Torus Knot Polynomials and Susy Wilson Loops

We give, using an explicit expression obtained in (Jones V, Ann Math 126:335, 1987 ), a basic hypergeometric representation of the HOMFLY polynomial of ( n , m ) torus knots, and present a number of equivalent expressions, all related by Heine’s transformations. Using this result, the ( m , n ) ↔ (...

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Bibliographic Details
Published in:Letters in mathematical physics Vol. 104; no. 12; pp. 1535 - 1556
Main Authors: Giasemidis, Georgios, Tierz, Miguel
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.12.2014
Subjects:
Complex Systems
Geometry
Group Theory and Generalizations
Mathematical and Computational Physics
Physics
Physics and Astronomy
Theoretical
15B52
HOMFLY polynomial
Random matrix models
Basic hypergeometric functions
57M27
33D15
ISSN:0377-9017, 1573-0530
Online Access:Get full text
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Summary:We give, using an explicit expression obtained in (Jones V, Ann Math 126:335, 1987 ), a basic hypergeometric representation of the HOMFLY polynomial of ( n , m ) torus knots, and present a number of equivalent expressions, all related by Heine’s transformations. Using this result, the ( m , n ) ↔ ( n , m ) symmetry and the leading polynomial at large N are explicit. We show the latter to be the Wilson loop of 2d Yang–Mills theory on the plane. In addition, after taking one winding to infinity, it becomes the Wilson loop in the zero instanton sector of the 2d Yang–Mills theory, which is known to give averages of Wilson loops in N = 4 SYM theory. We also give, using matrix models, an interpretation of the HOMFLY polynomial and the corresponding Jones–Rosso representation in terms of q -harmonic oscillators.
ISSN:0377-9017
1573-0530
DOI:10.1007/s11005-014-0724-z