Non-Commutative Calabi-Yau Manifolds

Abstract

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).

11 pages, latex. Minor corrections and some clarifications added

References in corpus (5)

• Marginal and Relevant Deformations of N=4 Field Theories and Non-Commutative Moduli Spaces of Vacua
• Discrete Torsion, AdS/CFT and duality
• Lectures on Non-BPS Dirichlet branes
• Non-Commutative Moduli Spaces, Dielectric Tori and T-duality
• D-branes on Orbifolds with Discrete Torsion And Topological Obstruction

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• A Matrix Model for ν_{k_1k_2}=\frac{k_1+k_2}{k_1 k_2} Fractional Quantum Hall States
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