Root systems and del Pezzo surfaces revisited

After talking with my good friend Tom and reading the paper by Iqbal, Neitzke, and Vafa, I believe that I have access to two interpretations of the appearance of root systems inside the Picard lattice of del Pezzo surfaces. The first is pleasingly simple and touches on my recent exploration of the MMP. For del Pezzo surfaces X=dP_d of degree d less than 8, the (-1)-curves generate the Picard lattice of X and in particular are the extremal rays generating the Mori cone. Since X is Fano, -K_X is ample and so K_X pairs negatively with every curve on X by Kleiman’s criterion for ampleness. This argument shows that, instead of being only partly locally polyhedral, the Mori cone of a Fano variety is globally polyhedral and contained within the halfspace on which K_X is negative. In particular, it only meets the orthogonal hyperplane to K_X at the origin. The root systems constructed last time are exactly the shortest integral vectors in this hyperplane, and their Weyl reflections can be verified to preserve the Mori cone and hence permute the (-1)-curves while preserving intersections: that is, induce automorphisms of their incidence graph.

Having only gingerly dabbled in string theory before, I took the physics interpretation largely on faith with reassurance from the computational similarities that arise. However, it is interesting to see how caring about algebraic automorphisms of del Pezzo surfaces naturally arises in what is really only a smooth context from wanting a fixed Planck length, and how an explicit identification between the Picard lattice of X and the coweight lattice of the corresponding root system emerges from regarding the relevant moduli problem in string theory.

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