Characteristic classes in $TMF$ of level ${\Gamma }_1(3)$

Abstract

Let $TM{F}_1(n)$ be the spectrum of topological modular forms equipped with a ${\Gamma }_1(n)$-structure. We compute the $K(2)$-local $TM{F}_1(3)$-cohomology of $BString$ and $BSpin$: 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 $LSpin$.