The Symplectic-Orthogonal Penner Models

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 $F^{SP\mathrm{/}SO}(\mu )=1\mathrm{/}2F(\mu )\mp F^{NO}(\mu )$, 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.