The Geometry of (Super) Conformal Quantum Mechanics

TL;DR

This paper explores the geometric conditions under which N-particle quantum mechanics models exhibit conformal and superconformal symmetries, linking these symmetries to specific geometric structures on the target space.

Contribution

It characterizes the geometric conditions for conformal and superconformal symmetry in sigma models with torsion, extending known symmetries to various supergroups.

Findings

SL(2,R) symmetry exists iff the geometry admits a closed homothetic Killing vector.

Superconformal extensions to Osp(1|2), SU(1,1|1), and D(2,1;α) are achieved under specific geometric conditions.

Examples illustrating these symmetry conditions are provided.

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;\alpha), which involve a triplet of complex structures and SU(2) x SU(2) isometries, are derived. Examples are given.