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

TL;DR

This paper computes the $K(2)$-local $TMF_1(3)$-cohomology of classifying spaces for String and Spin groups, introducing new classes that generalize Pontryagin classes, with applications to constructing cohomology classes from loop group representations.

Contribution

It explicitly computes the $K(2)$-local $TMF_1(3)$-cohomology of $B{ ext{String}}$ and $B{ ext{Spin}}$, introducing new classes generalizing Pontryagin classes.

Findings

Power series ring generators for cohomology of $B{ ext{String}}$ and $B{ ext{Spin}}$

Explicit construction of classes generalizing Pontryagin classes

Application to constructing $TMF(3n)$-cohomology classes from loop group representations

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 $L{\mathit Spin}$.