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

1. Problem Statement

The moduli space of 3-dimensional $\mathcal{N}=3$ supersymmetric gauge theories is significantly more complex than that of $\mathcal{N}=4$ theories.

• Complexity of Branches: While $\mathcal{N}=4$ moduli spaces typically consist of two distinct branches (Higgs and Coulomb) distinguished by the $SU(2)_H \times SU(2)_C$ R-symmetry, $\mathcal{N}=3$ moduli spaces are generally the intersection of multiple hyperKähler cones. These cones are often parametrized by dressed monopole operators rather than just mesonic operators.
• Lack of Symmetry Decomposition: The R-symmetry of $\mathcal{N}=3$ theories is only $SU(2)_R$. Unlike the $\mathcal{N}=4$ case, there is no clear decomposition of the R-symmetry group to distinguish operators belonging to different branches.
• Limitations of Existing Methods: Standard field-theoretic approaches using the superconformal index (SCI) rely on taking limits of fugacities to isolate specific branches. However, without the extra $U(1)_A$ axial symmetry present in $\mathcal{N}=4$ (or the specific $SU(2)$ decomposition), it is difficult to project out unwanted operators to isolate a single branch in $\mathcal{N}=3$ theories.
• Goal: The authors aim to develop a systematic, field-theoretic prescription to compute the Hilbert series (which counts gauge-invariant operators) for specific branches of the moduli space of 3d $\mathcal{N}=3$ quiver gauge theories, including those with unitary, orthosymplectic, and star-shaped gauge groups, and those arising from Type IIB brane configurations with generic $(p,q)$ fivebranes.

2. Methodology

The core innovation of the paper is a prescription for extracting the Hilbert series of specific branches by taking a specific limit of the superconformal index.

A. The Prescription

1. Introduction of Auxiliary Fugacities: The authors introduce an auxiliary axial fugacity, denoted as $a$ (or $z$ in the limit), which is not a symmetry of the full theory. However, on specific branches of the moduli space, the vacuum expectation values (VEVs) of operators that break this symmetry vanish, making $a$ an effective symmetry for that branch.
2. Charge Assignment: Fields are assigned specific axial charges ($+1, -1, 0$) based on the branch being studied. This assignment is derived from:
   • Brane intuition: Identifying the motion of D3-branes along specific types of fivebranes (D5, NS5, or $(1,k)$ fivebranes).
   • Field theory rules: Identifying minimal sequences of gauge nodes where Chern-Simons (CS) levels sum to zero.
3. The Limit: The superconformal index $I(x, a)$ is redefined using the substitution:
   $$x = zt, \quad a = z^{-1/2}t^{1/2}$$
   The Hilbert series $H(t)$ for a specific branch is obtained by taking the limit:
   $$H(t) = \lim_{z \to 0} I(x, a) \Big|_{(2.13)}$$
   • Mechanism: In this limit, operators not belonging to the target branch acquire positive powers of $z$ and are suppressed (projected out). Operators belonging to the branch have $z^0$ weight and survive.
   • Goodness Condition: The limit is well-defined provided the theory satisfies specific "goodness" conditions (e.g., $k + F \ge N$ for local nodes) ensuring no unitarity bound violations.

B. Branch Identification Rules

The authors provide rules to identify which dressed monopole operators form a branch:

• Minimal Sequences: A branch corresponds to a minimal sequence of consecutive gauge nodes whose CS levels sum to zero.
• Maximal Branches: A maximal branch is formed by the union of non-overlapping minimal sequences.
• Fugacity Assignment:
  • For a sequence of nodes forming a branch, bifundamentals within the sequence get charge $+1$ (or $-1$ depending on convention), while those connecting to outside nodes get $-1$ (or $+1$).
  • Adjoint chirals are assigned charges to ensure superpotential consistency or become massive.

3. Key Contributions and Results

A. Unitary Quivers (Linear and Circular)

The prescription is applied to various unitary quivers ($U(N)$ gauge groups):

• Linear Quivers: Analyzed theories like $U(N)_k \times U(N)_{-k} \times U(N)_k$. The authors successfully isolated the NS5-branch, $(1,k)$-branch, and D5-branch.
  • Result: The Hilbert series for the NS5-branch matches the Coulomb branch of a magnetic quiver (e.g., $U(1)$ or $U(2)$ with specific flavors).
  • Geometry: Identified geometries such as orbifolds $\mathbb{C}^2/\mathbb{Z}_n$.
• Circular Quivers: Applied to circular quivers with CS levels.
  • Result: Confirmed that the NS5 and $(1,k)$ branches correspond to the symmetric product of the Coulomb branch of the corresponding magnetic quiver (e.g., $\text{Sym}^N(\mathbb{C}^2/\mathbb{Z}_{k+F})$).
• Unequal Ranks: Successfully handled quivers with unequal ranks (e.g., $U(1) \times U(2) \times U(1)$), demonstrating the robustness of the method.

B. Orthosymplectic Quivers

The method was generalized to theories with $SO(N)$ and $USp(2N)$ gauge groups, which are crucial for Type IIB brane setups involving orientifolds.

• Global Forms: The authors carefully distinguished between different global forms of the gauge group (e.g., $SO(N)$ vs. $O(N)$ vs. $Spin(N)$) by tracking $\mathbb{Z}_2$ magnetic and charge conjugation symmetries ($\zeta$ and $\chi$).
• Ambiguity Resolution: In previous literature, magnetic quivers for certain orthosymplectic branches were ambiguous (e.g., whether the dual was $SO(2)$ SQCD or $USp(2)$ SQCD). This paper resolves these ambiguities by showing:
  • The original theory with $SO(2)$ nodes corresponds to $SO(2)$ SQCD.
  • The theory with $O(2)$ nodes (gauged charge conjugation) corresponds to $USp(2)$ SQCD.
• Specific Examples: Detailed computations for $USp(2) \times SO(2) \times USp(2)$ and $SO(2) \times USp(4) \times SO(2)$ quivers, providing explicit Hilbert series and identifying the correct magnetic quivers.

C. Generalized Brane Configurations ($T[U(N)]$ Blocks)

The authors extended the prescription to theories involving $T[U(N)]$ blocks, which arise from generic $(p,q)$ fivebranes in Type IIB string theory.

• Non-Lagrangian Blocks: These theories are often non-Lagrangian or quasi-Lagrangian. The authors demonstrated how to assign fugacities to the $T[U(N)]$ building blocks to isolate branches.
• Example: The $T_{(p,q)}[3]$ theory was analyzed, confirming that the D5-branch in these theories can be parametrized by dressed monopole operators, a feature not present in standard $\mathcal{N}=4$ D5-branches.

D. Geometric Branches of Affine Dynkin Quivers

The paper investigates the geometric branch of affine Dynkin quivers (e.g., affine $A_n$ and $D_n$).

• Definition: This branch corresponds to the motion of M2-branes probing singularities (e.g., $\mathbb{C}^2/\Gamma$).
• Method: A specific limit of the index was derived where all magnetic fluxes are equal ($m_i = m$).
• Results:
  • For circular Abelian quivers, the geometric branch was shown to be isomorphic to the hypertoric manifold $H_{n+1} // \ker(\beta)$.
  • For non-Abelian circular quivers (e.g., $U(N)$), the Hilbert series was found to be a deviation from the symmetric product of the Abelian geometric branches, involving Hall-Littlewood polynomials.
  • Affine $D_n$: The geometric branch was confirmed to be $\mathbb{C}^2/\tilde{D}_n \times \mathbb{C}^2/\mathbb{Z}_F$, matching expectations from M-theory.

4. Significance

• Unified Field-Theoretic Approach: The paper provides a powerful, purely field-theoretic tool to study moduli spaces of $\mathcal{N}=3$ theories without relying solely on brane constructions, which can be difficult to construct for complex or non-Lagrangian theories.
• Resolution of Duality Ambiguities: By precisely tracking global symmetries (charge conjugation and magnetic $\mathbb{Z}_2$), the authors clarify the relationship between different global forms of gauge groups and their dual magnetic quivers, resolving discrepancies in the literature.
• Extension to Non-Lagrangian Theories: The ability to handle $T[U(N)]$ blocks and generic $(p,q)$ fivebranes extends the reach of Hilbert series computations to a broader class of 3d SCFTs.
• Geometric Branch Prediction: The successful computation of geometric branches for affine quivers, including cases with Chern-Simons levels, offers new predictions for the geometry of M2-brane probes and validates the correspondence between field theory limits and geometric singularities.

In summary, the paper establishes a robust "limit prescription" for the superconformal index that acts as a projector onto specific branches of the moduli space in 3d $\mathcal{N}=3$ theories, successfully validating known results and providing new insights into the structure of orthosymplectic and geometric branches.