The identification of the extended refined open partition function and the Kontsevich-Penner matrix model

TL;DR

This work establishes a deep link between open intersection theory and Kontsevich-Penner matrix models by constructing the extended refined open partition function and proving its identification with the Kontsevich-Penner model under the Miwa parametrization $s_i=2^i i!\operatorname{tr}\Lambda^{-2i-2}$ for all $N\ge 1$. Using the Harish-Chandra–Itzykson–Zuber formula and the Weierstrass transform, the authors transform $\mathcal Z^o$ into a form revealing the extended model $\mathcal Z_N^{o,ext}$ and the complex integral structure needed to match $Z_N$. They prove the $N=1$ case, $Z_1=\mathcal Z_1^{o,ext}$, via operational calculus and then outline the path to general $N$, including the Virasoro constraints for $\tau^o$ and the transformation to a Miwa-based tau function $\widetilde{\tau}$. The paper also discusses the challenges for $N\ge 2$ in the refined complex matrix model, highlighting obstacles that prevent a direct extension of the $N=1$ method, yet solidifies the conceptual bridge between open intersection theory and Kontsevich-Penner matrix models with potential for computational leverage on open invariants.

Abstract

The open intersection theory has been initiated by R. Pandharipande, J. P. Solomon and R. J. Tessler. In the scope of matrix model theory, A. Buryak and R. J. Tessler have constructed a matrix model $\mathcal{Z}^o$ for the open partition function based on a Kontsevich type combinatorial formula for the open intersection numbers found by R. J. Tessler. In this paper, using the Harish-Chandra-Itzykson-Zuber formula and operational calculus, we transform $\mathcal{Z}^o$ into another simple form, and define the matrix model $\mathcal{Z}_N^{o,ext,s}$ for the extended refined open partition function from it. The expression of $\mathcal{Z}_N^{o,ext,s}$ will immediately lead us to the Kontsevich-Penner matrix model $Z_N$ under the Miwa parametrization $s_i=2^ii!\operatorname{tr} Λ^{-2i-2}$. Hence it confirms the identification between the two models for general $N\geq 1$.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Lemma 3
• Proof 1
• Proof 2: of $Z_1=\mathcal{Z}_1^{o,ext}$
• Proof 3: of Theorem \ref{['T2']}