Abstract
We study actions of spherical twists on 2-Calabi–Yau categories with a Bridgeland stability condition. In these categories, we describe how to reduce the phase spread of a spherical object using stable spherical twists. In 2-Calabi–Yau quiver categories, we describe how to construct all spherical stable objects by applying simple spherical twists to the simple objects. As an application of this idea, we prove that for 2-Calabi–Yau categories associated to ADE quivers, all spherical objects lie in the braid group orbit of a simple object. We also give a new proof of the fact that the space of Bridgeland stability conditions is connected for these categories.
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Acknowledgements
We are grateful to Fabian Haiden, Ailsa Keating, Alexander Polishchuk, and Catharina Stroppel for discussions that prompted us to write this paper. We are indebted to Tom Bridgeland for his encouragement and for discussions and suggestions about phase improvement. We thank the anonymous referee for carefully reading the draft and suggesting a number of improvements to the presentation.
Funding
This work was supported by the Australian Research Council [DE180101360 to A.D., FT180100069 to A.M.L.].
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Bapat, A., Deopurkar, A. & Licata, A.M. Spherical objects and stability conditions on 2-Calabi–Yau quiver categories. Math. Z. 303, 13 (2023). https://doi.org/10.1007/s00209-022-03172-8
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DOI: https://doi.org/10.1007/s00209-022-03172-8