Non-Commutative Calabi-Yau Manifolds

Details Non-Commutative Calabi-Yau Manifolds Non-Commutative Calabi-Yau Manifolds

2000-09-26

arxiv.org/abs/hep-th/0009209v2

Phys.Lett. B499 (2001) 207-214

Physics

High Energy Physics

High Energy Physics - Theory

11 pages, latex. Minor corrections and some clarifications added

Scientific paper

10.1016/S0370-2693(01)00005-3

We discuss aspects of the algebraic geometry of compact non-commutative Calabi-Yau manifolds. In this setting, it is appropriate to consider local holomorphic algebras which can be glued together into a compact Calabi-Yau algebra. We consider two examples: a toroidal orbifold T^6/Z_2 x Z_2, and an orbifold of the quintic in CP_4, each with discrete torsion. The non-commutative geometry tools are enough to describe various properties of the orbifolds. First, one describes correctly the fractionation of branes at singularities. Secondly, for the first example we show that one can recover explicitly a large slice of the moduli space of complex structures which deform the orbifold. For this example we also show that we get the correct counting of complex structure deformations at the orbifold point by using traces of non-commutative differential forms (cyclic homology).

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