Article

# Superconformal Calogero models as a gauged matrix mechanics

02/2010;

Source: arXiv

**ABSTRACT**

We present basics of the gauged superfield approach to constructing

N-superconformal multi-particle Calogero-type systems developed in

arXiv:0812.4276, arXiv:0905.4951 and arXiv:0912.3508. This approach is

illustrated by the multi-particle systems possessing SU(1,1|1) and

D(2,1;\alpha) supersymmetries, as well as by the model of new N=4

superconformal quantum mechanics.

N-superconformal multi-particle Calogero-type systems developed in

arXiv:0812.4276, arXiv:0905.4951 and arXiv:0912.3508. This approach is

illustrated by the multi-particle systems possessing SU(1,1|1) and

D(2,1;\alpha) supersymmetries, as well as by the model of new N=4

superconformal quantum mechanics.

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**ABSTRACT:**In spacetime dimension two, pure Yang-Mills possesses no physical degrees of freedom, and consequently it admits a supersymmetric extension to couple to an arbitrary number, $ \mathcal{N} $ say, of Majorana-Weyl gauginos. This results in $ \left( {\mathcal{N},0} \right) $ super Yang-Mills. Further, its dimensional reduction to mechanics doubles the number of supersymmetries, from $ \mathcal{N} $ to $ \mathcal{N} + \mathcal{N} $ , to include conformal supercharges, and leads to a superconformal Yang-Mills quantum mechanics with symmetry group $ {\mathbf{OSp}}\left( {\mathcal{N}|{2},\mathbb{R}} \right) $ . We comment on its connection to $ Ad{{S}_{{2}}} \times {{S}^{{\mathcal{N}{ } - {1}}}} $ and reduction to a supersymmetric Calogero model. - [Show abstract] [Hide abstract]

**ABSTRACT:**We discuss the superconformal quantum mechanics arising from the M2-branes. We begin with a comprehensive review on the superconformal quantum mechanics and emphasize that conformal symmetry and supersymmetry in quantum mechanics contain a number of exotic and enlightening properties which do not occur in higher dimensional field theories. We see that superfield and superspace formalism is available for $\mathcal{N}\le 8$ superconformal mechanical models. We then discuss the M2-branes with a focus on the world-volume descriptions of the multiple M2-branes which are superconformal three-dimensional Chern-Simons matter theories. Finally we argue that the two topics are connected in M-theoretical construction by considering the multiple M2-branes wrapped around a compact Riemann surface and study the emerging IR quantum mechanics. We establish that the resulting quantum mechanics realizes a set of novel $\mathcal{N}\ge 8$ superconformal quantum mechanical models which have not been reached so far. Also we discuss possible applications of the superconformal quantum mechanics to mathematical physics.