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