On a long exact sequence of groups of equivalence classes of 2d N =(0,1) SQFTs

Strongly motivated by a mathematical result by Lin and Yamashita [LY24], we describe a long exact sequence formed by groups of equivalence classes of two-dimensional 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.