To gauge or to double gauge? Matrix models, global symmetry, and black hole cohomologies

Title: To gauge or to double gauge? Matrix models, global symmetry, and black hole cohomologies

Problem Statement
The paper addresses the complexity of analyzing the Hilbert space structure and dynamics of matrix models, which are proposed as non-perturbative definitions of quantum gravity (e.g., the BFSS and BMN models). While these models are rich in degrees of freedom, their full analysis is often intractable. Previous work has shown that gauging a symmetry simplifies the dynamics by projecting onto singlet sectors, but the structure of the Hilbert space in these sectors remains complex due to trace relations and the interplay between gauge and global symmetries. Specifically, the authors aim to understand the "double-gauged" scenario: matrix models where a global symmetry is also gauged (projected onto singlets), effectively treating the global symmetry as a gauge symmetry. This is contrasted with the "To gauge or not to gauge?" debate, where it was observed that at strong coupling, non-singlets decouple. The authors investigate whether this double projection further simplifies the Hilbert space structure and the spectrum of operators, particularly in the context of identifying "non-graviton" operators related to supersymmetric black hole microstates in $AdS_5$.

Methodology
The authors employ a combination of invariant theory, partition function analysis, and superconformal index techniques:

1. Molien-Weyl Formula: They utilize the Molien-Weyl formula to compute partition functions for bosonic matrix models. This involves integrating the extended partition function over the gauge group (to enforce gauge invariance) and subsequently over the global symmetry group (to enforce the singlet projection).
2. Hironaka Decomposition: The resulting partition functions are analyzed using Hironaka decomposition to identify the ring of invariants. This allows the authors to distinguish between primary invariants (algebraically independent generators) and secondary invariants (quadratically reducible generators), accounting for trace relations.
3. Gauge Fixing and Singular Value Decomposition (SVD): To interpret the partition functions, the authors perform explicit gauge fixing. They map the multi-matrix configurations to simpler forms using SVD, reducing the degrees of freedom to a smaller set of effective matrices (e.g., a single real symmetric matrix).
4. BMN Sector Analysis: For the supersymmetric case, they focus on the 1/16 BPS subsector of $\mathcal{N}=4$ $SU(N)$ Super Yang-Mills (SYM) theory, specifically the BMN sector. They compute the superconformal index restricted to $SU(3)_R$ singlets.
5. Graviton vs. Non-Graviton Separation: They explicitly construct the space of "graviton" operators (chiral primaries and their descendants) and subtract their contribution from the full index to isolate the "non-graviton" index, which is conjectured to contain black hole microstates.

Key Contributions and Results

1. Bosonic Matrix Models ($U(2)$ and $SU(2)$ with $SO(d)$ global symmetry):

• Reduction to Single-Matrix Models: The authors demonstrate that the $SO(d)$ singlet sector of $SU(2)$ matrix models with $d \ge 4$ is effectively described by an $SO(3)$ gauged matrix model of a single real symmetric $3 \times 3$ matrix $B$. For $d=3$, an additional secondary invariant (squared determinant) appears.
• $U(2)$ Models: For $U(2)$ models with $d \ge 5$, the singlet sector reduces to an $SO(3)$ model involving a symmetric matrix, an antisymmetric matrix (or vector), and a scalar. For $d=4$, an additional secondary invariant appears.
• Simplification of Dynamics: This reduction implies that the Hamiltonian in the singlet sector simplifies significantly. For instance, the bosonic part of the BFSS model in the $SO(d)$ singlet sector reduces to a particle moving in a three-dimensional potential, $H = \frac{1}{2}(p_x^2 + p_y^2 + p_z^2) + \frac{1}{2}(x^2y^2 + y^2z^2 + z^2x^2)$.
• Kinematic Validity: The authors note that while partition functions were computed for free models, the identification of invariants relies on kinematic trace relations, suggesting these structural simplifications extend to interacting models.

2. BMN Sector of $\mathcal{N}=4$ SYM ($SU(N)$ with $SU(3)_R$ global symmetry):

• Singlet Projection: The authors project the 1/16 BPS sector onto $SU(3)_R$ singlets. They find that this projection drastically simplifies the spectrum of graviton operators.
• Absence of Singlet Gravitons ($N=2$): For $SU(2)$ gauge theory, they prove that there are no $SU(3)_R$ singlet graviton operators. The singlet graviton index is trivial ($1$). Consequently, the singlet projection of the full index isolates the non-graviton (black hole) cohomologies directly.
• Simplified Graviton Sector ($N=3$): For $SU(3)$ gauge theory, the singlet graviton sector is extremely simple, consisting of only two non-trivial operators (at orders $t^{18}$ and $t^{24}$). This allows for a closed-form expression of the singlet graviton index.
• Closed-Form Non-Graviton Indices: By subtracting the simplified graviton index from the full BMN index, the authors derive closed-form expressions for the non-graviton index for $N=2$ and $N=3$. These expressions reveal structural features of the spectrum, such as infinite towers of cohomologies generated by operators like $\text{tr}(f^2)$ and $\text{tr}(f^3)$.
• $N=4$ Case: For $SU(4)$, the authors identify new singlet graviton operators (including a purely bosonic one at $t^{12}$) that were absent in lower $N$ cases due to trace relations. While a full closed form is not provided, they provide the series expansion and identify the new graviton contributions.

Significance and Claims
The paper claims that "double gauging" (projecting onto global symmetry singlets) is a powerful tool for simplifying the analysis of matrix models and supersymmetric gauge theories.

• Structural Simplification: The primary claim is that the Hilbert space structure simplifies in the singlet sector, often reducing complex multi-matrix systems to effective single-matrix models.
• Black Hole Microstates: In the context of the BMN sector, the authors argue that the scarcity of $SU(3)_R$ singlet gravitons (especially for $N=2, 3$) makes the singlet sector an ideal setting to study non-graviton operators. These operators are expected to correspond to the microstates of supersymmetric $AdS_5$ black holes.
• Closed Forms: The work provides the first closed-form expressions for the non-graviton index in the $SU(3)_R$ singlet sector for $N=2$ and $N=3$, offering a tractable alternative to the numerical Gröbner basis methods previously required for finite $N$.
• Future Directions: The authors suggest that these simplifications could facilitate numerical studies and quantum simulation of holographic protocols (e.g., black hole formation and evaporation) on near-term quantum computers, as the reduced Hamiltonians are more amenable to simulation than full matrix models. They also note that the status of the one-loop exactness conjecture for $SU(N)$ remains unclear, and the singlet sector provides a simpler setting to investigate potential lifting of states due to quantum corrections.