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First-order deformations of freely generated vertex algebras

Vladimir Kovalchuk, Fei Qi

TL;DR

The paper solves the problem of classifying first-order vertex-algebraic deformations for grading-restricted vertex algebras freely generated by positive-weight elements by computing $H^2_{1/2}(V,V)$ via Huang’s cohomology. It develops a general, generator-based cocycle framework in which a cocycle is determined by its singular part on generators, uses a generating-function technique to extend to all pairs, and proves that in the freely generated case all complementary solutions are coboundaries, yielding $H^2_{1/2}(V,V)=H^2_{\infty}(V,V)$. This leads to explicit first-order deformations for Virasoro, universal affine, Heisenberg, and $W_3^c$ VOAs, highlighting how central-charge and level deformations capture the nontrivial deformations. The results provide a practical algorithm for deformation classification and illuminate convergence properties essential for composing Gerstenhaber-like structures in this context, with potential extensions beyond freely generated algebras.

Abstract

We solve the problem of how to classify the first-order vertex-algebraic deformations for any grading-restricted vertex algebra $V$ that is freely generated by homogeneous elements of positive weights. We approach by computing the second cohomology $H^2_{1/2}(V, V)$ constructed by Yi-Zhi Huang. We start with the cocycle on two generators and show that its cohomology class is completely determined by its singular part. To extend the cocycle to any pair of elements in $V$, we take a generating function approach, formulate the cocycle equation, and show that all the complementary solutions are coboundaries. Then we use a very general procedure to construct a particular solution. The procedure applies to vertex algebras that are not freely generated. As a by-product, we show that $H^2_{1/2}(V, V) = H^2_\infty(V, V)$. Using these results, we explicitly determine the first-order deformations of the universal Virasoro VOA $Vir_c$, universal affine VOA $V^l(\mathfrak{g})$, Heisenberg VOA $V^l(\mathfrak{h})$, and the universal Zamolodchikov VOA $W_3^c$.

First-order deformations of freely generated vertex algebras

TL;DR

The paper solves the problem of classifying first-order vertex-algebraic deformations for grading-restricted vertex algebras freely generated by positive-weight elements by computing via Huang’s cohomology. It develops a general, generator-based cocycle framework in which a cocycle is determined by its singular part on generators, uses a generating-function technique to extend to all pairs, and proves that in the freely generated case all complementary solutions are coboundaries, yielding . This leads to explicit first-order deformations for Virasoro, universal affine, Heisenberg, and VOAs, highlighting how central-charge and level deformations capture the nontrivial deformations. The results provide a practical algorithm for deformation classification and illuminate convergence properties essential for composing Gerstenhaber-like structures in this context, with potential extensions beyond freely generated algebras.

Abstract

We solve the problem of how to classify the first-order vertex-algebraic deformations for any grading-restricted vertex algebra that is freely generated by homogeneous elements of positive weights. We approach by computing the second cohomology constructed by Yi-Zhi Huang. We start with the cocycle on two generators and show that its cohomology class is completely determined by its singular part. To extend the cocycle to any pair of elements in , we take a generating function approach, formulate the cocycle equation, and show that all the complementary solutions are coboundaries. Then we use a very general procedure to construct a particular solution. The procedure applies to vertex algebras that are not freely generated. As a by-product, we show that . Using these results, we explicitly determine the first-order deformations of the universal Virasoro VOA , universal affine VOA , Heisenberg VOA , and the universal Zamolodchikov VOA .
Paper Structure (35 sections, 27 theorems, 301 equations)

This paper contains 35 sections, 27 theorems, 301 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Grading-restricted vertex algebras
  4. Freely generated vertex algebras and standard filtration
  5. $E$- and $R$-notations
  6. First-order deformation
  7. 2-cocycles and 2-coboundaries
  8. The cocycle equation and its complementary solution
  9. Regular part of the cocycle on the generators
  10. Generating function approach
  11. Multivariable cocycle equation
  12. The base case
  13. Checking the convergence
  14. The $n$-variable cocycle equation
  15. The ansatz for the solution the $n$-variable cocycle equation
  16. ...and 20 more sections

Key Result

Proposition 3.1

Fix $a, b\in S$. Then there exists a homogeneous linear map $\phi: V\to V$ (depending on $a, b$) such that $\phi$ commutes with $D$, and

Theorems & Definitions (73)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.9
  • Proposition 3.1
  • proof
  • Remark 3.2
  • ...and 63 more