Graded D-branes and skew categories

I describe extended gradings of open topological field theories in twodimensions in terms of skew categories, proving a result which alows one totranslate between the formalism of graded open 2d TFTs and equivariant cycliccategories. As an application of this formalism, I describe the open 2d TFT ofgraded D-branes in Landau-Ginzburg models in terms of an equivariant cyclicstructure on the triangulated category of `graded matrix factorizations'introduced by Orlov. This leads to a specific conjecture for the Serre functoron the latter, which generalizes results known from the minimal and Calabi-Yaucases. I also give a description of the open 2d TFT of such models whichmanifestly displays the full grading induced by the vector-axial R-symmetrygroup.Comment: 37 page