What is a good, simple argument as to why Chern-Simons theory' is renormalisable? Any good books/references dealing with this effectively? Why does the $\beta$-function vanish? Thanks!


From nLab:renormalization: Of theories in BV-CS form:

In (Costello 07) a comparatively simple renormalization procedure is given that applies to theories that are given by action functionals which can be given in the form

$$ S(\phi) = \langle \phi , Q \phi \rangle + I(\phi) $$


  • the fields ϕ are sections of a graded field bundle E on which Q is a differential, ⟨−,−⟩ a compatible antibracket pairing such that (E,Q,⟨⟩) is a free field theory (as discussed there) in BV-BRST formalism;

  • I is an interaction that is at least cubic.

These are action functionals that are well adapted to BV-BRST formalism and for which there is a quantization to a factorization algebra of observables.

Most of the fundamental theories in physics are of this form, notably Yang-Mills theory. In particular also all theories of infinity-Chern-Simons theory-type coming from binary invariant polynomials are perturbatively of this form, notably ordinary Chern-Simons theory.

For a discussion of just the simple special case of 3d Chern-Simons theory see (Costello 11, chapter 5.4 and 5.14).


  1. The setup

  2. Operator (heat) kernels and propagators

  3. The renormalization group operator

  4. The path integral

  5. Renormalized action

  6. Renormalization

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    $\begingroup$ Costello's book is certainly not a good reading for a beginner, as well as referring to general BV-BRST technique. $\endgroup$ – John Aug 31 '13 at 21:30
  • $\begingroup$ John, maybe renormalization as such is not for beginners? I think Costello's account is about the best that there is. A beginner might take it as incentive to study and learn more and eventually cease to be a beginner and become an expert. $\endgroup$ – Urs Schreiber Sep 1 '13 at 17:22
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    $\begingroup$ It is of course a matter of taste, Collins' book is the best but maybe out-of-date, there is no Chern-Simons there. Costello aims at mathematical rigour, which adds a few useful information but complicates the book a lot. $\endgroup$ – John Sep 1 '13 at 17:43
  • $\begingroup$ Not sure what you mean. (Out of date?? And of course there is Chern-Simons in the book, see the explicit links above.) But let's leave it at that. Maybe somebody else has what you are after. $\endgroup$ – Urs Schreiber Sep 1 '13 at 18:08
  • $\begingroup$ Do not understand, in the Collins "Renormalization" there is no CS there, but the book itself is great $\endgroup$ – John Sep 1 '13 at 18:49

I think Witten computed the path integral of Chern-Simons theory from Quantum Field Theory and Jones Polynomial

The coupling constant $k$ takes integer values so that the classical action is well-defined for non-trivial bundles. The quantization shifts $k$ by some integer for topological reasons. Specifically, the Faddeev-Popov determinant explicitly depends on the metric, which breaks the topological invariant. In order to overcome this problem, one adds a counter-term, which is proportional to the gravitation Chern-Simons action

$$I[g]=\int_{M}\mathrm{Tr}(\omega\wedge d\omega+\frac{2}{3}\omega\wedge\omega\wedge\omega),$$

to the quantum effective action, so that it cancels out the metric dependence from the gauge fixing. However, this will introduce a new quantum anomaly, called framing anomaly. To be specific, connected three dimensional orientable manifolds are parallelizable (i.e. their tangent bundles are trivial). Choosing a nowhere vanishing frame on the spacetime manifold is a framing. However, there are distinct homotopy classes of framings. Adding the above gravitational Chern-Simons action would introduce the dependence on choices of framings. The final path integral of Chern-Simons theory will therefore be dependent on framings, if you want to preserve topological invariance.

The theory is topological, and its coupling is still quantized, and therefore, doesn't make sense to talk about the $\beta$-function.


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