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

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

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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).

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See

1. The setup

2. Operator (heat) kernels and propagators

3. The renormalization group operator

4. The path integral

5. Renormalized action

6. Renormalization