An Itzykson-Zuber-like Integral and Diffusion for Complex Ordinary and Supermatrices

In 1980, Itzykson and Zuber presented their result on a certain integral over the unitary group which had great impact in several areas of mathematical physics. Let U be a matrix parameterizing the unitary group U(N) with the invariant Haar measure dμ(U). Moreover, consider two diagonal matrices x and y with entries xn and yn, respectively, where n = 1, . . . , N . The Itzykson-Zuber integral can then be written in the form ∫ dμ(U) exp ( i trUxUy ) = det [exp(ixnym)]n,m=1,...,N ∆N (x)∆N (y) , (1.1)