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The theory of Topological Modular Forms suggests the existence of deformationinvariants for two-dimensional supersymmetric field theories that are morerefined than the standard elliptic genus. In this note we give a physicaldefinition of some of these invariants. The theory of mock modular forms makesa surprise appearance, shedding light on the integrality properties of somewell-known examples.

Mock modularity and a secondary elliptic genus