  Similarly to the system Hamiltonian, a subsystem's reduced density matrix is
composed of blocks characterized by symmetry quantum numbers (charge sectors).
We present a geometric approach for extracting the contribution of individual
charge sectors to the subsystem's entanglement measures within the replica
trick method, via threading appropriate conjugate Aharonov-Bohm fluxes through
a multi-sheet Riemann surface. Specializing to the case of 1+1D conformal field
theory, we obtain general exact results for the entanglement entropies and
spectrum, and apply them to a variety of systems, ranging from free and
interacting fermions to spin and parafermion chains, and verify them
numerically. We find that the total entanglement entropy, which scales as $\ln
L$, is composed of $\sqrt{\ln L}$ contributions of individual subsystem charge
sectors for interacting fermion chains, or even $\mathcal{O} (L^0)$
contributions when total spin conservation is also accounted for. We also
explain how measurements of the contribution to the entanglement from separate
charge sectors can be performed experimentally with existing techniques.
