NC geometry and fractional branes

Details NC geometry and fractional branes NC geometry and fractional branes

Oct 2003

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2003cqgra..20.4447s&link_type=abstract

Classical and Quantum Gravity, Volume 20, Issue 20, pp. 4447-4472 (2003).

Mathematics

Algebraic Geometry

2

Scientific paper

Considering complex n-dimension Calabi Yau homogeneous hypersurfaces Script Hn with discrete torsion and using the Berenstein and Leigh algebraic geometry method, we study fractional D-branes that result from stringy resolution of singularities. We first develop the method introduced by Berenstein and Leigh (Preprint hep-th/0105229) and then build the non-commutative (NC) geometries for orbifolds Script O = Script Hn/Zn+2n with a discrete torsion matrix tab = exp[i2pi/n+2(etaab - etaba)], etaab in SL(n, Z). We show that the NC manifolds Script O(nc) are given by the algebra of functions on the real (2n + 4) fuzzy torus Script Tbetaij2(n+2) with deformation parameters betaij = exp i2pi/n+2[(etaab-1 - etaba-1)qai qbj] with qai being charges of Znn+2. We also give graphic rules to represent Script O(nc) by quiver diagrams which become completely reducible at orbifold singularities. It is also shown that regular points in these NC geometries are represented by polygons with (n + 2) vertices linked by (n + 2) edges while singular ones are given by (n + 2) non-connected loops. We study the various singular spaces of quintic orbifolds and analyse the varieties of fractional D-branes at singularities as well as the spectrum of massless fields. Explicit solutions for the NC quintic Script Q(nc) are derived with details and general results for complex n-dimension orbifolds with discrete torsion are presented.

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