On hidden broken nonlinear superconformal symmetry of conformal mechanics and nature of double nonlinear superconformal symmetry

TL;DR

The paper reveals a hidden nonlinear superconformal symmetry in conformal mechanics for positive integer parameters, which is broken at the state level due to boundary conditions, and relates this to a similar broken double nonlinear superconformal symmetry.

Contribution

It uncovers a hidden nonlinear superconformal symmetry in conformal mechanics and connects it to the previously observed double nonlinear superconformal symmetry, highlighting their broken nature.

Findings

Hidden nonlinear superconformal symmetry exists for positive integer parameters.

The symmetry involves odd integrals forming spin representations and is broken at the state level.

The double nonlinear superconformal symmetry shares the same broken characteristic.

Abstract

We show that for positive integer values $l$ of the parameter in the conformal mechanics model the system possesses a hidden nonlinear superconformal symmetry, in which reflection plays a role of the grading operator. In addition to the even $so(1,2)\oplus u(1)$-generators, the superalgebra includes $2l+1$ odd integrals, which form the pair of spin-$(l+{1/2})$ representations of the bosonic subalgebra and anticommute for order $2l+1$ polynomials of the even generators. This hidden symmetry, however, is broken at the level of the states in such a way that the action of the odd generators violates the boundary condition at the origin. In the earlier observed double nonlinear superconformal symmetry, arising in the superconformal mechanics for certain values of the boson-fermion coupling constant, the higher order symmetry is of the same, broken nature.