Attractor invariants, brane tilings and crystals

Mozgovoy, Sergey; Pioline, Boris

doi:10.5802/aif.3682

Attractor invariants, brane tilings and crystals
[Invariants d’attracteurs, pavages branaires et crystaux]

Mozgovoy, Sergey ^1,² ; Pioline, Boris ³

¹ School of Mathematics Trinity College Dublin Dublin 2, Ireland
² Hamilton Mathematics Institute Trinity College Dublin Dublin 2, Ireland
³ Laboratoire de Physique Théorique et Hautes Energies, Sorbonne Université and CNRS UMR 7589 Campus Pierre et Marie Curie 4 place Jussieu 75005, Paris, France

Annales de l'Institut Fourier, Online first, 84 p.

Résumé
Abstract

Les états liés supersymétriques de D-branes sur une variété de Calabi-Yau $𝒳$ tridimensionnelle sont comptés par les invariants de Donaldson-Thomas généralisés $Ω_{Z} (γ)$ . Ceux-ci dépendent du caractère de Chern (ou charge électromagnétique) $γ \in H^{*} (𝒳)$ et d’une condition de stabilité (ou charge centrale) $Z$ . Ces invariants DT se déduisent des invariants d’attracteur $Ω_{*} (γ)$ , un cas particulier d’invariant DT où $Z = Z_{γ}$ est une perturbation générique de la condition d’auto-stabilité. Difficiles à calculer en général, ces invariants deviennent tractables dans le cas où $𝒳$ est une résolution crépante d’une variété de Calabi-Yau torique singulière, associée à un pavage périodique, et ainsi à un carquois avec potentiel. Nous passons en revue des résultats connus et des conjectures sur les invariants DT raffinés, encadrés ou non, et calculons explicitement les invariants d’attracteur pour une classe de variétés Calabi-Yau qui inclut l’espace total du fibré canonique sur une surface projective torique régulière, et la résolution crépante des quotients $ℂ^{3} / G$ . Dans tous ces cas, nous vérifions que $Ω_{*} (γ) = 0$ pour tous les vecteurs $γ$ à l’exception de ceux associés à une représentation simple du carquois et ceux dans le noyau de la forme d’Euler antisymétrisée. Sur la base de calculs explicites en dimension basse, nous conjecturons la valeur de tous les invariants d’attracteur, donnant ainsi une solution au problème du calcul des invariants DT pour toute condition de stabilité. Nous calculons également les invariants DT raffinés non-commutatifs et vérifions l’accord avec le comptage des cristaux fondus dans la limite non-raffinée.

Supersymmetric D-brane bound states on a Calabi–Yau threefold $𝒳$ are counted by generalized Donaldson–Thomas invariants $Ω_{Z} (γ)$ , depending on a Chern character (or electromagnetic charge) $γ \in H^{*} (𝒳)$ and a stability condition (or central charge) $Z$ . Attractor invariants $Ω_{*} (γ)$ are special instances of DT invariants, where $Z$ is the attractor stability condition $Z_{γ}$ (a generic perturbation of self-stability), from which DT invariants for any other stability condition can be deduced. While difficult to compute in general, these invariants become tractable when $𝒳$ is a crepant resolution of a singular toric Calabi–Yau threefold associated to a brane tiling, and hence to a quiver with potential. We survey some known results and conjectures about framed and unframed refined DT invariants in this context, and compute attractor invariants explicitly for a variety of toric Calabi–Yau threefolds, in particular when $𝒳$ is the total space of the canonical bundle of a smooth projective surface, or when $𝒳$ is a crepant resolution of $ℂ^{3} / G$ . We check that in all these cases, $Ω_{*} (γ) = 0$ unless $γ$ is the dimension vector of a simple representation or belongs to the kernel of the skew-symmetrized Euler form. Based on computations in small dimensions, we predict the values of all attractor invariants, thus potentially solving the problem of counting DT invariants of these threefolds in all stability chambers. We also compute the non-commutative refined DT invariants and verify that they agree with the counting of molten crystals in the unrefined limit.

Reçu le : 2021-08-07
Révisé le : 2022-10-05
Accepté le : 2023-05-16
Première publication : 2025-02-10

DOI : 10.5802/aif.3682

Classification : 14F05, 14J60, 14N10, 16G20
Keywords: Quivers, Donaldson–Thomas invariants, Toric Calabi–Yau threefolds
Mots-clés : invariants de Donaldson-Thomas, variété de Calabi-Yau

Affiliations des auteurs :

Mozgovoy, Sergey ^{1, 2} ; Pioline, Boris ³

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Mozgovoy, Sergey; Pioline, Boris. Attractor invariants, brane tilings and crystals. Annales de l'Institut Fourier, Online first, 84 p.

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