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 $R=\beta$.Comment: 17 pages. arXiv admin note: text overlap with arXiv:1209.081

SO/SpSO/Sp Chern-Simons Gauge Theories At Large NN, SO/SpSO/Sp Penner Models And The Gauge Group Volumes

We construct a deformed $SO/Sp$ Penner generating function responsible for the close connection between $SO/Sp$ Chern-Simons gauge theories at large $N$ and the $SO/Sp$ Penner models. This construction is then shown to follow from a sector of a Chern-Simons gauge theory with coupling constant $\lambda$. The free energy and its continuum limit of the perturbative Chern-Simons gauge theory are obtained from the Penner model. Finally, asymptotic expansions for the logarithm of the gauge group volumes are given for every genus $g\geq 0$ and shown to be equivalent to the continuum limits of the $SO/Sp$ Chern-Simons gauge theories and the $SO/Sp$ Penner modelsComment: 19 pages; Progress of Theoretical Physics, Vol. 127, No. 2, February 201

Perturbative Chern-Simons Theory From The Penner Model

We show explicitly that the perturbative SU(N) Chern-Simons theory arises naturally from two Penner models, with opposite coupling constants. As a result computations in the perturbative Chern-Simons theory are carried out using the Penner model, and it turns out to be simpler and transparent. It is also shown that the connected correlators of the puncture operator in the Penner model, are related to the connected correlators of the operator that gives the Wilson loop operator in the conjugacy class.Comment: 7 Pages, Published Versio