Knots, links and branes at large N

Abstract

We consider Wilson loop observables for Chern-Simons theory at large N and its topological string dual and extend the previous checks for this duality to the case of links. We find an interesting structure involving representation/spin degeneracy of branes ending on branes which features in the large N dual description of Chern-Simons theory. This leads to a refinement of the integer invariants for links and knots. We illustrate our results with explicit computations on the Chern-Simons side.

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N = 1 U (N ) gauge theory in d = 4, and ending on the resolved conifold geometry where the branes have disappeared and been replaced by flux.…”

Open string instantons and relative stable morphisms

“…According to the general principle of virtual moduli cycles, the expected (virtual) number of maps in M rel g,µ (W r ) should be (3)(4)(5)(6)(7)(8)(9)(10) [ Extending V to M rel g,µ (P 1 ) needs more work. There is an obvious extension as follows: Let f ∈ M rel g,µ (P 1 ) be any relative stable morphism with domain C and distinguished divisor D d as before.…”

Lagrangians for the Gopakumar–Vafa conjecture

“…By the way, this construction is identical to that given by Equation (5.3) of [9] when applied to the hyperkähler rotation of the zero locus of a holomorphic function in C 2 .…”

“…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.…”

“…Even so, it is a good bet, verified in part by Katz and Liu [7], Labastida, Marino and Vafa [9] and Aganagic, Klemm and Vafa [1], that such a correspondence exists and that it is mediated by a suitable Lagrangian 3-manifold sitting in O(−1) ⊕ O(−1). To be specific, the knot should determine the Lagrangian, and then a knot invariant should come as a suitable count of compact, holomorphic curves with boundary on the Lagrangian.…”