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

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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.

edited Apr 23, 2015

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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.

answered Apr 23, 2015 by (4,660 points)
edited Apr 23, 2015 by 40227

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!

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