Motivic degree zero Donaldson–Thomas invariants

By K. Behrend, Jim Bryan, Balázs Szendrői. 2009

179 citations recorded by the source corpus.

Given a smooth complex threefold X, we define the virtual motive $[\operatorname{Hilb}^{n}(X)]_{\operatorname {vir}}$ of the Hilbert scheme of n points on X. In the case when X is Calabi–Yau, $[\operatorname{Hilb}^{n}(X)]_{\operatorname{vir}}$ gives a motivic refinement of the n-point degree zero Donaldson–Thomas invariant of X. The key example is X=ℂ3, where the Hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety, and its virtual motive is defined in terms of the Denef–Loeser motivic nearby fiber. A crucial technical result asserts that if a function is equivariant with respect to a suitable torus action, its motivic nearby fiber is simply given by the motivic class of a general fiber. This allows us to compute the generating function of the virtual motives $[\operatorname{Hilb}^{n} (\mathbb{C}^{3})]_{\operatorname{vir}}$ via a direct computation involving the motivic class of the commuting variety. We then give a formula for the generating function for arbitrary X as a motivic exponential, generalizing known results in lower dimensions. The weight polynomial specialization leads to a product formula in terms of deformed MacMahon functions, analogous to Göttsche’s formula for the Poincaré polynomials of the Hilbert schemes of points on surfaces.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f9c6737b29cb0dda36f3a4edc1889f06d84512db

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In Discrete Mathematics and Combinatorics

1. L. Gottsche
2. Ben Davison
3. Vivek V. Shende
4. Marko Stosic
5. Yaping Yang
6. Sergei Gukov
7. D. Joyce
8. Yinan Song

In Geometry and Topology

1. Yinan Song
2. Marko Stosic
3. Ben Davison
4. Vivek V. Shende
5. L. Gottsche
6. D. Joyce
7. Yaping Yang
8. Sergei Gukov

In Mathematical Physics

1. Marko Stosic
2. Ben Davison
3. Yinan Song
4. L. Gottsche
5. D. Joyce
6. Vivek V. Shende
7. Yaping Yang
8. Sergei Gukov

Citations in the corpus

• Refined curve counting on complex surfaces (2012, background): “(For further discussions of motivic DT theory, see Kontsevich and Soibelman [42], Behrend, Bryan and Szendrői [6], Choi, Katz and Klemm [15], and Bussi, Joyce and Meinhardt [11].”
• The integrality conjecture and the cohomology of preprojective stacks (2016, background): “ → X×C∗ gives a faithful functor MMHM(X) → MHM(X) — one should think of this functor as “forgetting monodromy.” There is an embedding of monoidal categories τ: MHM(X) → MMHM(X) defined by (16) τ= •⊠QC∗[1]. One should think of this functor as turning a mixed Hodge module into a monodromic mixed Hodge ”
• The cohomological Hall algebra of a preprojective algebra (2014, background): “R(J C W,v),Q) to the critical CoHA M v∈NI H∗ c,Gv(Rep(Γ,v),ϕtrWv) ∨ with multiplication defined in [KoSo11]. 46 Y. YANG AND G. ZHAO It has been predicted, by the calculation of generating functions in [BBS13], that the critical CoHA can be described in terms of CHGv(R(JC W,v),Q). In this sense, Theor”
• A theory of generalized Donaldson–Thomas invariants (2008, result): “These quiver analogues of Mα,nstp (τ ′), P Iα,n(τ ′) are not new, similar things have been studied in quiver theory by Nakajima, Reineke, Szendrői and other authors for some years [24, 82–85, 88, 89, 99].”
• Homological algebra of knots and BPS states (2011, background): “which can be identified with the set of integers Z. Another feature familiar to the practitionersof the refined / motivic DonaldsonThomas theory is that HBPS can jump as one varies stability conditions [16, 17, 18, 19, 20, 21]. Thus, in a closely related type IIA superstring compactification on a Calab”