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Limits of the Superconformal Index and the Moduli Space of 3d $\mathcal{N}=3$ Theories

Riccardo Comi, Sebastiano Garavaglia, William Harding, Noppadol Mekareeya

TL;DR

The paper introduces a systematic, index-based method to extract Hilbert series for the moduli space branches of 3d ${ m N}=3$ quiver gauge theories by taking carefully engineered limits of the superconformal index, aided by an auxiliary axial fugacity ${a}$. This prescription separates branches—such as mesonic (D5) and various monopole-driven (1,k) branches—through specific weightings and limiting procedures, and it connects these field-theoretic results to Type IIB brane setups and magnetic quivers. The authors apply the method to a wide class of theories, including unitary, orthosymplectic, star-shaped quivers, and affine Dynkin quivers, obtaining explicit Hilbert series and confirming them against known results while providing new predictions for unexplored cases. The work yields concrete Hilbert series for Higgs-like and Coulomb-like branches, clarifies the role of global forms (e.g., SO vs O), and offers a versatile toolkit for analyzing geometric branches of affine quivers, with potential impact on both mathematical and brane-engineered QFT models. Overall, this framework enhances our ability to count gauge-invariant operators on complex moduli spaces in 3d supersymmetric theories and links them to clear brane-picture interpretations and magnetic-quiver data.

Abstract

We compute the Hilbert series of three-dimensional $\mathcal{N}=3$ quiver gauge theories by taking a specific limit of the superconformal index. Our approach introduces auxiliary fugacities associated with symmetries which, while not present in the full theory, arise as effective symmetries on specific branches of the moduli space. By evaluating the index in a limit governed by these parameters, we successfully isolate the Hilbert series of the desired branches. We validate our results against the literature and provide several new extensions. We focus primarily on linear and circular quivers with unitary gauge groups, which originate from Type IIB brane configurations involving generic $(p,q)$ fivebranes. We further generalise this approach to star-shaped and orthosymplectic $\mathcal{N}=3$ quivers. Finally, we investigate the geometric branches of affine Dynkin quivers, demonstrating agreement with known results, while offering new predictions for unexplored cases.

Limits of the Superconformal Index and the Moduli Space of 3d $\mathcal{N}=3$ Theories

TL;DR

The paper introduces a systematic, index-based method to extract Hilbert series for the moduli space branches of 3d quiver gauge theories by taking carefully engineered limits of the superconformal index, aided by an auxiliary axial fugacity . This prescription separates branches—such as mesonic (D5) and various monopole-driven (1,k) branches—through specific weightings and limiting procedures, and it connects these field-theoretic results to Type IIB brane setups and magnetic quivers. The authors apply the method to a wide class of theories, including unitary, orthosymplectic, star-shaped quivers, and affine Dynkin quivers, obtaining explicit Hilbert series and confirming them against known results while providing new predictions for unexplored cases. The work yields concrete Hilbert series for Higgs-like and Coulomb-like branches, clarifies the role of global forms (e.g., SO vs O), and offers a versatile toolkit for analyzing geometric branches of affine quivers, with potential impact on both mathematical and brane-engineered QFT models. Overall, this framework enhances our ability to count gauge-invariant operators on complex moduli spaces in 3d supersymmetric theories and links them to clear brane-picture interpretations and magnetic-quiver data.

Abstract

We compute the Hilbert series of three-dimensional quiver gauge theories by taking a specific limit of the superconformal index. Our approach introduces auxiliary fugacities associated with symmetries which, while not present in the full theory, arise as effective symmetries on specific branches of the moduli space. By evaluating the index in a limit governed by these parameters, we successfully isolate the Hilbert series of the desired branches. We validate our results against the literature and provide several new extensions. We focus primarily on linear and circular quivers with unitary gauge groups, which originate from Type IIB brane configurations involving generic fivebranes. We further generalise this approach to star-shaped and orthosymplectic quivers. Finally, we investigate the geometric branches of affine Dynkin quivers, demonstrating agreement with known results, while offering new predictions for unexplored cases.
Paper Structure (48 sections, 303 equations)

This paper contains 48 sections, 303 equations.

Table of Contents

  1. Introduction
  2. The prescription
  3. Moduli space of 3d $\mathcal{N}=4$ theories from the index
  4. Higgs branch limit:
  5. Coulomb branch limit:
  6. The prescription for 3d $\mathcal{N}=3$ theories
  7. Mesonic branch or the D5-branch.
  8. Monopole branches or the $(1,k)$-branch.
  9. Comments on the relation with Type IIB brane setups
  10. Unitary quivers
  11. Linear quiver with $\mathop{\mathrm{U}}\nolimits(N)_k \times \mathop{\mathrm{U}}\nolimits(N)_{-k} \times \mathop{\mathrm{U}}\nolimits(N)_k$ gauge group
  12. Abelian case: $N=1$
  13. The case of $N=2$
  14. The $T_{(p,q)}[3]$ theory and its generalisation
  15. The Abelian case
  16. ...and 33 more sections