Abstract
Superconformal indices of four-dimensional \( \mathcal{N} \) = 1 gauge theories factorize into holomorphic blocks. We interpret this as a modular property resulting from the combined action of an SL(3, ℤ) and SL(2, ℤ) ⋉ ℤ2 transformation. The former corresponds to a gluing transformation and the latter to an overall large diffeomorphism, both associated with a Heegaard splitting of the underlying geometry. The extension to more general transformations leads us to argue that a given index can be factorized in terms of a family of holomorphic blocks parametrized by modular (congruence sub)groups. We find precise agreement between this proposal and new modular properties of the elliptic Γ function. This leads to our conjecture for the “modular factorization” of superconformal lens indices of general \( \mathcal{N} \) = 1 gauge theories. We provide evidence for the conjecture in the context of the free chiral multiplet and SQED and sketch the extension of our arguments to more general gauge theories. Assuming the validity of the conjecture, we systematically prove that a normalized part of superconformal lens indices defines a non-trivial first cohomology class associated with SL(3, ℤ). Finally, we use this framework to propose a formula for the general lens space index.
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Acknowledgments
We are grateful to Miranda Cheng, Abhijit Gadde, and Finn Larsen for conversations about this work. We also thank an anonymous referee for several critical remarks on v2 of this paper. VJ is supported by the South African Research Chairs Initiative of the Department of Science and Innovation and the National Research Foundation. YL was supported by the UCAS program of special research associate, the internal funds of the KITS, and the Chinese Postdoctoral Science Foundation. YL is also supported by a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). WL is supported by NSFC No. 11875064, No. 11947302, and the Max-Planck Partnergruppen fund; she is also grateful for the hospitality of AEI Potsdam, where part of this work was done. SvL is in part supported by the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa. Opinions expressed and conclusions arrived at are those of the author and are not necessarily to be attributed to the CoE-MaSS.
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Jejjala, V., Lei, Y., van Leuven, S. et al. Modular factorization of superconformal indices. J. High Energ. Phys. 2023, 105 (2023). https://doi.org/10.1007/JHEP10(2023)105
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DOI: https://doi.org/10.1007/JHEP10(2023)105