Feynman integrals, Calabi-Yau geometries and integrable systems
Albrecht Klemm (Bonn U.)

Recently it has been realized that the parameter dependence of
Feynman integrals in dimensional regularisation can be calculated
explicitly using period-- and chain integrals of suitably chosen
Calabi-Yau motives, where the transcendentality weight of
the motive is proportional to the dimension of the Calabi Yau
geometry and the loop order of the Feynman graphs. We exemplify
this for the Banana graphs, the Ice Cone graphs and the Train Track graphs
in two dimensions. In the latter case there is a calculational very useful
relation between the differential realisation of the
Yangian symmetries and the Picard-Fuchs system of compact
Calabi-Yau spaces M as well as between the physical correlations
functions and the quantum volume of the manifolds W that are the
mirrors to M.