# Zn Orbifolds of Vertex Operator Algebras

2021

Dissertation

Ph.D.

## Organizational Unit

College of Natural Science and Mathematics, Mathematics

Andrew R. Linshaw

Nikolaos Galatos

Schuyler Van Engelenburg

## Keywords

Algebra, Heisenberg, Operator, Orbifold, Vertex, VOA

## Abstract

Given a vertex algebra V and a group of automorphisms of V, the invariant subalgebra VG is called an orbifold of V. This construction appeared first in physics and was also fundamental to the construction of the Moonshine module in the work of Borcherds. It is expected that nice properties of V such as C2-cofiniteness and rationality will be inherited by VG if G is a finite group. It is also expected that under reasonable hypotheses, if V is strongly finitely generated and G is reductive, VG will also be strongly finitely generated. This is an analogue of Hilbert’s theorem on the finite generation of classical invariant rings, and it is known in certain classes of vertex algebras such as free field algebras, affine vertex algebras, and W-algebras. Unfortunately, the proof is nonconstructive and it is difficult to give explicit strong generating sets. Finding strong generators is very useful for understanding the representation theory of vertex algebras since strong generators give rise to generators for their Zhu algebras.

There are two main results in this thesis. First, the rank n Heisenberg algebra H(n) has full automorphism the orthogonal group. It is known to be strongly finitely generated, but no upper bound is known for how many generators are needed. We show that it is if type W(2; 4; ... ; 2N) for some N satisfying n2+3n <= 2N <= 2n2+4n. This means that it has a minimal strong generating set consisting of one field in each weight 2; 4; ... ; 2N. Second, we consider the Zn orbifold of the rank 2 Heisenberg algebra H(2). For n = 2 and n = 3, the structure of this orbifold was already understood, but it is considerably more difficult to find the minimal strong generating type for all n. We will show that for all n >= 3, H(2)Zn is of type W(2; 3; 4; 5; n2; (n + 2)2).

This strong generating set immediately gives rise to generators for Zhu’s associative algebra A(H(2)Zn) and Zhu’s commutative algebra RH(2)Zn . It is an interesting question how the orbifold functor interacts with the passage to Zhu’s commutative algebra. In particular, if V is a vertex algebra and G is a finite group of automorphisms of V, there is an induced action of G on RV, and we have a map RVG ! (RV)G. In general, this map is not an isomorphism but it an interesting question whether it is an isomorphism at the level of reduced rings, that is, after quotienting by the nilradical. Using the strong generating set for H(2)Z3 , we will prove that this is the case in this example.

More generally, if RVG and (RV)G become isomorphic as reduced rings, they must have the same Krull dimension. For strongly finitely generated vertex algebras, this can be viewed as a generalization of the conjecture that taking G-invariants preserves C2-cofiniteness, since C2-cofiniteness of V means that RV has Krull dimension zero.

## Publication Statement

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Daniel Graybill

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