Strong-Coupling Fermion-Parity- and Symmetry-Based Indicators for Topological Superconductors with Crystal Symmetries

While the classification of magnetic and nonmagnetic topological phases with crystal symmetries is largely complete, the classification of topological (crystalline) superconductors (TSCs) is still in its relative infancy, owing to the absence of momentum-space (Wilson-loop-like) topological invariants for TSC states in arbitrarily complicated models. Partial progress has been made through ``refined'' symmetry-based indicators for unconventional SC states, but largely in the weak-pairing regime. Building from the recently introduced framework of U(1)-spin-resolved crystalline topology in insulators, we formulate complete momentum-space invariants for strong-coupling 10-fold-way and layer- and wire-constructable crystalline TSCs in Classes D and DIII via a complementary framework of U(1)-charge-resolved topology. Through this, we introduce a new method - termed fermion-parity- and symmetry-based-indicators - that combines momentum-space fermion parity and symmetry eigenvalues to efficiently identify TSCs with arbitrary pairing, spin, orbital, and sublattice structures. Our analysis reveals intriguing relationships between normal-state topological (crystalline) insulators and TSCs, allowing new progress towards unraveling their surface anomalies.