Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 795Ω 1 :Ω 1 · Ω 1 · Ω 1 ∈ Ω 3 = Ω 1· (dQ) 2 ⊕ Ω 2· dω ⊕ Ω 2· dλ (4.314)Now let us repeat our calculation: the dimension of the l.h.s. of thisexpression is 5g − 5 that is the dimension of the space of holomorphic3–differentials. The dimension of the first space in expansion of ther.h.s. is g, the second one is 3g − 4 and the third one is 2g − 4. Sinceg + (3g − 4) + (2g − 4) = 6g − 8 is greater than 5g − 5 (unless g ≤ 3),formula (4.314) does not define the unique expansion of the triple productof Ω 1 and, therefore, the associativity spoils.The situation can be improved if one considers the curves with additionalinvolutions. As an example, let us consider the family of hyper–elliptic curves: y 2 = P ol 2g+2 (λ). In this case, there is the involution,σ : y −→ −y and Ω 1 is spanned by the σ−odd holomorphic 1–forms xi−1 dxy,i = 1, ..., g. Let us also note that both dQ and dω are σ−odd, while dλ isσ−even. This latter fact means that dλ can be only meromorphic unlessthere are punctures on the surface (which is, indeed, the case in the presenceof the mass hypermultiplets). Thus, formula (6.6) can be replaced bythat without dλΩ 2 + = Ω 1 − · dQ ⊕ Ω 1 − · dω, (4.315)where we have expanded the space of holomorphic 2–forms into the partswith definite σ−parity: Ω 2 = Ω 2 + ⊕ Ω 2 −, which are manifestly given bythe differentials xi−1 (dx) 2y, i = 1, ..., 2g − 1 and xi−1 (dx) 22 y, i = 1, ..., g − 2respectively. Now it is easy to understand that the dimensions of the l.h.s.and r.h.s. of (4.315) coincide and are equal to 2g − 1.Analogously, in this case, one can check the associativity. It is given bythe expansionΩ 3 − = Ω 1 − · (dQ) 2 ⊕ Ω 2 + · dω,where both the l.h.s. and r.h.s. have the same dimension: 3g − 2 = g +(2g − 2). Thus, the algebra of holomorphic 1–forms on hyper–elliptic curveis really associative [Mironov (1998)].