U(N) torus link invariants in the large N limit from the matrix model approach

In this paper we study <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>U</a:mi><a:mo stretchy="false">(</a:mo><a:mi>N</a:mi><a:mo stretchy="false">)</a:mo></a:math> colored HOMFLY-PT polynomials of torus links in the double scaling limit (polynomial variable <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>q</e:mi><e:mo stretchy="false">→</e:mo><e:mn>1</e:mn></e:math>, <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>N</h:mi><h:mo stretchy="false">→</h:mo><h:mi>∞</h:mi></h:math> keeping <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>q</k:mi><k:mi>N</k:mi></k:msup></k:math> fixed). We show that, in this limit, the colored HOMFLY-PT polynomial of any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mo stretchy="false">(</m:mo><m:mi>L</m:mi><m:mi>α</m:mi><m:mo>,</m:mo><m:mi>L</m:mi><m:mi>β</m:mi><m:mo stretchy="false">)</m:mo></m:math> torus link can be expressed in terms of the colored HOMFLY-PT polynomial of <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mo stretchy="false">(</q:mo><q:mi>L</q:mi><q:mo>,</q:mo><q:mi>L</q:mi><q:mo stretchy="false">)</q:mo></q:math> 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:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>U</u:mi><u:mo stretchy="false">(</u:mo><u:mi>N</u:mi><u:mo stretchy="false">)</u:mo></u:math> matrix model. We determine the leading and subleading contribution to some of the correlators at large <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>N</y:mi></y:math> from the matrix model approach and find that they match exactly with those obtained from the corresponding colored HOMFLY-PT polynomials. Published by the American Physical Society 2024