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 topological 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 symmetric under finite global symmetry $G$ (acting faithfully), all the irreducible representations of $G$ appear in the spectra of the untwisted sector; the growth of states is Cardy like and proportional to the ''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.