Categorical and K-theoretic Donaldson–Thomas theory of $\mathbb {C}^3$ (part II)
Forum of Mathematics, Sigma, Tome 11 (2023)
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Quasi-BPS categories appear as summands in semiorthogonal decompositions of DT categories for Hilbert schemes of points in the three-dimensional affine space and in the categorical Hall algebra of the two-dimensional affine space. In this paper, we prove several properties of quasi-BPS categories analogous to BPS sheaves in cohomological DT theory.We first prove a categorical analogue of Davison’s support lemma, namely that complexes in the quasi-BPS categories for coprime length and weight are supported over the small diagonal in the symmetric product of the three-dimensional affine space. The categorical support lemma is used to determine the torsion-free generator of the torus equivariant K-theory of the quasi-BPS category of coprime length and weight.We next construct a bialgebra structure on the torsion free equivariant K-theory of quasi-BPS categories for a fixed ratio of length and weight. We define the K-theoretic BPS space as the space of primitive elements with respect to the coproduct. We show that all localized equivariant K-theoretic BPS spaces are one-dimensional, which is a K-theoretic analogue of the computation of (numerical) BPS invariants of the three-dimensional affine space.
@article{10_1017_fms_2023_103,
author = {Tudor P\u{a}durariu and Yukinobu Toda},
title = {Categorical and {K-theoretic} {Donaldson{\textendash}Thomas} theory of $\mathbb {C}^3$ (part {II)}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {11},
year = {2023},
doi = {10.1017/fms.2023.103},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.103/}
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Tudor Pădurariu; Yukinobu Toda. Categorical and K-theoretic Donaldson–Thomas theory of $\mathbb {C}^3$ (part II). Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.103
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