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.
Highlights
Closed intersection numbers, follows from the matrix integral representation (1.1)
We continue our investigation of the Kontsevich-Penner model, which describes intersection theory on moduli spaces both for open and closed curves
For arbitrary N we show that the string and dilaton equations completely specify the solution of the KP hierarchy
Summary
We prove that the tau-function τ1 coincides with the conjectural generating function of open intersection numbers of [12]. We show that the residue formula for the generation function, proved in [12], follows from the determinant expression (2.12) for the tau-functions of the KP hierarchy. This type of relations appears to be universal for tau-functions. We proved the following statement: for any tau-function τ and arbitrary series Φ1(z) = 1+O(z−1) the residue (3.9) gives a tau-function of the KP hierarchy, corresponding to the point of the Sato Grassmannian.
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