Abstract: The theory of topological modular forms leads to many interesting constraints and predictions for two-dimensional quantum field theories, and some of them might have interesting implications for the swampland program. In this talk, I will show that a conjecture by Segal, Stolz and Teichner requires the constant term of the partition function of a bosonic holomorphic CFTs to be divisible by specific integers determined by the central charge. We verify this constraint in large classes of physical examples, and rule out the existence of an infinite set of “extremal CFTs”, including those with central charges c = 48, 72, 96 and 120.