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There is no better definition than what Wikipedia offers - in general, a topological excitation is a (field) state, i.e. a localized quantity since fields depend on spacetime, whose integral is a topological invariant. One prime example are Yang-Mills theories in 4D, where the integral $\int \mathrm{Tr}(F\wedge F)$, as essentially the second Chern class of ...


2

OP asks (v1): How one can know the gauge field emerging from the local gauge invariance is actually the EM field? Assuming that OP is pondering about gauging theoretical models (rather than concerned with our actual world and phenomenological inputs) then the answer is: One cannot know. For starters, the gauge group $G$ could be different than $U(1)$. ...


1

The Lagrangian for GR is $$ L \propto \int R \sqrt{-g} \, d^4 x $$ where $R$ is the Ricci scalar $$ R = R^\mu_\mu = R^{\mu \nu}_{\mu \nu} $$ So, this is a scalar which is related linearly to all the components of the Riemann tensor, and is a second-order differential of the metric $g$ of the form $$ R \sim g \partial^2 g + (\partial g)^2 $$ This is ...



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