The Geometry of (Super) Conformal Quantum Mechanics

Abstract

N-particle quantum mechanics described by a sigma model with an N-dimensional target space with torsion is considered. It is shown that an SL(2,R) conformal symmetry exists if and only if the geometry admits a homothetic Killing vector $D^a$ whose associated one-form $D_a$ is closed. Further, the SL(2,R) can always be extended to Osp(1|2) superconformal symmetry, with a suitable choice of torsion, by the addition of N real fermions. Extension to SU(1,1|1) requires a complex structure I and a holomorphic U(1) isometry $D^a{I}_a{}^b{\partial }_b$. Conditions for extension to the superconformal group D(2,1;α), which involve a triplet of complex structures and SU(2) x SU(2) isometries, are derived. Examples are given.

23 pages harvmac. Conventions simplified; typos corrected; references added

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