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Generation of Matrix Models by W-operators

A. Morozov, Sh. Shakirov

TL;DR

The paper develops a unifying scheme to generate matrix-model partition functions by exponentiating $W$-operators from the $\hat{W}^{(3)}$ algebra and acting on simple generating functions. It provides explicit realizations for the external-field Hermitian model via $\hat{W}_2$, the Gaussian Hermitian model via $\hat{W}_{-2}$, and the Hurwitz–Kontsevich model via $\hat{W}_0$, and it derives both matrix-integral and character-expansion representations. Through Faddeev–Popov tricks and Virasoro constraints, it links these $W$-operators to traditional matrix-model techniques, including Miwa transformations and Calogero-type structures. The results offer a powerful algebraic route to construct and analyze matrix models, with potential connections to DV theories and generalized Kontsevich–type integrals.

Abstract

We show that partition functions of various matrix models can be obtained by acting on elementary functions with exponents of W-operators. A number of illustrations is given, including the Gaussian Hermitian matrix model, Hermitian model in external field and the Hurwitz-Kontsevitch model, for which we suggest an elegant matrix-model representation. In all these examples, the relevant W-operators belong to the W_3-algebra.

Generation of Matrix Models by W-operators

TL;DR

The paper develops a unifying scheme to generate matrix-model partition functions by exponentiating -operators from the algebra and acting on simple generating functions. It provides explicit realizations for the external-field Hermitian model via , the Gaussian Hermitian model via , and the Hurwitz–Kontsevich model via , and it derives both matrix-integral and character-expansion representations. Through Faddeev–Popov tricks and Virasoro constraints, it links these -operators to traditional matrix-model techniques, including Miwa transformations and Calogero-type structures. The results offer a powerful algebraic route to construct and analyze matrix models, with potential connections to DV theories and generalized Kontsevich–type integrals.

Abstract

We show that partition functions of various matrix models can be obtained by acting on elementary functions with exponents of W-operators. A number of illustrations is given, including the Gaussian Hermitian matrix model, Hermitian model in external field and the Hurwitz-Kontsevitch model, for which we suggest an elegant matrix-model representation. In all these examples, the relevant W-operators belong to the W_3-algebra.

Paper Structure

This paper contains 14 sections, 143 equations.

Table of Contents

  1. Introduction
  2. Hermitian Matrix Model in external field and operator $\hat{W}_{2}$
  3. Gaussian Hermitian Model and operator $\hat{W}_{-2}$
  4. The main relation, eq.(\ref{['Gauss']})
  5. Eq.(\ref{['Gauss']}) from Faddeev-Popov trick
  6. Eq.(\ref{['Gauss']}) from Virasoro constraints
  7. Non-Gaussian models and operators $\hat{W}_{-p}$
  8. Hurwitz-Kontsevich Model and operator $\hat{W}_{0}$
  9. Approximate matrix integral
  10. Corrections
  11. Exact matrix integral
  12. Character expansion
  13. Appendix: $GL(n)$ characters as eigenfunctions of ${\hat{W}}_0$
  14. Conclusion