Question

Can topological degrees of freedoms interact?

Answer 1

For concreteness, I can't understand why the following term will not be allowed in a Lagrangian.

L∝μph ϵabc Ad∂bAc δgcd
where, Aμ is a topologically massive photon with mass μph and δgcd is a perturbation to the topologically massive graviton.

I understand that a Chern-Simons mass term is independent of the metric, but this does not make it obvious why perturbations of the metric will also be spared from taking part in interactions.

EDIT (an hour later)

I hoped - invariance under coordinate transformations could perhaps rule out the proposed term from the Lagrangian. Instead, allowed terms are:

L∝μph ϵabd Aa∂bAc δgcd
The plot thickens...

Answer 2

Premise

For concreteness, I can't understand why the following term will not be allowed in a Lagrangian.

L∝μph ϵabc Ad∂bAc δgcd

where, Aμ is a topologically massive photon with mass μph and δgcd is a perturbation to the topologically massive graviton.

I understand that a Chern-Simons mass term is independent of the metric, but this does not make it obvious why perturbations of the metric will also be spared from taking part in interactions.

EDIT (an hour later)

I hoped - invariance under coordinate transformations could perhaps rule out the proposed term from the Lagrangian. Instead, allowed terms are:

L∝μph ϵabd Aa∂bAc δgcd