D-branes, categories and N=1 supersymmetry

Abstract

We show that boundary conditions in topological open string theory on Calabi-Yau manifolds are objects in the derived category of coherent sheaves, as foreseen in the homological mirror symmetry proposal of Kontsevich. Together with conformal field theory considerations, this leads to a precise criterion determining the BPS branes at any point in CY moduli space, completing the proposal of Π-stability.

Anomalies and graded coisotropic branes

“…This gives the desired bundle E C , whose transition function with respect to the open cover {U α , U β } is given by (3) h αβ = ϕ α · ϕ −1 β = ϕ · R ·φ ′ −1 . In particular, one can check that ψ + and R a ψ − glue smoothly into a single section χ ∈ Γ(E C ), with the gluing condition restricted to ∂Σ being precisely the boundary condition (2). Similarly one can show that ρ + and R b ρ − glue smoothly into a section η ∈ Γ(E C ⊗ K C ), with K C being the canonical bundle of Σ C .…”

“…One promising proposal to understand this mirror phenomenon in mathematical terms is the Homological Mirror Symmetry (HMS) conjecture [7], which interprets mirror symmetry as the equivalence of two triangulated categories: the bounded derived category of coherent sheaves D b (X) on the one hand, and the derived Fukaya category DF (X) on the other hand. It was later argued by Douglas [2] (see also [1]) that the derived category D b (X) corresponds to the category of topological B-branes. It is therefore tempting to regard the HMS conjecture as a mathematical re-phrasing of the physical statement that mirror symmetry exchanges A-branes and B-branes.…”

D-brane monodromies from a matrix-factorization perspective

“…As explained in refs. [35,36] this ambiguity is not important as long as we analyze the physics of a single brane but becomes relevant for the analysis of open strings stretching between different branes. Thus in order to keep track of this ambiguity, we assign to each brane an integer, n, and denote the graded brane by P [n].…”

“…The U(1) R-symmetry representation for the matrix factorization (4.33) becomes 35) whereas the Z 3 equivariant representation turns out to be Similarly to the analysis of the 'short' brane, S 3 , using eq. (2.21) we map these bosonic open-string states to fermionic open-string states stretching between the anti-D2-brane,X[−2], and the 'long' brane, L 3 .…”

On the boundary coupling of topological Landau-Ginzburg models

“…This is potentially important since it seems to allow for a simple realization of the framework of [7,8,9,10,11,12,13,14,15,16,17] in such theories.…”