Higher rank K-theoretic Donaldson-Thomas Theory of points

Abstract

We 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 rank r K-theoretic Donaldson-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 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 ${{\mathbb {A}}}^3$ in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.

The local motivic DT/PT correspondence

“…In higher rank, the starting point of motivic DT theory is the observation that is a global critical locus [1, Theorem 2.6]. This has also been exploited to prove a plethystic formula (the Awata–Kanno conjecture in String Theory) for the partition function of higher rank K‐theoretic DT invariants [13].…”

Framed sheaves on projective space and Quot schemes

“…We emphasise this since it is the starting point of higher rank Donaldson-Thomas theory of points in all its flavours: enumerative [2,25], motivic [6,24], K-theoretic [13].…”

“…can be embedded in a smooth quasiprojective variety U r ,n,m , called the non-commutative Quot scheme in [2,13], as follows. Consider the m-loop quiver, i.e.…”

“…1). This construction is called r -framingfor m = 3 it has some relevance in motivic Donaldson-Thomas theory [6,7] and K-theoretic Donaldson-Thomas theory [13]. It is also performed with care in [15] in the r = 1 case and in [14] for arbitrary r .…”

“…2 with a conjecture suggesting that Oprea's obstruction theory might take a very explicit form (Conjecture 2.12). Quot schemes also received a lot of attention lately in enumerative geometry [13,21,23,25], and in the context of motivic invariants [9,17,24].…”