# A Z 2-orbifold model of the symplectic fermionic vertex by Abe T.

By Abe T.

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2 Proof We see that φ1 (τ )2d = SSF(θ )+ (τ ) + SSF(θ )+ (τ ), φ2 (τ )2d = SSF(θ )+ (τ ) − SSF(θ )− (τ ) and φ3 (τ )2d = 2d SSF + (τ ) + SSF − (τ ) , η(τ )2d = SSF + (τ ) − SSF − (τ ). 6) prove the proposition. 3 The automorphism group of SF + We determine the automorphism group of SF + in this section. We first recall that the group of all linear isomorphisms of h which preserve the skew-symmetric bilinear form · , · is the symplectic group Sp(2d, C). We extend the action of Sp(2d, C) on h to SF by the properties g(1) = 1, 1 r g(ψ(−n · · · ψ(−n 1) r) 1) = (g(ψ 1 ))(−n1 ) · · · (g(ψ r ))(−nr ) 1 for any g ∈ Sp(2d, C), ψ i ∈ h and ni ∈ Z>0 .

Thus ψ(0) acts trivially on the quotient SF[r]/SF[r + 1]. Therefore, SF[r]/SF[r + 1] ∼ = SF ⊗ (r (h)/r+1 (h)) as left SF-modules. Since dim(r (h)/r+1 (h)) = 2d r , SF[r]/SF[r + 1] is a direct sum copies of SF as an SF-module. of 2d r We note that SF is naturally an SF + -module. We determine the space (SF) of singular vectors in SF as an SF + -module. If u ∈ (SF) ∩ SF[r] and u ∈ / SF[r + 1] for 788 T. Abe some r, then u + SF[r + 1] is a nonzero singular vector of SF[r]/SF[r + 1]. Since this quotient is a direct sum of copies of SF + and SF − , we see that u + SF[r + 1] is a sum of eigenvectors for L0 of weight 0 or 1.

This implies that the linear maps γ and γg from SF − to the contragredi∞ − ∗ − ent SF + -module (SF − ) = n=1 (SF n ) ⊂ D(SF ) defined by γ (u) = (u, · ) and + γg (u) = (u, · )g respectively are SF -module isomorphisms. Hence γ −1 ◦ γg = αg idSF − for some αg ∈ C − {0}. This proves the proposition. 6, we can assume that ( · , · )g = ( · , · ) if necessary by multiplying a suitable scalar to fg . Hence we have fg (ψ), fg (φ) = ψ, φ for any ψ, φ ∈ h(∼ = SF − 1 ). − Therefore, the restriction of fg to SF 1 gives an element of Sp(2d, C).