Polynomial Invariants for Torus Knots¶and Topological Strings

Abstract

We make a precision test of a recently proposed conjecture relating Chern-Simons gauge theory to topological string theory on the resolution of the conifold. First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern-Simons gauge theory, and then we evaluate these vevs in arbitrary irreducible representations of SU (N ) for torus knots. We find complete agreement with the predictions derived from the target space interpretation of the string amplitudes. We also show that the structure of the free energy of topological open string theory gives further constraints on the Chern-Simons vevs. Our work provides strong evidence towards an interpretation of knot polynomial invariants as generating functions associated to enumerative problems.

The Topological Vertex

“…The f (R 1 ,···,R L ) introduced in (7.18) can be extracted from Z (R 1 ,···,R L ) through a procedure spelled out in detail in [33,34,42]. One has, for example, 25) It is also convenient to introduce the quantities 26) in such a way that…”

Mirror symmetry and aG2flop

“…There is now a large body of evidence supporting this conjecture including highly non-trivial exact computations to all orders in N [3] [4][5] [6] [7]. 2 On the other hand, this duality was embedded in type IIA superstrings [9] where it was interpreted as a geometric transition starting with N D6 branes wrapped over S 3 of the conifold geometry, which gives an N = 1 U (N ) gauge theory in d = 4, and ending on the resolved conifold geometry where the branes have disappeared and been replaced by flux.…”

Lagrangians for the Gopakumar–Vafa conjecture

“…In this regard, notice that any given 2πN periodic map γ to C with {γ(θ + 2πk)} 0≤k<N distinct at all values of θ can be homotoped through such maps to one that obeys (8). In particular, such a homotopy does not change the braid isotopy class of the corresponding braid.…”

“…Subsequently, the scope of the conjecture was expanded by Ooguri and Vafa [11]. Successful tests of the have been made, for example, by Labastida and Marino [8], Ramadevi and Sarkar [12], Labastida, Marino and Vafa [9] and Aganagic, Klemm and Vafa [1]. In the mean time, Faber and Pandarhapande [2], Katz and Liu [7] and Li and Song [10] have considered the mathematical foundations for the conjecture and verified certain parts of it.…”