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The pure Abelian Chern--Simons theory has zero metric stress. For a source written as $-c\int \star J\wedge A$, holding the covariant components of the one-forms $J,A$ fixed makes the source equal to $-c\int \sqrt{-g}{g}^{\mu \nu }{J}_{\mu }{A}_{\nu }{d}^{3}x$. Variation using $\delta \sqrt{-g}=-\sqrt{-g}{g}_{\mu \nu }\delta g^{\mu \nu }/2$ gives the displayed stress-energy tensor. If the independent source is instead the fixed two-form $\mathcal{J}=\star J$, the source action is metric-independent and its stress vanishes. The variational data must therefore be stated.

Let $\mathcal{J}=\star J$ be the current $(d-1)$-form on an oriented $d$-dimensional spacetime. For a slab bounded by two spacelike hypersurfaces ${\Sigma }_1,{\Sigma }_2$ and a side boundary $B$, the Stokes theorem givesThus $Q(\Sigma )={\int }_{\Sigma }\mathcal{J}$ is conserved if the side flux vanishes, for example for a spatially compact current or appropriate falloff at infinity. With side flux it instead obeys the corresponding charge-balance equation. This is the differential-form version of current conservation.

For the Abelian Chern--Simons theory, take $A\mapsto A+d\chi$ with a smooth compactly supported gauge parameter. The change in the gauge-field term is a boundary term because $d\chi \wedge dA=d(\chi dA)$. For the source,Discarding the boundary terms, the action variation isGauge invariance for every such $\chi$ and $c\ne 0$ therefore gives $d\star J=0$. If $c=0$, the current decouples and gauge invariance places no condition on it. On a spacetime with boundary, boundary conditions or boundary degrees of freedom are also needed to handle the discarded terms; on a nontrivial compact gauge bundle, large-gauge invariance is an additional global issue.

For the field equation, useThe one-form $\delta A$ commutes with the two-form $\mathcal{J}$ under the wedge product, soup to the boundary term. ConsequentlyApplying $d$ and using $d^2=0$ again gives current conservation for $c\ne 0$. The factor two comes from varying both occurrences of $A$.

Orient a spatial region $D$ so its boundary is $\gamma$, and define $Q_D={\int }_D\mathcal{J}$. Applying Stokes theorem and the field equation yieldsFor a U(1) connection, in a convention with unit minimal charge, the gauge-invariant quantity is its holonomy $e^{i\Phi }$, the Wilson-loop or Aharonov-Bohm phase. A large gauge transformation can change the chosen representative of $\Phi$ by $2\pi n$. Therefore $\Phi$ is a phase angle modulo $2\pi$, rather than an absolute gauge-invariant real number. A particle of charge $q$ has phase $e^{iq\Phi }$ in the corresponding normalization. The flux-charge relation thus attaches a gauge phase to enclosed charge.

The metric variation requires specifying the independent source. The pure Chern-Simons term $A\wedge dA$ has no metric dependence, hence contributes zero stress-energy tensor. If the one-forms named in the question have fixed covariant components $A_{\mu },J_{\mu }$, the Hodge star in the source does depend on the metric:Using $\delta \sqrt{-g}=-\sqrt{-g}{g}_{\mu \nu }\delta g^{\mu \nu }/2$ givesUnder that literal fixed-one-form convention,There is another common convention in the topological source theory: hold the conserved two-form $\mathcal{J}=\star J$, equivalently the vector current density, fixed as the metric varies. Then both $A\wedge dA$ and $\mathcal{J}\wedge A$ are metric-independent, and $T_{\mu \nu }=0$ for this action. These are different variations, not contradictory calculations. The Chern-Simons source stress convention explains why a source prescription is necessary; a dynamical matter source would contribute its own action and stress as well.

Represent quasiparticle worldlines by conserved currents $j_I^{\alpha }$ and add the minimal-coupling termUp to the corresponding sign convention for charge, variation of the Abelian Chern--Simons theory givesFor a static quasiparticle of integer gauge-charge vector $q$, integration over a small surrounding disk yields the Chern--Simons flux attachment relationA quasiparticle of charge $q^{\prime }$ carried once around this flux acquires the Aharonov--Bohm phaseso

The constituent-particle current has charge vector $t=(1,\dots ,1{)}^T$. Its integral over the same disk isFor $K=I_n$ and $q=e_J$, this gives $N=1$. The self-exchange angle is ${\theta }_{qq}=\pi e_J^T{e}_J=\pi$, so the excitation is a fermion with one unit of microscopic particle number and may be identified with an electron.

Varying the Abelian Chern--Simons theory with respect to $a_{\alpha }^I$ givesFor a stationary quasiparticle of charge vector $q$, the time component implies that its enclosed gauge flux isup to the orientation convention. Coupling a particle of charge $p$ to this flux produces the full-braid phaseAn exchange is half of the corresponding full braid. Reversing the orientation conjugates the phase.