260324 Thorsten Schimannek

After giving a brief introduction to Calabi-Yau manifolds and classical mirror symmetry, I will focus on some of the special properties arising for torus fibered CY 3-folds.
My goal is to first describe how mirror symmetry can be used to obtain ``Gopakumar-Vafa polynomials", that capture the multiplicities of singular fibers in terms of the Chern classes of bundles that determine the structure of the torus fibration.
This will lead us to conjecture subtle constraints on the possible multiplicities that, from a physical perspective, are related to the cancellation of Dai-Freed anomalies in six-dimensional supergravity.
I will outline how these constraints connect to cobordism groups, to the APS eta-invariant on lens spaces and a quadratic refinement of the corresponding torsion pairing in cohomology.