Is there a relationship between fusion and S^1-equivariance for spinors on loop space?

Question

A while ago (officially in 1987), Witten conjectured that string structures on a manifold $M$ correspond to "$S^1$-equivariant" (or more precisely $\mathrm{Diff}^+(S^1)$-equivariant) spin structures on its free loop space $\mathcal{L}M$. This conjecture was based on a heuristic computation of the index of a hypothetical $S^1$-equivariant Dirac operator. In 2005, Stolz and Teichner conjectured that the necessary additional structure on the spin structure of $\mathcal{L}M$ is a fusion product, and a proof of this has recently been given by Ludewig.

What I'm wondering about is the relationship between Witten's "$S^1$-equivariance" and the fusion product. Is there an interpretation of the fusion product as a kind of equivariance? A priori, fusion is a locality condition for the spinor bundle that allows the definition of a super $2$-vector bundle on $M\times M$. But for Witten's computation to work as he outlined, it would be necessary for there to actually be an $S^1$-equivariant Dirac operator, and for it to lift to spectra, one would expect some kind of genuine $S^1$-equivariant version of $MSpin$ to describe this structure (since the K-theory trace of this operator should land in $KO_{S^1}(\mathcal{L}M)$, whose completion is the Tate K-theory of $M$ by Kitchloo-Morava). Is this actually the case?

Answer 1

Matthias' paper that you cite is about the spinor bundle on loop spaces, and Matthias proves there that a manifold $M$ admits a string structure if and only if $LM$ admits a fusive spinor bundle.

The correspondence between string structures on $M$ and spin structures on $LM$ is more basic than looking at spinor bundles (i.e., a structure that builds on a spin structure). The first result in that respect is Theorem 1.4 in my 2012 paper Spin structures on loop spaces that characterize string manifolds: a spin manifold $M$ is string if and only if its loop space $LM$ has a fusive spin structure.

A better statement is to make the category of string structures on $M$ equivalent to a category fusive spin structures on $LM$, but this is more difficult: the main result (Theorem A) of my 2014 paper String geometry vs. spin geometry on loop spaces says that the category of string structures on $M$ is equivalent to a category of thin-homotopy-equivariant fusive spin structures on $LM$.

This might be helpful to understand the $S^1$-equivariance: it has nothing to do with the fusion product - it is induced by the thin-homotopy-equivariant structure (Prop. 3.1.4 in the last reference).

The picture might become a bit clearer when upgrading to geometric string structures on $M$ (string structures with string connections) and geometric spin structures on $LM$ - which we have to do anyway when we want to consider Dirac operators. Then, there is another equivalence of categories $$Spin^{\nabla_{sf}}_{fus}(LM) \cong String^{\nabla}(M)$$ where now on the loop space side we consider fusive spin structures equipped with superficial spin connections. The passage from the setting with connections to the setting without computes a thin-homotopy-equivariant structure from such a connection, basically by parallel transport along homotopies between elements of $Diff^{+}(S^1)$ and $id_{S^1}$ (Prop. 4.1.4 in the last reference).

Maybe it is also good to remark that the appearance of all these structures is not at all special for string structures or spin structures on loop spaces. They all appear in the characterization of the image of transgression. For instance, the line bundles on $LM$ that are transgressions of bundle gerbes on $M$, are equipped with fusion products and thin-homotopy-equivariant structures, and transgressive connections have the above-mentioned property of being superficial; see Transgression to Loop Spaces and its Inverse, II.

In short: the spin structures on $LM$ that correspond to string structures on $M$ are $Diff^+(S^1)$-equivariant, but this is independent of the fusion product.