Chern–Simons theory on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.gif" display="inline" overflow="scroll"><mml:mi>L</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> lens spaces and Gopakumar–Vafa duality

Brini, Andrea; Seminara, Domenico; Tanzini, Alessandro

doi:10.1016/j.geomphys.2009.11.006

Cited by 19 publications

(33 citation statements)

References 29 publications

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“…The matrix model in (4.2) can also be obtained by localizing the N = 2 U (N 1 +N 2 ) Chern-Simons theory on the squashed lens space S 3 √ ν /Z 2 with squashing parameter √ ν [62,70,71]. Selecting the particular vacuum that breaks the gauge group to U (N 1 ) × U (N 2 ), and then continuing it to the supermatrix version as done for the ABJ(M) matrix model [72][73][74], we land on (4.2). In fact, considering Chern-Simons theory with supergroup U (N 1 |N 2 ) is (at least perturbatively) equivalent to the lens space interpretation by rewriting its matrix model in the two-cut form and taking the analytic continuation N 2 → −N 2 .…”

Section: Properties and Relations With Other Matrix Modelsmentioning

confidence: 99%

A matrix model for the latitude Wilson loop in ABJM theory

Bianchi

Mauri

Penati

et al. 2018

J. High Energ. Phys.

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113

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In ABJ(M) theory, we propose a matrix model for the exact evaluation of BPS Wilson loops on a latitude circular contour, so providing a new weak-strong interpolation tool. Intriguingly, the matrix model turns out to be a particular case of that computing torus knot invariants in U(N 1 |N 2 ) Chern-Simons theory. At weak coupling we check our proposal against a three-loop computation, performed for generic framing, winding number and representation. The matrix model is amenable of a Fermi gas formulation, which we use to systematically compute the strong coupling and genus expansions. For the fermionic Wilson loop the leading planar behavior agrees with a previous string theory prediction. For the bosonic operator our result provides a clue for finding the corresponding string dual configuration. Our matrix model is consistent with recent proposals for computing Bremsstrahlung functions exactly in terms of latitude Wilson loops. As a by-product, we extend the conjecture for the exact B θ 1/6 Bremsstrahlung function to generic representations and test it with a four-loop perturbative computation. Finally, we propose an exact prediction for B 1/2 at unequal gauge group ranks.

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Section: Properties and Relations With Other Matrix Modelsmentioning

confidence: 99%

A matrix model for the latitude Wilson loop in ABJM theory

Bianchi

Mauri

Penati

et al. 2018

J. High Energ. Phys.

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113

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“…It would be interesting to generalize our discussion to other manifolds. For most spaces the topological string dual of CS theory is not known, but for Lens spaces the GV duality is discussed in [75,76], and the 3d N = 2 partition function is also known, so it should be possible to repeat all of our computations.…”

Section: Discussionmentioning

confidence: 99%

A string dual for partially topological Chern-Simons-matter theories

Aharony¹,

Feldman²,

Honda³

2019

J. High Energ. Phys.

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We consider a string dual of a partially topological U(N ) Chern-Simons-matter (PTCSM) theory recently introduced by Aganagic, Costello, McNamara and Vafa. In this theory, fundamental matter fields are coupled to the Chern-Simons theory in a way that depends only on a transverse holomorphic structure on a manifold; they are not fully dynamical, but the theory is also not fully topological. One description of this theory arises from topological strings on the deformed conifold T * S 3 with N Lagrangian 3-branes and additional coisotropic 'flavor' 5-branes. Applying the idea of the Gopakumar-Vafa duality to this setup, we suggest that this has a dual description as a topological string on the resolved conifold O (−1) ⊕ O (−1) → CP 1 , in the presence of coisotropic 5-branes. We test this duality by computing the annulus amplitude on the deformed conifold and the disc amplitude on the resolved conifold via equivariant localization, and we find an agreement between the two. We find a small discrepancy between the topological string results and the large N limit of the partition function of the PTCSM theory arising from the deformed conifold, computed via field theory localization by a method proposed by Aganagic et al. We discuss possible origins of the mismatch. b 40 9.3 The O(N ) free energy for Im(b 2 ) > 0 41 9.4 The O(N ) free energy for Im(b 2 ) < 0 44 -i -9.5 The O(N ) free energy for real and irrational b 2 45 9.6 The O(N ) free energy for rational b 2 45 10 Conclusions and future directions 46 A Derivations of the O(N ) free energy for rational b 2 49 A.1 The limit to rational b 2 from complex b 2 49 A.2 A direct computation at rational b 2 51References 53 7 Note that a Lagrangian 3-brane is a particular case of a coisotropic brane with rank(σ) = 0. 8 See [46-51] for a mathematical review.

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“…That a dual curve counting theory exists is encouraged by the successful test of this proposal for the case of L(p, 1) lens spaces in [5,53]. The case of more general 3-manifolds was considered in [15,16,25,27], and we review it below.…”

Section: The Gov Correspondence For Clifford-klein 3-manifoldsmentioning

confidence: 99%

“…The singular fiber X [0] above µ = 0 is the singular quadric det A = 0: this is the singular conifold of Example 2.1. The latter admits a canonical toric minimal resolution (25) π :…”

Section: Introductionmentioning

confidence: 99%

“…A further natural direction would be to seek the broadest generalisation of the correspondence in its original form beyond the basic case of the three-sphere; this would require to provide a description of the string dual of Chern-Simons both in terms of Gromov-Witten theory and of the Eynard-Orantin theory on a specific spectral curve setup. This program was initiated in [5] (see also [25,52,53]) for the case of lens spaces, and what is presumably its widest frame of validity has been recently described in [15,27].This short survey is meant as an overview of results contained in [15,27], putting them into context and highlighting their implications for quantum invariants and Calabi-Yau enumerative geometry alike. It is structured as follows: we first give a lightning review of the three-way relation between matrix models, intersection theory on M g,n , and the Eynard-Orantin recursion.…”

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confidence: 99%

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On the Gopakumar–Ooguri–Vafa correspondence for Clifford–Klein 3-manifolds

Brini¹

2018

Proceedings of Symposia in Pure Mathematics

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Gopakumar, Ooguri and Vafa famously proposed the existence of a correspondence between a topological gauge theory on one hand -U(N ) Chern-Simons theory on the three-sphere -and a topological string theory on the other -the topological A-model on the resolved conifold. On the physics side, this duality provides a concrete instance of the large N gauge/string correspondence where exact computations can be performed in detail; mathematically, it puts forward a triangle of striking relations between quantum invariants (Reshetikhin-Turaev-Witten) of knots and 3-manifolds, curve-counting invariants (Gromov-Witten/Donaldson-Thomas) of local Calabi-Yau 3-folds, and the Eynard-Orantin recursion for a specific class of spectral curves. I here survey recent results on the most general frame of validity of this correspondence and discuss some of its implications. 1 arXiv:1711.07826v1 [hep-th] 20 Nov 2017asymptotic expansion of the Reshetikhin-Turaev invariant associated to the quantum group U q (sl N ) at large N and fixed q N equates the formal Gromov-Witten potential of X in the genus expansion. This has a B-model counterpart due to recent developments in higher genus toric mirror symmetry a [19,42], where the same genus expansion can be phrased in terms of the Eynard-Orantin topological recursion [41] on the Hori-Iqbal-Vafa mirror curve of X [58,59].The GOV correspondence has had a profound impact for both communities involved. In Gromov-Witten / Donaldson-Thomas theory, it has laid the foundations of the use of large N dualities to solve the topological string on toric backgrounds [4,6,77] and to obtain all-genus results that went well beyond the existing localisation computations at the time, as well as some striking results for the intersection theory on moduli spaces of curves [75] and, via the relation of Chern-Simons theory to random matrices, an embryo of the remodeling proposal [19,71,73]. In the other direction, the integral structure of BPS invariants leads to non-trivial constraints for the structure of quantum knot invariants [66,68,74].Since the original correspondence of [47,81] was confined to the case where the gauge group is the unitary group U(N ), the base manifold is S 3 , and the knot is the trivial knot, a natural question to ask is whether a similar connection could be generalised to other classical gauge groups [17,18,89], knots other than the unknot [24,65,66] (see also [3] for a significant generalisation, in a rather different setting), as well as categorified/refined invariants of various types [7,51]. A further natural direction would be to seek the broadest generalisation of the correspondence in its original form beyond the basic case of the three-sphere; this would require to provide a description of the string dual of Chern-Simons both in terms of Gromov-Witten theory and of the Eynard-Orantin theory on a specific spectral curve setup. This program was initiated in [5] (see also [25,52,53]) for the case of lens spaces, and what is presumably its widest frame of validity has been recently ...

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Chern–Simons theory on L(p,q) lens spaces and Gopakumar–Vafa duality

Cited by 19 publications

References 29 publications

A matrix model for the latitude Wilson loop in ABJM theory

A matrix model for the latitude Wilson loop in ABJM theory

A string dual for partially topological Chern-Simons-matter theories

On the Gopakumar–Ooguri–Vafa correspondence for Clifford–Klein 3-manifolds

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