N=2 supersymmetric extension of l-conformal Galilei algebra

N=2 supersymmetric extension of the l-conformal Galilei algebra is constructed. A relation between its representations in flat spacetime and in Newton-Hooke spacetime is discussed. An infinite-dimensional generalization of the superalgebra is given.Comment: V4: 8 pages, references and acknowledgements adde

Dynamical realizations of N=1 l-conformal Galilei superalgebra

Dynamical systems which are invariant under N=1 supersymmetric extension of the l-conformal Galilei algebra are constructed. These include a free N=1 superparticle which is governed by higher derivative equations of motion and an N=1 supersymmetric Pais-Uhlenbeck oscillator for a particular choice of its frequencies. A Niederer-like transformation which links the models is proposed.Comment: 12 pages. New material and references added. Published versio

Canonical Coherent States for the Relativistic Harmonic Oscillator

In this paper we construct manifestly covariant relativistic coherent states on the entire complex plane which reproduce others previously introduced on a given $SL(2,R)$ representation, once a change of variables $z\in C\rightarrow z_D \in$ unit disk is performed. We also introduce higher-order, relativistic creation and annihilation operators, \C,\Cc, with canonical commutation relation [\C,\Cc]=1 rather than the covariant one [\Z,\Zc]\approx Energy and naturally associated with the $SL(2,R)$ group. The canonical (relativistic) coherent states are then defined as eigenstates of \C. Finally, we construct a canonical, minimal representation in configuration space by mean of eigenstates of a canonical position operator.Comment: 11 LaTeX pages, final version, shortened and corrected, to appear in J. Math. Phy

Modular Invariance on the Torus and Abelian Chern-Simons Theory

The implementation of modular invariance on the torus as a phase space at the quantum level is discussed in a group-theoretical framework. Unlike the classical case, at the quantum level some restrictions on the parameters of the theory should be imposed to ensure modular invariance. Two cases must be considered, depending on the cohomology class of the symplectic form on the torus. If it is of integer cohomology class $n$, then full modular invariance is achieved at the quantum level only for those wave functions on the torus which are periodic if $n$ is even, or antiperiodic if $n$ is odd. If the symplectic form is of rational cohomology class $\frac{n}{r}$, a similar result holds --the wave functions must be either periodic or antiperiodic on a torus $r$ times larger in both direccions, depending on the parity of $nr$. Application of these results to the Abelian Chern-Simons is discussed.Comment: 24 pages, latex, no figures; title changed; last version published in JM

Logarithmic Correlators in Non-relativistic Conformal Field Theory

We show how logarithmic terms may arise in the correlators of fields which belong to the representation of the Schrodinger-Virasoro algebra (SV) or the affine Galilean Conformal Algebra (GCA). We show that in GCA, only scaling operator can have a Jordanian form and rapidity can not. We observe that in both algebras logarithmic dependence appears along the time direction alone.Comment: 18 pages, no figures,some errors correcte

The Maximal Invariance Group of Newtons's Equations for a Free Point Particle

The maximal invariance group of Newton's equations for a free nonrelativistic point particle is shown to be larger than the Galilei group. It is a semi-direct product of the static (nine-parameter) Galilei group and an $SL(2,R)$ group containing time-translations, dilations and a one-parameter group of time-dependent scalings called {\it expansions}. This group was first discovered by Niederer in the context of the free Schr\"odinger equation. We also provide a road map from the free nonrelativistic point particle to the equations of fluid mechanics to which the symmetry carries over. The hitherto unnoticed $SL(2, R)$ part of the symmetry group for fluid mechanics gives a theoretical explanation for an observed similarity between numerical simulations of supernova explosions and numerical simulations of experiments involving laser-induced implosions in inertial confinement plasmas. We also give examples of interacting many body systems of point particles which have this symmetry group.Comment: Plain TeX File: 15 Page

Nonrelativistic conformal field theories

We study representations of the Schr\"odinger algebra in terms of operators in nonrelativistic conformal field theories. We prove a correspondence between primary operators and eigenstates of few-body systems in a harmonic potential. Using the correspondence we compute analytically the energy of fermions at unitarity in a harmonic potential near two and four spatial dimensions. We also compute the energy of anyons in a harmonic potential near the bosonic and fermionic limits.Comment: 26 pages, 9 figures; added a comment on the convergence of epsilon expansion

On Schr\"odinger superalgebras

We construct, using the supersymplectic framework of Berezin, Kostant and others, two types of supersymmetric extensions of the Schr\"odinger algebra (itself a conformal extension of the Galilei algebra). An `$I$-type' extension exists in any space dimension, and for any pair of integers $N_+$ and $N_-$. It yields an $N=N_++N_-$ superalgebra, which generalizes the N=1 supersymmetry Gauntlett et al. found for a free spin-\half particle, as well as the N=2 supersymmetry of the fermionic oscillator found by Beckers et al. In two space dimensions, new, `exotic' or `$IJ$-type' extensions arise for each pair of integers $\nu_+$ and $\nu_-$, yielding an $N=2(\nu_++\nu_-)$ superalgebra of the type discovered recently by Leblanc et al. in non relativistic Chern-Simons theory. For the magnetic monopole the symmetry reduces to \o(3)\times\osp(1/1), and for the magnetic vortex it reduces to \o(2)\times\osp(1/2).Comment: On Schr\"odinger superalgebras, no figurs. Published versio

Metrics with Galilean Conformal Isometry

The Galilean Conformal Algebra (GCA) arises in taking the non-relativistic limit of the symmetries of a relativistic Conformal Field Theory in any dimensions. It is known to be infinite-dimensional in all spacetime dimensions. In particular, the 2d GCA emerges out of a scaling limit of linear combinations of two copies of the Virasoro algebra. In this paper, we find metrics in dimensions greater than two which realize the finite 2d GCA (the global part of the infinite algebra) as their isometry by systematically looking at a construction in terms of cosets of this finite algebra. We list all possible sub-algebras consistent with some physical considerations motivated by earlier work in this direction and construct all possible higher dimensional non-degenerate metrics. We briefly study the properties of the metrics obtained. In the standard one higher dimensional "holographic" setting, we find that the only non-degenerate metric is Minkowskian. In four and five dimensions, we find families of non-trivial metrics with a rather exotic signature. A curious feature of these metrics is that all but one of them are Ricci-scalar flat.Comment: 20 page

Deformed relativistic and nonrelativistic symmetries on canonical noncommutative spaces

We study the general deformed conformal-Poincare (Galilean) symmetries consistent with relativistic (nonrelativistic) canonical noncommutative spaces. In either case we obtain deformed generators, containing arbitrary free parameters, which close to yield new algebraic structures. We show that a particular choice of these parameters reproduces the undeformed algebra. The modified coproduct rules and the associated Hopf algebra are also obtained. Finally, we show that for the choice of parameters leading to the undeformed algebra, the deformations are represented by twist functions.Comment: 9 pages, LaTeX, shortened, version appearing in Phys. Rev.