Landau-Ginzburg to Calabi-Yau dictionary for D-branes

Abstract

Based on work by Orlov, we give a precise recipe for mapping between B-type D-branes in a Landau-Ginzburg orbifold model (or Gepner model) and the corresponding large-radius Calabi-Yau manifold. The D-branes in Landau-Ginzburg theories correspond to matrix factorizations and the D-branes on the Calabi-Yau manifolds are objects in the derived category. We give several examples including branes on quotient singularities associated to weighted projective spaces. We are able to confirm several conjectures and statements in the literature.

D-brane monodromies from a matrix-factorization perspective

“…The inverse large radius monodromy of the brane, X 2 , adjusts the equivariant representation of the brane, M C (X 2 ), 26) whereas the matrix factorization,…”

“…All those described ingredients are captured in a graded category [37,38,26,39,22], where the objects are matrix factorizations, the morphisms between objects are open-string states, and finally the shift functor is the operator, T . For us it is important to note that in the category of matrix factorizations of topological B-banes, in addition to the gauge equivalences (2.2), two matrix factorizations are also equivalent if they only differ by blocks of trivial matrix factorizations [37,15,38] …”

Matrix Factorizations, D-branes and their deformations

“…This is a very non-trivial mathematical statement, since we know that the Btype D-branes are described by a certain category D b (Coh(X W )) of coherent sheaves associated with X W , which has in general a very complicated structure (in particular for Calabi-Yau threefolds). As mentioned in the introduction, this statement has been proven in [25][26][27] by constructing a category of matrix factorizations associated with W , and showing that it is equivalent to…”

“…As we will explain in more detail below, the objects, which correspond to the D-branes (respective boundary conditions of open string world-sheets), are represented by matrix factorizations [23,24], and the maps between them are represented by certain matrix valued, moduli dependent open-string vertex operators. That this simple physical model faithfully represents the abstract mathematical notion of the category of coherent sheaves is highly non-trivial, and has been proven recently [25] to quite some generality (see also [26,27]). The key point is that the relevant category of topological B-type D-branes on a Calabi-Yau manifold described by W = 0, is isomorphic to a certain category of matrix factorizations of W [28,29].…”

“…By mirror symmetry the latter coincide with the Kähler variables of the A-model. As already mentioned, this map is given by (26) for the closed string modulus, and by certain theta functions for the open string moduli:…”