Collective field theory of gauged multi-matrix models: Integrating out off-diagonal strings

Technical Summary: Collective Field Theory of Gauged Multi-Matrix Models

Problem Statement
The paper addresses the challenge of generalizing the collective field formalism from single-matrix quantum mechanics to gauged multi-matrix models. While single-matrix models (e.g., in the large $N$ limit) successfully describe emergent spacetime geometries and are equivalent to two-dimensional string theory, extending this to systems with two or more $N \times N$ Hermitian matrices is analytically intractable. The complexity arises because gauge-invariant quantities in multi-matrix models involve traces over infinitely many combinatorial "words" of constituent matrices, leading to coupled Schwinger-Dyson equations that cannot be solved analytically. The authors aim to derive an analytical effective Hamiltonian for a toy two-matrix model to better understand the emergence of spatial dimensions in these gauge theories.

Methodology
The authors employ a specific approximation scheme for a gauged two-matrix model with $U(N)$ symmetry, featuring a BFSS-like interaction term. The core strategy involves a distinct ordering of operations compared to standard collective field approaches:

1. Gauge Fixing: The $U(N)$ gauge symmetry is fixed by setting the gauge field $A(t)=0$ and diagonalizing one of the matrices (denoted $X$). This reduces the system to diagonal elements ($\lambda_i$) and off-diagonal elements, while the second matrix ($Y$) remains in a gauge-fixed form with diagonal elements ($\rho_i$) and off-diagonal strings ($Y_{ij}$).
2. Integrating Out Off-Diagonal Strings: Before introducing collective field variables, the authors integrate out the off-diagonal elements of the second matrix ($Y_{ij}$) via a path integral. This step treats the off-diagonal strings as complex harmonic oscillators with frequencies dependent on the separation of the eigenvalues of $X$.
3. Mass Term Regularization: The authors find that integrating out massless off-diagonal strings results in a time-non-local effective action (due to the Green's function scaling as $|s-\tau|$). To obtain a time-local potential suitable for a tractable effective field theory (EFT), they introduce a small mass term ($m^2$) to the $Y$ matrix. This breaks the symmetry between $X$ and $Y$ but allows for the expansion and truncation of the non-local logarithmic potential.
4. Collective Field Transformation: The resulting effective action for the $2N$ diagonal degrees of freedom (where $N$ are associated with the fermionic-like $\lambda_i$ due to the Vandermonde determinant and $N$ with bosonic $\rho_i$) is transformed into a collective field theory. The discrete variables are mapped to a continuous collective field $\phi(x, y)$ representing the eigenvalue density in a $(2+1)$-dimensional spacetime.

Key Results

• Time-Local Effective Action: By adding a mass term to the $Y$ matrix, the authors successfully derive a time-local effective action. In the limit of large mass $m$, the non-local interaction terms are truncated to a leading-order local potential proportional to $(\lambda_i - \lambda_j)^2$.
• Emergent $(2+1)$-Dimensional Field Theory: The collective field formalism applied to the integrated-out system yields a $(2+1)$-dimensional collective field action. The spatial dimensions emerge from the eigenvalue densities of the two matrices ($\lambda$ and $\rho$).
• Fermionic-Bosonic Structure: The analysis reveals that the effective theory describes $N$ interacting fermions (arising from the diagonal elements of the diagonalized matrix $X$ and the Vandermonde determinant) and $N$ bosons (from the diagonal elements of $Y$). The off-diagonal strings, upon integration, mediate interactions between these fermions.
• Recovery of Single Matrix Limit: The derived collective field Hamiltonian correctly reduces to the well-known single-matrix quantum mechanical collective field Hamiltonian in the appropriate limit, validating the consistency of the approach.
• Cosmological Constant Artifact: The integration of off-diagonal modes with a mass term generates a constant additive term in the effective action, interpreted as a cosmological constant arising from the infrared cutoff provided by the mass. The authors note this is an artifact of the regularization method and drop it for the collective field analysis.

Significance and Claims
The paper claims to provide a novel analytical approach to multi-matrix models by prioritizing the integration of off-diagonal degrees of freedom before the introduction of collective variables. This method bypasses the intractability of solving coupled Wilson loop equations directly. The authors emphasize that their findings are consistent with exact "bootstrap" results for similar models, suggesting the validity of their approximation scheme.

The work demonstrates that a two-matrix quantum mechanical model naturally gives rise to a field theory in an ambient $(2+1)$-dimensional spacetime, offering a concrete mechanism for how spatial dimensions can emerge from gauge theories. The authors acknowledge that the introduction of the mass term breaks the direct correspondence to the full BFSS model and introduces non-locality in the massless limit, framing their results as a "toy model" exploration of the general method's viability rather than a complete solution to the BFSS model. Future work is proposed to address the covariant addition of mass terms and the generalization to more matrices.