  We introduce a family of quantum R\'enyi fidelities and discuss their
symmetry resolution. We express the symmetry-resolved fidelities as Fourier
transforms of charged fidelities, for which we derive exact formulas for
Gaussian states. These results also yield a formula for the total fidelities of
Gaussian states, which we expect to have applications beyond the scope of this
paper. We investigate the total and symmetry-resolved fidelities in the XX spin
chain, and focus on (i) fidelities between thermal states, and (ii) fidelities
between reduced density matrices at zero temperature. Both thermal and reduced
fidelities can detect the quantum phase transition of the XX spin chain.
Moreover, we argue that symmetry-resolved fidelities are sensitive to the inner
structure of the states. In particular, they can detect the phase transition
through the reorganisation of the charge sectors at the critical point. This a
main feature of symmetry-resolved fidelities which we expect to be general. We
also highlight that reduced fidelities can detect quantum phase transitions in
the thermodynamic limit.
