  We build a connection between topology of smooth 4-manifolds and the theory
of topological modular forms by considering topologically twisted
compactification of 6d (1,0) theories on 4-manifolds with flavor symmetry
backgrounds. The effective 2d theory has (0,1) supersymmetry and, possibly, a
residual flavor symmetry. The equivariant topological Witten genus of this 2d
theory then produces a new invariant of the 4-manifold equipped with a
principle bundle, valued in the ring of equivariant weakly holomorphic
(topological) modular forms. We describe basic properties of this map and
present a few simple examples. As a byproduct, we obtain some new results on 't
Hooft anomalies of 6d (1,0) theories and a better understanding of the relation
between 2d (0,1) theories and TMF spectra.
