Fiber functor

Abstract: In this paper we establish an equivalence between the category of graded D-branes of type B in Landau–Ginzburg models with homogeneous superpotential W and the triangulated category of singularities of the fiber of W over zero. The main result is a theorem that shows that the graded triangulated category of singularities of the cone over a projective variety is connected via a fully faithful functor to the bounded derived category of coherent sheaves on the base of the cone. This implies that the category of graded D-branes of type B in Landau–Ginzburg models with homogeneous superpotential W is connected via a fully faithful functor to the derived category of coherent sheaves on the projective variety defined by the equation W = 0.

Abstract: For a braided tensor category C and a subcategory K there is a notion of centralizer CC(K), which is a full tensor subcategory of C. A pre-modular tensor category [7] is known to be modular in the sense of Turaev iff the center Z2(C) ≡ CC(C) (not to be confused with the center Z1 of a tensor category, related to the quantum double) is trivial, i.e. consists only of multiples of the tensor unit, and dim C 6 0. Here dim C = P i d(Xi) 2 , the Xi being the simple objects. We prove several structural properties of modular categories. Our main technical tool is the following double centralizer theorem. Let C be a modular category and K a full tensor subcategory closed w.r.t. direct sums, subobjects and duals. Then CC(CC(K)) = K

Abstract: We study the category O of representations of the rational Cherednik algebra AW attached to a complex reflection group W.W e construct an exact functor, called Knizhnik-Zamolodchikov functor: O → HW -mod, where HW is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equiv- alence between O/Otor, the quotient of O by the subcategory of AW -modules supported on the discriminant, and the category of finite-dimensional HW - modules. The standard AW -modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of "cells", provided W is a Weyl group and the Hecke algebra HW has equal parameters. We prove that the category O is equivalent to the module category over a finite dimensional algebra, a generalized "q-Schur algebra" associated to W.