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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: Lie algebras and their representations; 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.

Type de document :
Article dans une revue
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Soumis le : vendredi 11 décembre 2020 - 11:47:38
Dernière modification le : mercredi 20 octobre 2021 - 03:19:02

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Igor Reider. Configurations of points and strings. Journal of Geometry and Physics, 2011, 61 (7), pp.1158 - 1180. ⟨10.1016/j.geomphys.2010.09.012⟩. ⟨hal-03054055⟩



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