Friday, January 11, 2013
Motivic Mirror Symmetry
Rolf Schimmrigk's new paper on the arxiv demonstrates that the Omega-motives of exactly solvable mirror pairs of Calabi-Yau threefolds are isomorphic, as expected by mirror symmetry since the L-functions of the Omega-motives of such pairs are predicted to be identical.
String theory suggests a relation between two-dimensional conformal field theory and the geometry of Calabi-Yau varieties. This comes in the form of relating automorphic forms derived from pure or mixed motives that arise in Calabi-Yau varieties to modular forms that come from Kac-Moody algebras on the worldsheet. Such relations allow an arithmetic link between the spacetime and worldsheet theory, giving a means to pass from the extra-dimensional geometry to the worldsheet and back.
In the framework of M-theory, it would be of interest to generalize the Calabi-Yau results to G2 manifolds and other higher dimensional compactification spaces.