Higher rank K-theoretic Donaldson-Thomas Theory of points

We exploit the critical structure on the Quot scheme Quot(A3) (O-circle plus 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 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 (C*)(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 A(3) in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.

Automated summary

By utilizing the critical structure and obstruction theory on Quot(A3), the study computes K-theoretic DT invariants of the local Calabi-Yau 3-fold with rank r, proving conjectures proposed in string theory. The invariants are shown to be independent of equivariant parameters, leading to further reductions and proofs of additional conjectures. Additionally, a mathematical definition of the chiral elliptic genus is formulated, allowing for the definition and resolution of elliptic DT invariants in arbitrary rank.