The Landau-Ginzburg to Calabi-Yau Dictionary for D-Branes

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.Comment: 24 pages, refs added + minor correctio

Permutation branes and linear matrix factorisations

All the known rational boundary states for Gepner models can be regarded as permutation branes. On general grounds, one expects that topological branes in Gepner models can be encoded as matrix factorisations of the corresponding Landau-Ginzburg potentials. In this paper we identify the matrix factorisations associated to arbitrary B-type permutation branes.Comment: 43 pages. v2: References adde

Integrability of the N=2 boundary sine-Gordon model

We construct a boundary Lagrangian for the N=2 supersymmetric sine-Gordon model which preserves (B-type) supersymmetry and integrability to all orders in the bulk coupling constant g. The supersymmetry constraint is expressed in terms of matrix factorisations.Comment: LaTeX, 19 pages, no figures; v2: title changed, minor improvements, refs added, to appear in J. Phys. A: Math. Ge

D-branes in Toroidal Orbifolds and Mirror Symmetry

We study D-branes extended in T^2/Z_4 using the mirror description as a tensor product of minimal models. We describe branes in the mirror both as boundary states in minimal models and as matrix factorizations in the corresponding Landau-Ginzburg model. We isolate a minimal set of branes and give a geometric interpretation of these as D1-branes constrained to the orbifold fixed points. This picture is supported both by spacetime arguments and by the explicit construction of the boundary states, adapting the known results for rational boundary states in the minimal models. Similar techniques apply to a larger class of toroidal orbifolds.Comment: 30 pages, 2 figure

Rigidity and defect actions in Landau-Ginzburg models

Studying two-dimensional field theories in the presence of defect lines naturally gives rise to monoidal categories: their objects are the different (topological) defect conditions, their morphisms are junction fields, and their tensor product describes the fusion of defects. These categories should be equipped with a duality operation corresponding to reversing the orientation of the defect line, providing a rigid and pivotal structure. We make this structure explicit in topological Landau-Ginzburg models with potential x^d, where defects are described by matrix factorisations of x^d-y^d. The duality allows to compute an action of defects on bulk fields, which we compare to the corresponding N=2 conformal field theories. We find that the two actions differ by phases.Comment: 53 pages; v2: clarified exposition of pivotal structures, corrected proof of theorem 2.13, added remark 3.9; version to appear in CM

Moduli Webs and Superpotentials for Five-Branes

We investigate the one-parameter Calabi-Yau models and identify families of D5-branes which are associated to lines embedded in these manifolds. The moduli spaces are given by sets of Riemann curves, which form a web whose intersection points are described by permutation branes. We arrive at a geometric interpretation for bulk-boundary correlators as holomorphic differentials on the moduli space and use this to compute effective open-closed superpotentials to all orders in the open string couplings. The fixed points of D5-brane moduli under bulk deformations are determined.Comment: 41 pages, 1 figur

D-brane superpotentials and RG flows on the quintic

The behaviour of D2-branes on the quintic under complex structure deformations is analysed by combining Landau-Ginzburg techniques with methods from conformal field theory. It is shown that the boundary renormalisation group flow induced by the bulk deformations is realised as a gradient flow of the effective space time superpotential which is calculated explicitly to all orders in the boundary coupling constant.Comment: 24 pages, 1 figure, v2:Typo in (3.14) correcte

Defect Perturbations in Landau-Ginzburg Models

Perturbations of B-type defects in Landau-Ginzburg models are considered. In particular, the effect of perturbations of defects on their fusion is analyzed in the framework of matrix factorizations. As an application, it is discussed how fusion with perturbed defects induces perturbations on boundary conditions. It is shown that in some classes of models all boundary perturbations can be obtained in this way. Moreover, a universal class of perturbed defects is constructed, whose fusion under certain conditions obey braid relations. The functors obtained by fusing these defects with boundary conditions are twist functors as introduced in the work of Seidel and Thomas.Comment: 46 page

Matrix Factorizations and Homological Mirror Symmetry on the Torus

We consider matrix factorizations and homological mirror symmetry on the torus T^2 using a Landau-Ginzburg description. We identify the basic matrix factorizations of the Landau-Ginzburg superpotential and compute the full spectrum, taking into account the explicit dependence on bulk and boundary moduli. We verify homological mirror symmetry by comparing three-point functions in the A-model and the B-model.Comment: 41 pages, 9 figures, v2: reference added, minor corrections and clarifications, version published in JHE

Triangle-generation in topological D-brane categories

Tachyon condensation in topological Landau-Ginzburg models can generally be studied using methods of commutative algebra and properties of triangulated categories. The efficiency of this approach is demonstrated by explicitly proving that every D-brane system in all minimal models of type ADE can be generated from only one or two fundamental branes.Comment: 34 page