1-forms over the moduli space of irreducible connections defined by the spectrum of Dirac operators

Abstract

Let M ( P) be the moduli space of irreducible connections of a G-principal bundle P over a closed Riemannian spin manifold M. Let D A be the Dirac operator of M coupled to a connection A of P and f a smooth function on M. We consider a smooth variation A( u) of A with tangent vector ω and denote T ω:= d du ( D A ( u) − f) ( u=0. The coefficients of the asymptotic expansion of trace ( T ω · e - t( D A − f) 2 ) near t=0 define 1-forms a ( k) f , k=0, 1, 2, … on M ( P). In this paper we calculate aa (0) f , a (1) f , a (2) f and study some of their properties. For instance using the 1-form a (2) f for suitable functions f we obtain a foliation of codimension 5 of the space of G-instantons of S 4.