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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 ...


1

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 ...


1

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 ...


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.



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