Framed knots at large N

TL;DR

This paper explores the framing dependence of Wilson loops in large N U(N) Chern-Simons theory, deriving new formulas for topological string amplitudes and intersection numbers, and proposing a refined understanding of large N dualities.

Contribution

It provides explicit formulas for topological string amplitudes and intersection numbers, and suggests that large N dualities are better formulated in U(N) gauge theories.

Findings

Derived closed-form expressions for topological string amplitudes.

Established new intersection number formulas on moduli spaces.

Provided evidence favoring U(N) over SU(N) in large N dualities.

Abstract

We study the framing dependence of the Wilson loop observable of U(N) Chern-Simons gauge theory at large N. Using proposed geometrical large N dual, this leads to a direct computation of certain topological string amplitudes in a closed form. This yields new formulae for intersection numbers of cohomology classes on moduli of Riemann surfaces with punctures (including all the amplitudes of pure topological gravity in two dimensions). The reinterpretation of these computations in terms of BPS degeneracies of domain walls leads to novel integrality predictions for these amplitudes. Moreover we find evidence that large N dualities are more naturally formulated in the context of U(N) gauge theories rather than SU(N).