One-Point Functions of Loops and Constraints Equations of the Multi-Matrix Models at finite N

TL;DR

This paper derives explicit formulas for one-point functions in multi-matrix models at finite N using differential operators, revealing underlying algebraic structures and symmetry constraints.

Contribution

It introduces a method to compute one-point functions via differential operators derived from Schwinger-Dyson equations, uncovering $W_{1+ abla}$-like algebraic structures.

Findings

One-point functions expressed through differential operators.

Operators satisfy $W_{1+ abla}$-like algebra.

Partition function constraints reveal model symmetries.

Abstract

We derive one-point functions of the loop operators of Hermitian matrix-chain models at finite $N$ in terms of differential operators acting on the partition functions. The differential operators are completely determined by recursion relations from the Schwinger-Dyson equations. Interesting observation is that these generating operators of the one-point functions satisfy $W_{1+\infty}$-like algebra. Also, we obtain constraint equations on the partition functions in terms of the differential operators. These constraint equations on the partition functions define the symmetries of the matrix models at off-critical point before taking the double scaling limit.