Topological automorphic forms are a generalization of topological modular forms: where the latter come with the moduli space of elliptic curves, topological automorphic forms are associated to a given Shimura variety. Moreover, just as topological modular forms refine to the tmf-spectrum representing the corresponding cohomology theory, so every Shimura variety induces a cohomology theory $taf$.
chromatic level $n =$ | 1 | 2 | $\geq 3$ |
---|---|---|---|
cohomology theory/spectrum $E =$ | KO | TMF | TAF |
algebraic group | $GL_1$ | $GL_2$ | $U(1,n-1)$ |
geometric object | multiplicative group | elliptic curve | Shimura variety |
FQFT | superparticle | heterotic superstring | ?? |
The definition is due to
An introductory survey is in
Lecture notes include
Mark Behrens, Topological Automorphic Forms, Lecture series (2011) (lecture notes)
Doug Ravenel, Seminar on topological automorphic forms (2009-11) (web)