Higher rank K-theoretic Donaldson-Thomas Theory of points

Abstract

AbstractWe 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 rankr K-theoreticDonaldson-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 tocohomologicalinvariants, we compute the corresponding DT invariants, proving a conjecture of Szabo. Reducing further toenumerativeDT invariants, we solve the higher rank DT theory of a pair$(X,F)$, whereFis an equivariant exceptional locally free sheaf on a projective toric$3$-foldX.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 defineelliptic DT invariantsof${{\mathbb {A}}}^3$in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.