Categories of Massless D-Branes and del Pezzo Surfaces

TL;DR

This paper introduces a new concept of 'Q-masslessness' for objects in derived categories, relating to massless D-branes, and explores examples involving del Pezzo surfaces in Calabi-Yau threefolds.

Contribution

It defines 'Q-masslessness' using monodromy and spherical functors, and characterizes the category of Q-massless objects for certain del Pezzo surfaces as fractional Calabi-Yau categories.

Findings

Q-massless objects are studied via monodromy and spherical functors.

For del Pezzo surfaces as hypersurfaces in weighted P3, the category is fractional Calabi-Yau.

Examples include rational surfaces in Calabi-Yau threefolds.

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 Kahler 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 P3, the category of Q-massless objects is a "fractional Calabi-Yau" category of graded matrix factorizations.