The Index Bundle for a Family of Dirac-Ramond Operators

String theoretic considerations imply the existence of a Dirac-like operator, known as the Dirac-Ramond operator, on the free loop space of a closed string manifold. We study the index bundle of the Dirac-Ramond operator associated with a family π : Z → X of closed spin manifolds. We work instead with a formal version of the operator, the usual Dirac operator twisted by a certain formal q-series of vector bundles. Its index bundle is an element of K(X)[[q]]. In the case where the total space Z is a string manifold, we show that the Chern character of this index bundle has certain modular properties. We then use the modularity to derive some explicit formulas for the Chern character of this index bundle. We also show that these formulas identify the index bundle with an L(E₈) bundle in a special case.