Higher rank K-theoretic Donaldson-Thomas Theory of points

Abstract

We exploit the critical structure on the Quot scheme QuotA3(O⊕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 A3. 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 r > 1, that the invariants do not depend on the equivariant parameters of the framing torus (*)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 A3 in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.