Noncommutative K3 Surfaces

TL;DR

This paper explores noncommutative deformations of K3 surfaces derived from toroidal orbifolds and quartic hypersurfaces, revealing moduli space dimensions and matrix representations with implications for string theory singularities.

Contribution

It constructs new noncommutative K3 surfaces via algebraic deformations and provides explicit matrix representations, extending previous work on Calabi-Yau threefolds.

Findings

Moduli space dimension is 18 for both complex and noncommutative deformations.

Explicit matrix representation of noncommutative K3 in quartic variables.

Fractionation of branes occurs at singularities due to discrete torsion.

Abstract

We consider deformations of a toroidal orbifold $T^4/Z_2$ and an orbifold of quartic in $CP^3$. In the $T^4/Z_2$ case, we construct a family of noncommutative K3 surfaces obtained via both complex and noncommutative deformations. We do this following the line of algebraic deformation done by Berenstein and Leigh for the Calabi-Yau threefold. We obtain 18 as the dimension of the moduli space both in the noncommutative deformation as well as in the complex deformation, matching the expectation from classical consideration. In the quartic case, we construct a $4 \times 4$ matrix representation of noncommutative K3 surface in terms of quartic variables in $CP^3$ with a fourth root of unity. In this case, the fractionation of branes occurs at codimension two singularities due to the presence of discrete torsion.