Large matrices and Virasoro Conjecture

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 $\mathrm{G}{\mathrm{L}}_{\mathrm{\infty }}$ action on 1 as an element of the ring $\mathbb{C}[x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.}]$ 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.