A New Construction of the Penner Model

The free energy of the Penner model is shown to be closely related to the integral over the two diagonalizing unitary matrices of a complex rectangular matrix. Matrix models1 can be solved if and only if the original integrals over matrix elements can be reduced to integrals over eigenvalues2,3. In these models, the “angular” integrals over the diagonalizing unitary matrices result in a numerical factor which is usually and properly ignored, since it contains no useful information and can be absorbed into the normalization of the partition function. Here we point out a surprising fact: the numerical factor resulting from the angular integrals for a complex matrix with N ′ rows and N columns is closely related to the partition function for the Penner model4−7, which in turn is closely related to the moduli space of Riemann surfaces. We will give a precise statement and proof of the relationship, but unfortunately are unable to provide a physical or intuitive explanation of it. Let us begin with a brief review of the Penner model. It is defined by the partition function4,5 Z(t, N) = ξ(N) ∫ Dφ eNtTr[φ+log(1−φ)] (1) where φ is a hermitian N ×N matrix, t is a positive real parameter, Dφ indicates separate integration over the real and imaginary parts of each matrix element of φ, and ξ(N) is a normalization constant (which we will not be interested in). The free energy F (t, N) = logZ(t, n) of this model can be expanded as F (t, N) = ∞ ∑