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.
On the Definition of the Partition Function in Quantum Regge Calculus
