Abstract
The fundamental heterotic string has a tower of BPS states whose supersymmetric index has an exponential growth in the charges. We construct the saddle-point of the gravitational path integral corresponding to this index. The saddle-point configuration is a supersymmetric rotating non-extremal Euclidean black hole. This configuration is singular in the two-derivative theory. We show that the addition of higher-derivative terms in four-dimensional \( \mathcal{N} \) = 2 supergravity resolves the singularity. In doing so, we extend the recently-developed “new attractor mechanism” to include the effect of higher-derivative terms. Remarkably, the one-loop, four-derivative F-term contribution to the prepotential leads to a precise match of the gravitational and microscopic index. We also comment, using the effective theory near the horizon, on the possibility of a string-size near-extremal black hole. Our results clarify the meaning of different descriptions of this system in the literature. The thermal state transitions to a winding condensate and a gas of strings without ever reaching a small black hole, while the index is captured by the rotating Euclidean black hole solution and is constant and thus smoothly connected to the microscopic ensemble.
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Acknowledgments
It is a pleasure to thank Pietro Benetti-Genolini, Jan Boruch, Matt Heydeman, Luca Iliesiu, Gabriel Lopes Cardoso, Juan Maldacena, Thomas Mohaupt, Ashoke Sen, and Edward Witten for interesting and useful discussions. Y.C. acknowledges the KITP program “What is String Theory? Weaving Perspectives Together”, during which this work is completed. S.M. acknowledges the support of the J. Robert Oppenheimer Visiting Professorship at the Institute for Advanced Study, Princeton and the STFC grants ST/T000759/1, ST/X000753/1. The work of GJT is supported by the University of Washington and the DOE award DE-SC0024363.
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Chen, Y., Murthy, S. & Turiaci, G.J. Gravitational index of the heterotic string. J. High Energ. Phys. 2024, 41 (2024). https://doi.org/10.1007/JHEP09(2024)041
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DOI: https://doi.org/10.1007/JHEP09(2024)041