A NEW CONSTRUCTION OF THE PENNER MODEL
Mark Srednicki · 1992
ABSTRACT: 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 models 1 can be solved if and only if the original integrals over matrix elements can be reduced to integrals over eigenvalues 2,3. 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 model 4−7, 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 e Nt Tr[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) = log Z(t, n) of this model can be expanded as