-vertex theory with genus $g$ and $e$ external legs.
• Figure 3: Quiver diagram for the mirror of $T_N$. Each node represents a unitary group of the labelled rank and the overall $U(1)$ is modded out.
• Figure 4: The moduli spaces of $k$$E_6$, $E_7$ and $E_8$ instantons on $\mathbb{C}^2$ can be realized using the Coulomb branch of quiver diagrams (a), (b) and (c) respectively. Each node represents a unitary group of the labelled rank and the overall $U(1)$ is modded out in each diagram.
• Figure 5: Quiver for the mirror of the $A_2$ theory on a circle times a torus with one maximal puncture. The overall $U(1)$ is factored out.