Abstract
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 [∆ − δ, ∆ + δ] for large ∆ at fixed δ. Consequently we prove that given any unitary modular invariant 2D 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.
Highlights
Of Torus partition function with/without possible insertion of some operators
This idea can be generalized to discrete symmetries by thinking of inserting topological defect lines (TDL) while doing the path integral over the relevant manifold to define the grand canonical partition function
One can allow non invertible TDLs and define grand canonical partition functions
Summary
The untwisted sectors can be divided into two pieces: even and odd, named as P and Q This is obtained when the TDL corresponding to the Z2 symmetry is extended along the spatial direction. Z2 is non-anomalous, one can have even and odd states in the twisted sector as well, we call them R and S respectively Gauging this Z2 symmetry lands us onto the theory D. It would be interesting to explore other aspects of modular bootstrap for example bounding the dimension of lowest nontrivial Virasoro primary, constructing the extremal functionals [37,38,39,40,41,42,43,44,45] in presence of TDLs. As a technique, we generalize the application of Tauberian formalism in context of CFT beyond S modular invariant partition functions. In section B, we review the derivation of spin selection rule for anomalous global symmetry
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