Gravitational Chern-Simons theory for bosons and fermions

Question

Q1: What is the difference of boson and fermions for their Gravitational Chern-Simons theory?

I suppose in general if the metric is not flat, we have vierbein ${e_{\hat{b}}}^{\nu}$, with $$ g_{\mu\nu} {e_{\hat{a}}}^{\mu} {e_{\hat{b}}}^{\nu} = \eta_{\hat{a}\hat{b}}, $$ where $g_{\mu\nu}$ is curved and $\eta_{\hat{a}\hat{b}}$ is Lorentzian flat. With the spin-connection is $$ \omega^{\hat{c}}_{\hat{b}\mu} = {e^{\hat{c}}}_{\nu}\partial_\mu {e_{\hat{b}}}^{\nu} + {\Gamma^\nu}_{\sigma\mu} {e^{\hat{c}}}_{\nu} {e_{\hat{b}}}^{\sigma}, $$ where ${\Gamma^\nu}_{\sigma\mu}$ are the Christoffel symbols.

SET-UP: Now let us imagine there are some matter fields bosons $\phi$ or fermions $\psi$ coupling to the spacetime metric, and we integrate out the bosons $\phi$ or fermions $\psi$ to get the effective actions involving gravitational Chern-Simons action in 2+1D.

So a 2+1D gravitational Chern-Simons action can be (spin connection $\omega$): $$ S=\int\omega\wedge\mathrm{d}\omega + \frac{2}{3}\omega\wedge\omega\wedge\omega \tag{1} $$

I am sure this works for fermions.

Q2: however, if bosons have spin 0 or spin 1, do we still have spin-connection? I suppose still yes?

Q3: or, do we have a 2+1D gravitational Chern-Simons action for bosons to be ( the connection $\Gamma$): $$ S=\int\Gamma\wedge\mathrm{d}\Gamma + \frac{2}{3}\Gamma\wedge\Gamma\wedge\Gamma \tag{2} $$

Q4: what are the differences between Eq.1 and Eq.2 in terms of background matter fields? (suppose the metric $g_{\mu\nu}$ is coupled to matters, either bosons or fermions.)

Please feel free give Reference. And make sure that your answer is to my point Q1-4.

Answer 1

Even though standard, there is an unfortunate confusion caused by the way the term "spin connection" is used in the physics literature.

The formula that you give relating the Christoffel symbols to the $\omega$-thing with hatted indices is a formula that relates two different but equivalent local component-incarnations of the Levi-Civita connection. It so happends that the components expressed in $\omega$-form naturally lend themselves better to the components-description of the minimal coupling term of the connection to fermions, but intrinsically, meaning up to gauge equivalences relating different local component expressions, the connection $\nabla$ which is expressed locally by "$\Gamma$" is equivalent to that locally expressed by "$\omega$", these are just two different local incarnations of what intrinsically is the same physical thing.

The real question is this: on an oriented manifold, the Levi-Civita connection $\nabla$ a priori is (or gives equivalently, namely on the associated frame bundle) an $SO(n)$-principal connection. For eventually coupling fermions to this, one needs to ask this to lift (in the sense of G-structures) to a $Spin(n)$-principal connection.

But notice that in your question above no actual fermions ever appear. What you are really asking is for the Chern-Simons functional $CS(\nabla)$ on $SO(n)$-principal connections or on $Spin(n)$-principal connections.

Check out "SO(n)-Chern-Simons theory" and the like, and maybe restate your question then.