K-Rational D-Brane Crystals

TL;DR

This paper explores how discrete D-brane configurations can be used to reconstruct certain Calabi-Yau geometries, linking string theory, algebraic number theory, and conjectures like Birch and Swinnerton-Dyer.

Contribution

It introduces the concept of K-arithmetic D-crystals and connects D-brane configurations with motivic L-functions and conjectures in number theory.

Findings

D-brane configurations can determine Calabi-Yau geometry in simple cases

K-arithmetic D-crystals relate to conjectures of Birch and Swinnerton-Dyer

Heegner D-crystals encode geometric information through L-functions

Abstract

In this paper the problem of constructing spacetime from string theory is addressed in the context of D-brane physics. It is suggested that the knowledge of discrete configurations of D-branes is sufficient to reconstruct the motivic building blocks of certain Calabi-Yau varieties. The collections of D-branes involved have algebraic base points, leading to the notion of K-arithmetic D-crystals for algebraic number fields K. This idea can be tested for D0-branes in the framework of toroidal compactifications via the conjectures of Birch and Swinnerton-Dyer. For the special class of D0-crystals of Heegner type these conjectures can be interpreted as formulae that relate the canonical Neron-Tate height of the base points of the D-crystals to special values of the motivic L-function at the central point. In simple cases the knowledge of the D-crystals of Heegner type suffices to uniquely determine the geometry.