A duality between string topology and the fusion product in equivariant K-theory

Let $G$ be a compact Lie group. Let $G \to E \to M$ be a principal $G$-bundle over a closed manifold $M$, and $G \to Ad(E) \to M$ its adjoint bundle. In this paper we describe a new Frobenius algebra structure on $h_*(Ad(E)),$ where $h_*$ is an appropriate generalized homology theory. Recall that a Frobenius algebra has both a product and a coproduct. The product in this new Frobenius algebra is induced by the string topology product. In particular, the product can be defined when $G$ is any topological group and in the case that $E$ is contractible it is precisely the Chas-Sullivan string product on $H_*(LM).$ We will show that the coproduct is induced by completing the untwisted Freed-Hopkins-Teleman fusion product. Indeed, when $M$ is replaced by $BG$ and $h_*$ is $K$-theory the coproduct is the completion of the Freed-Hopkins-Teleman fusion structure. We will then show that this duality between the string and fusion products is realized by a Spanier-Whitehead duality between certain Thom spectra of virtual bundles over $Ad(E)$.

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Published 1 January 2007