Elliptically fibered Calabi-Yau threefolds and string-string duality

With Dr. Anda Degeratu I am investigating aspects of the heterotic - type IIA string-string duality from a mathematical point of view. The project was initiated by Prof. John McKay and could relate a result on pointlike instantons by Aspinwall and Morrison to McKay's conjecture about a connection between the affine Dynkin data of the E-series and the conjugacy classes of the Monster and its offspring. Currently our main focus lies on the investigation of constraints arising in Calabi-Yau geometry from the conjectured duality, like heterotic anomaly cancellation.

We have completed an extended review article summarizing the necessary preliminary steps,

Anda Degeratu, Katrin Wendland
Friendly giant meets pointlike instantons? On a new conjecture by John McKay
  • Moonshine - The First Quarter Century and Beyond, A Workshop on the Moonshine Conjectures and Vertex Algebras
  • LMS Lecture Notes Series no. 372 (2010), 55-127
  • Augsburg preprint number 037/2007

In this work, we also discover a previously unnoticed correction to the count of neutral massless hypermultiplets. We discuss the implications of the anomaly cancellation condition for the geometry of elliptically fibered Calabi-Yau 3-folds, and we give a list of representative examples which serve to discuss a number of gaps in the literature.

We expect to find a new index theorem for special elliptically fibered Calabi-Yau threefolds, which includes gauge theory data associated to the degenerate fibers. We can already prove a weaker prediction, which we derive from the conjectured duality, and which relates intersection numbers in the base of our elliptically fibered Calabi-Yau threefolds to indices of twisted Dirac operators on K3. Currently, we are working on an alternative version of the Tate algorithm, in order to find intrinsic explanations for the gauge theory data associated to degenerate fibers according to the physics predictions.