Abstract
In this note we review a construction of a BPS Hilbert space in an effective supersymmetric quiver theory with 4 supercharges. We argue abstractly that this space contains elements of an equivariant generalized cohomology theory \( {E}_G^{\ast}\left(-\right) \) of the quiver representation moduli space giving concretely Dolbeault cohomology, K-theory or elliptic cohomology depending on the spacial slice is compactified to a point, a circle or a torus respectively, and something more amorphous in other cases. Furthermore BPS instantons — basic contributors to interface defects or a Berry connection — induce a BPS algebra on the BPS Hilbert spaces representing Fourier-Mukai transforms on the quiver representation moduli spaces descending to an algebra over \( {E}_G^{\ast}\left(-\right) \) as its representation. In the cases when the quiver describes a toric Calabi-Yau three-fold (CY3) the algebra is a respective generalization of the quiver BPS Yangian algebra discussed in the literature, in more general cases it is given by an abstract generalized cohomological Hall algebra.
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Acknowledgments
The author would like to thank Wei Li and Masahito Yamazaki for collaborations on works [9, 54, 63, 107] preceding this paper and fruitful comments on the draft. Also the author would like to thank Mikhail Kapranov, Nikita Kolganov, Andrei Mironov, Viktor Mishnyakov, Gregory W. Moore and Alexei Morozov for illuminating comments at various stages of this project. This work is supported by the Russian Science Foundation (Grant No.20-12-00195).
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Galakhov, D. BPS states meet generalized cohomology. J. High Energ. Phys. 2023, 59 (2023). https://doi.org/10.1007/JHEP07(2023)059
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DOI: https://doi.org/10.1007/JHEP07(2023)059