Stolz–Teichner's TMF classification conjecture for two-dimensional supersymmetric quantum field theories

The group $cpi_d\operatorname{TMF}$ is the degree-$d$ part of topological modular forms, and $cpi_0$ denotes connected components of the space of theories. A two-dimensional $cmathcal{N}=(0,1)$ supersymmetric quantum field theory has gravitational anomaly $2(c_R-c_L)$, where $c_R$ and $c_L$ are its right- and left-moving central charges. Stolz–Teichner's conjecture. The simplified classification conjecture proposes an equivalence

$$\pi_{d} \operatorname{TMF} \simeq \pi_0 \left\{ 2d \ \mathcal{N}=(0,1)\ \operatorname{SQFTs\ with\ gravitational\ anomaly}\ 2(c_R-c_L)=d \right\}.$$

This predicts that topological modular forms classify two-dimensional minimally supersymmetric quantum field theories by gravitational anomaly. Compactifications of six-dimensional $(1,0)$ superconformal field theories on smooth four-manifolds provide evidence for the conjecture, while the general geometric and field-theoretic construction remains open.