Higher rank K-theoretic Donaldson-Thomas theory of points

Abstract

We exploit the critical locus structure on the Quot scheme $\mathrm{Quot}_{\mathbb A^3}(\mathscr O^{\oplus r},n)$, 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.