Notes on the Mathai-Quillen formalism and characteristic classes

These notes arose out of a desire (compulsion?) to understand this Mathai-Quillen form as a partition function, in the sense that statistical physics suggests, and through it to study the characteristic classes of E. The aim was to discover if it is possible to extract gaussian-shaped representatives of these characteristic classes as moments or correlation functions of this partition function, and to localize their periods to the vanishing set of a typical section. Such a localization is described by a well-known formula of Raoul Bott (8). The usual route of contact between Bott's formula and localization of characteristic dieren tial forms is through equivariant cohomology, and the framework in which this connection is made is quite beautiful and general. However, it was hoped in these notes to study whether the connection may be established from a dieren t perspective, through the computation of specialized path-integral-like traces in the Cliord algebra. Though the present document exists only an unnished record of study, and doubt- less contains errors of logic and omission, one can give an informal, but believable, geometric argument for the existence of the gaussian shaped representatives of coho- mology classes of M, by appealing to the Mathai-Quillen construction: A cohomology class of M is Poincar e dual to some degeneracy cycle in homology, perhaps represented by a submanifold Z. The Mathai-Quillen Thom representative N for the normal bundle N of Z pulls back by the dieomorphism of N with a tubular neighborhood of Z M to a gaussian-shaped form, which may be sharply peaked about a representative Z of the dual cycle to , representing any cohomology class . If 1 and 2 are characteristic classes such that 1 ^ 2 is an element of the top cohomology of M, then representatives of the dual homology classes (Z1) and (Z2) are complementary in dimension, and the generic intersection of a pair Z1; Z2 of their