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Automorphism group schemes of lattice vertex operator algebras

Scott Carnahan, Hayate Kobayashi

TL;DR

This work constructs a scheme-theoretic description of the automorphism groups of lattice vertex operator algebras over arbitrary commutative rings. It introduces a split reductive group G_L over b Z, whose Lie algebra matches the weight-1 space (V_L)_1, and a finite flat group scheme O( ilde{L}) arising from a mu_2-cover of the lattice, such that Aut V_L is (up to a controlled intersection) the product G_L morph O( ilde{L}). The intersection is the Tits subgroup, and the quotient Aut V_L/G_L is isomorphic to the outer automorphism group Aut L / W_L of the lattice. Over rings where det L is invertible, Aut V_L descends to a base-change of this product, yielding a uniform, modular description of lattice VOA symmetries across bases. This ties lattice symmetries to explicit reductive group data and clarifies how the lattice’s outer automorphisms interact with VOA automorphisms, with broad implications for extending these structures to other rings and lattices.

Abstract

Given a positive definite even lattice and a commutative ring, there is a standard construction of a lattice vertex algebra over the commutative ring, and this is promoted to a vertex operator algebra when the determinant of the lattice is invertible. We describe the groups of automorphisms of these vertex algebras and vertex operator algebras as affine group schemes, showing in particular that each is an extension of an explicitly described split reductive group of ADE type by the outer automorphism group of the lattice.

Automorphism group schemes of lattice vertex operator algebras

TL;DR

This work constructs a scheme-theoretic description of the automorphism groups of lattice vertex operator algebras over arbitrary commutative rings. It introduces a split reductive group G_L over b Z, whose Lie algebra matches the weight-1 space (V_L)_1, and a finite flat group scheme O( ilde{L}) arising from a mu_2-cover of the lattice, such that Aut V_L is (up to a controlled intersection) the product G_L morph O( ilde{L}). The intersection is the Tits subgroup, and the quotient Aut V_L/G_L is isomorphic to the outer automorphism group Aut L / W_L of the lattice. Over rings where det L is invertible, Aut V_L descends to a base-change of this product, yielding a uniform, modular description of lattice VOA symmetries across bases. This ties lattice symmetries to explicit reductive group data and clarifies how the lattice’s outer automorphisms interact with VOA automorphisms, with broad implications for extending these structures to other rings and lattices.

Abstract

Given a positive definite even lattice and a commutative ring, there is a standard construction of a lattice vertex algebra over the commutative ring, and this is promoted to a vertex operator algebra when the determinant of the lattice is invertible. We describe the groups of automorphisms of these vertex algebras and vertex operator algebras as affine group schemes, showing in particular that each is an extension of an explicitly described split reductive group of ADE type by the outer automorphism group of the lattice.

Paper Structure

This paper contains 16 sections, 37 theorems, 9 equations.

Table of Contents

  1. Review of vertex algebras over commutative rings
  2. Vertex algebras
  3. Lattice vertex algebras
  4. Reductive group schemes
  5. Fundamentals
  6. The action of the reductive part
  7. Symmetries of the Mewtwo cover
  8. The cover of a lattice
  9. Orthogonal transformations
  10. Action of orthogonal transformations on the lattice vertex algebra
  11. The automorphism group scheme
  12. Basic properties
  13. Local Conjugation
  14. Normalizer of the distinguished torus
  15. Main Theorem
  16. ...and 1 more sections

Key Result

Lemma 1.1.7

Let $V$ be a $\mathbb{Z}$-graded vertex algebra over a commutative ring $R$. Then, $\operatorname{\underline{Aut}}_{V/R}$ is a group sheaf for the faithfully flat topology.

Theorems & Definitions (92)

  • Definition 1.1.1
  • Remark 1.1.2
  • Definition 1.1.3
  • Definition 1.1.4
  • Definition 1.1.5
  • Remark 1.1.6
  • Lemma 1.1.7
  • proof
  • Lemma 1.1.8
  • proof
  • ...and 82 more