A class of vertex operator algebras generated by Virasoro vectors
Runkang Feng
TL;DR
This work studies a class of simple OZ-type vertex operator algebras generated by a family of Virasoro vectors ω^{ij} with central charges c_m, proving that the entire VOA is uniquely determined by its Griess algebra V_2 and determining its automorphism group Aut(V) as the symmetric group ${\mathfrak S}_n$. It shows that V_2 forms a Matsuo algebra with parameters α = h^{(m)}_{m+1,1} and β = c_m when G = ${\mathfrak S}_n$, and establishes a concrete generating set for V via iterated modes of the ω^{ij}. The paper also provides necessary conditions for unitarity by analyzing the σ-invariant Hermitian form, obtaining sharp bounds on m depending on n, and connects these positivity conditions to C_2-cofiniteness. Overall, it extends the structural understanding of OZ-type VOAs generated by Virasoro vectors and links their symmetry to explicit algebraic objects and fusion rules, with implications for realizations and unitary theories.
Abstract
In this paper, we study a class of simple OZ-type vertex operator algebras $V$ generated by simple Virasoro vectors $ω^{ij}=ω^{ji}$, $1\leq i<j\leq n$, $n\geq 3$. We prove that $V$ is uniquely determined by its Griess algebra $V_2$. The automorphism group of $V$ is also determined. Furthermore, we give the necessary conditions for $V$ to be unitary.