Exceptional Collections

An exceptional collection is a nice basis of D-branes. Among other purposes, these collections are useful for deriving the low energy gauge theory description of a stack of D-branes at a Calabi-Yau singularity. For more details, here are the slides to an introductory talk I gave in October, 2005 at Texas A&M.

The purpose of this web page is to make available some Maple code that I wrote for determining whether a collection of line bundles on a toric surface is strongly exceptional or not. The code takes as input the toric fan and line bundles. Using the Kawamata-Viehweg vanishing theorem, it tries to determine if the collection is strongly exceptional or not.

Click here to download a Maple library containing the routines and here for a Maple worksheet demonstrating the use of the library files.

I also have a few short Quicktime movies (12 seconds) that I can show. In this paper, in order to relate the exceptional collection method of deriving gauge theories to the dimer method, we presented an algorithm for converting an exceptional collection of line bundles on a toric surface into a Beilinson quiver on a torus. The algorithm produces a quiver which depends on the symplectic coordinates of the surface. The movies animate this coordinate dependence. The first movie animates the Beilinson quiver on two dimensional complex projective space, CP^2.

CP^2 movie: (mov format) (mp4 format)

The second animates the quiver on the blow up of CP^2 at a point.

dP_1 movie: (mov format) (mp4 format)

These last two movies illustrate the difference between a quiver constructed for the third del Pezzo using the canonical (not Kaehler-Einstein) metric and the numerical Kaehler-Einstein metric.

canonical metric: (mov format)

Kaehler-Einstein metric: (mov format)

The movies are primitive because it's tedious to draw using Mathematica. The arrow directions are left to the imagination and the nodes are unlabeled. Node one is always in the lower left hand corner. I made the movies to make sure the quivers had no "knots" on the torus, but they're also kind of fun to watch.

Here is a quiver for L^{263}, described in greater detail here. This example was used to demonstrate an algorithm for producing an exceptional collection on a toric surface generated by four rays.