Abstract
With the use of mathematical techniques of tropical geometry, it was shown by Mikhalkin some twenty years ago that certain Gromov-Witten invariants associated with topological quantum field theories of pseudoholomorphic maps can be computed by going to the tropical limit of the geometries in question. Here we examine this phenomenon from the physics perspective of topological quantum field theory in the path integral representation, beginning with the case of the topological sigma model before coupling it to topological gravity. We identify the tropicalization of the localization equations, investigate its geometry and symmetries, and study the theory and its observables using the standard cohomological BRST methods. We find that the worldsheet theory exhibits a nonrelativistic structure, similar to theories of the Lifshitz type. Its path-integral formulation does not require a worldsheet complex structure; instead, it is based on a worldsheet foliation structure.
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Acknowledgments
We wish to thank Ori Ganor and Oleg Viro for illuminating discussions. E.A. and A.F.V. are thankful to the organizers of the IAS workshop String Amplitudes at Finite α′ (February 2023), Nima Arkani-Hamed, Lorenz Eberhardt and Sebastian Mizera, for the opportunity to attend the workshop, and for their hospitality and illuminating discussions therein. This work has been supported by NSF grant PHY-2112880. K.-I.E. also acknowledges support from the NSF Quantum Leap Challenge Institute program through grant OMA-2016245.
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Albrychiewicz, E., Ellers, KI., Valiente, A.F. et al. Tropological sigma models. J. High Energ. Phys. 2024, 135 (2024). https://doi.org/10.1007/JHEP06(2024)135
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DOI: https://doi.org/10.1007/JHEP06(2024)135