Large N Duality, Lens Spaces and the Chern-Simons Matrix Model

Halmagyi, Nick; Okuda, Takuya; Yasnov, Vadim

doi:10.1088/1126-6708/2004/04/014

Cited by 34 publications

(57 citation statements)

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“…They are necessary if one wants to obtain full analytic results when the geometry is something more complicated than S 3 . Note that matrix models in the lens space case have already been studied (with loop equations) [13], but if one desires an all order result as in [8], the knowledge of orthogonal polynomials is then necessary. After the construction of the biorthogonal Stieltjes-Wigert polynomials in Section 2, we discuss some of their mathematical properties in Section 3.…”

Section: From the Point Of View Of Topological Strings This Describementioning

confidence: 99%

Chern-Simons matrix models and Stieltjes-Wigert polynomials

Dolivet

Tierz

2007

Journal of Mathematical Physics

108

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Abstract. Employing the random matrix formulation of Chern-Simons theory on Seifert manifolds, we show how the Stieltjes-Wigert orthogonal polynomials are useful in exact computations in Chern-Simons matrix models. We construct a biorthogonal extension of the Stieltjes-Wigert polynomials, not available in the literature, necessary to study Chern-Simons matrix models when the geometry is a lens space. We also discuss several other results based on the properties of the polynomials: the equivalence between the StieltjesWigert matrix model and the discrete model that appears in q-2D Yang-Mills and the relationship with Rogers-Szegö polynomials and the corresponding equivalence with an unitary matrix model. Finally, we also give a detailed proof of a result that relates quantum dimensions with averages of Schur polynomials in the Stieltjes-Wigert ensemble.

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Section: From the Point Of View Of Topological Strings This Describementioning

confidence: 99%

Chern-Simons matrix models and Stieltjes-Wigert polynomials

Dolivet

Tierz

2007

Journal of Mathematical Physics

108

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“…We give here some information about the Chern-Simons quantities that appear in the text, mainly in (20). For more information, see [5] and references therein.…”

Section: Appendix a Notation And Chern-simons Quantitiesmentioning

confidence: 99%

Discrete and oscillatory matrix models in Chern–Simons theory

Haro

Tierz

2005

Nuclear Physics B

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Abstract. We derive discrete and oscillatory Chern-Simons matrix models. The method is based on fundamental properties of the associated orthogonal polynomials. As an application, we show that the discrete model allows to prove and extend the recently found relationship between Chern-Simons theory and q-deformed 2dYM. In addition, the equivalence of the Chern-Simons matrix models gives a complementary view on the equivalence of effective superpotentials in N = 1 gauge theories.

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“…The idea is to extend the group Γ = Z p action to T * S 3 , pull it through the transition to O(−1) ⊕ O(−1) and then resolve the resulting quotient singularity (see Halmagyi, Okuda and Yasnov [70] for details). The dual quotient is a bundle over CP 1 fibered…”

Section: Construction Of Large N Dualsmentioning

confidence: 99%

“…Computationally only the L(p, 1) example has been considered so far [4,70]. Here the Chern-Simons partition function is known (see for example Hansen and Takata [71], Garoufalidis and Mariño [63] and Turaev [152]) and the Gromov-Witten free energy is computable via the topological vertex since the dual is toric.…”

Section: Construction Of Large N Dualsmentioning

confidence: 99%

“…The duality statement is more complicated than in the S 3 case and one has to split the Chern-Simons partition function into contributions from different (gauge classes of) flat connections before comparing to the Gromov-Witten side [4]. In addition to the analysis in [4], there is an indirect check of the duality based on mirror symmetry in [70]. In general, the precise meaning of the duality for more general manifolds remains an open question.…”

Section: Construction Of Large N Dualsmentioning

confidence: 99%

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Introduction to the Gopakumar–Vafa Large N Duality

Auckly¹,

Koshkin²

2007

Geometry and Topology Monographs

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Gopakumar-Vafa Large N Duality is a correspondence between Chern-Simons invariants of a link in a 3-manifold and relative Gromov-Witten invariants of a 6-dimensional symplectic manifold relative to a Lagrangian submanifold. We address the correspondence between the Chern-Simons free energy of S 3 with no link and the Gromov-Witten invariant of the resolved conifold in great detail. This case avoids mathematical difficulties in formulating a definition of relative Gromov-Witten invariants, but includes all of the important ideas.There is a vast amount of background material related to this duality. We make a point of collecting all of the background material required to check this duality in the case of the 3-sphere, and we have tried to present the material in a way complementary to the existing literature. This paper contains a large section on Gromov-Witten theory and a large section on quantum invariants of 3-manifolds. It also includes some physical motivation, but for the most part it avoids physical terminology.

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Large N Duality, Lens Spaces and the Chern-Simons Matrix Model

Abstract: We demonsrate that the spectral curve of the matrix model for Chern-Simons theory on the Lens space S 3 /Z p is precisely the Riemann surface which appears in the mirror to the blownup, orbifolded conifold. This provides the first check of the A-model large

Cited by 34 publications

References 20 publications

Chern-Simons matrix models and Stieltjes-Wigert polynomials

Chern-Simons matrix models and Stieltjes-Wigert polynomials

Discrete and oscillatory matrix models in Chern–Simons theory

Introduction to the Gopakumar–Vafa Large N Duality

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