On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions

The complex Dotsenko-Fateev integral In(σ; τ ; θ) is the Selberg beta integral, where real variables in the integrand ∏| x k | σ-1 |1- x k | τ-1 ∏| x k -x 1 | 2θ are replaced by complex variables and an integration over a cube is replaced by an integration over the whole C n . According Dotsenko, Fateev, and Aomoto, such integral is a product of Gamma functions. We dene a family of beta integrals over spaces C m × C m+1 × ...× C n , which for m = n gives the integral In (with three additional integer parameters.