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There are various programs (which start with Segal's survey, I believe), all of which I know nothing about, to interpret elliptic cohomology classes in terms of von Neumann algebras, loop group representations, conformal field theories, ...

 

I can understand why a geometric interpretation of elliptic cohomology would be desirable, but it's mystifying to me why researchers in this area are concentrating on these specific objects.

There are various programs (which start with Segal's survey, I believe), all of which I know nothing about, to interpret elliptic cohomology classes in terms of von Neumann algebras, loop group representations, conformal field theories, ...

I can understand why a geometric interpretation of elliptic cohomology would be desirable, but it's mystifying to me why researchers in this area are concentrating on these specific objects.

Loop group representations. This is the analogue of $G$-equivariant K-theory having something to do with the representation theory of $G$. One picture of tmf whose accuracy I can't comment on is that it should look heuristically like $K(ku)$, the cohomology theory presented by (Bass-Dundes-Rognes) 2-vector bundles. So $G$-equivariant tmf should look heuristically like $G$-equivariant 2-vector bundles, which over a point should look heuristically like representations of $G$ on (suitably dualizable) 2-vector spaces.

Loop group representations. This is the analogue of $G$-equivariant K-theory having something to do with the representation theory of $G$. One picture of tmf whose accuracy I can't comment on is that it should look heuristically like $K(ku)$, the cohomology theory presented by (Baas-Dundas-Rognes) 2-vector bundles. So $G$-equivariant tmf should look heuristically like $G$-equivariant 2-vector bundles, which over a point should look heuristically like representations of $G$ on (suitably dualizable) 2-vector spaces.

Loop group representations. This is the analogue of $G$-equivariant K-theory having something to do with the representation theory of $G$. One picture of tmf whose accuracy I can't comment on is that it should look at least a bit like $K(ku)$, the cohomology theory presented by (Bass-Dundes-Rognes) 2-vector bundles. So $G$-equivariant tmf should at least look a bit like something presented by $G$-equivariant 2-vector bundles, which over a point should at least look a bit like representations of $G$ on (suitably dualizable) 2-vector spaces.

Whatever that means, such a thing ought to have a "character" which, rather than being a class function on $G$, or equivalently a function on the adjoint quotient $G/G$, is instead an equivariant vector bundle on $G/G$. Now $G/G$ looks at least morally like the classifying stack $BLG$ of the loop group of $LG$, and hence an equivariant vector bundle on $G/G$ looks at least morally like a representation of $LG$. Freed-Hopkins-Teleman makes this precise. The non-equivariant version of this story is the story about tmf having something to do with ($S^1$-equivariant?) K-theory of the free loop space $LX$.

The historical motivation for generalizing this to $2$-dimensional conformal rather than topological field theories is, I think, to explain the modularity properties of the Witten genus? But again, don't trust me to have the specifics right here. (I guess it's $2$-dimensional topological rather than conformal field theories over $X$ that look more like $2$-vector bundles.)

Loop group representations. This is the analogue of $G$-equivariant K-theory having something to do with the representation theory of $G$. One picture of tmf whose accuracy I can't comment on is that it should look heuristically like $K(ku)$, the cohomology theory presented by (Bass-Dundes-Rognes) 2-vector bundles. So $G$-equivariant tmf should look heuristically like $G$-equivariant 2-vector bundles, which over a point should look heuristically like representations of $G$ on (suitably dualizable) 2-vector spaces.

Whatever that means, such a thing ought to have a "character" which, rather than being a class function on $G$, or equivalently a function on the adjoint quotient $G/G$, is instead an equivariant vector bundle on $G/G$. Now $G/G$ looks heuristically like the classifying stack $BLG$ of the loop group $LG$, and hence an equivariant vector bundle on $G/G$ looks heuristically like a representation of $LG$. Freed-Hopkins-Teleman makes this precise. The non-equivariant version of this story is Witten's story about tmf having something to do with ($S^1$-equivariant?) K-theory of the free loop space $LX$.

The historical motivation for generalizing this to $2$-dimensional conformal rather than topological field theories is, I think, to explain the modularity properties of the Witten genus. But again, don't trust me to have the specifics right here. (I guess it's $2$-dimensional topological rather than conformal field theories over $X$ that look more like $2$-vector bundles.)