Abstract
Geometric engineering is a collection of tools developed to establish dictionaries between local singularities in string theory and (supersymmetric) quantum fields. Extended operators and defects, as well as their higher quantum numbers captured by topological symmetries, can be encoded within geometric engineering dictionaries. In this paper we revisit and clarify aspects of these techniques, with special emphasis on ’t Hooft anomalies, interpreted from the SymTFT perspective as obstructions to the existence of Neumann boundary conditions. These obstructions to gauging higher symmetries are captured via higher link correlators for the SymTFT on spheres. In this work, we give the geometric engineering counterpart of this construction in terms of higher links of topological membranes. We provide a consistency check in the context of 5D SCFTs with anomalous 1-form symmetries, where we give two independent derivations of the anomaly in terms of higher links, one purely field theoretical and the other purely geometrical. Along the way, we also recover the construction of non-invertible duality defects in 4D \( \mathcal{N} \) = 4 SYM from a geometric engineering perspective.
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Acknowledgments
We thank Bobby Acharya, Jonathan Heckman, Inaki Garcia Etxebarria, and Xiao-Gang Wen for discussions. MDZ thanks in particular Dan Freed, David Jordan, Ibou Bah, and Nytia Kitchloo, as well as Federico Bonetti and Ruben Minasian for discussions on closely related ongoing projects. In particular, a different more refined mathematical model for geometric engineering at infinity is currently being developed in a collaboration driven by David Jordan. MDZ also acknowledges a discussion with Fabio Apruzzi at the Simons Collaboration on Global Categorical Symmetries in November 2023. SM thanks Elias Riedel Garding and Azeem Hasan for discussions and clarifications. The work of MDZ has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 851931). MDZ and SM also acknowledge support from the Simons Foundation (grant #888984, Simons Collaboration on Global Categorical Symmetries). RM is supported by a Knut and Alice Wallenberg postdoctoral scholarship in mathematics and was supported in part by DOE grant DE-SC1019775 during the earlier stages of this project.
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Del Zotto, M., Meynet, S.N. & Moscrop, R. Remarks on geometric engineering, symmetry TFTs and anomalies. J. High Energ. Phys. 2024, 220 (2024). https://doi.org/10.1007/JHEP07(2024)220
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DOI: https://doi.org/10.1007/JHEP07(2024)220