In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish a connection of these categories with D-branes in Landau-Ginzburg models It seems that two different types of categories can be associated with singularities (or singularities of maps). Categories of the firs...

We introduce a class of analytic sheaves in a Banach space X, that we shall call cohesive sheaves. Cohesion is meant to generalize the notion of coherence from finite dimensional analysis. Accordingly, we prove the analog of Cartan’s Theorems A and B for cohesive sheaves on pseudoconvex open subsets Ω ⊂ X, provided X has an unconditional basis.

We propose local versions of monotonicity for Boolean and pseudoBoolean functions: say that a pseudo-Boolean (Boolean) function is p-locally monotone if none of its partial derivatives changes in sign on tuples which differ in less than p positions. As it turns out, this parameterized notion provides a hierarchy of monotonicities for pseudo-Boolean (Boolean) functions. Local monotonicities are ...

We apply the methods of Căldăraru to construct a twisted FourierMukai transform between a pair of holomorphic symplectic four-folds. More precisely, we obtain an equivalence between the derived category of coherent sheaves on a certain four-fold and the derived category of twisted sheaves on its ‘mirror’ partner. As corollaries, we show that the two spaces are connected by a one-parameter famil...

Using multiple point spaces some new examples of perverse sheaves on images of maps are described. Furthermore, suppose f : X ! Y is a nite and proper map of complex analytic manifolds of dimension n and n + 1 such that every multiple point space is non-singular and has the dimension expected of a generic map. Then we can describe the composition series for the constant sheaf on the image in th...

The aim of this paper is to prove the existence of large complete subvarieties in moduli spaces of rank two stable sheaves with arbitrary c1 and sufficiently large c2, on algebraic surfaces. Then we study the restriction of these sheaves to curves of high degree embedded in the surface. In the final section we gives a relation with the spin strata defined by Pidstrigach and Tyurin.

We study the Verlinde bundles of generalized theta functions constructed from moduli spaces of sheaves over abelian surfaces. In degree 0, the splitting type of these bundles is expressed in terms of indecomposable semihomogeneous factors. Furthermore, Fourier-Mukai symmetries of the Verlinde bundles are found, consistently with strange duality. Along the way, a transformation formula for the t...

Recently there is a surge of research interest in the construction of stable vector bundles on Calabi-Yau manifolds motivated by questions from string theory. An interesting aspect of the moduli spaces of stable sheaves on Calabi-Yau manifolds is their relation to the higher dimensional gauge theory studied by Donaldson, R. Thomas and Tian et al. [D-T, Tho, Tia]. A holomorphic Casson invariant ...