What is the mathematical basis for the Gopakumar-Vafa gauge/geometry correspondence?

Question

I am a mathematician trying to understand the reasoning behind the proposed duality between large-N Chern-Simons on $S^3$ and the closed string A model on the resolved conifold. This was established by Gopakumar-Vafa in http://arxiv.org/abs/hep-th/9811131 where both theories were matched at the level of the partition function (Gromov-Witten potential in the case of the A model). Later, Ooguri-Vafa extended the computation to Wilson loop observables. To recap, the argument for the duality goes something like this: Witten demonstrated an equivalence between large N Chern-Simons on $S^3$ and the open string A model on $T^*S^3$ (the deformed conifold). This involves an appropriate choice of boundary conditions (i.e. introduction of D-branes) on the Lagrangian $S^3$ and therefore an identity with open Gromov-Witten theory. Loseley, the Gopakumar-Vafa conjecture is therefore that open Gromov-Witten theory on the deformed conifold is equivalent to closed Gromov-Witten theory on the resolved conifold. The explanation for this, as presented in thier paper, relies heavily on an analogy with the AdS/CFT  correspondence and how the large N limit of branes relates to the conifold transition. I understand little of these topics (AdS/CFT and D-brane dynamics) and so was hoping for a mathematical explanation for why the two theories should be equivalent.

Answer 1

I don't think that there is a known nice mathematical explanation in the sense I guess you would like, i.e. different from simply computing both sides and checking that they match. To support this point of view: at the beginning of the following video

Kontsevich explains that the large $N$ asymptotics of the volumes of the unitary groups $U(N)$ can be naturally expressed in terms of Euler characteristics of moduli spaces of Riemann surfaces and claims that there is no known intrinsic satisfactory explanation. Such relation is a special case of the Gopakumar-Vafa correspondence and of course Kontsevich is aware of that and so what he is asking for is something more and I don't think that someone knows the answer.

Nevertheless, for someone not comfortable with the AdS/CFT point of view, they are maybe more approachable explanations in the physics context. For example, in this paper:

http://arxiv.org/abs/hep-th/0205297

Ooguri and Vafa give a physical derivation of the correspondence from the worldsheet point of view whereas in this paper:

http://arxiv.org/abs/hep-th/0011256

Atiyah, Maldacena and Vafa give a physical derivation from the "dual" point of view, i.e. from the spacetime point of view. This paper tries to reduce the correspondence to a geometric statement (a flop in 7 dimensions) and so is maybe more accessible to a mathematician.

Comments

Thanks, that's useful stuff. I have just come across a 2004 paper by Liu and Yau (http://arxiv.org/abs/math/0411038 ) that treats the correspondence from an entirely mathematical point of view but they do little other than check agreement between open/closed computations. I guess you're right and I will need to learn some AdS/CFT!

Answer 2

There is now a rigorous mathematical explanation for why the knot invariants are computed by holomorphic curve counts in the resolved conifold. 

Basically the point is that the skein relations (universal identities between Wilson lines in Chern-Simons theory) turn out to be identical with the boundary terms obstructing the definition of open Gromov-Witten theory.  Thus it is possible to `couple' these and cancel the boundary terms by defining a skein-valued holomorphic curve count. Combined with standard methods in symplectic geometry for carrying holomorphic curves across the conifold transition, one can show that the HOMFLYPT polynomial is a count of curves in the resolved conifold. 

Some further work on this gives the calculation for a single toric brane and the total contribution of a string stretching between two branes.

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What we are still missing is an explanation of how to think of the partition function of Chern-Simons theory (i.e. without any knots) in terms of holomorphic curves before the geometric transition, i.e. in the cotangent bundle.   

Said differently, we understand why open strings stretching from some other lagrangian L and the zero section S become, across the conifold transition, open strings stretching from L to itself.  What we don't understand is why open strings stretching from S to itself become closed strings on the other side.  This is because we don't have a sensible definition of a moduli space of open strings stretching from S to itself -- there are no nonconstant holomorphic curves in a cotangent bundle.  Likely if we had any definition at all, we could apply the same ideas.