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Non-Holomorphic Cycles and Non-BPS Black Branes

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Abstract

We study extremal non-BPS black holes and strings arising in M-theory compactifications on Calabi–Yau threefolds, obtained by wrapping M2 branes on non-holomorphic 2-cycles and M5 branes on non-holomorphic 4-cycles. Using the attractor mechanism we compute the black hole mass and black string tension, leading to a conjectural formula for the asymptotic volumes of connected, locally volume-minimizing representatives of non-holomorphic, even-dimensional homology classes in the threefold, without knowledge of an explicit metric. In the case of divisors we find examples where the volume of the representative corresponding to the black string is less than the volume of the minimal piecewise-holomorphic representative, predicting recombination for those homology classes and leading to stable, non-BPS strings. We also compute the central charges of non-BPS strings in F-theory via a near-horizon \(AdS_3\) limit in 6d which, upon compactification on a circle, account for the asymptotic entropy of extremal non-supersymmetric 5d black holes (i.e., the asymptotic count of non-holomorphic minimal 2-cycles).

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Notes

  1. Indices on \(t^I\) are raised and lowered with a Kronecker delta function, which we use for ease of presentation.

  2. We take n to be large, in which case the change in horizon moduli under the shift in black string charge is negligible.

  3. These line bundles are not themselves effective [49], but when combined with the hyperplanes they generate all effective line bundles.

  4. In this calculation we only consider the leading-order central charge and so do not include corrections to the field strength due to gravitational Chern-Simons terms.

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Acknowledgements

We are grateful to Naomi Gendler, Liam McAllister, and John Stout for useful discussions. The work of C.L. was partially supported by the Alfred P. Sloan Foundation Grant No. G-2019-12504 and by DOE Grant DE-SC0013607. The work of of A. S. was partially supported by the US grants: NSF DMS-1607871, NSF DMS-1306313, Simons 38558, as well as Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. No 14.641.31.0001. The research of C.V. was partially supported by the National Science Foundation under Grant No. NSF PHY-2013858 and by a grant from the Simons Foundation (602883, CV). The work of S.-T. Y. was partially supported by the US grants: NSF DMS-0804454, NSF PHY1306313, and Simons Foundation 38558.

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Authors and Affiliations

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

    Cody Long & Cumrun Vafa

  2. Center for Mathematical Sciences and Applications, Harvard University, Cambridge, MA, 02139, USA

    Cody Long, Artan Sheshmani & Shing-Tung Yau

  3. Institut for Matematik, Aarhus Universitet, 8000, Aarhus C, Denmark

    Artan Sheshmani

  4. Laboratory of Mirror Symmetry, National Research University Higher School of Economics, Russian Federation, Moscow, Russia, 119048

    Artan Sheshmani

  5. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Shing-Tung Yau

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Long, C., Sheshmani, A., Vafa, C. et al. Non-Holomorphic Cycles and Non-BPS Black Branes. Commun. Math. Phys. 399, 1991–2043 (2023). https://doi.org/10.1007/s00220-022-04587-4

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