SU(1,1∣N)SU(1,1|N) superconformal mechanics with fermionic gauge symmetry

We study superpaticle models with fermionic gauge symmetry on the coset spaces of the $SU(1,1|N)$ supergroup. We first construct $SU(1,1|N)$ supersymmetric extension of a particle on $AdS_2$ possessing the $\kappa$-symmetry. Including angular degrees of freedom and extending this model to a superparticle on the $AdS_2\times \mathbb{CP}^{N-1}$ background with two-form flux, one breaks the $\kappa$-symmetry down to a fermionic gauge symmetry with one parameter. A link of the background field configuration to the near horizon black hole geometries is discussed.Comment: V2: 17 pages, presentation improved, the version to appear in JHE

Ricci-flat spacetimes with l-conformal Galilei symmetry

Ricci-flat metrics of the ultrahyperbolic signature which enjoy the l-conformal Galilei symmetry are constructed. They involve the AdS_2-metric in a way similar to the near horizon black hole geometries. The associated geodesic equations are shown to describe a second order dynamical system for which the acceleration generators are functionally independent.Comment: V2: refs. added, the version to appear in PL

Super 0--brane action on the coset space of D(2,1;α)D(2,1;\alpha) supergroup

The super 0-brane action on the coset space of $D(2,1;\alpha)$ supergroup is constructed which involves a set of parameters related by $\kappa$-symmetry. It describes a massive superparticle propagating near the horizon of the extreme Reissner-Nordstr$\"o$m-AdS-dS black hole which is linked to the recently constructed $D(2,1;\alpha)$-superparticle [JHEP 1703 (2017) 054] by a canonical transformation.Comment: 16 page

Krein signature for instability of PT\mathcal{PT}-symmetric states

Krein quantity is introduced for isolated neutrally stable eigenvalues associated with the stationary states in the $\mathcal{PT}$-symmetric nonlinear Schr\"{o}dinger equation. Krein quantity is real and nonzero for simple eigenvalues but it vanishes if two simple eigenvalues coalesce into a defective eigenvalue. A necessary condition for bifurcation of unstable eigenvalues from the defective eigenvalue is proved. This condition requires the two simple eigenvalues before the coalescence point to have opposite Krein signatures. The theory is illustrated with several numerical examples motivated by recent publications in physics literature

On OSp(N∣2)OSp(N|2) superconformal mechanics

Superparticle models with $OSp(N|2)$ supersymmetry group are studied. We first consider the $N=4$ case and construct the models with $\kappa$-symmetry on the coset spaces of the $OSp(4|2)$ supergroup. In addition, within the canonical formalism we present an $OSp(4|2)$ superparticle model with semi-dynamical angular variables. For generic $N$ we construct a superparticle model on $AdS_2\times S^{N-1}$ with the reduced $\kappa$-symmetry. It is demonstrated that the Hamiltonian of this model has the same structure as the one for the $N=4$ case because additional fermions contribute to the second-class constraints only.Comment: Minor changes compared to v