Generalized Penner model and the Gaussian beta ensemble

In this paper, a new expression for the partition function of the generalized Penner model given by Goulden, Harer and Jackson is derived. The Penner and the orthogonal Penner partition functions are special cases of this formula. The parametrized Euler characteristic $\xi^s_g(\gamma)$ deduced from our expression of the partition function is shown to exhibit a contribution from the orbifold Euler characteristic of the moduli space of Riemann surfaces of genus $g$, with $s$ punctures, for all parameters $\gamma$ and $g$ odd. The other contributions for $g$ even are linear combinations of the Bernoulli polynomials at rational arguments. It turns out that the free energy coefficients of the generalized Penner model in the continuum limit, are identical to those coefficients in the large $N$ expansion of the Gaussian $\beta$-ensemble. Moreover, the duality enjoyed by the generalized Penner model, is also the duality symmetry of the Gaussian $\beta$-ensemble. Finally, a shift in the 't Hooft coupling constant required by the refined topological string, would leave the Gaussian $\beta$-ensemble duality intact. This duality is identified with the remarkable duality of the $c=1$ string at radius $R=\beta$.