nLab analytically continued Chern-Simons theory

Idea

Where ordinary 3d Chern-Simons theory is given by an action functional with values in the circle group $\mathbb{R}/\mathbb{Z}$ on a space of special unitary group-principal connections, its “analytic continuation”(Gukov 03, Witten 10) instead is defined on complex special linear group-principal connections and its values are elements in $\mathbb{C}/\mathbb{Z}$ (see also at Chern-Simons theory with complex gauge group).

The Wilson line quantum observables of analytically continued Chern-Simons theory are accordingly analytic continuations of knot invariants (Garoufalidis 07).

Properties

Discussion of the phase space with its complex symplectic form is in Gukov 03, section 2.2

Related concepts

• Chern-Simons theory with complex gauge group

• complex volume, Cheeger-Simons classes

• volume conjecture

• holomorphic Chern-Simons theory

• Spin Chern-Simons theory

• analytically continued Wess-Zumino-Witten theory

References

• Sergei Gukov, Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial, Commun.Math.Phys. 255 (2005) 577-627 (arXiv:hep-th/0306165)

• Stavros Garoufalidis, Chern-Simons theory, analytic continuation and arithmetic (arXiv:0711.1716)

• Edward Witten, Analytic Continuation Of Chern-Simons Theory, Chern–Simons Gauge Theory, 20, 347-446. (arXiv:1001.2933)

• Tudor Dimofte, Quantum Riemann Surfaces in Chern-Simons Theory (arXiv:1102.4847)

• Edward Witten, Two Lectures On The Jones Polynomial And Khovanov Homology (arXiv:1401.6996)