Manuscript Oai280-A@1 · Oai280-A.tex · proposition · 27038823

Proposition: A conformal norm identity

Stage 5: conformal energy and gravitational energy › The central-charge norm identity · lines 1566–1585 · proof included or linked

Oai280-A@1#prop:centralcharge

For a normalized unitary VOA,

c_{V} = 2∥ ν ∥^{2} .

If it is of CFT type and $c_{V} = 0$ , it is the scalar VOA.

Proof. Since $ν = L_{- 2} Ω$ , unitarity and the Virasoro relation give

∥ ν ∥^{2} = ⟨ Ω, L_{2} L_{- 2} Ω ⟩ = ⟨ Ω, (4 L_{0} + c_{V} /2) Ω ⟩ = c_{V} /2.

If $c_{V} = 0$ , positivity gives $ν = 0$ , hence $L_{0} = 0$ . CFT type then forces $V = V_{0} = C Ω$ .

Displayed formulas in this passage

unlabeled · line 1569
$c_{V} = 2∥ ν ∥^{2} .$
unlabeled · line 1577
$∥ ν ∥^{2} = ⟨ Ω, L_{2} L_{- 2} Ω ⟩ = ⟨ Ω, (4 L_{0} + c_{V} /2) Ω ⟩ = c_{V} /2.$