Higher rank K-theoretic Donaldson-Thomas theory of points

Abstract

We exploit the critical locus structure on the Quot scheme ${\mathrm{Quot}}_{{\mathbb{A}}^3}({\mathcal{O}}^{\oplus r},n)$, in particular the associated symmetric obstruction theory, in order to define rank $r$ K-theoretic Donaldson-Thomas invariants of the 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 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 vector bundle on a projective toric $3$-fold $X$. Finally, we give a mathematical definition of the chiral elliptic genus studied in physics by Benini-Bonelli-Poggi-Tanzini. This allows us to define elliptic DT invariants of ${\mathbb{A}}^3$ in arbitrary rank, and to study their first properties.

v2. Replaced appendix with Theorem 6.5. Minor changes and corrections following referee's comments. 50 pages. Accepted for publication in Forum Math. Sigma

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• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• Double nested Hilbert schemes and the local stable pairs theory of curves
• A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds
• Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi-Yau 4-folds
• Equivariant Segre and Verlinde invariants for Quot schemes
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