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Wall-Crossing Holomorphic Anomaly and Mock Modularity of Multiple M5-Branes

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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 \({{\mathcal{N}}=4}\) topological Yang–Mills theory on \({{\mathbbm{P}}^2}\) 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.

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Authors and Affiliations

  1. Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, USA

    Murad Alim

  2. Institute for Theoretical Physics and Spinoza Institute, Utrecht University, 3508 TD, Utrecht, The Netherlands

    Babak Haghighat

  3. Bethe Center for Theoretical Physics, University of Bonn, Nussallee 12, 53115, Bonn, Germany

    Babak Haghighat, Albrecht Klemm, Marco Rauch & Thomas Wotschke

  4. Arnold Sommerfeld Center for Theoretical Physics, LMU, Theresienstr. 37, 80333, Munich, Germany

    Michael Hecht

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  1. Murad Alim
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Alim, M., Haghighat, B., Hecht, M. et al. Wall-Crossing Holomorphic Anomaly and Mock Modularity of Multiple M5-Branes. Commun. Math. Phys. 339, 773–814 (2015). https://doi.org/10.1007/s00220-015-2436-3

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