The spectral curve of the lens space matrix model

Halmagyi, Nick; Yasnov, Vadim

doi:10.1088/1126-6708/2009/11/104

Cited by 69 publications

(139 citation statements)

References 23 publications

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“…In simple examples the integrals can be done directly in terms of known functions. In the case of ABJM theory, the matrix model is very similar to that of the lens space Chern-Simons matrix model for which the spectral curve is known [21][22][23]. This solution was used in [24] to calculate the expectation value of Wilson loops and in [16] to calculate the S 3 partition function.…”

Section: Localization In the Boundary Gauge Theory And The Airy Functionmentioning

confidence: 99%

Localization in supergravity and quantum AdS 4 /CFT3 holography

Dabholkar

Drukker

Gomes

2014

J. High Energ. Phys.

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Abstract:We compute the quantum gravity partition function of M-theory on AdS 4 ×X 7 by using localization techniques in four-dimensional gauged supergravity obtained by a consistent truncation on the Sasaki-Einstein manifold X 7 . The supergravity path integral reduces to a finite dimensional integral over two collective coordinates that parametrize the localizing instanton solutions. The renormalized action of the off-shell instanton solutions depends linearly and holomorphically on the "square root" prepotential evaluated at the center of AdS 4 . The partition function resembles the Laplace transform of the wave function of a topological string and with an assumption about the measure for the localization integral yields an Airy function in precise agreement with the computation from the boundary ABJM theory on a 3-sphere. Our bulk quantum gravity computation is nonperturbatively exact in four-dimensional Planck length but ignores corrections due to brane-instantons.

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Section: Localization In the Boundary Gauge Theory And The Airy Functionmentioning

confidence: 99%

Localization in supergravity and quantum AdS 4 /CFT3 holography

Dabholkar

Drukker

Gomes

2014

J. High Energ. Phys.

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“…The present work generalizes the analysis of the model r = 2 [1,18,31] relevant to study lens spaces, and the invariant of fiber knots in lens spaces is equal to invariants of torus knots in S 3 . Chern-Simons theory on M at large N is dual to type A open topological strings on T * M [49], and through geometric transitions, this can sometimes be related to closed topological strings on another target space X M .…”

Section: Perspectives In Topological Stringsmentioning

confidence: 65%

“…Chern-Simons theory on M at large N is dual to type A open topological strings on T * M [49], and through geometric transitions, this can sometimes be related to closed topological strings on another target space X M . This program has been completed for M = S 3 [29] and the lens spaces Z p 2 \S 3 /Z p 1 [1,18,31] and X M is obtained by cyclic quotient of the resolved conifold in both cases. At the level of the spectral curve, this just amounts to perform a fractional framing transforming on the mirror curve of the resolved conifold.…”

Section: Perspectives In Topological Stringsmentioning

confidence: 99%

“…, a r themselves are topological invariants of M. This is not true anymore in the presence of one or two exceptional fibers, i.e., for r 2 there exist homeomorphisms which do not preserve fibers and change (a 1 , a 2 ). r = 1, 2 give lens spaces, and since they are well understood from the point of view of Chern-Simons theory [18,31], we shall assume r 3 in the following.…”

Section: Classificationmentioning

confidence: 99%

“…The only difference in the result is a rescaling of the Newton polygon, and now the coefficients inside the Newton polygon depend on the collection of filling fractions l = N h. This dependence is in general transcendental, since the I are periods of the 1-form ln y d ln x on the spectral curve. For lens spaces, this analysis has been explicitly performed in [31], and it would be interesting to extend it to the general Seifert geometry.…”

Section: Non-trivial Flat Connectionsmentioning

confidence: 99%

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Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces

Borot

Eynard

Weiße

2016

Sel. Math. New Ser.

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We study in detail the large N expansion of SU(N ) and SO(N )/Sp(2N ) Chern-Simons partition function Z N (M) of 3-manifolds M that are either rational homology spheres or more generally Seifert fibered spaces. This partition function admits a matrix model-like representation, whose spectral curve can be characterized in terms of a certain scalar, linear, non-local Riemann-Hilbert problem (RHP). We develop tools necessary to address a class of such RHPs involving finite subgroups of PSL 2 (C). We associate with such problems a (maybe infinite) root system and describe the relevance of the orbits of the Weyl group in the construction of its solutions. These techniques are applied to the RHP relevant for Chern-Simons theory on Seifert spaces. When π 1 (M) is finite-i.e., for manifolds M that are quotients of S 3 by a finite isometry group of type ADE-we find that the Weyl group associated with the RHP is finite and the spectral curve is algebraic and can be in principle computed. We then show that the large N expansion of Z N (M) is computed by the topological recursion. This has consequences for the analyticity properties of SU/SO/Sp perturbative invariants of knots along fibers in M.Mathematics Subject Classification 14Hxx · 45Exx · 51Pxx · 57M27 · 60B20 · 81T45

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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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The spectral curve of the lens space matrix model

Cited by 69 publications

References 23 publications

Localization in supergravity and quantum AdS 4 /CFT3 holography

Localization in supergravity and quantum AdS 4 /CFT3 holography

Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces

The volume conjecture and topological strings

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