Abstract
We propose an upper bound on the Atiyah-Singer index in the effective action of string theory. For E8 × \( {E}_8^{\prime } \) and SO(32) heterotic string theories on smooth Calabi-Yau threefolds with line bundles, we find that the tadpole cancellation and supersymmetry conditions lead to an upper bound on the generation number of quarks and leptons as well as Higgs doublets. By taking into account the observed value of four-dimensional gauge couplings and the supergravity approximation, we explicitly evaluate the bound on favorable complete intersection Calabi-Yau threefolds. The bound can be extended to Calabi-Yau threefolds in the Kreuzer-Skarke database. We also put the upper bound on the Atiyah-Singer index in Type IIB/F-theory compactifications.
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Acknowledgments
The authors would like to thank Andrei Constantin for useful discussions. This work was supported in part by Kyushu University’s Innovator Fellowship Program (S.N.), JSPS KAKENHI Grant Numbers JP20K14477 (H.O.), JP 21J20739 (M.T.), JP22J12877 (K.I.) and JP23H04512 (H.O).
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Ishiguro, K., Kai, T., Nishimura, S. et al. Upper bound on the Atiyah-Singer index from tadpole cancellation. J. High Energ. Phys. 2024, 200 (2024). https://doi.org/10.1007/JHEP01(2024)200
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DOI: https://doi.org/10.1007/JHEP01(2024)200