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We study the index bundle of the Dirac-Ramond operator associated with afamily $\pi: Z \to X$ of compact spin manifolds. We view this operator as theformal twisted Dirac operator $\dd \otimes\bigotimes_{n=1}^{\infty}S_{q^n}TM_{\C}$ so that its index bundle is an elementof $K(X)[[q]]$. When $p_1 (Z) = 0$, we derive some explicit formulas for theChern character of this index bundle using its modular properties. We also usethe modularity to identify our index bundle with an $L(E_8)$ bundle in aspecial case.

The Index Bundle for a Family of Dirac-Ramond Operators