arXiv:1108.1299v2 [hep-th] 17 Aug 2011 On the geometry of conformal mechanics K. Andrzejewski ∗ , J. Gonera † Department of Theoretical Physics and Computer Sciences, University of  L´ od´ z, Pomorska 149/153, 90-236  L´ od´ z, Poland Abstract A geometric picture of conformally invariant mechanics is presented. Al- though the standard form of the model is recovered, the careful analysis of global geometry of phase space leads to the conclusion that, in the attractive case, the singularity related to the phenomenon of ”falling on the center” is spurious. This opens new possibilities concerning both the interpretation and quantization of the model. Moreover, similar modification seem to be relevant in supersymmetric and multidimensional generalization of conformal mechanics. Conformal mechanics [1] and its supersymmetric extensions [2] provide simple yet non- trivial examples of conformally invariant theories. There appeared numerous papers dealing with various aspects of these models [3]. An interesting geometrical interpre- tation of the conformal mechanics in terms of nonlinear sigma model on the SO(2, 1) group has been given in the nice paper by Ivanov et. al. [4] (see [5] for supersymmetric counterpart of such approach). The authors used the method of nonlinear realizations [6] together with exponential parameterization of SO(2, 1) group to define a covariant dynamics on group manifold (actually – on appropriate coset space). The dynamics results from setting all Cartan forms, except one, equal to zero. This leads, through the so-called inverse Higgs phenomenon [7], to the equation of motion which, by a sim- ple change of variables, is shown to be equivalent to standard equation of conformal mechanics. The methods developed by Ivanov, Krivonos and Leviant may be applied in other contexts. For example, it has been used in Ref. [8] to construct Galilean conformally invariant dynamics in arbitrary space-time dimension. One of the important ingredients of the approach proposed in Ref. [4] is the use of exponential parameterization of the SO(2, 1) group. However, it is known that, in the case of semisimple group, the exponential parameterization does not provide the map covering the whole group. For this reason some global topological properties on conformal dynamics can escape our attention. * E-mail: k-andrzejewski@uni.lodz.pl † E-mail: jgonera@uni.lodz.pl 1