Can the path-integral of Abelian Chern-Simons theory be valuated exactly?

$$\int \mathcal{D}[A] \exp\left\{{\frac{i}{2\pi}\int A\wedge dA}\right\}$$

I found Witten's paper "Quantum Field Theory and Jones Polynomial" very hard to understand. Is there any pedagogical way to find out the gauge fixing of the above action? Is it possible to perform the path-integral non-perturbatively?

  • $\begingroup$ I think so, from what I have read the rigorous approach uses quantum groups. $\endgroup$ Commented Feb 28, 2018 at 15:34
  • $\begingroup$ Thank you. I don't know anything about quantum groups. Could you give me a link? $\endgroup$
    – Valac
    Commented Feb 28, 2018 at 15:48
  • $\begingroup$ The literature on quantum groups are highly mathematical and often the physical motivation is missing so I'm not sure what to recommend; but have a look at Kocks Frobenius Algebras & 2D Topological Field Theories, this tackles a much easier case; again the treatment here is mathematical, but at least the treatment is pedagogical and quite straight-forward from what I remember. Its aimed at undergraduate mathematics students. $\endgroup$ Commented Feb 28, 2018 at 17:21
  • $\begingroup$ you can evaluate it with standard path integral methods exactly, but what are the boundary conditions you are interested in? $\endgroup$ Commented Feb 28, 2018 at 19:43
  • $\begingroup$ How to perform the integral exactly? What are the gauge fixing term and ghost term? $\endgroup$
    – Valac
    Commented Feb 28, 2018 at 22:24

1 Answer 1


A conventional path integral quantization of the Chern-Simons theory is performed by Alvarez Gaumé. The author applies gauge fixing, Faddeev-Popov construction and computation of the one loop effective action. The main result is the level renormalization at one loop $k\rightarrow k+c_v$ . ($c_v$ is the Coxeter number of the gauge group).

Further perturbative results were obtained by: Giavarini Martin and Ruiz Ruiz. (They use a different regularization scheme by adding a Yang-Mills term and taking the large topological mass limit). They show that the level renormalization does not change at the two-loop level and compute the Wilson loop expectation value.

Nonperturbative results of the Chern-Simons theory can better be obtained by means of canonical quantization. In many cases the gauge redundancy can be removed exactly and only a finite number of degrees of freedom are left, which can in tern be canonically quantized. Please see for example the following review by Dunne (section 3), where the quantization in the case when the space time manifold is $T^2 \times \mathbb{R}$ is performed.


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