  Irrational invariants from D-brane superpotentials are pursued on the mirror
quintic, systematically according to the degree of a representative curve.
Lines are completely understood: the contribution from isolated lines vanishes.
All other lines can be deformed holomorphically to the van Geemen lines, whose
superpotential is determined via the associated inhomogeneous Picard-Fuchs
equation. Substantial progress is made for conics: the families found by
Mustata contain conics reducible to isolated lines, hence they have a vanishing
superpotential. The search for all conics invariant under a residual Z2
symmetry reduces to an algebraic problem at the limit of our computational
capabilities. The main results are of arithmetic flavor: the extension of the
moduli space by the algebraic cycle splits in the large complex structure limit
into groups each governed by an algebraic number field. The expansion
coefficients of the superpotential around large volume remain irrational. The
integrality of those coefficients is revealed by a new, arithmetic twist of the
di-logarithm: the D-logarithm. There are several options for attempting to
explain how these invariants could arise from the A-model perspective. A
successful spacetime interpretation will require spaces of BPS states to carry
number theoretic structures, such as an action of the Galois group.
