Rational RG flow, extension, and Witt class

TL;DR

This work proposes that rational RG flows preserving a surviving pre-modular fusion subcategory $\mathcal{S}_{UV}$ and ending in an IR subcategory $\mathcal{S}_{IR}$ are in one-to-one correspondence with completely $(\mathcal{S}_{UV}\boxtimes\mathcal{S}_{IR})'$-anisotropic representatives of the Witt class $[\mathcal{S}_{UV}\boxtimes\mathcal{S}_{IR}]$, realized by maximal connected étale algebras. A universal half-integer constraint on conformal dimensions, $h^{UV}_j+h^{IR}_j\in\tfrac{1}{2}\mathbb{Z}$, and a double braiding relation between UV and IR braidings are derived from the extended vertex operator (super)algebra structure associated with RG domain walls. The conjecture is tested across unitary and non-unitary minimal models, various WZW examples, and the $E$-type minimal model $(A_{10},E_6)$, consistently yielding completely anisotropic Witt representatives such as Toric Code, Vect$_{\mathbb{C}}$, and Ising-typed constructions. These results provide a practicable framework to classify and solve rational RG flows via categorical and VOA extensions, linking infrared data to a unique Witt-class representative and offering analytic constraints on conformal data. The findings highlight a principled mechanism for emergent symmetry and energy minimization in RG endpoints and demonstrate the predictive power of Witt-equivalence-based classification in RCFTs.

Abstract

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 $(A_{10},E_6)\to M(4,3)$.