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7

In the absence of magnetic monopoles, Maxwell's equations are $$\text d F = 0,$$ $$\star\text d\star F = J_e,$$ where $J$ is the 4- current 1-form due to electric charges. For cohomological reasons, from the first equation one can asserts that there exists a 1-form $A$ such that $F = \text d A$, and $A$ is the interpreted as the 4-potential $(\phi,\mathbf ... 4 The mistake is that$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 You want to compute the integral Z = \int d [A] e^{iS} and since it has a gauge symmetry, there are multiple values of A that generating the same Action S, since S(A_g)=S(A_{g'}) for the two different choices of gauge g,g'. Now you have a gauge-fixing condition like \partial^\mu A_\mu = 0. The integral which contains enough physical ... 2 I) The gauge transformation of the real gauge field V reads \exp(\tilde{V}) ~=~e^Xe^Ve^Y, \qquad X~:=~i\Omega^{\dagger}, \qquad Y~:=~-i\Omega. \tag{1}$$Keeping only linear orders in \Omega, the BCH formula reads$$\tilde{V}~=~B({\rm ad} V)X+V+B(-{\rm ad} V)Y~=~V+\frac{1}{2}[V,Y-X]+B_+({\rm ad} V)(X+Y),\tag{2} $$where$$ ... 1 Choosing a gauge usually means implicitly performing a gauge transformation such that a given condition holds, such as choosing a gauge\partial_\mu A^\mu = 0$in electrodynamics. A local trivialization of a$G$-principal bundle$G \to P \overset{\pi}{\to} M$is given by an open covering$\{U_i\}$of$M$and diffeomorphisms$\{\phi_i : U_i \times G \to ...

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Wess-Zumino gauge is a particular choice of gauge where the vector superfield has a particular form and has less components than the generic vector super field. So if i'm free to make a gauge transformation i can choose the components of the chiral super field $\Omega$ in a manner that the sum of the $\theta$ (or any other "$\theta$ component" i want to ...

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ADE refers to the ADE classification, which refers to simply-laced simple Dynkin diagrams and corresponding Lie algebra and Lie group. refers to finite subgroups $\Gamma$ of $SU(2)$, which is related to orbifolds $M/\Gamma$, i.e. manifolds with singularities. See also elementary catastrophes. An ADE gauge theory means that the gauge group is an ADE ...

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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.

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