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Next, we pass to the case of orientifold groups Γ = Z2 Zm with m = 2, 4. The restriction of the obstruction 3-cocycle to the orbifold group Z4 is u n,n ,n = (−1)kn n +n −[n +n ] 4 . It is not trivializable if k is odd, see [15]. On the other hand, its further restriction to Z2 ⊂ Z4 is trivial for all k. In order to proceed further, we note that the scalar product tr τz −n 0 e1 takes values in integers if n is even and in half-integers if n is odd. It follows that, for k even, only the terms ∆± in bγ ,γ contribute to u γ ,γ ,γ if m = 4.
Using (69), we obtain the relations: − wκ wzn wκ−1 wzn = e2π i ∆n , wzn wκ wzn wκ−1 = e2π i ∆n , + where ⎧ 0 for n = 0, ⎪ ⎪ ⎪ ⎪ ⎨± 1 (e ± e ) for n = 1, r 4 1 ∆± n = 1 ⎪± e1 for n = 2, ⎪ 2 ⎪ ⎪ ⎩ 1 ± 4 (e1 ∓ er ) for n = 3. Together with (70), they are all that is needed to find bγ ,γ for γ , γ in the maximal orientifold group Γ = Z2 Z4 . We may set bn,n = bn,n = bn,n = bn,n = n+n −[n+n ] 4 n −n−[n −n] 4 e1 , e1 + ∆− n, n −n−[n −n] 4 + s e1 + ∆+[n 0 +n] 32 K. Gaw¸edzki, R. R. Suszek, K. Waldorf for n, n = 0, 1, 2, 3.
Suppose (H1)–(H9) and that ε∗ > 0 is a sufficiently small number. 4) where p(x, y), q(x, y), r (x, y) are real polynomials of degree (2N + 1) satisfying | p(x, y)| + |q(x, y)| + |r (x, y)| = O(x 2 + y 2 ) as (x, y) → (0,√0) and αm,n (ω), βm,n (ω), γm,n (ω) ∈ Ha (Rd ; R2 ) ∩ L 2c (Hω∗ ) with 0 < a < inf ω∈K ω − λ(ω). Proof. 10). 4). 10) are given by real linear expressions of z, z¯ and f, ω , f, σ ∂ω ω and f, σ3 ξ . Hence it follows that p(x, y), q(x, y), r (x, y) are real polynomials and αm,n (ω), βm,n (ω), γm,n (ω) ∈ Ha (Rd ; R2 ).