Higher rank motivic Donaldson–Thomas invariants of via wall-crossing, and asymptotics

Abstract

We compute, via motivic wall-crossing, the generating function of virtual motives of the Quot scheme of points on ${\mathbb{A}}^3$ , generalising to higher rank a result of Behrend–Bryan–Szendrői. We show that this motivic partition function converges to a Gaussian distribution, extending a result of Morrison.

Framed motivic Donaldson–Thomas invariants of small crepant resolutions

“…However, all the motivic invariants studied here will live in the subring  ℂ ⊂  μ ℂ of classes carrying the trivial μ-action, so we will not be concerned with the subtle structure of this larger ring. which is also the one used in [6,8]. We decided to adopt the conventions in [15,16] to keep close to the original formulae.…”

“…Building on the case of 𝑌 = 𝔸 3 , fully explored in [5][6][7]22], a virtual motive for the Quot scheme Quot 𝑌 (𝐹, 𝑛) was defined in [26, Def. ] can be computed using the cut-and-paste relations, after noticing that 𝑈 𝜎 𝑖 ≃ 𝔸 3 and 𝑈 𝜎 𝑖,𝑖+1 ≃ 𝔸 2 × ℂ * .…”

“…For instance, $$\begin{equation} {\big [\operatorname{GL}_k\big ]}_{\operatorname{\mathrm{vir}}} = {{\left(-{\mathbb {L}}^{\frac{1}{2}}\right)}}^{-k^2}\cdot \big [\operatorname{GL}_k\big ]. \end{equation}$$Remark Our definition of $$[\operatorname{crit}(f)]_{\operatorname{\mathrm{vir}}}$$ differs from the original one [2, § 2.8], which is also the one used in [6, 8]. We decided to adopt the conventions in [15, 16] to keep close to the original formulae.…”

“…vir ⋅ 𝑠 𝑛 , originating from the critical locus structure on Quot 𝔸 3 ( 𝒪 ⊕𝑟 , 𝑛 ) , is studied in detail in [5,6,22]. The series 𝖣𝖳 points 𝑟 (𝑌, 𝑠) was introduced and computed for all 3-folds 𝑌 in [26, § 4], generalising the 𝑟 = 1 case corresponding to Hilb 𝑛 𝑌 [2].…”

“…The series $$\mathsf {DT}_r^{\operatorname{points}}\big ({\mathbb {A}}^3,s\big ) = \sum _{n \ge 0} {\big [{\operatorname{Quot}_{{\mathbb {A}}^3}}\big (\mathcal O^{\oplus r},n\big )\big ]}_{\operatorname{\mathrm{vir}}}\cdot s^n$$, originating from the critical locus structure on $${\operatorname{Quot}_{{\mathbb {A}}^3}}\big (\mathcal O^{\oplus r},n\big )$$, is studied in detail in [5, 6, 22]. The series $$\mathsf {DT}_r^{\operatorname{points}}(Y,s)$$ was introduced and computed for all 3‐folds Y in [26, § 4], generalising the $$r=1$$ case corresponding to $${\operatorname{Hilb}^n}Y$$ [2].…”