Topological Modular Forms Constrain Threshold States in Extremal $\mathcal{N}=1$ AdS$_3$ Gravity

Abstract: We identify a topological constraint on extremal $\mathcal{N}=1$ AdS$_3$ spectra that is not implied by genus-one modular covariance of the three even spin structure partition functions. Assuming the expected topological modular forms divisibility of the Ramond Witten index, we show that for infinitely many odd $n$ at central charge $c=12n$, an extremal holomorphic $\mathcal{N}=1$ superconformal field theory requires an odd number of Neveu--Schwarz primaries at the BTZ threshold. Thus the strict ansatz with no threshold primaries is excluded for this infinite family.