Abstract
Application of the supersymmetric localization method to theories on anti-de Sitter spacetime has received recent interest, yet still remains as a challenging problem. In this paper, we focus on (global) Euclidean AdS2, on which we consider an Abelian đť’© = (2, 2) theory and implement localization computation to obtain the exact partition function. For comparison, we also revisit the theory on S2 and perform a parallel computation. We refine the notion of equivariant supersymmetry and use appropriate functional integration measure. For AdS2 we choose a supersymmetric boundary condition which is compatible with the principle of variation. To evaluate the 1-loop determinant about the localization saddle, we use index theory and fixed point formula, where we pay attention to the effect of zero modes and their superpartners. The existence of fermionic superpartner of 1-form boundary zero modes is proven. Obtaining the 1-loop determinant requires expansion of the index that presents an ambiguity, which we resolve using boundary condition. The resulting partition function reveals an overall dependence on the size of the background manifold, AdS2 as well as S2, as a sum of two types of contributions: a local one from local conformal anomaly through the index computation and a global one coming from zero modes. This overall size dependence matches with the perturbative 1-loop evaluation using heat kernel method.
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01 September 2023
An Erratum to this paper has been published: https://doi.org/10.1007/JHEP09(2023)003
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Acknowledgments
This work is supported by an appointment to the JRG Program at the APCTP through the Science and Technology Promotion Fund and Lottery Fund of the Korean Government, by the Korean Local Governments - Gyeongsangbuk-do Province and Pohang City, and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1F1A1048531). We wish to specially thank Ashoke Sen for in-depth discussions that helped us improve our understanding of the result. We also thank Francesco Benini, Dongwook Ghim, Rajesh Gupta, Christopher Herzog, Kazuo Hosomichi, Sungjoon Kim, Bum-Hoon Lee, Sungjay Lee, Sameer Murthy and Daisuke Yokoyama for many interesting and useful discussions related to the topics discussed in this paper. We also wish to specially thank Hee-Joong Chung and Rak-Kyeong Seong for their hospitality in Jeju National University and UNIST respectively during the initial stages of the project.
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ArXiv ePrint: 2302.10370
The original online version of this article was revised: Because of an error of the publisher on the article web-page on Springer Link, in title and abstract đť’© = (2, 2) was incorrectly replaced by đť’© = (2).
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Lezcano, A.G., Jeon, I. & Ray, A. Supersymmetric localization: đť’© = (2, 2) theories on S2 and AdS2. J. High Energ. Phys. 2023, 56 (2023). https://doi.org/10.1007/JHEP07(2023)056
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DOI: https://doi.org/10.1007/JHEP07(2023)056