Local index theorem for projective families

Abstract

Abstract. We give a superconnection proof of the Mathai-Melrose-Singer index theorem for the family of twisted Dirac operators [28,29].

Analysis of twisted supercharge families on product manifolds

“…In 1.5 we formulate a superconnection formalism for the odd twisted K-theory in the twisted cohomology. The analogous construction in the even twisted K-theory was done in [3]. We end the section 1 by solving an index character for the odd supercharges in the twisted cohomology.…”

“…We can also compute the index in the twisted cohomology theory following the superconnection technique of [3]. An even superconnection is a descent datum A " pA OE , ∇ λ , Ω OE q where A OE are locally defined even superconnections, which are odd elements in…”

“…Then we use the local supercurvatures associated to A OE to define the global curvature form Now we apply the supertrace in the even formalism. This defines a cocycle in the twisted cohomology determined by the twisted differential d`H, and its twisted cohomology class is independent on the choice of the superconnection, see [3].…”

“…The construction is essentially based on the theory of projective representations of the loop algebra of the circle group, lt " lup1q, on Fock spaces. Projective families of Dirac operators have been used for analytic realizations of the even twisted K-theory elements, see [8], [9], [3]. Using the methods of [9] and [3], one can realize the even twisted K-theory classes with decomposable twisting in terms of projective families of pseudodifferential operators.…”

“…The methods to develop characteristic classes in [7] will be extended to the even theory and the characteristic classes of the indexes of the suspended supercharges will be computed. In addition, by applying the methods of [3], both odd and even index characters for the supercharges will be computed in the twisted cohomology theory as well.…”