nLab Chern-Simons theory as topological string theory

Idea

Under mapping the Feynman diagrams of perturbative Chern-Simons theory on a 3-manifold $X^3$ via 't Hooft double line notation to open string worldsheets, its quantum observables (Wilson loop Vassiliev knot invariants) are equivalent to those of an open topological string theory with target space the cotangent bundle $T^\ast X^3$ with D3-branes. (Witten 92)

For $X^3 = S^3$ the 3-sphere, the large N limit is also dual, in a topological string/TQFT-version of AdS/CFT duality, to a closed topological string theory with target space the blow-up of the conifold (Gopakumar-Vafa 98).

References

Review:

• Marcos Marino, Chern-Simons Theory and Topological Strings, Rev. Mod. Phys. 77:675-720, 2005 (arXiv:hep-th/0406005)

Direct open topological string theory

The original article:

• Edward Witten, Chern-Simons Gauge Theory As A String Theory, Prog. Math. 133: 637-678, 1995 (arXiv:hep-th/9207094)

Dual closed topological string theory

• Rajesh Gopakumar, Cumrun Vafa, On the Gauge Theory/Geometry Correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415-1443 (arXiv:hep-th/9811131)

• S. Sinha, Cumrun Vafa, $SO$ and $Sp$ Chern-Simons at Large $N$ (arXiv:hep-th/0012136)

Including Wilson loop observables:

• Hirosi Ooguri, Cumrun Vafa, Knot Invariants and Topological Strings, Nucl. Phys. B 577:419-438, 2000 (arXiv:hep-th/9912123)