Matrix model as a mirror of Chern-Simons theory

By Mina Aganagic, A. Klemm, Marcos Mariño, C. Vafa. 2002

342 citations recorded by the source corpus.

Using mirror symmetry, we show that Chern-Simons theory on certain manifolds such as lens spaces reduces to a novel class of hermitean matrix models, where the measure is that of unitary matrix models. We show that this agrees with the more conventional canonical quantization of Chern-Simons theory. Moreover, large-N dualities in this context lead to computation of all genus A-model topological amplitudes on toric Calabi-Yau manifolds in terms of matrix integrals. In the context of type-IIA superstring compactifications on these Calabi-Yau manifolds with wrapped D6 branes (which are dual to M-theory on G2 manifolds) this leads to engineering and solving F-terms for = 1 supersymmetric gauge theories with superpotentials involving certain multi-trace operators.

Subfields: Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/d37deae2ccb3132b8190a16ffb194ce3c0e44f59

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Citations in the corpus

Nonperturbative effects and nonperturbative definitions in matrix models and topological strings (2008, methodology): “The nonperturbative answer, which is the Chern–Simons partition function on S3/Zp, is given by [63, 1] ZCS(N, gs) = ∑ N1+···+Np=N ζN11 · · · ζ Np p Z(N1, · · · , Np), (6.21) where ζj = exp (πik̂ p (j − 1)2 ) , j = 1, · · · , p (6.22) – 44 – and k̂ = k +N is the shifted coupling constant of Chern–Sim”
Knot Homology from Refined Chern-Simons Theory (2011, methodology): “It easily follows [47] that the right hand side of (2.”
A-polynomial, B-model, and quantization (2011, neutral): “23) have been explicitly constructed in [18, 19, 20, 21, 22, 23, 24].”
The Topological Vertex (2003, methodology): “In the closed string case the vertex can be checked, in principle to all genera, by comparison with mirror B-model calculations using holomorphic anomaly [8], see [41,42,27], as well as against A-model localisation calculations using the techniques of [16,43,44].”
Summing up all genus free energy of ABJM matrix model (2011, methodology): “The ABJM matrix model is equivalent to the so-called Lens space matrix model [23, 24], which computes the partition function of the U(N1) × U(N2) branch of the Chern-Simons theory on the Lens space L(2, 1) = S(3)/Z2, by the analytic continuation N2 → −N2 [22].”
Gravitational corrections in supersymmetric gauge theory and matrix models (2002, methodology): “It can be shown that for this model the propagators can be chosen to vanish except for S [18].”
Chern-Simons theory and topological strings (2004, methodology): “This idea was suggested by Aganagic, Mari˜no, and Vafa (2004) and Iqbal (2002), and was developed into a complete set of rules by Aganagic, Klemm, Mari˜no, and Vafa (2003) .”
Open string amplitudes and large order behavior in topological string theory (2006, background): “Notice that the “effective” matrix model description introduced here is rather different from the multi-matrix models of [53, 3].”
Fivebranes and 3-manifold homology (2016, background): “However their unrefined limit reproduces blocks Za in the CS partition function with an extra factor: Ẑ(unred)a |t=−1 = Ẑa (q; q)∞ (6.50) The denominator can be interpreted as the contribution of point-like instantons in twisted N = 4 SYM on R+ ×M3 realizing CS on its boundary.”
Exact results in ABJM theory from topological strings (2009, methodology): “Many results of [13] were rederived in the paper [14, 19] by using standard matrix model techniques, which we now review.”
Liouville Field Theory — A decade after the revolution (2004, background): “(62)The exact mirror of finite s is also known [205], [206].”
Lectures on localization and matrix models in supersymmetric Chern–Simons-matter theories (2011, methodology): “These models were introduced in [72], and the case p = 2 was extensively studied in [4].”
Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant (2012, methodology): “These results have been obtained by a standard matrix model technique after the analytic continuation [28] to the lens space L(2, 1) = S(3)/Z2 matrix model [39, 40], which is obtained from the pure Chern-Simons theory on L(2, 1).”
Nonperturbative aspects of ABJM theory (2011, methodology): “Through this procedure, the full three dimensional path integral gets reduced to a finite dimensional integral closely related to – 1 – the matrix model describing topological Chern–Simons on the lens space S3/Z2 [3, 4], as it was pointed out in [5, 6].”
Non-perturbative effects and the refined topological string (2013, background): “This is certainly natural from the point of view of the duality between this topological string and ChernSimons theory on RP3 [22], and the relation between the Chern-Simons matrix model on RP3 and the ABJM matrix model [20].”
Holomorphic anomaly in gauge theories and matrix models (2006, methodology): “In this Appendix we give some details of the matrix model calculations following the approach in [8, 7].”
Black Holes, q-Deformed 2d Yang-Mills, and Non-perturbative Topological Strings (2004, background): “Since the YM path integral localizes to configurations where Φ is covariantly constant,so in particular Φ and A commute, integrating over the angles gives (see [24] for details.”
Topological Strings from Quantum Mechanics (2014, background): “In the case of ABJM or local P1 × P1, this is largely “explained” by the fact that the canonical partition function ZX(N, ~) has an explicit matrix model representation, whose ’t Hooft expansion involves the standard topological string [12, 69].”