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Contrary to popular belief, it is not necessary to choose a gauge to quantize a gauge theory. It is just convenient, since the non-gauge-fixing approaches are often difficult to implement for all but the simplest cases. Gauge theories are, in the Hamiltonian picture, certain kinds of constrained Hamiltonian systems. Dirac's canonical quantization procedure, ...

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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 ...

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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 ...

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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 ...

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$SU(N)$ 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 ...

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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 reads $$S[A]~=~\int_M\mathbb{L}.$$ The exterior derivative $\mathrm{d}\mathbb{L}$ of a CS form is (also by definition) the Lie algebra trace of a polynomial of the ...

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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 ...

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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 ...

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In the textbook of TASI 2009, section "Introduction to extra dimension" i can find the answer as follows. They state that $5D$ or higher dimensional gauge coupling has a negative mass dimension, so the 5d or higher dimensional gauge theory is non-renormalizable.

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