Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Deformation rigidity of some simple affine VOAs

Andrew R. Linshaw, Fei Qi

TL;DR

This paper proves deformation rigidity for a class of simple affine VOAs by showing all first-order deformations are trivial. The authors adapt and extend obstruction-vanishing techniques to both positive integral levels and the non-integral admissible level $-4/3$ for $\frak{sl}_2$, demonstrating regularity of $Y_1$ via analysis of commutator relations, weight considerations, and singular vectors. The results indicate that neither $C_2$-cofiniteness nor rationality is necessary for deformation rigidity and motivate a broader conjecture: all simple affine VOAs not coinciding with their universal counterparts $V^k(\mathfrak g)$ should exhibit rigidity. These findings connect deformation theory with structure theory of affine VOAs and expand the known scope of Huang’s deformation rigidity conjecture.

Abstract

In this paper, we prove that simple affine vertex operator algebras with positive integral levels admit only trivial first-order deformations. Therefore, the deformation rigidity conjecture of strongly rational vertex operator algebras holds for these cases. We also show that the same holds simple affine vertex operator algebra of $\mathfrak{sl}_2$ at the non-integral admissible level $-4/3$. Therefore, neither $C_2$-cofiniteness nor rationality is a necessary condition for deformation rigidity of VOAs. We conjecture that the same should hold for every simple affine VOA that does not coincide with the corresponding universal affine VOA.

Deformation rigidity of some simple affine VOAs

TL;DR

This paper proves deformation rigidity for a class of simple affine VOAs by showing all first-order deformations are trivial. The authors adapt and extend obstruction-vanishing techniques to both positive integral levels and the non-integral admissible level for , demonstrating regularity of via analysis of commutator relations, weight considerations, and singular vectors. The results indicate that neither -cofiniteness nor rationality is necessary for deformation rigidity and motivate a broader conjecture: all simple affine VOAs not coinciding with their universal counterparts should exhibit rigidity. These findings connect deformation theory with structure theory of affine VOAs and expand the known scope of Huang’s deformation rigidity conjecture.

Abstract

In this paper, we prove that simple affine vertex operator algebras with positive integral levels admit only trivial first-order deformations. Therefore, the deformation rigidity conjecture of strongly rational vertex operator algebras holds for these cases. We also show that the same holds simple affine vertex operator algebra of at the non-integral admissible level . Therefore, neither -cofiniteness nor rationality is a necessary condition for deformation rigidity of VOAs. We conjecture that the same should hold for every simple affine VOA that does not coincide with the corresponding universal affine VOA.
Paper Structure (15 sections, 2 theorems, 42 equations)

This paper contains 15 sections, 2 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. First-order deformations
  4. Equivalent first-order deformations
  5. Simple affine VOA
  6. Triviality of a regular $Y_1$
  7. The positive integral level case
  8. The mode $f^{def}(1)$
  9. The terms with $e^{def}(-1)$
  10. The term with $h^{def}(0)$
  11. Proof of $c=0$
  12. Affine $\mathfrak{sl}_2$ with admissible level $-4/3$
  13. $f^{def}(1)$-action on the singular vector
  14. $h^{def}(1)$-actions on the singular vector
  15. Proof of $c=0$

Key Result

Theorem 3.1

If for every $a, b\in {\mathfrak g}$, $Y_1(a(-1)\mathbf{1}, x)b(-1)\mathbf{1}$ is regular, i.e., it contains no negative powers, then $Y_1$ defines a trivial first-order deformation.

Theorems & Definitions (4)

  • Remark 2.1
  • Theorem 3.1
  • proof
  • Theorem 4.1