Large\bf $ w_{1+\infty} $--type constraints in two--matrix and Kontsevich model--different approach

TL;DR

This paper employs $Q$-polynomials to derive finite-$N$ $w$-constraints in two-matrix and Kontsevich models, revealing their Lie algebra structure and relation to matrix potential powers.

Contribution

It introduces a new approach using $Q$-polynomials to derive and analyze $w$-constraints in matrix models at finite $N$, connecting them to Lie algebra structures.

Findings

Constraints form a Lie algebra structure.

Number of constraints limited by potential power.

Constraints reduce to known $W$-constraints in special cases.

Abstract

The technique of $Q$-polinomials are used to derive the $w$- constraints in the two-matrix and Kontsevich-like model at finite $N$. These constraints are closed and form Lie algebra. They are associated with the matrices, $\lambda ^n{\partial}_\lambda^m$ with $n,m\geq 0$. In the case of two-matrix model they can be reduced to the $W$-constraints of \cite{8}. For the case of Kontsevich-like model and two-matrix model with the finite polinomial potential, the number of constraints are limited by the power of the finite matrix potential i.e. the spin of $w$-s coincide with that power. This statement is the natural consequence of the form of constraints.