Communications in Number Theory and Physics

Volume 6 (2012)

Number 2

On the arithmetic of D-brane superpotentials: lines and conics on the mirror quintic

Pages: 279 – 337



Johannes Walcher (Departments of Mathematics and Statistics, and Physics, McGill University, Montreal)


Irrational invariants from D-brane superpotentials are pursued on the mirror quintic,systematically according to the degree of a representative curve. Lines are completelyunderstood: the contribution from isolated lines vanishes. All other lines canbe deformed holomorphically to the van Geemen lines, whose superpotential isdetermined via the associated inhomogeneous Picard–Fuchs equation. Substantialprogress is made for conics: the families found by Musta\c{t}\v{a} contain conicsreducible to isolated lines, hence they have a vanishing superpotential. The searchfor all conics invariant under a residual $\zet_2$ symmetry reduces to an algebraic problemat the limit of our computational capabilities. The main results are of arithmeticflavor: the extension of the moduli space by the algebraic cycle splits in the largecomplex structure limit into groups each governed by an algebraic number field. Theexpansion coefficients of the superpotential around large volume remain irrational.The integrality of those coefficients is revealed by a new, arithmetic twist ofthe di-logarithm: the D-logarithm. There are several options for attempting to explainhow these invariants could arise from the A-model perspective. A successful spacetimeinterpretation will require spaces of BPS states to carry number theoretic structures,such as an action of the Galois group.

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