topological modular form

$\mathrm{\Omega }$-spectrum: a sequence of topological spaces $E_n$, equipped with weak homotopy equivalences $s_n:E_n\to \mathrm{\Omega }{E}_{n+1}$. Namely, it induces isomorphisms: $s_n^{\ast }:{\pi }_{\ast }(E_n)={\pi }_{\ast }(\mathrm{\Omega }{E}_{n+1})={\pi }_{\ast +1}(E_{n+1})$. Thom spectrum & cobordism

Then, Brown's representability theorem gives a cohomology: $E^n(X)=li{m}_{k\to \mathrm{\infty }}[{\mathrm{\Sigma }}^{k}X,E_{n+k}]$ (homotopy classes), where ${\mathrm{\Sigma }}^{k}X$ is the suspension. eg. For Abelian group, Eilenberg-Maclane spectrum gives the singular cohomology. The elliptic cohomology $El{l}_{C/R}$ is given by the elliptic spectrum $Spec(\mathbb{R})\to {\mathcal{M}}_{ell}$, where ${\mathcal{M}}_{ell}$ is the moduli stack of elliptic curves. $El{l}_{C/R}$ is a presheaf ($X$ is the first variable, contravariant; functor: limit has universal property, homomorphism) over (coefficient) ${\mathcal{M}}_{ell}$.

$E_{C/R}$ isn't a sheaf, but can be sheafified to a sheaf of $E_{\mathrm{\infty }}$-ring spectrum (image) on ${\mathcal{M}}_{ell}$. Its global section is TMF.

Conjecture: the theory space of 2d, $\mathcal{N}=1$, anomaly d theories is TMF (spectrum). Witten genus can map its 0th homotopy group to weak modular forms, which is an isomorphism when no anomaly. Witten genus can be viewed as the Witten index (1d, $\mathcal{N}=1$) on loop space.
stable homotopy group

Seiberg-Witten