Coisotropic branes, noncommutativity, and the mirror correspondence

Aldi, Marco; Zaslow, Eric

doi:10.1088/1126-6708/2005/06/019

Cited by 19 publications

(31 citation statements)

References 5 publications

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“…None of these assertions really require the analysis in the present paper, and indeed fuller and more direct explanations have been given in [9], following a variety of earlier clues and examples [4][5][6][7][8]. In our brief and somewhat cavalier explanation here, we have omitted some key details (involving the conditions for the space of (B cc , B L ) strings to have a hermitian structure, the role of the flat Chan-Paton line bundle of B L , etc.)…”

Section: Recovering the Hilbert Spacementioning

confidence: 91%

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A new look at the path integral of quantum mechanics

Witten

2010

Surveys in Differential Geometry

122

220

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Abstract. The Feynman path integral of ordinary quantum mechanics is complexified and it is shown that possible integration cycles for this complexified integral are associated with branes in a two-dimensional A-model. This provides a fairly direct explanation of the relationship of the A-model to quantum mechanics; such a relationship has been explored from several points of view in the last few years. These phenomena have an analog for ChernSimons gauge theory in three dimensions: integration cycles in the path integral of this theory can be derived from N = 4 super YangMills theory in four dimensions. Hence, under certain conditions, a Chern-Simons path integral in three dimensions is equivalent to an N = 4 path integral in four dimensions.

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Section: Recovering the Hilbert Spacementioning

confidence: 91%

“…It has been known from various points of view [4][5][6][7][8][9] that there is a relationship between the A-model and quantization. In the present paper, we make a new and particularly direct proposal for what the key relation is: the most basic coisotropic A-brane gives a new integration cycle in the Feynman integral of quantum mechanics.…”

Section: Introductionmentioning

confidence: 99%

A new look at the path integral of quantum mechanics

Witten

2010

Surveys in Differential Geometry

122

220

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“…The brane construction that we will need in this paper relies on aspects of the two-dimensional topological A-model that are not novel [10][11][12][13][14] but are perhaps also not well known. We will here summarize the facts that will be used later in the paper, without attempting full explanations.…”

Section: Coisotropic A-branesmentioning

confidence: 99%

“…Our basic idea is to map this problem to a brane construction in two dimensions. Under certain conditions, a two-dimensional A-model admits unusual branes [10] whose existence brings noncommutativity into the A-model in several related ways [11][12][13][14]. Of most direct relevance to us is an A-brane construction that leads to quantization of finite-dimensional classical phase spaces [14].…”

Section: Introductionmentioning

confidence: 99%

The omega deformation, branes, integrability and Liouville theory

Nekrasov

Witten

2010

J. High Energ. Phys.

220

381

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Abstract:We reformulate the Ω-deformation of four-dimensional gauge theory in a way that is valid away from fixed points of the associated group action. We use this reformulation together with the theory of coisotropic A-branes to explain recent results linking the Ω-deformation to integrable Hamiltonian systems in one direction and Liouville theory of two-dimensional conformal field theory in another direction.

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“…One of the most interesting dualities involving Lagrangian branes in the A-model is mirror symmetry. The original construction [77] of this duality was generalized to include Lagrangian branes in [59,60], and to the simplest configuration of coisotropic branes in [78], but an understanding of the duality for more general 5-brane setups is lacking.…”

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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Coisotropic branes, noncommutativity, and the mirror correspondence

Cited by 19 publications

References 5 publications

A new look at the path integral of quantum mechanics

A new look at the path integral of quantum mechanics

The omega deformation, branes, integrability and Liouville theory

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

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