High Energy Modular Bootstrap, Global Symmetries and Defects.

We derive Cardy-like formulas for the growth of operators in different sectors of unitary $2$ dimensional CFT in the presence of topologial defect lines by putting an upper and lower bound on the number of states with scaling dimension in the interval $[\Delta-\delta,\Delta+\delta]$ for large $\Delta$ at fixed $\delta$. Consequently we prove that given any unitary modular invariant $2$D CFT and symmetric under finite global symmetry $G$ (acting faithfully), all the irreducible representations of $G$ appear in the spectra of the untwisted sector; their growth is Cardy like and proportional to "square" of the dimension of the irrep. In the Schwarzian limit, the result matches onto that of JT gravity with a bulk gauge theory. If the symmetry is non-anomalous, the result applies to any sector twisted by a group element. For $c>1$, the statements are true for Virasoro primaries. Furthermore, the results are applicable to large c CFTs. We also extend our results for the continuous $U(1)$ group.