$U(N)$ Torus Link Invariants in the Large $N$ limit from Matrix Model Approach

Abstract

In this paper we study $U(N)$ colored HOMFLY-PT polynomials of torus links in the double scaling limit (polynomial variable $q\to 1$, $N\to \infty$ keeping $q^N$ fixed). We show that, in this limit, the colored HOMFLY-PT polynomial of any $(L\alpha ,L\beta )$ torus link can be expressed in terms of the colored HOMFLY-PT polynomial of $(L,L)$ torus link. Using the connection between matrix models and the Chern-Simons field theoretic invariants, we show that the colored torus link invariants are uniquely expressed in terms of connected correlation functions of operators in $U(N)$ matrix model. We determine the leading and subleading contribution to some of the correlators at large $N$ from the matrix model approach and find that they match exactly with those obtained from the corresponding colored HOMFLY-PT polynomials.

arXiv comment35 pages, 1 figure, 1 table

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