Large matrices and Virasoro Conjecture

TL;DR

This paper reviews key aspects of Witten's conjecture proof, explores the KP hierarchy via the boson-fermion correspondence, and conjectures a link between nonlinear sigma models and planar graph theories to derive Virasoro constraints.

Contribution

It introduces a novel conjecture connecting nonlinear sigma models on Kähler manifolds with planar graph theories, aiding in deriving Virasoro constraints.

Findings

Analysis of the GL_infinity action on KP solutions

Conjecture of equivalence between sigma models and planar graph theories

Derivation of Virasoro constraints assuming the conjecture

Abstract

In this paper, we first review one of difficult parts of the proof of Witten's conjecture by Kontsevich that had not been emphasized before. In the derivation of the KdV equations, we review the boson-fermion correspondence method \cite{K} to show that the trajectory of $\rm GL_\infty$ action on 1 as an element of the ring $\mathbb{C}[x_1,x_2,...]$ yields the solutions of KP hierarchies. Then we consider the corresponding theory in which the target manifold is a K\"{a}hler manifold. We conjecture that this nonlinear sigma model is equivalent to a "planar graph" theory. Assuming the conjecture holds, we are able to get the Virasoro constraints in the Virasoro conjecture.