Matrix model as a mirror of Chern-Simons theory

Aganagic, Mina; Klemm, Albrecht; Mariño, Marcos; Vafa, Cumrun

doi:10.1088/1126-6708/2004/02/010

Cited by 261 publications

(669 citation statements)

References 41 publications

(118 reference statements)

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“…This is the mirror of the computation of [22] and was discussed in the mirror setup in [29]. The effect on the free energy of the probe at v upon integrating out the stretched strings between branes at v i and the probe is…”

Section: B-branes and Closed String B-modelmentioning

confidence: 88%

The Topological Vertex

Aganagic

Klemm

Mariño

et al. 2004

Commun. Math. Phys.

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665

1,547

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We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kähler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum KodairaSpencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.

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Section: B-branes and Closed String B-modelmentioning

confidence: 88%

The Topological Vertex

Aganagic

Klemm

Mariño

et al. 2004

Commun. Math. Phys.

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665

1,547

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show abstract

“…The relation between t i andt i will be given below. The Picard-Fuchs equations as well as explicit expressions for the periods in terms of various expansion parameters can be found in [1,18].…”

Section: Topological String In the Associated Local Calabi-yau Geometmentioning

confidence: 99%

“…Generally, the amplitudes F (g) for g > 1 are defined recursively in terms of all amplitudes with lower g and the propagators S ij as defined in [3]. It can be shown that for this model the propagators can be chosen to vanish except for S tt [18]. It can be determined from the g = 0, 1 amplitudes by using the simplification of the holomorphic anomaly equation which occurs at genus one in the case with only one non-zero propagator:…”

Section: Topological String In the Associated Local Calabi-yau Geometmentioning

confidence: 99%

“…For F (1) we obtain from (2.12) and (2.15) We have calculated the h (g) i with the topological B-model methods described in [18], albeit at another expansion point, up to genus 3. These expressions can be found in appendix A.…”

Section: Topological String In the Associated Local Calabi-yau Geometmentioning

confidence: 99%

“…can be exponentiated and included in the action, as in [18]. This generates an interaction term between the two eigenvalue bands…”

Section: Perturbative Calculationmentioning

confidence: 99%

See 2 more Smart Citations

Gravitational corrections in supersymmetric gauge theory and matrix models

Klemm¹,

Mariño²,

Theisen³

2003

J. High Energy Phys.

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140

208

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Gravitational corrections in N = 1 and N = 2 supersymmetric gauge theories are obtained from topological string amplitudes. We show how they are recovered in matrix model computations. This provides a test of the proposal by Dijkgraaf and Vafa beyond the planar limit. Both, matrix model and topological string theory, are used to check a conjecture of Nekrasov concerning these gravitational couplings in Seiberg-Witten theory. Our analysis is performed for those gauge theories which are related to the cubic matrix model, i.e. pure SU (2) SeibergWitten theory and N = 2 U (N ) SYM broken to N = 1 via a cubic superpotential. We outline the computation of the topological amplitudes for the local Calabi-Yau manifolds which are relevant for these two cases.

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The volume conjecture and topological strings

Dijkgraaf

Fuji²

2009

Fortschritte der Physik

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In this paper, we discuss a relation between Jones-Witten theory of knot invariants and topological open string theory on the basis of the volume conjecture. We find a similar Hamiltonian structure for both theories, and interpret the AJ conjecture as the D-module structure for a D-brane partition function. In order to verify our claim, we compute the free energy for the annulus contributions in the topological string using the Chern-Simons matrix model, and find that it coincides with the Reidemeister torsion in the case of the figure-eight knot complement and the SnapPea census manifold m009.The hyperbolic three-manifold M is given bywhere Γ is a torsion-free and discrete subgroup of P SL(2; C) with the action (2.2).The volume of an ideal tetrahedron T αβγ is computed directly by using the metric (2.1) [33].The function Λ(θ) is called the Lobachevsky function. The volume formula can also be rewritten in terms of the Bloch-Wigner function D(z)(2.9)The gluing condition along each edge imposes constraints on the face angles of the ideal tetrahedra.(2.10)Furthermore, we should also consider the gluing condition for the boundary of the knot complement. In order to realize the torus as a boundary, the face angles must satisfy the completeness conditions along the meridian and the longitude.Mostow's rigidity theorem [34] implies that these conditions are solved uniquely with Im z i > 0 for all i. By summing all volumes for the ideal tetrahedra, we are able to determine the hyperbolic volume uniquely for each hyperbolic three-manifold.

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Matrix model as a mirror of Chern-Simons theory

Cited by 261 publications

References 41 publications

The Topological Vertex

The Topological Vertex

Gravitational corrections in supersymmetric gauge theory and matrix models

The volume conjecture and topological strings

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