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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 $U(1)$. ...

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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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They are not analogous. $R_{abcd}$ is just Riemann Tensor and $R_{abcd}R^{abcd}$ is Riemann Tensor squared. Mathematically they must be squared since having a Single term Riemann Tensor / Ricci Tensor in gravitational action doesn't make sense. Physically speaking they are modification of Einstein Hilbert action. They are curvature not field one shall ...

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I have been able to determine the answer to this question myself. If $A$ is a flat connection, then $$\partial_z A_{\bar z} - \partial_{\bar z} A_z - i [ A_z , A_{\bar z} ] = 0 \implies A_z = i U^{-1} \partial_z U$$ for some scalar $U \in {\cal G}$. Now, we wish to invert the equation $$D_z \phi = f(z,{\bar z})$$ To do this, we note  D_z \phi = U^{-1} ...

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You can find the Hamiltonian of the potentials if you know the Lagrangian, or you can calculate the energy in reference to the work done on charges in an electric field, which the electric potential sort of comes about secondarily, because it is related to the work. The calculation is done here. As for the equivalence between the two, are you familiar with ...

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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 ...

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