Configurations of points and strings

Abstract

Let X be a smooth projective variety of dimension n ≥ 2 . It is shown that a finite configuration of points on X subject to certain geometric conditions possesses rich inner structure. On the mathematical level this inner structure is a variation of Hodge-like structure. As a consequence one can attach to such point configurations: (i) Lie algebras and their representations; (ii) a Fano toric variety whose hyperplane sections are Calabi–Yau varieties. These features imply that the points cease to be zero-dimensional objects and acquire dynamics of linear operators “propagating” along the paths of a particular trivalent graph. Furthermore, following particular linear operators along the “shortest” paths of the graph, one creates, for every point of the configuration, a distinguished hyperplane section of the Fano variety in (ii), i.e. the points “open up” to become Calabi–Yau varieties. Thus one is led to a picture which is very suggestive of quantum gravity according to string theory.