Topological manipulations, like gauging a finite symmetry, produce new quantum field theories from known ones. It is natural to ask how effective they are at moving one around theory space. I will sketch an appealing conjectural answer to this question in the context of 2d rational conformal field theories, which leverages ideas and techniques from 3d topological field theory. I will then present a variety of partial results in the direction of this conjecture, and physically motivate the discussion by situating it in broader quantum field theory lore.

Conformal field theories in (1+1)D are key actors in many dramas in physics and mathematics. Their classification has therefore been an important and long-standing problem. In this talk, I will describe the main ideas behind the classification of (most) "small" bosonic CFTs - theories with low central charge (less than 24) and few primary operators (less than 5). I will then highlight two applications of this result. First, I will describe how it can be used in tandem with bosonization and fermionization techniques to establish the classification of chiral fermionic CFTs with central charge less than 23. Second, I will showcase how it can be used to bootstrap the generalized global symmetries of chiral bosonic CFTs. Talk based on arXiv:2208.05486 [hep-th] (joint work with Sunil Mukhi) and arXiv:2303.16921 [hep-th].