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On the gauge theory. Geometry correspondence. (English) Zbl 1026.81029
Vafa, Cumrun (ed.) et al., Winter school on mirror symmetry, vector bundles and Lagrangian submanifolds. Proceedings of the winter school on mirror symmetry, Cambridge, MA, USA, January 1999. Providence, RI: American Mathematical Society (AMS). AMS/IP Stud. Adv. Math. 23, 45-63 (2001).
The authors propose a new duality: they show that the large $$N$$ limit of $$\text{SU}(N)$$ Chern-Simons theory on the 3-sphere $$S^3$$ is exactly the same as an $$N=2$$ topological closed string theory blow up of the conifold geometry. The $$\text{SU}(N)$$ Chern-Simons theory on $$S^3$$ arises from the A-model open string theory on the Calabi-Yau manifold $$T^*S^3$$ with Dirichlet boundary conditions on $$S^3$$. The authors compare the partition functions of the Chern-Simons and the closed string theories and find a strikingly exact match for all values of the ’t Hooft coupling constant $$\lambda$$ and to all orders of $$1/N$$. They discuss a possibility to derive the duality from the 2-d linear sigma model, considered by E. Witten, that is an $$N=2$$ supersymmetric $$U(1)$$ gauge theory, whose low energy dynamics description reduces to the usual nonlinear sigma model on the $$S^2$$ blown up version of the conifold. The authors believe that the linear sigma model approach can be also useful in deriving the AdS/CFT correspondence.
For the entire collection see [Zbl 0980.00027].

##### MSC:
 81T10 Model quantum field theories 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 81T13 Yang-Mills and other gauge theories in quantum field theory 83E30 String and superstring theories in gravitational theory 83C47 Methods of quantum field theory in general relativity and gravitational theory