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The vacuum Verma module has a level-one null vector

Statement

For every $c \in C$ , the vacuum Verma module $M (c, 0)$ has a nonzero singular vector $L_{- 1} v_{0}$ at grade one. Its Shapovalov norm is zero, it lies in the radical, and $\det G_{1} (c, 0) = 0$ . This is an algebraic null-vector example; it asserts no positivity or Hilbert-space structure.

Facts & Assumptions

Given: The algebraic module $M (c, 0)$ and its normalized form.

[F1]

Virasoro algebra in the $L_{m}$ convention: $[L_{m}, L_{n}] = (m - n) L_{m + n} + (m^{3} - m) δ_{m + n, 0} C / 12$ .

[F2]

Virasoro Verma modules and contravariant forms: $L_{n} v_{0} = 0$ for $n > 0$ , $L_{0} v_{0} = 0$ , and $C v_{0} = c v_{0}$ ; the normalized form is contravariant and its radical is the unique maximal proper submodule.

[F3]

PBW basis and grades of Virasoro Verma modules: each grade has its partition basis and $L_{- 1} v_{0}$ is the nonzero basis vector for the unique partition of grade one.

Proof

1.1F1F2F3algebra

Put $w = L_{- 1} v_{0}$ . It is nonzero by [F3] and has grade one because $[L_{0}, L_{- 1}] = L_{- 1}$ by [F1]. For $n = 1$ , $L_{1} w = [L_{1}, L_{- 1}] v_{0} = 2 L_{0} v_{0} = 0$ . For every $n \geq 2$ , $L_{n} w = [L_{n}, L_{- 1}] v_{0} = (n + 1) L_{n - 1} v_{0} = 0$ , since $n - 1 > 0$ . Thus $w$ is singular. Also $L_{0} w = w$ , so it is not a second grade-zero vector.

2.1F1F2F3step 1.1algebra∎

Contravariance gives $⟨ w, w ⟩ = ⟨ v_{0}, L_{1} L_{- 1} v_{0} ⟩ = 0$ , because $L_{1} L_{- 1} v_{0} = 2 L_{0} v_{0} = 0$ . If $z$ is homogeneous of grade $n$ , then $L_{0} w = w$ , $L_{0} z = n z$ , and contravariance under $L_{0}$ gives $(1 - n) ⟨ w, z ⟩ = 0$ . Thus $w$ is orthogonal to every grade $n \neq 1$ . Grade one is spanned by $w$ by [F3], and its self-pairing is zero, so $w$ is orthogonal to the whole algebraic direct sum and lies in the radical. At general highest weight $h$ , the grade-one Gram matrix is the one-entry matrix $(2 h)$ , so at $h = 0$ its determinant is zero for every central charge $c$ . A zero norm here is a bilinear-form statement and uses no positivity assumption.

The vacuum Verma module has a level-one null vector

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources