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Title: Chern-Simons matrix models and Stieltjes-Wigert polynomials

Abstract

Employing the random matrix formulation of Chern-Simons theory on Seifert manifolds, we show how the Stieltjes-Wigert orthogonal polynomials are useful in exact computations in Chern-Simons matrix models. We construct a biorthogonal extension of the Stieltjes-Wigert polynomials, not available in the literature, necessary to study Chern-Simons matrix models when the geometry is a lens space. We also study the relationship between Stieltjes-Wigert and Rogers-Szegoe polynomials and the corresponding equivalence with a unitary matrix model. Finally, we give a detailed proof of a result that relates quantum dimensions with averages of Schur polynomials in the Stieltjes-Wigert ensemble.

Authors:
;  [1];  [2]
  1. Laboratoire de Physique Theorique de L'Ecole Normale Superieure, 24 rue L'Homond 75231, Paris Cedex 05 (France)
  2. (IEEC/CSIC), Campus UAB, Facultat de Ciencies, Torre C5-Parell-2A Planta, E-08193 Bellaterra, Barcelona (Spain)
Publication Date:
OSTI Identifier:
20929644
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 2; Other Information: DOI: 10.1063/1.2436734; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; CALCULATION METHODS; GEOMETRY; MATHEMATICAL SPACE; MATRICES; POLYNOMIALS; RANDOMNESS

Citation Formats

Dolivet, Yacine, Tierz, Miguel, and Institut d'Estudis Espacials de Catalunya. Chern-Simons matrix models and Stieltjes-Wigert polynomials. United States: N. p., 2007. Web. doi:10.1063/1.2436734.
Dolivet, Yacine, Tierz, Miguel, & Institut d'Estudis Espacials de Catalunya. Chern-Simons matrix models and Stieltjes-Wigert polynomials. United States. doi:10.1063/1.2436734.
Dolivet, Yacine, Tierz, Miguel, and Institut d'Estudis Espacials de Catalunya. Thu . "Chern-Simons matrix models and Stieltjes-Wigert polynomials". United States. doi:10.1063/1.2436734.
@article{osti_20929644,
title = {Chern-Simons matrix models and Stieltjes-Wigert polynomials},
author = {Dolivet, Yacine and Tierz, Miguel and Institut d'Estudis Espacials de Catalunya},
abstractNote = {Employing the random matrix formulation of Chern-Simons theory on Seifert manifolds, we show how the Stieltjes-Wigert orthogonal polynomials are useful in exact computations in Chern-Simons matrix models. We construct a biorthogonal extension of the Stieltjes-Wigert polynomials, not available in the literature, necessary to study Chern-Simons matrix models when the geometry is a lens space. We also study the relationship between Stieltjes-Wigert and Rogers-Szegoe polynomials and the corresponding equivalence with a unitary matrix model. Finally, we give a detailed proof of a result that relates quantum dimensions with averages of Schur polynomials in the Stieltjes-Wigert ensemble.},
doi = {10.1063/1.2436734},
journal = {Journal of Mathematical Physics},
number = 2,
volume = 48,
place = {United States},
year = {Thu Feb 15 00:00:00 EST 2007},
month = {Thu Feb 15 00:00:00 EST 2007}
}
  • The authors report on a study of the expectation values of Wilson loops in D = 3 Chern-Simons theory. The general skein relations (of higher orders) are derived for these expectation values. The authors show that the skein relations for the Wilson loops carrying the fundamental representations of the simple Lie algebras SO(n) and Sp(n) are sufficient to determine invariants for all knots and links and that the resulting link invariants agree with Kauffman polynomials. The polynomial for an unknotted circle is identified to the formal characters of the fundamental representations of these Lie algebras.
  • We obtain the skein relations for link polynomials from the Chern-Simons theories on SO({ital N}) and Sp(2{ital N}), using Witten's method which he has used in order to obtain such a relation for the Jones polynomial in the case of SU({ital N}). We show that for SO({ital N}) and Sp(2{ital N}) these polynomials are related to the Dubrovnik polynomial.
  • Kauffman knot polynomial invariants are discovered in classical abelian Chern-Simons field theory. A topological invariant t{sup I(L)} is constructed for a link L, where I is the abelian Chern-Simons action and t a formal constant. For oriented knotted vortex lines, t{sup I} satisfies the skein relations of the Kauffman R-polynomial; for un-oriented knotted lines, t{sup I} satisfies the skein relations of the Kauffman bracket polynomial. As an example the bracket polynomials of trefoil knots are computed, and the Jones polynomial is constructed from the bracket polynomial.
  • The gauge dependence of matrix elements of the Chern-Simons current {ital K}{sup {mu}} can be computed exactly in the (1+1)-dimensional Schwinger model. The matrix elements are gauge dependent even in the forward direction, i.e., when the momentum transfer vanishes. The Kogut-Susskind dipole mechanism is derived using functional-integral methods.
  • In this paper, it is shown that a gauge theory for complex scalar field with up to sextic self-interactions and a Chern-Simons term in 2 + 1 dimensions has solitons which may become bubbles of the stable broken-symmetry phase in a medium of the symmetric one producing the first order phase transition. In the non-relativistic limit scale invariance prevents the determination of an optimal bubble size. Possible extensions to 3 + 1 dimensions of bubbles of string type are indicated.