Twisted vertex operator representation of $𝔰 𝔩 2 𝕂 ˆ$

Let $ˆ 𝔞 1 = 𝔰 𝔩 2 𝕂 ˆ [𝜎 1]$ be the $𝜎 1$ -twisted affine Lie algebra of $𝔰 𝔩 2 𝕂$ , and $𝑉$ be the corresponding $ℤ + 12$ -natural Heisenberg module on $𝔥 = 𝕂 𝛼$ . Defining

𝛼 (𝑧) \pm = \sum 𝑛 \in \pm (ℕ 0 + 12) 𝛼 (𝑛) 𝑧 - 𝑛 𝛼 (𝑧) = \sum 𝑛 \in ℤ + 12 𝛼 (𝑛) 𝑧 - 𝑛 = 𝛼 (𝑧) + + 𝛼 (𝑧) - 𝐸 \pm (𝛼, 𝑧) = e - 𝐷 - 1 𝛼 (𝑧) \pm

we construct the twisted vertex operator

𝑋 ℤ + 12 (𝛼, 𝑧) = 𝐸 - (- 𝛼, 𝑧) 𝐸 + (- 𝛼, 𝑧) 2 ⟨ 𝛼, 𝛼 ⟩ = : e 𝐷 - 1 𝛼 (𝑧) : 2 - ⟨ 𝛼, 𝛼 ⟩

where the second expression used the Normal ordered product. Skipping over a lot of detail¹, the representation of $˜ 𝔥 [- 1]$ on $𝑉$ extends to precisely two irreducible representations

𝜋 \pm : ˜ 𝔞 1 \to E n d 𝑉 𝑥 𝛼 1 (𝑧) \mapsto 𝑋 (𝛼 1, 𝑧)

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1988. Vertex operator algebras and the Monster, §3, pp. 61ff. ↩

Twisted vertex operator representation of 𝔰𝔩2⁡𝕂ˆ

Footnotes

Twisted vertex operator representation of $𝔰 𝔩 2 𝕂 ˆ$