When we impose discrete symmetries in the standard model we have Dai-Freed global anomalies. However, interestingly if we introduce three right-handed neutrinos we can have an anomaly-free discrete Z 4 gauge symmetry. This Z 4 symmetry should be spontaneously broken down to the Z 2 symmetry to generate the heavy Majorana masses for the right-handed neutrinos. We show that this symmetry breaking naturally generates topological inflation, which is consistent with the CMB observations at present and predicts a significant tensor mode with scalar-tensor ratio r &gt; 0.03. The right-handed neutrinos play an important role in reheating processes. The reheating temperature is as high as ∼ 108 GeV, and non-thermal leptogenesis successfully takes place.

Japan

Masahiro Kawasaki

China

Tsutomu Yanagida

Dai-Freed anomaly in the standard model and topological inflation

In this paper, a novel discrete algebra is presented which follows by combining the SU(2) Lie-Poisson bracket with the discrete Frenet equation. Physically, the construction describes a discrete piecewise linear string in <math><msup><mi>R</mi><mn>3</mn></msup></math>. The starting point of our derivation is the discrete Frenet frame assigned at each vertex of the string. Then the link vector that connects the neighboring vertices is assigned the SU(2) Lie-Poisson bracket. Moreover, the same bracket defines the transfer matrices of the discrete Frenet equation which relates two neighboring frames along the string. The procedure extends in a self-similar manner to an infinite hierarchy of Poisson structures. As an example, the first descendant of the SU(2) Lie-Poisson structure is presented in detail. For this, the spinor representation of the discrete Frenet equation is employed, as it converts the brackets into a computationally more manageable form. The final result is a nonlinear, nontrivial, and novel Poisson structure that engages four neighboring vertices.

Sweden

Jin Dai

Greece

Theodora Ioannidou

France

Antti J. Niemi

SU(2) Lie-Poisson algebra and its descendants

Vortices in a Bose-Einstein condensate are modelled as spontaneously symmetry breaking minimum energy solutions of the time dependent Gross-Pitaevskii equation, using the method of constrained optimization. In a non-rotating axially symmetric trap, the core of a single vortex precesses around the trap center and, at the same time, the phase of its wave function shifts at a constant rate. The precession velocity, the speed of phase shift, and the distance between the vortex core and the trap center, depend continuously on the value of the conserved angular momentum that is carried by the entire condensate. In the case of a symmetric pair of identical vortices, the precession engages an emergent gauge field in their relative coordinate, with a flux that is equal to the ratio between the precession and shift velocities.

Julien Garaud

Russia

Antti Niemi

Vortex precession and exchange in a Bose-Einstein condensate

Classical Hamiltonian systems with conserved charges and those with constraints often describe dynamics on a pre-symplectic manifold. Here we show that a pre-symplectic manifold is also the proper stage to describe autonomous energy conserving Hamiltonian time crystals. We explain how the occurrence of a time crystal relates to the wider concept of spontaneously broken symmetries; in the case of a time crystal, the symmetry breaking takes place in a dynamical context. We then analyze in detail two examples of timecrystalline Hamiltonian dynamics. The first example is a piecewise linear closed string, with dynamics determined by a Lie-Poisson bracket and Hamiltonian that relates to membrane stability. We explain how the Lie-Poisson brackets descents to a time-crystalline pre-symplectic bracket, and we show that the Hamiltonian dynamics supports two phases; in one phase we have a time crystal and in the other phase time crystals are absent. The second example is a discrete one dimensional model of a Hamiltonian chain. It is obtained by a reduction from the Q-ball Lagrangian that describes time dependent nontopological solitons. We show that a time crystal appears as a minimum energy domain wall configuration, along the chain.

Switzerland

Anton Alekseev

Provenance of classical Hamiltonian time crystals

Anomalies can be elegantly analyzed by means of the Dai-Freed theorem. In this framework it is natural to consider a refinement of traditional anomaly cancellation conditions, which sometimes leads to nontrivial extra constraints in the fermion spectrum. We analyze these more refined anomaly cancellation conditions in a variety of theories of physical interest, including the Standard Model and the SU(5) and Spin(10) GUTs, which we find to be anomaly free. Turning to discrete symmetries, we find that baryon triality has a ℤ9 anomaly that only cancels if the number of generations is a multiple of 3. Assuming the existence of certain anomaly-free ℤ4 symmetry we relate the fact that there are 16 fermions per generation of the Standard model — including right-handed neutrinos — to anomalies under time-reversal of boundary states in four-dimensional topological superconductors. A similar relation exists for the MSSM, only this time involving the number of gauginos and Higgsinos, and it is non-trivially, and remarkably, satisfied for the SU(3) × SU(2) × U(1) gauge group with two Higgs doublets. We relate the constraints we find to the well-known Ibañez-Ross ones, and discuss the dependence on UV data of the construction. Finally, we comment on the (non-)existence of K-theoretic θ angles in four dimensions.

United Kingdom

Germany

Iñaki García-Etxebarria

Belgium

Netherlands

Miguel Montero

Dai-Freed anomalies in particle physics

We describe a physics derivation of theorems due to Dai and Freed about the Atiyah-Patodi-Singer eta-invariant which is important for anomalies and topological phases of matter. This is done by studying a massive fermion. The key role is played by the wave function of the ground state in the Hilbert space of the fermion in the large mass limit. The ground state takes values in the determinant line bundle and has nontrivial Berry phases which characterize the low energy topological phases.

Kazuya Yonekura

Dai-Freed theorem and topological phases of matter