# What are the details of the renormalization of Chern-Simons theory?

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!

-

## 1 Answer

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

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

where

• 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).

See

-
Costello's book is certainly not a good reading for a beginner, as well as referring to general BV-BRST technique. –  John Aug 31 '13 at 21:30
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. –  Urs Schreiber Sep 1 '13 at 17:22
It is of course a matter of taste, Collins' book is the best but maybe out-of-date, there is no Chern-Simons there.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. –  John Sep 1 '13 at 17:43
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 (Out of date??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. –  Urs Schreiber Sep 1 '13 at 18:08
Do not understand, in the Collins "Renormalization" there is no CS there, but the book itself is great –  John Sep 1 '13 at 18:49