Equivalence Relations on Vertex Operator Algebras, I: Genus

In this first of a series of two papers, we investigate two different equivalence relations obtained by generalizing the notion of genus of even lattices to the setting of vertex operator algebras (or two-dimensional chiral algebras). The bulk genus equivalence relation was defined in arXiv:math/0209333 and groups (suitably regular) vertex operator algebras according to their modular tensor category and central charge. Hyperbolic genus arXiv:2004.01441 tests isomorphy after tensoring with a hyperbolic plane vertex algebra. Physically, two rational chiral algebras are said to belong to the same bulk genus if they live on the boundary of the same 2+1d topological quantum field theory; they belong to the same hyperbolic genus if they can be related by current-current exactly marginal deformations after tensoring a non-chiral compact boson. As one main result, we prove the conjecture that the hyperbolic genus defines a finer equivalence relation than the bulk genus. This is based on a new, equivalent characterization of the hyperbolic genus that uses the maximal lattice inside a vertex operator algebra and its commutant (or coset). We discuss the implications of these constructions for the classification of rational conformal field theory. In particular, we propose a program for (partially) classifying $c=32$, holomorphic vertex operator algebras (or chiral conformal field theories), and obtain novel lower bounds, via a generalization of the Smith-Minkowski-Siegel mass formula, on the number of vertex operator algebras at higher central charges. Finally, we conjecture a Siegel-Weil identity which computes the "average" torus partition function of an ensemble of chiral conformal field theories defined by any hyperbolic genus, and interpret this formula physically in terms of disorder-averaged holography.