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Descent Theory for Vertex Algebras

Robin Mader, Terry Gannon, Arturo Pianzola

TL;DR

The paper develops a descent framework for vertex algebras over differential rings, enabling base change, twisted forms, and cohomological classifications. It introduces $R$-vertex algebras, arc algebras, and vertex schemes to realize automorphism data as affine vertex group schemes, and applies faithful flat descent and non-abelian cohomology to classify twisted forms of affine and Heisenberg vertex algebras over Laurent polynomial bases. The main results show that twisted forms of $V(\mathfrak g,\ell)$ are governed by Out$(\mathfrak g)$ (via loop algebras) and that isotrivial forms of Heisenberg VOAs correspond to finite-order orthogonal transformations, with a cohomological Li correspondence for twisted modules emerging from descent theory. The framework provides a conceptual route, grounded in Galois and continuous cohomology, to understand forms, pullbacks of twisted modules, and their interrelations in vertex algebra theory.

Abstract

Vertex algebras can be defined over any differential commutative ring. We develop the general descent theory for vertex algebras over such bases. We apply this to the classification of twisted forms of affine and Heisenberg vertex algebras, and to reinterpret and generalize a correspondence of Li.

Descent Theory for Vertex Algebras

TL;DR

The paper develops a descent framework for vertex algebras over differential rings, enabling base change, twisted forms, and cohomological classifications. It introduces -vertex algebras, arc algebras, and vertex schemes to realize automorphism data as affine vertex group schemes, and applies faithful flat descent and non-abelian cohomology to classify twisted forms of affine and Heisenberg vertex algebras over Laurent polynomial bases. The main results show that twisted forms of are governed by Out (via loop algebras) and that isotrivial forms of Heisenberg VOAs correspond to finite-order orthogonal transformations, with a cohomological Li correspondence for twisted modules emerging from descent theory. The framework provides a conceptual route, grounded in Galois and continuous cohomology, to understand forms, pullbacks of twisted modules, and their interrelations in vertex algebra theory.

Abstract

Vertex algebras can be defined over any differential commutative ring. We develop the general descent theory for vertex algebras over such bases. We apply this to the classification of twisted forms of affine and Heisenberg vertex algebras, and to reinterpret and generalize a correspondence of Li.

Paper Structure

This paper contains 27 sections, 22 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Vertex rings
  3. Vertex rings and the canonical Hasse-Schmidt derivation
  4. Vertex algebras over commutative vertex rings
  5. Filtered vertex algebras
  6. Automorphisms of vertex algebras
  7. Extension of scalars
  8. Vertex group schemes
  9. Arc algebras
  10. Vertex schemes and groups
  11. Vacuum subgroups
  12. Representability of $\operatorname{\mathbf{Aut}}(V)$
  13. Finitely generated vertex algebras
  14. Heisenberg and universal affine vertex algebras
  15. Descent theory
  16. ...and 12 more sections

Key Result

Theorem 2.2

A commutative vertex ring $R$ is a commutative unital ring with respect to the product $rs = r_{-1} s$, for $r,s \in R$, with $1_R = \mathbf 1.$ Furthermore, the ring $R$ admits an iterative Hasse-Schmidt derivation on this product (Equation (HasseSchmidtVA) for $n = -1$). Conversely, any commutativ

Theorems & Definitions (42)

  • Definition 2.1
  • Theorem 2.2: Cf. Mason and Bor
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Example 3.3
  • Example 3.4
  • ...and 32 more