---
tag: JN73KE96YB
kind: concept
name: Topological modular forms
msc: [55N34, 55P43, 11F11]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_modular_forms
same_as:
  wikidata: Q7825037
  wikipedia: Topological modular forms
---

## Summary

Topological modular forms is a [[ring spectrum|BAW0T68GWK]], written $\mathrm{tmf}$, that collects all [[elliptic cohomology|XH68V2BP4K]] theories into one universal object.
Goerss, Hopkins and Miller built it as the global sections of a sheaf of $E_\infty$ ring spectra on the moduli stack of elliptic curves.
Its homotopy groups map to the ring of integral [[modular forms|GWKN5VG78A]], and this map is an isomorphism after tensoring with $\Q$.
The torsion in $\pi_* \mathrm{tmf}$ detects many elements of the stable homotopy groups of spheres, and $\mathrm{tmf}$ receives the Witten genus of string manifolds.

## Explanations

### L1 human

Each elliptic curve gives an elliptic cohomology theory, and modular forms are functions on the space of all elliptic curves.
Topological modular forms glue all the elliptic cohomology theories together in a homotopy-theoretic way.
The result is a single cohomology theory whose coefficients are "modular forms, plus extra torsion".
For example, $\pi_3 \mathrm{tmf} \cong \Z/24$, which matches the stable homotopy group $\pi_3^s \cong \Z/24$ of the sphere.
No classical modular form explains this group, so $\mathrm{tmf}$ sees more than modular forms do.