We exploit the critical structure on the Quot scheme $\text {Quot}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$ , in particular the associated symmetric obstruction theory, in order to study rank r K-theoretic Donaldson-Thomas (DT) invariants of the local Calabi-Yau $3$ -fold ${{\mathbb {A}}}^3$ . We compute the associated partition function as a plethystic exponential, proving a conjecture proposed in string theory by Awata-Kanno and Benini-Bonelli-Poggi-Tanzini. A crucial step in the proof is the fact, nontrival if $r>1$ , that the invariants do not depend on the equivariant parameters of the framing torus $({{\mathbb {C}}}^\ast )^r$ . Reducing from K-theoretic to cohomological invariants, we compute the corresponding DT invariants, proving a conjecture of Szabo. Reducing further to enumerative DT invariants, we solve the higher rank DT theory of a pair $(X,F)$ , where F is an equivariant exceptional locally free sheaf on a projective toric $3$ -fold X. As a further refinement of the K-theoretic DT invariants, we formulate a mathematical definition of the chiral elliptic genus studied in physics. This allows us to define elliptic DT invariants of ${{\mathbb {A}}}^3$ in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.

In 2008, Klemm–Pandharipande defined Gopakumar–Vafa type invariants of a Calabi–Yau 4-folds $X$ using Gromov–Witten theory. Recently, Cao–Maulik–Toda proposed a conjectural description of these invariants in terms of stable pair theory. When $X$ is the total space of the sum of two line bundles over a surface $S$, and all stable pairs are scheme theoretically supported on the zero section, we express stable pair invariants in terms of intersection numbers on Hilbert schemes of points on $S$. As an application, we obtain new verifications of the Cao–Maulik–Toda conjectures for low-degree curve classes and find connections to Carlsson–Okounkov numbers. Some of our verifications involve genus zero Gopakumar–Vafa type invariants recently determined in the context of the log-local principle by Bousseau–Brini–van Garrel. Finally, using the vertex formalism, we provide a few more verifications of the Cao–Maulik–Toda conjectures when thickened curves contribute and also for the case of local $\mathbb{P}^3$.

Recently, Nekrasov discovered a new “genus” for Hilbert schemes of points on $${\mathbb {C}}^4$$ C 4 . We extend its definition to Hilbert schemes of curves and moduli spaces of stable pairs, and conjecture a K-theoretic DT/PT correspondence for toric Calabi–Yau 4-folds. We develop a K-theoretic vertex formalism, which allows us to verify our conjecture in several cases. Taking a certain limit of the equivariant parameters, we recover the cohomological DT/PT correspondence for toric Calabi–Yau 4-folds recently conjectured by the first two authors. Another limit gives a dimensional reduction to the K-theoretic DT/PT correspondence for toric 3-folds conjectured by Nekrasov–Okounkov. As an application of our techniques, we find a conjectural formula for the generating series of K-theoretic stable pair invariants of $$\text {Tot}_{{\mathbb {P}}^1}({\mathcal {O}}(-1) \oplus {\mathcal {O}}(-1) \oplus {\mathcal {O}})$$ Tot P 1 ( O ( - 1 ) ⊕ O ( - 1 ) ⊕ O ) . Upon dimensional reduction to the resolved conifold, we recover a formula which was recently proved by Kononov–Okounkov–Osinenko.

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