We define and investigate several notions of equivalence of VOAs. In doing so, we see that each notion of equivalence for even lattices (genus, Witt and rational equivalence) has two generalisations to VOAs, one being a more “classical” analogue, and one being a more honest “quantum” analogue.

We study the relations between these various notions. For example, we give a proof that two VOAs in the same hyperbolic genus (as defined by Moriwaki) are in the same bulk genus (as defined by Höhn). We also propose a programme for (partially) classifying c=32, holomorphic VOAs, and we conjecture a Siegel–Weil identity that computes the “average” torus partition function of an ensemble of chiral CFTs defined by any hyperbolic genus, and interpret this formula physically in terms of disorder-averaged holography.

We then study Witt and rational (or orbifold) equivalence. This boils down to the question when two VOAs can be related by topological manipulations (like conformal extensions and subalgebras). We argue that Witt equivalence is necessary for two theories to be related by topological manipulations, and we conjecture that it is also sufficient. We give proofs in various special cases.

We use the notion of Witt equivalence to argue, assuming the conjectural classification of unitary, c=1 RCFTs, that all of the finite global symmetries of the SU(2)_1 Wess-Zumino-Witten model are invertible. Finally, we 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 a Fibonacci symmetry.