Characteristic classes in $TMF$ of level $\Gamma_1(3)$

Abstract

Let $TMF_1(n)$ be the spectrum of topological modular forms equipped with a $\Gamma_1(n)$-structure. We compute the $K(2)$-local $TMF_1(3)$-cohomology of $B{\mathit String}$ and $B{\mathit Spin}$: both are power series rings freely generated by classes that we explicitly construct and which generalize the classical Pontryagin classes. As a first application of this computation, we show how to construct $TMF(3n)$-cohomology classes from stable positive energy representations of the loop groups $L{\mathit Spin}$.