Editing Goddard–Thorn theorem (section)

== Statement ==
This statement is that of Borcherds (1992).

Suppose that <math>V</math> is a [[unitary representation]] of the [[Virasoro algebra]] <math>\mathrm{Vir}</math>, so <math>V</math> is equipped with a [[non-degenerate]] [[bilinear form]] <math>(\cdot, \cdot)</math> and there is an [[algebra homomorphism]] <math>\rho: \mathrm{Vir} \rightarrow \mathrm{End}(V)</math> so that 
<math display = block>\rho(L_i)^\dagger = \rho(L_{-i})</math>
where the [[adjoint]] is defined with respect to the bilinear form, and 
<math display = block>\rho(c) = 24\mathrm{id}_V.</math>
Suppose also that <math>V</math> decomposes into a [[direct sum]] of [[eigenspace]]s of <math>L_0</math> with non-negative, integer eigenvalues <math>i \geq 0</math>, denoted <math>V^i</math>, and that each <math>V^i</math> is finite dimensional (giving <math>V</math> a <math>\mathbb{Z}_{\geq 0}</math>-[[graded vector space|grading]]). Assume also that <math>V</math> admits an action from a [[group (algebra)|group]] <math>G</math> that preserves this grading.

For the two-dimensional even [[unimodular lattice|unimodular]] [[Lorentzian metric|Lorentzian]] lattice II<sub>1,1</sub>, denote the corresponding [[vertex operator algebra#vertex operator algebra defined by an even lattice|lattice vertex algebra]] by <math>V_{II_{1,1}}</math>. This is a II<sub>1,1</sub>-graded algebra with a bilinear form and carries an action of the Virasoro algebra. 

Let <math>P^1</math> be the subspace of the vertex algebra <math>V \otimes V_{II_{1,1}}</math> consisting of vectors <math>v</math> such that <math>L_0 \cdot v = v, L_n \cdot v = 0</math> for <math>n > 0</math>. Let <math>P^1_r</math> be the subspace of <math>P^1</math> of degree <math>r \in II_{1,1}</math>. Each space inherits a <math>G</math>-action which acts as prescribed on <math>V</math> and trivially on <math>V_{II_{1,1}}</math>.

The [[quotient space (linear algebra)|quotient]] of <math>P^1_r</math> by the [[nullspace]] of its bilinear form is naturally [[isomorphism|isomorphic]] as a <math>G</math>-module with an invariant bilinear form, to <math>V^{1 - (r,r)/2}</math> if <math>r \neq 0</math> and <math>V^1 \oplus \mathbb{R}^2</math> if <math>r = 0</math>.

=== II<sub>1,1</sub> ===
The lattice II<sub>1,1</sub> is the rank 2 lattice with bilinear form
<math display = block>\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}.</math>
This is even, unimodular and integral with [[signature (linear algebra)|signature]] (+,-).