Motivic Donaldson–Thomas invariants of some quantized threefolds

Abstract

This paper is motivated by the question of how motivic Donaldson-Thomas invariants behave in families. We compute the invariants for some simple families of noncommutative Calabi-Yau threefolds, defined by quivers with homogeneous potentials. These families give deformation quantizations of affine three-space, the resolved conifold, and the resolution of the transversal An-singularity. It turns out that their invariants are generically constant, but jump at special values of the deformation parameter, such as roots of unity. The corresponding generating series are written in closed form, as plethystic exponentials of simple rational functions. While our results are limited by the standard dimensional reduction techniques that we employ, they nevertheless allow us to conjecture formulae for more interesting cases, such as the elliptic Sklyanin algebras.

The Hodge-Elliptic Genus, Spinning BPS States, and Black Holes

“…In §3, we saw that the natural field theory definition of the Hodge-elliptic genus has 'jumps' at points in moduli space with enhanced chiral algebra (while admitting a 'generic' answer that is constant away from such points). This implies a similar jumping phenomenon for the motivic Donaldson-Thomas invariants, which has been seen explicitly in computations on non-commutative Calabi-Yau threefolds [42]. In particular, varying in the complex or Kähler moduli space should correspond to varying the algebraic structure or the stability structure on the derived category, which may a priori lead to different motivic Donaldson-Thomas counts.…”

Brauer–Severi motives and Donaldson–Thomas invariants of quantized threefolds

“…We are interested in the (naive, equivariant) motives of the fibers of this functional which we denote by M W m,n (λ) = T r(W ) −1 (λ). Recall that to each isomorphism class of a complex variety X (equipped with a good action of a finite group of roots of unity) we associate its naive equivariant motive [X] which is an element in the ring Kμ 0 (Var C )[L −1/2 ] (see [4] or [3]) and is subject to the scissor-and product-relations k=0 (L n −L k ) and from [3, 2.2] that [A n , µ k ] = L n for a linear action of µ k on A n . This ring is equipped with a plethystic exponential Exp, see for example [2] and [4].…”

“…, m k are the dimensions of simple representations of R W and M i ∈ M C are motivic expressions without denominators, with M 1 the virtual motive of the scheme parametrizing (simple) 1-dimensional representations. Evidence for this conjecture comes from cases where the superpotential admits a cut and hence one can use dimensional reduction, introduced by A. Morrison in [12], as in the case of quantum affine three-space [3].…”

“…The purpose of this paper is to introduce an inductive procedure to test the conjectural exponential expressions given in [3] in other interesting cases such as the homogenized Weyl algebra and elliptic Sklyanin algebras. To this end we introduce the following quotient of the free necklace algebra on m variables…”

“…In section 4 we will compute the first two terms of U W (t) in the case of the quantized 3-space in a variety of ways. In the final section we repeat the computation for the homogenized Weyl algebra and compare it to the conjectured expression of [3]. In [11] we will compute the case of the elliptic Sklyanin algebras.…”