On the gauge theory/geometry correspondence. (English) Zbl 0972.81135

Gopakumar, Rajesh; Vafa, Cumrun

Adv. Theor. Math. Phys. 3, No. 5, 1415-1443 (1999).

Summary: The ’t Hooft expansion of SU\((N)\) Chern-Simons theory on \(S^3\) is proposed to be exactly dual to the topologically closed string theory on the \(S^2\) blow up of the conifold geometry. The \(B\)-field on the \(S^2\) has magnitude \(Ng_s=\lambda\), the ’t Hooft coupling. We are able to make a number of checks, such as finding exact agreement at the level of the partition function computed on both sides for arbitrary \(\lambda\) and to all orders in \(1/N\). Moreover, it seems possible to derive this correspondence from a linear sigma model description of the conifold. We propose a picture whereby a perturbative D-brane description, in terms of holes in the closed string worldsheet, arises automatically from the coexistence of two phases in the underlying U(1) gauge theory. This approach holds promise for a derivation of the AdS/CFT correspondence.

MSC:

81T30	String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T45	Topological field theories in quantum mechanics
81T13	Yang-Mills and other gauge theories in quantum field theory
83E30	String and superstring theories in gravitational theory
32J99	Compact analytic spaces
53C99	Global differential geometry

Keywords:

’t Hooft expansion; topologically closed string theory; conifold geometry; linear sigma model; D-brane; coexistence of two phases; AdS/CFT correspondence; Chern-Simons theory