Abstract
We investigate partition functions of the circular-quiver supersymmetric Chern-Simons theory which corresponds to the q-deformed Painlevé VI equation. From the partition functions with the lowest rank vanishing, where the circular quiver reduces to a linear one, we find 40 bilinear relations. The bilinear relations extend naturally to higher ranks if we regard these partition functions as those in the lowest order of the grand canonical partition functions in the fugacity. Furthermore, we show that these bilinear relations are a powerful tool to determine some unknown partition functions. We also elaborate the relation with some previous works on q-Painlevé equations.
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Acknowledgments
We are grateful to Giulio Bonelli, Yasuaki Hikida, Masazumi Honda, Yosuke Imamura, Hiroshi Itoyama, Hiroaki Kanno, Naotaka Kubo, Kimyeong Lee, Kazunobu Maruyoshi, Shun’ya Mizoguchi, Takahiro Nishinaka, Tadakatsu Sakai, Alessandro Tanzini, Yasuhiko Yamada, Shintarou Yanagida, Shuichi Yokoyama for valuable discussions and comments. The work of S. M. is supported by JSPS Grant-in-Aid for Scientific Research (C) #19K03829 and #22K03598. S. M. would like to thank Yukawa Institute for Theoretical Physics at Kyoto University for warm hospitality. Part of the exact values used to check the bilinear relations (4.6) was obtained by using the high-performance computing facility provided by Yukawa Institute for Theoretical Physics (Sushiki server). Preliminary results of this paper were presented in international conferences including “KEK Theory Workshop 2022” at Ibaraki, Japan, “6th International Conference on Holography, String Theory and Spacetime in Da Nang” at Danang, Vietnam and “Quantum Field Theories and Representation Theory” at Osaka, Japan. We are grateful to the organizers and also thank the participants for various valuable discussions.
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Moriyama, S., Nosaka, T. 40 bilinear relations of q-Painlevé VI from \( \mathcal{N} \) = 4 super Chern-Simons theory. J. High Energ. Phys. 2023, 191 (2023). https://doi.org/10.1007/JHEP08(2023)191
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DOI: https://doi.org/10.1007/JHEP08(2023)191