  A one-dimensional boundary of a two-dimensional topological superconductor
can host a number of topologically protected chiral modes. Combining two
topological superconductors with different topological indices, it is possible
to achieve a situation when only a given number of channels ($m$) are
topologically protected, while others are not and therefore are subject to
Anderson localization in the presence of disorder. We study transport
properties of such quasi-one-dimensional quantum wires with broken
time-reversal and spin-rotational symmetries (class D) and calculate the
average conductance, its variance and the third cumulant, as well as the
average shot noise power. The results are obtained for arbitrary wire length,
tracing a crossover from the diffusive Drude regime to the regime of strong
localization where only $m$ protected channels conduct. Our approach is based
on the non-perturbative treatment of the non-linear supersymmetric sigma model
of symmetry class D with two replicas developed in our recent publication [D.
S. Antonenko et al., Phys. Rev. B 102, 195152 (2020)]. The presence of
topologically protected modes results in the appearance of a topological
Wess-Zumino-Witten term in the sigma-model action, which leads to an additional
subsidiary series of eigenstates of the transfer-matrix Hamiltonian. The
developed formalism can be applied to study the interplay of Anderson
localization and topological protection in quantum wires of other symmetry
classes.
