By Abe T.
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Extra info for A Z 2-orbifold model of the symplectic fermionic vertex operator superalgebra
7 Let ψ, φ ∈ h and m, n ∈ Z>0 . Then g(ψ(−m) φ(−n) 1) = fg (ψ)(−m) fg (φ)(−n) 1. Proof First we assume that n = 1. For any u ∈ SF + , we see that (g(u), fg (ψ)(−m) fg (φ)) = −(fg (ψ)(m) g(u), fg (φ)) = (−1)m (g(u)(m) fg (ψ), fg (φ)), where the last identity follows from the skew symmetry formula ∞ a(n) b = (−1)kl i=0 k¯ (−1)n+1+i i L−1 b(n+i) a i! ¯l for a ∈ SF , b ∈ SF (k, l = 0, 1), n ∈ Z and the fact that L1 fg (φ) = fg (L1 φ) = 0. Since g(u)(m) fg (ψ) = fg (u(m) ψ) and the actions of fg and g preserve the bilinear form ( · , · ), we have (g(u), fg (ψ)(−m) fg (φ)) = (−1)m (u(m) ψ, φ) = (u, ψ(−m) φ) = (g(u), g(ψ(−m) φ)).
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