Séminaire du 3 octobre 2011, 14h00: Andrea
Sportiello, Milan University.
*
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unpublished material.)

Virginie Collette
Last modified: Tue Oct 4 16:24:27 CEST 2011