Mirror Symmetry of Branes and the Quest of Hyperbolic 3-manifolds

We discuss the computation of certain normal functions on the mirror quintic Calabi–Yau threefold in a 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 analyzed normal functions in this limits suggests a relation to hyperbolic 3-manifolds. This talk is based on work in progress with Dave Morrison and Johannes Walcher.