Mirror symmetry of branes and hyperbolic $3$-manifolds

We discuss the computation of normal functions between the van Geemen lines on the mirror quintic
Calabi-Yau threefold in a certain semi-stable degeneration limit. In this limit the normal functions are
described as elements of higher Chow groups. Physically this amounts to computing the domain wall
tension between certain B-branes on the mirror quintic in the large complex structure limit. By mirror
symmetry we expect that these normal functions/domain wall tensions have a geometric meaning on the
quintic Calabi-Yau threefold for suitable A-branes. As we discuss, the number theoretic structure of the
computed normal functions indicates that the relevant A-branes correspond to hyperbolic $3$-manifolds.