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Torus knot polynomials and Susy Wilson loops. (English) Zbl 1305.57025

Summary: We give, using an explicit expression obtained in [V. F. R. Jones, Ann. Math. (2) 126, 335–388 (1987; Zbl 0631.57005)], 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 \({ (m, n) \leftrightarrow (n, m)}\) 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 \({\mathcal{N}}\) = 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.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
15B52 Random matrices (algebraic aspects)

Citations:

Zbl 0631.57005
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References:

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