Senior Thesis

Department of Mathematics
Harvey Mudd College

Chris Pries

Harvey Mudd College Mathematics 2003

Thesis: Searching for Supersymmetric Cycles: A Quest for Cayley Manifolds in the Calabi-Yau 4-Torus

Thesis Advisor: Prof. Weiqing Gu
Second Reader: Prof. Jon Jacobsen

Awarded Chavin Prize for Best Thesis

Finding Super-Symmetric Cayley Cycles in Calabi-Yau Manifolds

Recent results of string theory have shown that while the traditional cycles studied in Calabi-Yau 4-manifolds preserve half the spacetime supersymmetry, the more general class of Cayley cycles are novel in that they preserve only one quarter of it. Moreover, Cayley cycles play a crucial role in understanding mirror symmetry on Calabi-Yau 4-manifolds and $\Spin{7}$-manifolds. Nonetheless, only very few nontrivial examples of Cayley cycles are known. In particular, it would be very useful to know interesting examples of Cayley cycles on the complex 4-torus. This thesis will develop key techniques for finding and constructing lattice periodic Cayley manifolds in Euclidean 8-space. These manifolds will project down to the complex 4-torus, yielding nontrivial Cayley cycles.