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.

arXiv commentv2. 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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• Rank $r$ DT theory from rank $1$
• Playing with the index of M-theory
• Noncommutative Instantons in Diverse Dimensions
• Tetrahedron instantons in Donaldson-Thomas theory
• On the M2-Brane Index on Noncommutative Crepant Resolutions
• Hyperquot schemes on curves: virtual class and motivic invariants
• Double nested Hilbert schemes and the local stable pairs theory of curves
• Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters
• 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
• Invariants of nested Hilbert and Quot schemes on surfaces
• Equivariant Segre and Verlinde invariants for Quot schemes