Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes

TL;DR

This work establishes a precise link between wall-crossing, mock modularity, and holomorphic anomalies for the elliptic genus of two M5-branes on a rigid divisor in a Calabi–Yau three-fold. By translating KS wall-crossing into Göttsche’s framework and applying Zwegers’ modular completion to the resulting indefinite theta-functions, the authors derive a holomorphic anomaly equation for rank two, {D}_2 Z^{(2)}_{P}, that factorizes through the square of the rank-one genus and lattice data. They validate the formalism on several b_2^+=1 surfaces (e.g., P^2, F_0, F_1, B_8) and connect the findings to E-string/half-K3 physics, while outlining natural extensions to higher rank and a contour description via meromorphic Jacobi forms. The results deepen the interplay between BPS counting, wall-crossing, and modular properties, with potential implications for refined invariants and OSV-type correspondences.

Abstract

Using wall-crossing formulae and the theory of mock modular forms we derive 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 modularity of an indefinite theta-function capturing the wall-crossing of BPS invariants associated to D4-D2-D0 brane systems. We show 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 show in examples that the information provided by the anomaly is enough to compute the BPS degeneracies for certain charges. We further speculate on a natural extension of the anomaly to higher D4-brane charge.