Introduction to integral discriminants
Abstract
The simplest partition function, associated with homogeneous symmetric forms S of degree r in n variables, is integral discriminant J_{nr}(S) = ∫e^{S(x1,...,xn)}dx_{1}...dx_{n}. Actually, Sdependence remains the same if e^{S} in the integrand is substituted by arbitrary function f(S), i.e. integral discriminant is a characteristic of the form S itself, and not of the averaging procedure. The aim of the present paper is to calculate J_{nr} in a number of nonGaussian cases. Using Ward identities — linear differential equations, satisfied by integral discriminants — we calculate J_{23},J_{24},J_{25} and J_{33}. In all these examples, integral discriminant appears to be a generalized hypergeometric function. It depends on several SL(n) invariants of S, with essential singularities controlled by the ordinary algebraic discriminant of S.
 Publication:

Journal of High Energy Physics
 Pub Date:
 December 2009
 DOI:
 10.1088/11266708/2009/12/002
 arXiv:
 arXiv:0903.2595
 Bibcode:
 2009JHEP...12..002M
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory
 EPrint:
 36 pages, 19 figures