CP-odd invariants for multi-Higgs models and applications with discrete symmetry

CP-odd invariants are useful for studying the CP properties of Lagrangians in any basis. We explain how to build basis invariants for the scalar sector, and how to distinguish CP-odd invariants from CP-even invariants. Up to a certain order, we use these methods to systematically build all the CP-odd invariants. The CP-odd invariants signal either explicit or spontaneous violation of CP. Making use of the CP-odd invariants, we determine the CP properties of potentials with 3 and with 6 Higgs fields arranged as triplets of specific discrete symmetries in the $\Delta(3n^2)$ or $\Delta(6n^2)$ series (inlcuding $A_4$, $S_4$, $\Delta(27)$ and $\Delta(54)$ as well as the cases for $n>3$).Comment: 9 pages. Contribution to the proceedings of DISCRETE 2016, Warsa

Higher Order Intersections in Low-Dimensional Topology

We show how to measure the failure of the Whitney trick in dimension 4 by constructing higher- order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants like Milnor, Sato-Levine and Arf invariants. We also define higher- order Sato-Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the non- triviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described. This article is an announcement and summary of results to be published in several forthcoming papers