The orbifold topological vertex

Bryan, Jenny; Cadman, Charles; Young, Ben

doi:10.1016/j.aim.2011.09.008

Cited by 59 publications

(133 citation statements)

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“…GW and DT theory have recently been defined for three dimensional orbifolds, i.e. spaces which are locally modeled by finite quotients of C 3 [1,2,4]. Moreover, the topological vertex algorithm has been generalized to three dimensional toric orbifolds in both GW theory [14] and DT theory [2].…”

Section: Context and Motivationmentioning

confidence: 99%

The loop Murnaghan–Nakayama rule

Ross

2013

J Algebr Comb

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Section: Context and Motivationmentioning

confidence: 99%

The loop Murnaghan–Nakayama rule

Ross

2013

J Algebr Comb

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“…He thanks to Jim Bryan for letting him know the result of [11] and recommending him to apply the vertex operator method in the setting of this paper. He also thanks to Osamu Iyama, Hiroaki Kanno, Hiraku Nakajima, Piotr Sulkowski, Yukinobu Toda, Masahito Yamazaki and Benjamin Young for useful comments.…”

Section: Acknowledgementsmentioning

confidence: 99%

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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“…Nonetheless, the actual perverse/ordinary comparison formula is of interest beyond the context of flops (for example in the setting of the crepant resolution conjecture for DonaldsonThomas invariants [2,3]). …”

Section: Explanationmentioning

confidence: 99%

Erratum to Donaldson–Thomas invariants and flops (J. reine angew. Math. 716 (2016), 103–145)

Calabrese

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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This is an erratum for [6] (which essentially coincides with [4]). The main result [6, Corollary 3.27] is correct. Indeed, if one assumes throughout to be working in [6, Situation 3.24] (namely, if only cares about flops) then the whole paper is fine as is. However, the key [6, Theorem 3.23] is incorrect as stated in its full generality. To reflect this, the arXiv version of the paper has been updated to [5].This erratum is divided in three sections. The first provides an explanation as to why some statements in [6] need to be modified. The second goes through the differences between [6] and [5]. The third is an appendix to [6], which is parallel to [5, Section 4].Acknowledgement. We would like to thank Jørgen Rennemo for spotting the mistake in [6] and an anonymous referee for very useful comments.

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The orbifold topological vertex

Cited by 59 publications

References 28 publications

The loop Murnaghan–Nakayama rule

The loop Murnaghan–Nakayama rule

Non-commutative Donaldson–Thomas theory and vertex operators

Erratum to Donaldson–Thomas invariants and flops (J. reine angew. Math. 716 (2016), 103–145)

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