One-matrix differential reformulation of two-matrix models

Abstract

Differential reformulations of field theories are often used for explicit computations. We derive a one-matrix differential formulation of two-matrix models, with the help of which it is possible to diagonalize the one-and two-matrix models using a formula by Itzykson and Zuber that allows diagonalizing differential operators with respect to matrix elements of Hermitian matrices. We detail the equivalence between the expressions obtained by diagonalizing the partition function in differential or integral formulation, which is not manifest at first glance. For one-matrix models, this requires transforming certain derivatives to variables. In the case of two-matrix models, the same computation leads to a new determinant formulation of the partition function, and we discuss potential applications to new orthogonal polynomials methods.

History of Mathematics in American Mathematical Textbooks of the Late 19th and Early 20th Centuries

“…The one-matrix differential formulation of two-matrix models was derived in Ref. [31]. For the complex multi-matrix model (1), it can be expressed as the following differential reformulation:…”