The Symplectic-Orthogonal Penner Models

TL;DR

This paper explores the relationships between symplectic, orthogonal, and Penner models, revealing that their free energies are closely related and share the same critical point, with implications for understanding moduli spaces of real algebraic curves.

Contribution

It identifies the generating function for the orbifold Euler characteristic of moduli spaces of real algebraic curves and shows the near equivalence of free energies in symplectic and orthogonal Penner models.

Findings

The generating function for orbifold Euler characteristic is explicitly formulated.

The free energies of symplectic and orthogonal Penner models are nearly identical.

Both models share the same critical point as the Penner model.

Abstract

The generating function for the orbifold Euler characteristic of the moduli space of real algebraic curves of genus $2g$ (locally orientable surfaces) with $n$ marked points $\chi^r(\mathfrak{M}_{2g,n})$, is identified with a simple formula. It is shown that the free energy in the continuum limit of both the symplectic and the orthogonal Penner models are almost identical, with the structure $F^{SP/SO}(\mu)=1/2F(\mu)\mp F^{NO}(\mu)$, where $F(\mu)$ is the Penner free energy and $F^{NO}(\mu)$ is the free energy contributions from the non-orientable surfaces. Both of these models have the same critical point as the Penner model.