Cayley-type identities: new combinatorial proofs and applications to non-perturbative quantum field theories.

The classic Cayley identity states that det D (det X)^s = s(s+1)...(s+n-1) (det X)^(s-1), where X=(x_{ij}) is an n-by-n matrix of indeterminates and D=(d/d x_{ij}) is the corresponding matrix of partial derivatives. We will give new, simple proofs of this identity, purely combinatorial, using Grassmann algebra (= exterior algebra) and Grassmann-Berezin integration. We will state some generalizations, both old and new, and sketch how the simplified proof approach suggests them. These identities are special cases of Bernstein-Sato identities, a tool for solving the problem of finding the analytic continuation of a certain distribution-valued analytic function of s on the right complex half-plane. This problem arises also in the rigorous treatment of a problem in theoretical physics: how to describe a quantum field theory with variables valued on the s-dimensional sphere. We will give a flavour of how and why the solution of this problem would be relevant in this field. (This talk is based on joint work with Sergio Caracciolo and Alan D. Sokal, arXiv:1105.6270 plus other unpublished material.)