Abstract
We calculate the helicity trace index B14 for \(\mathcal{N}\) = 8 pure D-brane black holes using various techniques of computational algebraic geometry and find perfect agreement with the existing results in the literature. For these black holes, microstate counting is equivalent to finding the number of supersymmetric vacua of a multi-variable supersymmetric quantum mechanics which in turn is equivalent to solving a set of multi-variable polynomial equations modulo gauge symmetries. We explore four different techniques to solve a set of polynomial equations, namely Newton Polytopes, Homotopy continuation, Monodromy and Hilbert series. The first three methods rely on a mixture of symbolic and high precision numerics whereas the Hilbert series is symbolic and admit a gauge invariant analysis. Furthermore, exploiting various exchange symmetries, we show that quartic and higher order terms are absent in the potential, which if present would have spoiled the counting. Incorporating recent developments in algebraic geometry focusing on computational algorithms, we have extended the scope of one of the authors previous works [1, 2] and presented a new perspective for the black hole microstate counting problem. This further establishes the pure D-brane system as a consistent model, bringing us a step closer to \(\mathcal{N}\) = 2 black hole microstate counting.
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Acknowledgments
We wish to thank Ashoke Sen for encouraging us to look beyond Mathematica and suggesting the twined case. A.C. would like to thank Swapnamay Mondal for introducing him to the works of Hanany & friends. The work of A.C. is supported by IIT Bhubaneswar Seed Grant SP-103. The work of S.M. is supported by fellowship from CSIR, Govt. of India.
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Chowdhury, A., Maji, S. Counting \(\mathcal{N}\) = 8 black holes as algebraic varieties. J. High Energ. Phys. 2024, 91 (2024). https://doi.org/10.1007/JHEP05(2024)091
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DOI: https://doi.org/10.1007/JHEP05(2024)091