Categorical and K-theoretic Donaldson-Thomas theory of $\mathbb{C}^3$ (part II)

TL;DR

This paper advances the understanding of categorical Donaldson-Thomas theory for ^3 by establishing support properties, constructing a bialgebra structure on K-theory, and analyzing BPS spaces, thereby connecting categorical and K-theoretic invariants.

Contribution

It proves a support lemma for quasi-BPS categories, constructs a bialgebra structure on their K-theory, and characterizes the primitive BPS spaces, providing new insights into categorical DT theory.

Findings

Support lemma for quasi-BPS categories supported on the small diagonal.

Construction of a bialgebra structure on equivariant K-theory.

Localized K-theoretic BPS spaces are one-dimensional.

Abstract

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.