Topological strings, quiver varieties, and Rogers–Ramanujan identities

Zhu, Shengmao

doi:10.1007/s11139-017-9976-4

Cited by 12 publications

(17 citation statements)

References 49 publications

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“…Fourth, the role and the meaning of quivers that we identify should be understood in all other systems related to or engineered by topological string theory, such as supersymmetric gauge theories, vortex counting, etc. Fifth, it is of interest to understand if there are relations between quivers that we identify in this paper, and other quivers identified in related contexts [31][32][33][34]. Sixth, it is tempting to relate the combinatorics of quivers that we identify to crystal models related to the topological vertex.…”

Section: A Brief Quantitative Summarymentioning

confidence: 98%

Topological strings, strips and quivers

Panfil

Sułkowski

2019

J. High Energ. Phys.

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We find a direct relation between quiver representation theory and open topological string theory on a class of toric Calabi-Yau manifolds without compact four-cycles, also referred to as strip geometries. We show that various quantities that characterize open topological string theory on these manifolds, such as partition functions, Gromov-Witten invariants, or open BPS invariants, can be expressed in terms of characteristics of the moduli space of representations of the corresponding quiver. This has various deep consequences; in particular, expressing open BPS invariants in terms of motivic Donaldson-Thomas invariants, immediately proves integrality of the former ones. Taking advantage of the relation to quivers we also derive explicit expressions for classical open BPS invariants for an arbitrary strip geometry, which lead to a large set of number theoretic integrality statements. Furthermore, for a specific framing, open topological string partition functions for strip geometries take form of generalized q-hypergeometric functions, which leads to a novel representation of these functions in terms of quantum dilogarithms and integral invariants. We also study quantum curves and A-polynomials associated to quivers, various limits thereof, and their specializations relevant for strip geometries. The relation between toric manifolds and quivers can be regarded as a generalization of the knots-quivers correspondence to more general Calabi-Yau geometries.

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Section: A Brief Quantitative Summarymentioning

confidence: 98%

Topological strings, strips and quivers

Panfil

Sułkowski

2019

J. High Energ. Phys.

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“…These are prototype and important examples of quivers, and properties of their moduli spaces were discussed in [15,16]. The relation of this family of quivers to LMOV invariants of framed unknot (equivalently extremal invariants of twist knots, or open topological string amplitudes for branes in C 3 geometry) was presented in [10], and discussed also in [11,12]. Consider now the generating function of the full unknot invariants (5.1) -or equivalently open topological string amplitudes for branes in the resolved conifold geometry.…”

Section: Unknotmentioning

confidence: 99%

“…in [2-4, 7, 8], as well as for some infinite families of knots and representations [9,10]. In particular, in [10] the relation of the framed unknot invariants (equivalently extremal invariants of twist knots, as well as open topological string amplitudes for branes in C 3 geometry) to motivic Donaldson-Thomas invariants of the m-loop quiver was found, which led to the proof of integrality of BPS numbers in those cases; this relation was then analyzed and discussed also in [11,12].…”

Section: Introductionmentioning

confidence: 99%

Knots-quivers correspondence

Kucharski

Reineke

Stošić³

et al. 2019

Advances in Theoretical and Mathematical Physics

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We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing knots. We identify various structural properties of quivers associated to knots, and identify such quivers explicitly in many examples, including some infinite families of knots, all knots up to 6 crossings, and some knots with thick homology. Moreover, based on these properties, we derive previously unknown expressions for colored HOMFLY-PT polynomials and superpolynomials for various knots. For all knots, for which we identify the corresponding quivers, the LMOV conjecture for all symmetric representations (i.e. integrality of relevant BPS numbers) is automatically proved.

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“…To illustrate the above structure, let us consider the first two knots in this series. The value p = 1 corresponds to (3,5) torus knot. In this case we only have B 1,1 matrix, which is built out of X 1,1 and Y 1,1 , and takes form (5.8)…”

Section: (3 3p + 2) Torus Knotsmentioning

confidence: 99%

“…For example for the (3,5) and (3,8) To obtain the standard form of the HOMFLY-PT polynomial -right-handed with the zero framing -the quiver matrix has to be transformed as explained in the section 2.2. For example the C (3,5) quiver in this case becomes The t degrees defined in (5.12) can be read off from the diagonal of this matrix. From quivers that we found above, extremal colored HOMLFY-PT polynomials for (3, 3p+2) torus knots can be determined using (2.25).…”

Section: (3 3p + 2) Torus Knotsmentioning

confidence: 99%

Donaldson-Thomas invariants, torus knots, and lattice paths

Panfil

Stošić²,

Sułkowski

2018

Phys. Rev. D

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In this paper, we find and explore the correspondence between quivers, torus knots, and combinatorics of counting paths. Our first result pertains to quiver representation theory-we find explicit formulas for classical generating functions and Donaldson-Thomas invariants of an arbitrary symmetric quiver. We then focus on quivers corresponding to ðr; sÞ torus knots and show that their classical generating functions, in the extremal limit and framing rs, are generating functions of lattice paths under the line of the slope r=s. Generating functions of such paths satisfy extremal A-polynomial equations, which immediately follows after representing them in terms of the Duchon grammar. Moreover, these extremal A-polynomial equations encode Donaldson-Thomas invariants, which provides an interesting example of algebraicity of generating functions of these invariants. We also find a quantum generalization of these statements, i.e. a relation between motivic quiver generating functions, quantum extremal knot invariants, and q-weighted path counting. Finally, in the case of the unknot, we generalize this correspondence to the full HOMFLY-PT invariants and counting of Schröder paths.

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Topological strings, quiver varieties, and Rogers–Ramanujan identities

Cited by 12 publications

References 49 publications

Topological strings, strips and quivers

Topological strings, strips and quivers

Knots-quivers correspondence

Donaldson-Thomas invariants, torus knots, and lattice paths

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