Open intersection numbers, Kontsevich-Penner model and cut-and-join operators

TL;DR

This paper demonstrates how the Kontsevich--Penner model encodes open intersection numbers, deriving constraints and operators that fully specify the related tau-functions within the KP hierarchy.

Contribution

It establishes the connection between the Kontsevich--Penner matrix integral and open intersection numbers, deriving Virasoro and W-constraints and constructing cut-and-join operators.

Findings

Proves the matrix integral describes open intersection numbers.

Derives complete Virasoro and W-constraints for the model.

Constructs cut-and-join operators from the constraints.

Abstract

We continue our investigation of the Kontsevich--Penner model, which describes intersection theory on moduli spaces both for open and closed curves. In particular, we show how Buryak's residue formula, which connects two generating functions of intersection numbers, appears in the general context of matrix models and tau-functions. This allows us to prove that the Kontsevich--Penner matrix integral indeed describes open intersection numbers. For arbitrary $N$ we show that the string and dilaton equations completely specify the solution of the KP hierarchy. We derive a complete family of the Virasoro and W-constraints, and using these constraints, we construct the cut-and-join operators. The case $N=1$, corresponding to open intersection numbers, is particularly interesting: for this case we obtain two different families of the Virasoro constraints, so that the difference between them describes the dependence of the tau-function on even times.