DERIVED CATEGORIES OF SMALl TORIC CALABI-YAU 3-FOLDS AND CURVE COUNTING INVARIANTS

Nagao, Kentaro

doi:10.1093/qmath/har025

Cited by 44 publications

(87 citation statements)

References 29 publications

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“…Proof A Z ı -semistable object V has the phase t if and only if .OEV / D 0 and so the genericity in this paper agrees with the one in [24]. Then the claim follows from [24, Proposition 2.10, Corollary 2.12].…”

Section: Stability Condition and Tiltingsupporting

confidence: 58%

“…Then we get a chamber structure in .K num .mod fin A /˝R/ ' R I . Proposition 1.9 The chamber structure coincides with the affine root chamber structure of type A L 1 .Proof A Z ı -semistable object V has the phase t if and only if .OEV / D 0 and so the genericity in this paper agrees with the one in [24]. Then the claim follows from [24, Proposition 2.10, Corollary 2.12].…”

mentioning

confidence: 78%

“…1 T-structure and chamber structure 1.1 Non-commutative and commutative crepant resolutions Let be a map from I h to f˙g. In [24], following Hanany and Vegh [15], we introduced a quiver with a potential A D .Q ; w /, which is a non-commutative crepant resolution of X . First, we set…”

Section: Notationsmentioning

confidence: 99%

“…Note that M ncDT . ; ∅; ∅/ is the moduli space we have studied in [24]. We define Euler characteristic version of open non-commutative Donaldson-Thomas invariants by the Euler characteristics of the connected components of M ncDT .…”

Section: Introductionmentioning

confidence: 99%

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Non-commutative Donaldson–Thomas theory and vertex operators

Nagao

2011

Geom. Topol.

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1509Non-commutative Donaldson-Thomas theory and vertex operators KENTARO NAGAO In [26], we introduced a variant of non-commutative Donaldson-Thomas theory in a combinatorial way, which is related to the topological vertex by a wall-crossing phenomenon. In this paper, we (1) provide an alternative definition in a geometric way, (2) show that the two definitions agree with each other and (3) compute the invariants using the vertex operator method, following Okounkov, Reshetikhin and Vafa [29] and Young [12]. The stability parameter in the geometric definition determines the order of the vertex operators and hence we can understand the wall-crossing formula in non-commutative Donaldson-Thomas theory as the commutator relation of the vertex operators. 14N35 IntroductionLet X WD˚x 1 x 2 D x 3 L C x 4 L « be an affine toric Calabi-Yau 3-fold, which corresponds to the trapezoid with height 1, with length L C edge at the top and L at the bottom. Let Y ! X be a crepant resolution of X . Note that Y has L C 2 affine lines as torus invariant closed subvarieties (L WD L C C L ). In other words, there are L C 2 open edges in the toric graph of Y . Given an .LC2/-tuple of Young diagrams . ; / D C ; ;.1=2/ ; : : : ; Let A be a non-commutative crepant resolution of the affine toric Calabi-Yau 3-fold X . We can identify the derived category of coherent sheaves on Y and the one of A -modules by a derived equivalence. A parameter gives a Bridgeland's stability condition of this derived category, and hence a core A of a t-structure on it (Definition 1.7). In fact, we have two specific parameters such that the corresponding t-structures coincide with the ones given by Y or A respectively. Given an element in A , we can restrict it to get a sheaf on the smooth locus X sm . Since the singular locus X sing is compact, it makes sense to study those elements in A which are isomorphic to I ; outside a compact subset of X , or in other words, those elements in A which have the same asymptotic behavior as I ; . We will study the moduli spaces of such objects as noncommutative analogues of M DT . ; /. In general the ideal sheaf I ; is not an element in A , however P ; WD H 0 A I ; is always in A . We will construct the moduli space M ncDT ; ; of quotients of P ; in A as a GIT quotient (Section 5.1). Note that M ncDT . ; ∅; ∅/ is the moduli space we have studied in [24]. We define Euler characteristic version of open non-commutative Donaldson-Thomas invariants by the Euler characteristics of the connected components of M ncDT . ; ; / 2 .The torus action on Y induces a torus action on the moduli M ncDT . ; ; /. We will compute the Euler characteristic by counting the number of torus fixed points. For a generic , the core A of the t-structure is isomorphic to the category of A -modules, where A is associated with a quiver with a potential. Hence, we can describe the torus fixed point set on M ncDT . ; ; / in terms of a crystal melting model (see Okounkov, Reshetikhin and Vafa [29] and Ooguri and Yamazaki [30]), which we have studied in [26]...

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Section: Stability Condition and Tiltingsupporting

confidence: 58%

mentioning

confidence: 78%

Section: Notationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Non-commutative Donaldson–Thomas theory and vertex operators

Nagao

2011

Geom. Topol.

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“…This meant that his proof was also something of a mystery to us. Nagao also found a proof for small crepant resolutions of affine toric Calabi-Yau 3-folds using all of Joyce's theory, and one using KontsevichSoibelman [18].…”

Section: Introductionmentioning

confidence: 99%

Hilbert schemes and stable pairs: GIT and derived category wall crossings

Stoppa¹,

Thomas²

2011

Bul. Soc. Math. France

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Abstract. -We show that the Hilbert scheme of curves and Le Potier's moduli space of stable pairs with one dimensional support have a common GIT construction. The two spaces correspond to chambers on either side of a wall in the space of GIT linearisations.We explain why this is not enough to prove the "DT/PT wall crossing conjecture" relating the invariants derived from these moduli spaces when the underlying variety is a 3-fold. We then give a gentle introduction to a small part of Joyce's theory for such wall crossings, and use it to give a short proof of an identity relating the Euler characteristics of these moduli spaces.When the 3-fold is Calabi-Yau the identity is the Euler-characteristic analogue of the DT/PT wall crossing conjecture, but for general 3-folds it is something different, as we discuss. Résumé (Schémas de Hilbert et paires stables : GIT et croisements de murs de caté-gories dérivées)Nous montrons que le schéma de Hilbert de courbes et l'espace de modules de Le Potier de paires stables à support à une dimension, ont une construction GIT commune. Les deux espaces correspondents aux chambres de par et d'autre d'un mur dans l'espace de linéarisations GIT.Nous expliquons pourquoi cela ne suffit pas pour prouver la « conjecture de croisement de murs DT/PT » qui relie les invariants dérivés de ces espaces de modules quand la variété sous-jacente est un 3-fold. Nous donnons, ensuite, une introduction simple à une petite partie de la théorie de Joyce sur les croisements de murs de ce type, Quand le 3-fold est de type Calabi-Yau, l'identité est le pendant, pour la caractéristique d'Euler, de la conjecture de croisement de murs DT/PT, mais dans le cas général elle s'avère être différente de celle-ci, comme nous l'expliquons.

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Framed motivic Donaldson–Thomas invariants of small crepant resolutions

Cazzaniga

Ricolfi

2022

Mathematische Nachrichten

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For an arbitrary integer r≥1$r\ge 1$, we compute r‐framed motivic DT and PT invariants of small crepant resolutions of toric Calabi–Yau 3‐folds, establishing a “higher rank” version of the motivic DT/PT wall‐crossing formula. This generalises the work of Morrison and Nagao. Our formulae, in particular their relationship with the r=1$r=1$ theory, fit nicely in the current development of higher rank refined DT invariants.

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DERIVED CATEGORIES OF SMALl TORIC CALABI-YAU 3-FOLDS AND CURVE COUNTING INVARIANTS

Cited by 44 publications

References 29 publications

Non-commutative Donaldson–Thomas theory and vertex operators

Non-commutative Donaldson–Thomas theory and vertex operators

Hilbert schemes and stable pairs: GIT and derived category wall crossings

Framed motivic Donaldson–Thomas invariants of small crepant resolutions

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