Modularity of Relatively Rational Vertex Algebras and Fusion Rules of Principal Affine W-Algebras

Abstract

We study modularity of the characters of a vertex (super)algebra equipped with a family of conformal structures. Along the way we introduce the notion of rationality and cofiniteness relative to such a family. We apply the results to determine modular transformations of trace functions on admissible modules over affine Kac-Moody algebras and, via BRST reduction, trace functions on regular affine W -algebras. * 1We prove the following result as Theorem 5.12. In fact we derive it from the stronger but more technical Proposition 5.11. Theorem 1.2. Let (V, ω) be a conformal vertex (super )algebra graded by integer conformal weights. Let h ∈ V be a current satisfying the OPE relation, and such that h 0 acts semisimply on V . Assume (V, ω) to be rational relative to h and cofinite relative to h, and write Irr(V, h) for the set of irreducible h-stable positive energy V -modules. Letwhere ρ is some representation of SL 2 (Z), is satisfied for all u ∈ V if it is satisfied for u = |0 .We make some remarks on the theorem and its proof. The essential idea of the proof is to apply Zhu's modularity theorem to the vertex algebra (V, ω(z)). However ω(z) equips V with noninteger conformal weights, and Zhu's theorem actually fails in this case. This situation is rectified in the reference [28], where it is shown that modular transformations map the trace functions F M to trace functions on particular twisted modules. The task becomes to relate trace functions on twisted and untwisted V -modules. This is achieved by the use of Li's shift operators ∆(u, z) (which appear explicitly in (1.2) above). The condition of relative cofiniteness is inspired by the work [7], and was used in [28].The transformation (1.2) was uncovered in the case of N = 2 superconformal vertex algebras in [13, Theorem 9.13 (b)], with h equal to the U (1) current of the N = 2 algebra. There the functions F M are shown to be flat sections of the bundle of conformal blocks over the universal elliptic curve, and (1.2) is derived from the geometry of this bundle.We also note that a result closely related to Theorem 1.2 was recently and independently obtained in [24] in the case of V rational and C 2 -cofinite (see also [26]).An important class of vertex algebras that are relatively cofinite and generically rational in the sense discussed above is afforded by the simple affine vertex algebras at admissible level.

Fusion categories for affine vertex algebras at admissible levels

“…Fusion rules of W-algebras are needed for the proof of the first main Theorem. They have been proven in the simply-laced case if a certain coprime property holds for the denominator of the level [32,33]. The proof was based on Verlinde's formula and I take the slightly different point of view that the map (2) is a ring isomorphism from the Grothendieck ring of the regular W-algebra to the endomorphism ring of the direct sum of all inequivalent simples (1).…”

“…Moreover it was shown that the Heisenberg coset is isomorphic to a regular W -algebra of type A. The proof relied on the fusion rules of regular W-algebras of type A and so in that work it could only be done if the coprime condition of [32,33] was satisfied. The exact same proof now works in general and so Corollary 1.…”

“…4 Ordinary modules for affine VOAs at admissible level I use notation and also many statements of the work of Arakawa and van Ekeren [33]. Let g be a simple Lie algebra and let k be a complex number then we denote the simple affine vertex operator algebra of g at level k by L k (g).…”

“…where the sum is over all inequivalent simple principal admissible level k modules and γ is the usual automorphy factor for Jacobi forms. Then for λ, µ ∈ P u−h ∨ + by [41,33] S χ…”

“…Here h is the Coxeter number, h ∨ the dual Coxeter number and r ∨ is the lacety of g. We take the notation of [33] for W-algebra modules, i.e. Here W + is the subgroup of automorphisms that preserve the coroot basis.…”