Chern-Simons Theory, Matrix Integrals, and Perturbative Three-Manifold Invariants

By Marcos Mariño. 2002

303 citations recorded by the source corpus.

The universal perturbative invariants of rational homology spheres can be extracted from the Chern-Simons partition function by combining perturbative and nonperturbative results. We spell out the general procedure to compute these invariants, and we work out in detail the case of Seifert spaces. By extending some previous results of Lawrence and Rozansky, the Chern-Simons partition function with arbitrary simply-laced group for these spaces is written in terms of matrix integrals. The analysis of the perturbative expansion amounts to the evaluation of averages in a Gaussian ensemble of random matrices. As a result, explicit expressions for the universal perturbative invariants of Seifert homology spheres up to order five are presented.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5e4d6821e48185a4bedb2402f31b9c2290dca8bf

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Citations in the corpus

Chern-Simons theory and topological strings (2004, unknown): “3) is crucial to guarantee integrality of the resulting amplitudes, as it was verified in Aganagic, Klemm, and Vafa (2002) and Mariño and Vafa (2002). If the open string amplitudes involves L boundaries, one has to specify L different framings, and (7.”
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].”
Torus Knots and Mirror Symmetry (2011, background): “oped in this paper to analyze the matrix model for torus knots will probably be very useful in order to understand the large Nlimit of the more general matrix models for Seifert spheres introduced in [37]. Such a large N limit would give a way to derive { 28 { the dual string geometries to Chern{Sim”
Instanton effects in ABJM theory from Fermi gas approach (2012, background): “1) (and its generalization to the ABJ model [21]) is related to the lens space matrix model [22, 23](5) by an analytic continuation of the rank of the gauge group, and it is further related to the topological string on local F0 via a large N duality.”
Fivebranes and 3-manifold homology (2016, background): “ e 2ˇi pk ˆ2 X n2Z N=pZ =S N e2ˇik n2 2p Z U(N) kCS[L(p;1)] n (6.1) where the contribution of the at connection labelled by n 2ZN=pZN (modulo permutation) can be represented by the following integral [78]: Z U(N) kCS[L(p;1)] n = 1 Q p k=1 N (n) k ! Z YN i=1 d˙ i 2ˇ e pki 4ˇ (˙ i 2ˇin=p)2 Y 6=j 2sinh”
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).”
Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c = 1 Matrix Models (2009, background): “The SU(N) Chern–Simons gauge theory on a generic three– manifold has been realized as a matrix model in [8]; see [58] for a review.”
Remodeling the B-Model (2007, methodology): “19) it just its two-point function [38], as expected from the identification of the recursive procedure with the theory of the “quantum” chiral boson on the mirror curve.”
Non-perturbative effects and the refined topological string (2013, result): “One of the first insights which made possible a precise quantitative understanding of the ABJM matrix model is its equivalence [20] to the matrix model describing Chern–Simons theory on RP3 [21], which is dual at large N to topological string theory on local P1 × P1 [22].”
Analytic Continuation Of Chern-Simons Theory (2010, methodology): “While it is impractical to give complete references on Chern-Simons theory, some contributions comparing the asymptotic or in some cases exact behavior of the Chern-Simons partition function to geometry, but not focusing on analytic continuation, are [22–34].”
Exact results for Wilson loops in superconformal Chern-Simons theories with matter (2009, methodology): “r, which can be written in a supersymmetric form [6]. Here we are able to use localization to recover some well known results on the reduction of the Chern-Simons partition function to a matrix model [7, 8], as well as reproducing some very simple knot invariants [9]. Another example is ABJM theory.”
Knot Homology from Refined Chern-Simons Theory (2011, methodology): “We derive a matrix integral expression for the S matrix of the theory, refining the Chern-Simons matrix model of [26,27].”
M5-branes from gauge theories on the 5-sphere (2012, background): “root 2sinh(πα(λ)) , (3.18) where Wis the Weyl group. One thus ﬁnds that the perturbative part of the partition function, with 16 SUSY, takes the form of the pure Chern-Simons partition function on S3 [24]. See also [25, 26] for some later studies of the same expression. For simplicity, let us ﬁrst c”
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].”
Nonperturbative effects and nonperturbative definitions in matrix models and topological strings (2008, methodology): “In this case, near the orbifold point in moduli space which is dual to the Gaussian point of Chern– Simons theory, the matrix model realization can be made explicit by using the matrix integral representation of Chern–Simons theory found in [60].”
Non-Abelian localization for Chern-Simons theory (2005, methodology): “In this case, we derive a formula originally obtained by Rozansky in [18] by again working empirically from the known formula for the partition function.”