Sphalerons, spectral flow, and anomalies

The topology of configuration space may be responsible in part for the existence of sphalerons. Here, sphalerons are defined to be static but unstable finite-energy solutions of the classical field equations. Another manifestation of the nontrivial topology of configuration space is the phenomenon of spectral flow for the eigenvalues of the Dirac Hamiltonian. The spectral flow, in turn, is related to the possible existence of anomalies. In this review, the interconnection of these topics is illustrated for three particular sphalerons of SU(2) Yang-Mills-Higgs theory.Comment: 35 pages with revtex4; invited paper for the August special issue of JMP on "Integrability, topological solitons and beyond

Anomalous Lorentz and CPT violation

If there exists Lorentz and CPT violation in nature, then it is crucial to discover and understand the underlying mechanism. In this contribution, we discuss one such mechanism which relies on four-dimensional chiral gauge theories defined over a spacetime manifold with topology $\mathbb{R}^3 \times S^1$ and periodic spin structure for the compact dimension. It can be shown that the effective gauge-field action contains a local Chern-Simons-like term which violates Lorentz and CPT invariance. For arbitrary Abelian $U(1)$ gauge fields with trivial holonomies in the compact direction, this anomalous Lorentz and CPT violation has recently been established perturbatively with a Pauli-Villars-type regularization and nonperturbatively with a lattice regularization based on Ginsparg-Wilson fermions.Comment: 8 pages; invited talk at the International Workshop on CPT and Lorentz Symmetry in Field Theory, University of the Algarve, Faro, Portugal, July 6-7, 2017; v4: published versio