An Itzykson–Zuber-like integral and diffusion for complex ordinary and supermatrices

We compute an analogue of the Itzykson–Zuber integral for the case of arbitrary complex matrices. The calculation is done for both ordinary and supermatrices by transferring the Itzykson–Zuber diffusion equation method to the space of arbitrary complex matrices. The integral is of interest for applications in quantum chromodynamics and the theory of two-dimensional quantum gravity.