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