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Two positivity bounds from Virasoro norm identities

Statement

Let $V$ be the Virasoro algebra with central element $C$ , represented by operators $L_{n}$ on an invariant complex inner-product space $D$ , with inner product conjugate-linear in the first variable and linear in the second. Assume $C = c I$ , the form is positive definite, and for all vectors used below the adjoint identity $⟨ L_{n} u, v ⟩ = ⟨ u, L_{- n} v ⟩$ holds. If $v \neq 0$ is a highest-weight vector of weight $h$ (so $L_{n} v = 0$ for $n > 0$ and $L_{0} v = h v$ ), then $h \in [0, \infty)$ . If $Ω \neq 0$ is a vacuum vector with $L_{n} Ω = 0$ for every $n \geq - 1$ , then $c \in [0, \infty)$ . These are only the two norm inequalities; no classification of unitary Virasoro representations is asserted.

Facts & Assumptions

Given: A representation of the fixed Virasoro algebra on the invariant domain, a positive-definite inner product, the stated adjoint identities, and the highest-weight and vacuum vectors.

[F1]

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

[F2]

Highest-weight vector and module for Virasoro: A highest-weight vector $v \neq 0$ satisfies $L_{n} v = 0$ for all $n > 0$ , $L_{0} v = h v$ , and $C v = c v$ ; the definition itself asserts neither unitarity nor irreducibility.

[A1]

The operators $L_{n}$ preserve the common domain, the inner product is positive definite, and the adjoint relation is valid on $(v, L_{- 1} v)$ and $(Ω, L_{- 2} Ω)$ .

[A2]

The vacuum vector is nonzero and obeys $L_{n} Ω = 0$ for every $n \geq - 1$ ; in particular $L_{0} Ω = L_{2} Ω = 0$ .

Proof

Given: [F1]–[F2], [A1]–[A2], and the indicated vectors.

1.1givenF1F2A1algebra

For the highest-weight vector, adjointness gives $∥ L_{- 1} v ∥^{2} = ⟨ v, L_{1} L_{- 1} v ⟩$ . Since $L_{1} v = 0$ , the operator product equals $[L_{1}, L_{- 1}] v$ ; [F1] gives $[L_{1}, L_{- 1}] = 2 L_{0}$ because $1^{3} - 1 = 0$ . Hence $∥ L_{- 1} v ∥^{2} = 2 h ∥ v ∥^{2}$ .

1.2givenF1A1A2algebra

For the vacuum, adjointness gives $∥ L_{- 2} Ω ∥^{2} = ⟨ Ω, L_{2} L_{- 2} Ω ⟩ = ⟨ Ω, [L_{2}, L_{- 2}] Ω ⟩$ because $L_{2} Ω = 0$ . By [F1], $[L_{2}, L_{- 2}] = 4 L_{0} + (6 / 12) C = 4 L_{0} + (c / 2) I$ on this representation; [A2] therefore gives $∥ L_{- 2} Ω ∥^{2} = (c / 2) ∥ Ω ∥^{2}$ .

2.1step 1.1A1algebra

The left side of step 1.1 is a nonnegative real number and $∥ v ∥^{2} > 0$ . Therefore $h = ∥ L_{- 1} v ∥^{2} / (2 ∥ v ∥^{2})$ is real and nonnegative.

3.1step 1.2A1algebra∎

Positivity and $∥ Ω ∥^{2} > 0$ make $c = 2 ∥ L_{- 2} Ω ∥^{2} / ∥ Ω ∥^{2}$ real and nonnegative. The two conclusions above give exactly the asserted bounds and no stronger representation-theoretic classification.

Two positivity bounds from Virasoro norm identities

Statement

Facts & Assumptions

Proof

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