The various spectra called topological modular forms have historical origins in a variety of areas of mathematics. One of the more enticing is that the associated homology theory serves as the natural target for the Witten genus on smooth compact manifolds with a string structure – that is, for Spin manifolds \(M\) with a chosen trivialization of the characteristic class \(\lambda(M)\) with \(2\lambda\) the first Pontrjagin class. For this and other related reasons, it’s important to calculate the \(TMF\) cohomology of \(BString\), the classifying space for String bundles.
In this paper, the authors calculate a variant of the \(TMF_0(3)\) cohomology of \(BString\), where \(TMF_0(3)\) is the spectrum obtained by the Hopkins-Miller technology from the moduli stack of smooth elliptic curves with a chosen subgroup of order \(3\). To be completely specific, they calculate the cohomology of \(BString\) with coefficients in a \(K(2)\)-local spectrum \(\widehat{TMF_0(3)}\), where \(K(2)\) is a height \(2\) Morava \(K\)-theory at the prime \(2\). The answer turns out to be surprisingly simple: there is an isomorphism of topological rings \[ \widehat{TMF_0(3)}^\ast [[\tilde{r},\pi_1,\pi_2,\ldots]] \cong \widehat{TMF_0(3)}^\ast BString \] where the classes \(\pi_i\) are versions of Pontrjagin classes and the class \(\tilde{r}\) is obtained by analyzing the cubical structure of a specific line bundle. Constructing and analyzing \(\tilde{r}\) is one of the most interesting parts of the paper.
The spectrum \(\widehat{TMF_0(3)}\) is obtained by taking the \(\mathbb{Z}/2\)-homotopy fixed points of a \(K(2)\)-local spectrum \(\widehat{TMF_1(3)}\) obtained from elliptic curves with a chosen point of order \(3\). The other key section of the paper analyzes the equivariant homotopy theory of \(\widehat{TMF_1(3)}\) and the \(\mathbb{Z}/2\)-action on \(\widehat{TMF_1(3)}^\ast BString\), previously calculated by the first author in [G. Laures, Trans. Am. Math. Soc. 368, No. 10, 7339–7357 (2016; Zbl 1345.55001)].