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Robust fractional charge localized at disclination defects has been recentlyfound as a topological response in $C_{6}$ symmetric 2D topological crystallineinsulators (TCIs). In this article, we thoroughly investigate the fractionalcharge on disclinations in $C_n$ symmetric TCIs, with or without time reversalsymmetry, and including spinless and spin-$\frac{1}{2}$ cases. We compute thefractional disclination charges from the Wannier representations in real spaceand use band representation theory to construct topological indices of thefractional disclination charge for all $2D$ TCIs that admit a (generalized)Wannier representation. We find the disclination charge is fractionalized inunits of $\frac{e}{n}$ for $C_n$ symmetric TCIs; and for spin-$\frac{1}{2}$TCIs, with additional time reversal symmetry, the disclination charge isfractionalized in units of $\frac{2e}{n}$. We furthermore prove that withelectron-electron interactions that preserve the $C_n$ symmetry and many-bodybulk gap, though we can deform a TCI into another which is topologicallydistinct in the free fermion case, the fractional disclination chargedetermined by our topological indices will not change in this process.Moreover, we use an algebraic technique to generalize the indices for TCIs withnon-zero Chern numbers, where a Wannier representation is not applicable. Withthe inclusion of the Chern number, our generalized fractional disclinationindices apply for all $C_n$ symmetric TCIs. Finally, we briefly discuss theconnection between the Chern number dependence of our generalized indices andthe Wen-Zee term.

Fractional disclination charge in two-dimensional $C_n-$symmetric topological crystalline insulators