Let me preface this question by saying that I wrote it at least in part to understand its statement. As such, I hope that the reader will excuse any mistakes.
$\DeclareMathOperator{\ind}{ind}\DeclareMathOperator{\Wit}{Wit}\newcommand{\Dirac}{D\hspace{-4pt}/\ }\newcommand{\Z}{\mathbb Z}\newcommand{\C}{\mathbb C}$In The index of the Dirac operator in loop space, Witten writes down an infinite series of tensor representations $R_n$ and argues that for a spin manifold $X$ equipped with a $S^1$-action, each of the equivariant indices $\ind^{S^1}(\Dirac\otimes R_n(TX))$ is a trivial (virtual) $S^1$-representation. The argument is as follows:
- Define $a_{n,k}$ as the multiplicity of the $k$-th power of the defining representation of $S^1$ in $\ind^{S^1}(\Dirac\otimes R_n(TX))$, so that the statement becomes $a_{n,k} = 0$ for $k\neq 0$
- Using a field integral expression for the equivariant index, he conjectured that $$ \Wit^{S^1}(X;q,w) = \sum_{n\ge 0,k\in \Z}a_{n,k}e^{2\pi i n\tau} e^{2\pi i k z} $$ converges absolutely for $0 < \Im z < \Im\tau$ and defines a Jacobi form
- In particular, for a fixed $\tau$, the resulting function of $z$ is holomorphic and periodic with respect to the lattice $\Z\oplus \tau\Z$, therefore constant
A mathematical proof can be given along these lines using $S^1$-equivariant elliptic cohomology, which assigns to any elliptic curve $E$ over $\C$ the global sections of a sheaf on $E$ whose stalk at $z$ is $H^{S^1}(X^{G_z})$, where $G_z = \Z/n$ for $z$ a point of (exact) order $n$, $G_0 = S^1$, and $G_z = e$ otherwise. The field integral can then be expressed as a transfer, i.e. a wrong-way map for $X\to \mathrm{pt}$, which was expressed via a Thom class by Rosu, making the last step in the proof rigorous.
Nowadays, there is a more refined version of equivariant elliptic cohomology, by Gepner and Meier following Lurie: Writing $\mathbf T$ for the group $S^1$, they assign to $X$ a quasicoherent sheaf $\mathcal{Ell}_{\mathbf T}(X)$ on the universal oriented elliptic curve. This leads me to the following question:
Question
For which $\mathbf T$-equivariant maps $f:X\to Y$ is there an induced transfer/wrong-way map $\mathcal{Ell}_{\mathbf T}(X)\to \mathcal{Ell}_{\mathbf T}(Y)$?
As stated, this question is of course not quite defined, so part of it is what the expected properties of the wrong-way map should be. For instance, I would expect a push-pull formula, and I would expect such a map to be defined by the choice of an equivariant String structure on the stable normal bundle of $f$ by an equivariant version of the String orientation of TMF described in Section 5.3 of Lurie's survey. It should also be compatible with the Borel-equivariant wrong-way map, which fixes it on the stalk at the identity section.
Assuming an answer to this question, I was wondering if there are known consequences of the existence of such a transfer. For the non-equivariant TMF-valued refinement of the Witten genus, applications include classification results for exotic 23-spheres, whereas rigidity of the complexified equivariant elliptic genus has geometric and number-theoretic implications.
