Rational RG flow, extension, and Witt class

Consider a renormalization group flow preserving a pre-modular fusion category $\mathcal S_1$. If it flows to a rational conformal field theory, the surviving symmetry $\mathcal S_1$ flows to a pre-modular fusion category $\mathcal S_2$ with monoidal functor $F:\mathcal S_1\to\mathcal S_2$. By clarifying mathematical (especially category theoretical) meaning of renormalization group domain wall/interface or boundary condition, we find the hidden extended vertex operator (super)algebra gives a unique (up to braided equivalence) completely $(\mathcal S_1\boxtimes\mathcal S_2)'$-anisotropic representative of the Witt equivalence class $[\mathcal S_1\boxtimes\mathcal S_2]$. The mathematical conjecture is supported physically, and passes various tests in concrete examples including non/unitary minimal models, and Wess-Zumino-Witten models. In particular, the conjecture holds beyond diagonal cosets. The picture also establishes the conjectured half-integer condition, which fixes infrared conformal dimensions mod $\frac12$. It further leads to the double braiding relation, namely braiding structures jump at conformal fixed points. As an application, we solve the flow from the $E$-type minimal model $(A_{10},E_6)\to M(4,3)$.