Mock modularity associated to wall crossing

Using wall-crossing formulae and the theory of mock modular forms we derived a holomorphic anomaly equation for the modified elliptic genus of two M5-branes wrapping a rigid divisor inside a Calabi-Yau manifold. The anomaly originates from restoring the modularity of an indefinite theta-function capturing the wall-crossing of BPS invariants associated with D4-D2-D0 brane systems. We showed the compatibility of this equation with anomaly equations previously observed in the context of N=4 topological Yang-Mills theory on P^2 and E-strings obtained from wrapping M5-branes on a del Pezzo surface. The non-holomorphic part is related to the contribution originating from bound-states of singly wrapped M5-branes on the divisor. We showed in examples that the information provided by the anomaly is enough to compute the BPS degeneracies for certain charges. We further speculated on a natural extension of the anomaly to a higher D4-brane charge.

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