New superconformal extensions of d=1 Calogero-type systems are obtained by gauging the U(n) isometry of matrix superfield models. We consider the cases of N=1, N=2 and N=4 one-dimensional supersymmetries. The bosonic core of the N=1 and N=2 models is the standard conformal A_{n-1} Calogero system, whereas the N=4 model is an extension of the U(2)-spin Calogero system.Comment: 5 pages; v2: minor clarifications, refs. added, version published in PR

We survey the salient features and problems of conformal and superconformal mechanics and portray some of its developments over the past decade. Both classical and quantum issues of single-and multiparticle systems are covered.

A new superconformal mechanics with OSp(4|2) symmetry is obtained by gauging the U(1) isometry of a superfield model. It is the one-particle case of the new N =4 super Calogero model recently proposed in arXiv:0812.4276 [hep-th]. Classical and quantum generators of the osp(4|2) superalgebra are constructed on physical states. As opposed to other realizations of N =4 superconformal algebras, all supertranslation generators are linear in the odd variables, similarly to the N =2 case. The bosonic sector of the component action is standard one-particle (dilatonic) conformal mechanics accompanied by an SU(2)/U(1) Wess-Zumino term, which gives rise to a fuzzy sphere upon quantization. The strength of the conformal potential is quantized.

We elaborate on a novel superconformal mechanics model possessing D(2, 1; α) symmetry and involving extra U(2) spin variables. It is the one-particle case of the N =4 superconformal matrix model recently proposed in arXiv:0812.4276 [hep-th], and it generalizes to arbitrary α =0 the OSp(4|2) superconformal mechanics of arXiv:0905.4951 [hep-th]. As in the latter case, the U(2) spin variables are described by a Wess-Zumino action and define the first Hopf map S 3 → S 2 in the target space. Upon quantization, they represent a fuzzy sphere. We find the classical and quantum generators of the D(2, 1; α) superalgebra and their realization on the physical states. The super wavefunction encompasses various multiplets of the SU(2) R and SU(2) L subgroups of D(2, 1; α), with fixed isospins. The conformal potential is determined by the external magnetic field in the Wess-Zumino term, whose strength is quantized like in the OSp(4|2) case. As a byproduct, we reveal new invariant subspaces in the enveloping algebra of D(2, 1; α) for our quantum realization.

We apply the nonlinear realizations method for constructing new Galilean conformal mechanics models. Our starting point is the Galilean conformal algebra which is a nonrelativistic contraction of its relativistic counterpart. We calculate Maurer-Cartan oneforms, examine various choices of the relevant coset spaces and consider the geometric inverse Higgs-type constraints which reduce the number of the independent coset parameters and, in some cases, provide dynamical equations. New Galilean conformally invariant actions are derived in arbitrary space-time dimension D = d+1 (no central charges), as well as in the special dimension D = 2+1 with one "exotic" central charge. We obtain new classical mechanics models which extend the standard (D = 0+1) conformal mechanics in the presence of d non-vanishing space dimensions.