Complex Geometry and Supersymmetry

I stress how the form of sigma models with (2, 2) supersymmetry differs depending on the number of manifest supersymmetries. The differences correspond to different aspects/formulations of Generalized K\"ahler Geometry.Comment: 9 pages, Proceedings of the Corfu Summer Institute 2011 School and Workshops on Elementary Particle Physics and Gravity September 4-18, 2011 Corfu, Greec

Complex geometry of quantum cones

The algebras obtained as fixed points of the action of the cyclic group $Z_N$ on the coordinate algebra of the quantum disc are studied. These can be understood as coordinate algebras of quantum or non-commutative cones. The following observations are made. First, contrary to the classical situation, the actions of $Z_N$ are free and the resulting algebras are homologically smooth. Second, the quantum cone algebras admit differential calculi that have all the characteristics of calculi on smooth complex curves. Third, the corresponding volume forms are exact, indicating that the constructed algebras describe manifolds with boundaries.Comment: 6 pages; submitted to the proceedings of Corfu Summer Institute 201