Branes, quivers and wave-functions

Kimura, Taro; Panfil, Miłosz; Sugimoto, Yuji; Sułkowski, Piotr

doi:10.48550/arxiv.2011.06783

Cited by 4 publications

(7 citation statements)

References 16 publications

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“…The third important aim of this work concerns threefolds without compact four-cycles: we show that for such manifolds open refined topological string amplitudes take form of generating series for symmetric quivers, and we identify corresponding quivers. Such a relation to quivers was originally found in the context of knots-quivers correspondence [26,27], its links with topological strings were further elucidated in [28,29], and (still in the unrefined case) it was generalized to Aganagic-Vafa branes in strip geometries [23,30]; related results are also discussed in [31,32]. Our results can be regarded as generalization of [23,30] to the refined case.…”

Section: Introductionsupporting

confidence: 65%

Refined open topological strings revisited

Cheng,

Sułkowski

2021

Preprint

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In this work we verify consistency of refined topological string theory from several perspectives. First, we advance the method of computing refined open amplitudes by means of geometric transitions. Based on such computations we show that refined open BPS invariants are non-negative integers for a large class of toric Calabi-Yau threefolds: an infinite class of strip geometries, closed topological vertex geometry, and some threefolds with compact four-cycles. Furthermore, for an infinite class of toric geometries without compact four-cycles we show that refined open string amplitudes take form of quiver generating series. This generalizes the relation to quivers found earlier in the unrefined case, implies that refined open BPS states are made of a finite number of elementary BPS states, and asserts that all refined open BPS invariants associated to a given brane are non-negative integers in consequence of their relation to (integer and non-negative) motivic Donaldson-Thomas invariants. Non-negativity of motivic Donaldson-Thomas invariants of a symmetric quiver is therefore crucial in the context of refined open topological strings. Furthermore, reinterpreting these results in terms of webs of five-branes, we analyze Hanany-Witten transitions in novel configurations involving lagrangian branes.

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Section: Introductionsupporting

confidence: 65%

Refined open topological strings revisited

Cheng,

Sułkowski

2021

Preprint

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“…These two seemingly different entities have been recently related by the so-called knots-quivers correspondence [1,2], which identifies various characteristics of knots with those of quivers and moduli spaces of their representations. The knotsquivers correspondence follows from properties of appropriately engineered brane systems in the resolved conifold that represent knots, thus it is intimately related to topological string theory and Gromov-Witten theory [3,4], and has been further generalized to branes in other Calabi-Yau manifolds [5,6], see also [7]. Other aspects and proofs (for two-bridge and arborescent knots and links) of the knots-quivers correspondence are discussed in [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Permutohedra for knots and quivers

Jankowski,

Kucharski,

Larraguível

et al. 2021

Preprint

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The knots-quivers correspondence states that various characteristics of a knot are encoded in the corresponding quiver and the moduli space of its representations. However, this correspondence is not a bijection: more than one quiver may be assigned to a given knot and encode the same information. In this work we study this phenomenon systematically and show that it is generic rather than exceptional. First, we find conditions that characterize equivalent quivers. Then we show that equivalent quivers arise in families that have the structure of permutohedra, and the set of all equivalent quivers for a given knot is parameterized by vertices of a graph made of several permutohedra glued together. These graphs can be also interpreted as webs of dual 3d N = 2 theories. All these results are intimately related to properties of homological diagrams for knots, as well as to multi-cover skein relations that arise in counting of holomorphic curves with boundaries on Lagrangian branes in Calabi-Yau three-folds.

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“…It was proved for all 2-bridge knots in [32] and for all arborescent knots in [33]. Generalizations and relations to various counts of BPS states were studied in [34][35][36][37][38]. Questions of uniqueness of quiver representations were studied in [39], and further discussed here in section 4.1.…”

Section: Relations Between Knots and Quiversmentioning

confidence: 99%

Knot homologies and generalized quiver partition functions

Ekholm¹,

Kucharski²,

Longhi³

2021

Preprint

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We conjecture a relation between generalized quiver partition functions and generating functions for symmetrically colored HOMFLY-PT polynomials and corresponding HOMFLY-PT homology Poincaré polynomials of a knot K. We interpret the generalized quiver nodes as certain basic holomorphic curves with boundary on the knot conormal L K in the resolved conifold, and the adjacency matrix as measuring their boundary linking. The simplest such curves are embedded disks with boundary in the primitive homology class of L K , other basic holomorphic curves consists of two parts: an embedded punctured sphere and a multiply covered punctured disk with boundary in a multiple of the primitive homology class of L K . We also study recursion relations for the partition functions connected to knot homologies. We show that, after a suitable change of variables, any (generalized) quiver partition function satisfies the recursion relation of a single toric brane in C 3 .

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Branes, quivers and wave-functions

Cited by 4 publications

References 16 publications

Refined open topological strings revisited

Refined open topological strings revisited

Permutohedra for knots and quivers

Knot homologies and generalized quiver partition functions

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