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Central Charges and Extra Dimensions in Supersymmetric Quantum Mechanics.pdf

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Central Charges and Extra Dimensions in Supersymmetric Quantum Mechanics

a r X i v : h e p - t h / 0 4 0 6 1 5 2 v 2 2 5 J u l 2 0 0 4 HWS-2004A06 DAMTP-2004-76 hep-th/0406152 Central Charges and Extra Dimensions in Supersymmetric Quantum Mechanics Michael Faux1 , David Kagan2 and Donald Spector1 1Department of Physics Hobart and William Smith Colleges Geneva, NY 14456, USA 2DAMTP, Center for Mathematical Sciences University of Cambridge Wilberforce Road, Cambridge, CB3 0WA, UK June 2004 Abstract We systematically include central charges into supersymmetric quantum mechanics formulated on curved Euclidean spaces, and explain how the background geometry manifests itself on states of the theory. In particular, we show in detail how, from the point of view of non-relativistic d = 1 world-line physics, one can infer the existence of target space dualities typically associated with string theory. We also explain in detail how the presence of a non-trivial supersymmetry central charge restricts the background geometry in which a particle may propagate. PACS: 11.30.Pb , 11.25.Tq , 03.65-w 1 Introduction As is well appreciated, supersymmetry [1] is a concept which not only provides elegant and useful solutions to interesting problems, such as the hierarchy problem in the standard model, but which also plays a key role in the structure of a variety of theories. For example, it appears as a required ingredient in consistent string theories [2], and also underlies the presence of shape invariance in exactly solvable systems in ordinary quantum mechanics [3]. As is also well appreciated, attempts to find more fundamental descriptions of nature frequently benefit from the inclusion of extra, less obvious, dimensions as part of our physical space. It is interesting to consider what the two ideas of supersymmetry and extra dimensions imply, at a basic level, when they are imposed simultaneously on ordinary, non-relativistic quantum mechanics. A conspicuous hallmark of extra dimensions is the appearance of central charges in the symmetry algebras o