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

We study a two-matrix toy model with a BFSS-like interaction term using the collective field formalism. The main technical simplification is obtained by gauge-fixing first, and integrating out the off-diagonal elements, before changing to the collective field variable. We show that the resulting $(2+1)$-dimensional collective field action has novel features with respect to non-locality, and that we need to add a mass term to get a time-local potential. As is expected, one recovers the single matrix quantum mechanical collective field Hamiltonian in the proper limit.

💡 Research Summary

The paper investigates a two‑matrix quantum mechanical toy model that mimics the bosonic sector of the BFSS matrix model, using the collective‑field formalism. The authors’ main technical innovation is to first fix the U(N) gauge completely, diagonalize one of the matrices (called X), and then integrate out the off‑diagonal components of the second matrix (Y) before introducing the collective field. By working in axial gauge (A(t)=0) and parametrising X = Ω λ Ω† with λ diagonal, the gauge redundancy is reduced to the Vandermonde determinant Δ(λ)=∏_{i<j}(λ_i−λ_j). After diagonalisation, the Euclidean action for N=2 is written in terms of the diagonal entries λ₁, λ₂, ρ₁, ρ₂ and the off‑diagonal complex variable Y₁₂. The off‑diagonal mode behaves as a harmonic oscillator with frequency ω = λ₁−λ₂, i.e. the distance between the two D0‑branes. Performing the Gaussian path integral over Y₁₂ yields a functional determinant det(−∂_τ² + (λ₁−λ₂)²/4). Expressed as a trace of a logarithm, this determinant generates a non‑local term in time: the Green’s function of the operator −∂_τ² is G(τ,σ)=−½|τ−σ|, and expanding the log leads to contributions proportional to ∫dτdσ |τ−σ|(λ₁−λ₂)²(τ)(λ₁−λ₂)²(σ). Hence the effective action for the diagonal degrees of freedom is intrinsically non‑local in time, reflecting the long‑range interaction mediated by the integrated‑out strings.

To obtain a tractable, time‑local effective theory, the authors introduce a small mass term −m² Tr Y² into the original Lagrangian. This mass term lifts both the off‑diagonal and diagonal Y‑modes, breaking the original X↔Y symmetry but rendering the logarithmic term regular and suppressing the non‑local contributions. With the mass term the effective action becomes S_eff = ∫dτ