Addington, N; Aspinwall, PS, *Categories of massless D-branes and del Pezzo surfaces*,
Journal of High Energy Physics, vol. 2013 no. 7
(August, 2013), Springer Nature [1305.5767v2], [doi] .
**Abstract:**

*In analogy with the physical concept of a massless D-brane, we define a notion of "Q-masslessness" for objects in the derived category. This is defined in terms of monodromy around singularities in the stringy Kähler moduli space and is relatively easy to study using "spherical functors". We consider several examples in which del Pezzo surfaces and other rational surfaces in Calabi-Yau threefolds are contracted. For precisely the del Pezzo surfaces that can be written as hypersurfaces in weighted P 3 , the category of "Q-massless" objects is a "fractional Calabi-Yau" category of graded matrix factorizations. © 2013 SISSA, Trieste, Italy.*