On SUSY curves

In this note we give a summary of some elementary results in the theory of super Riemann surfaces (SUSY curves)

Auslander-Buchweitz approximation theory for triangulated categories

We introduce and develop an analogous of the Auslander-Buchweitz approximation theory (see \cite{AB}) in the context of triangulated categories, by using a version of relative homology in this setting. We also prove several results concerning relative homological algebra in a triangulated category \T, which are based on the behavior of certain subcategories under finiteness of resolutions and vanishing of Hom-spaces. For example: we establish the existence of preenvelopes (and precovers) in certain triangulated subcategories of \T. The results resemble various constructions and results of Auslander and Buchweitz, and are concentrated in exploring the structure of a triangulated category \T equipped with a pair (\X,\omega), where \X is closed under extensions and $\omega$ is a weak-cogenerator in \X, usually under additional conditions. This reduces, among other things, to the existence of distinguished triangles enjoying special properties, and the behavior of (suitably defined) (co)resolutions, projective or injective dimension of objects of \T and the formation of orthogonal subcategories. Finally, some relationships with the Rouquier's dimension in triangulated categories is discussed.Comment: To appear at: Appl. Categor. Struct. (2011); 22 page

Lifting and restricting recollement data

We study the problem of lifting and restricting TTF triples (equivalently, recollement data) for a certain wide type of triangulated categories. This, together with the parametrizations of TTF triples given in "Parametrizing recollement data", allows us to show that many well-known recollements of right bounded derived categories of algebras are restrictions of recollements in the unbounded level, and leads to criteria to detect recollements of general right bounded derived categories. In particular, we give in Theorem 1 necessary and sufficient conditions for a 'right bounded' derived category of a differential graded(=dg) category to be a recollement of 'right bounded' derived categories of dg categories. In Theorem 2 we consider the particular case in which those dg categories are just ordinary algebras.Comment: 29 page

The Overall Coefficient of the Two-loop Superstring Amplitude Using Pure Spinors

Using the results recently obtained for computing integrals over (non-minimal) pure spinor superspace, we compute the coefficient of the massless two-loop four-point amplitude from first principles. Contrasting with the mathematical difficulties in the RNS formalism where unknown normalizations of chiral determinant formulae force the two-loop coefficient to be determined only indirectly through factorization, the computation in the pure spinor formalism can be smoothly carried out.Comment: 29 pages, harvmac TeX. v2: add reference

D-branes Wrapped on Fuzzy del Pezzo Surfaces

We construct classical solutions in quiver gauge theories on D0-branes probing toric del Pezzo singularities in Calabi-Yau manifolds. Our solutions represent D4-branes wrapped around fuzzy del Pezzo surfaces. We study the fluctuation spectrum around the fuzzy CP^2 solution in detail. We also comment on possible applications of our fuzzy del Pezzo surfaces to the fuzzy version of F-theory, dubbed F(uzz) theory.Comment: 1+42 pages, 9 figures v2: references added v3: statements on the structure of the Yukawa couplings weakened. published versio

Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space

We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d > 2 (for example, d = 3 is the quintic three-fold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the `d-dimensional pair of pants'; the introduction of the `relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an `orbifold' Fukaya category); a Morse-Bott model for the relative Fukaya category that allows one to make explicit computations; and the introduction of certain graded categories of matrix factorizations mirror to the relative Fukaya category.Comment: 133 pages, 17 figures. Changes to the argument ruling out sphere bubbling in the relative Fukaya category, and dealing with the behaviour of the symplectic form under branched covers. Other minor changes suggested by the referee. List of notation include