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The rule of the game is to use $A$ and $F=dA$ to write a topological action, and in $d+1$-space time dimension you need to come up with a gauge-invariant $d+1$-form which can then be integrated over the manifold to give you the action. Such an action does not depend on metric at all. Take $U(1)$ gauge field as an example. In $2+1$, the only thing you can ...
The mistake is that $W^a_\mu [\tau_a, \tau_b] W^{\mu b} = 0$ does not imply $[\tau_a, \tau_b] = 0$, this is true for any antisymmetric object $A_{ab}$. In your example, you can see this in the following way: Let $A_{ab}$ be antisymmetric: $A_{ab} = -A_{ba}$. Then: \begin{align} W^a_\mu A_{ab} W^{\mu b} &= W^b_\mu A_{ba} W^{\mu a} \qquad ... 2 Comments to the question (v4): By definition, the Lagrangian form \mathbb{L} of Chern-Simons (CS) theory (wrt. a Lie algebra valued one-form gauge field A) is a CS form, i.e. the CS action readsS[A]~=~\int_M\mathbb{L}.$$The exterior derivative \mathrm{d}\mathbb{L} of a CS form is (also by definition) the Lie algebra trace of a polynomial of the ... 2 SU(N) is the N-fold cover of PSU(N). They share the same Lie algebra, so the Yang-Mills action would look identical locally. The center of SU(N) is just Z_N. At the level of representations, the fundamental representation of SU(N) is a projective representation of PU(N), and only the adjoint ones are linear representations of PU(N). If the ... 2 Comment to the question (v7): In the context of an action formulation, if the Euler-Lagrange (EL) equations contain time-derivatives of a variable \phi^{\alpha}, then the variable \phi^{\alpha} is called a dynamical variable; else \phi^{\alpha} is an auxiliary variable. (Spatial derivatives are irrelevant for this classification.) Lagrange ... 1 Comments to the question (v3): On one hand, traditionally, the Batalin-Vilkovisky (BV) operator \Delta in Lagrangian BRST formulation encodes geometric data of the antisymplectic phase space for the model, specifically the antisymplectic structure [i.e. the so-called antibracket (\cdot,\cdot), or odd Poisson bracket] and a path integral volume density ... 1 if by "gauge transformation in the vielbein" you mean the local lorentz transformation that acts on one of the indices of the vielbein, then the answer is no. Because this local symmetry is in addition to the general coordinate transformation, and not part of it. In other words all veilbeins that are related by a gauged lorentz transformation correspond to ... 1 Take 2D SPT as an example. When the gauge fields are not dynamical, physically it really means that we are just modifying the Hamiltonian to a certain gauge field configuration (i.e. by changing the coupling of some of the terms, for example). These "gauge fields" are just extrinsic parameters in the Hamiltonian. To detect the SPT, we need to introduce ... 1 No, one can only conclude that the symmetrization$$[ \tau_a,\tau_b] + (a\leftrightarrow b)~=~0 is zero, which is indeed true because the commutator is antisymmetric.