On a long exact sequence of groups of equivalence classes of 2d $\mathcal{N}=(0,1)$ SQFTs

Strongly motivated by a mathematical result by Lin and Yamashita (arXiv:2412.02298), we describe a long exact sequence formed by groups of equivalence classes of two-dimensional $\mathcal{N}=(0,1)$ supersymmetric quantum field theories (SQFTs) with and without $SU(2)$ symmetry. As an application, we study chiral fermions in heterotic compactifications with $SU(2)$ symmetry of level one to four dimensions, and show that each even-dimensional irreducible representation of $SU(2)$ appears even times, assuming the conjectural relation between topological modular forms and SQFTs. This implies the absence of the Witten anomaly, but contains more information than that.