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

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.

28 pages

References in corpus (7)

• Generation of Matrix Models by W-operators
• Intersection theory on moduli of disks, open KdV and Virasoro
• The partition function of the extended $r$-reduced Kadomtsev-Petviashvili hierarchy
• Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra
• Open intersection numbers, matrix models and MKP hierarchy
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• On a conjectural solution to open KdV and Virasoro

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