On noncommutative Calabi–Yau hypersurfaces

Using the algebraic geometry method of Berenstein et al. [hep-th/0005087], we reconsider the derivation of the noncommutative quintic algebra Anc(5) and derive new representations by choosing different sets of Calabi–Yau charges {Cia}. Next we extend these results to higher d complex dimension noncommutative Calabi–Yau hypersurface algebras Anc(d+2). We derive and solve the set of constraint equations carrying the noncommutative structure in terms of Calabi–Yau charges and discrete torsion. Finally, we construct the representations of Anc(d+2) preserving manifestly the Calabi–Yau condition ∑iCia=0 and give comments on the noncommutative subalgebras.