Equivalence Relations on Vertex Operator Algebras, II: Witt Equivalence and Orbifolds

TL;DR

The work develops a unifying framework for relating strongly rational VOAs and 1+1d RCFTs through Witt equivalence, positing that matching central charges and Witt classes of Rep(V) are necessary for topological connections and conjecturing sufficiency. It introduces orbifold and interface equivalence as concrete realizations of topological manipulations, and shows that these notions imply Witt equivalence, with partial results tightening the link via inner orbifold equivalence. The authors relate these ideas to RCFT classification and deformation classes, proposing structure theorems under mild conjectures and highlighting the role of generalized symmetries, quantum Galois theory, and p-neighborhoods in organizing theories. They illustrate the framework with examples at low central charge, c=1 and c=24, and discuss implications for moonshine, SU(2)1, and Virasoro minimal models, while outlining future directions in proving conjectures and expanding the mathematical toolkit for gauging in categories of infinite rank topological lines.

Abstract

When can two strongly rational vertex operator algebras or 1+1d rational conformal field theories (RCFTs) be related by topological manipulations? For vertex operator algebras, the term "topological manipulations" refers to operations like passing to a conformal extension or restricting to a conformal subalgebra; for RCFTs, topological manipulations include operations like gauging (or orbifolding) a finite subpart of a generalized global symmetry or interpolating to a new theory via a topological line interface of finite quantum dimension. Inspired by results in the theory of even lattices and tensor categories, we say that two strongly rational vertex operator algebras are Witt equivalent if their central charges agree and if their modular tensor categories are Witt equivalent. Two RCFTs are said to be Witt equivalent if their central charges agree and if their associated 2+1d topological field theories can be separated by a topological surface. We argue that Witt equivalence is necessary for two theories to be related by topological manipulations. We conjecture that it is also sufficient, and give proofs in various special cases. We relate this circle of ideas to the problem of classifying RCFTs, and to lore concerning deformation classes of quantum field theories. We use the notion of Witt equivalence to argue, assuming the conjectural classification of unitary, $c=1$ RCFTs, that all of the finite symmetries of the $SU(2)_1$ Wess-Zumino-Witten model are invertible. We also sketch a "quantum Galois theory" for chiral CFTs, which generalizes prior mathematical literature by incorporating non-invertible symmetries; we illustrate this non-invertible Galois theory in the context of the monster CFT, for which we produce two Fibonacci lines. Finally, we discuss $p$-neighborhood of vertex operator algebras, which is a special topological manipulation related to $\mathbb{Z}_p$-orbifolding.

Figures (17)

• Figure 1: Interrelations between the various equivalence relations on strong-ly rational vertex operator algebras studied in our work. The solid arrows are rigorously established. The dashed arrows are conjectural.
• Figure 2: The 2+1d representation of an RCFT ${_V}\mathcal{H}_W$.
• Figure 3: The 2+1d representation of a topological line interface $I$ between two 1+1d RCFTs ${_V}\mathcal{H}_W$ and ${_{V'}}\mathcal{H}'_{W'}$.
• Figure 4: Witt equivalence of even lattices $L$ and $M$.
• Figure 5: Rational equivalence of even lattices $L$ and $M$.

Theorems & Definitions (117)

• Definition 2.1: Lattice Genus
• Definition 2.2: Witt Equivalence
• Definition 2.3: Witt Equivalence
• Definition 2.4: Rational Equivalence
• Remark 2.5
• Proposition 2.6
• proof
• Definition 2.7: Neighborhood
• Remark 2.8
• proof