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We study superconductors with $n$-fold rotational invariance both in thepresence and in the absence of spin-orbit interactions. More specifically, weclassify the non-interacting Hamiltonians by defining a series of $Z$-numbersfor the Bogoliubov-de Gennes (BdG) symmetry classes of the Altland-Zimbauerclassification of random matrices in $1$D, $2$D, and $3$D in the presence ofdiscrete rotational invariance. Our analysis emphasizes the important roleplayed by the angular momentum of the Cooper pairs in the system: for pairingsof nonzero angular momentum, the rotation symmetry may be representedprojectively, and a projective representation of rotation symmetry may haveanomalous properties, including the anti-commutation with the time-reversalsymmetry. In 1D and 3D, we show how an $n$-fold axis enhances the topologicalclassification and give additional topological numbers; in 2D, we establish arelation between the Chern number (in class D and CI) and the eigenvalues ofrotation symmetry at high-symmetry points. For each nontrivial class in 3D, wewrite down a minimal effective theory for the surface Majorana states.

Topological crystalline superconductors with linearly and projectively represented $C_{n}$ symmetry