  Conformal quantum mechanics has been proposed to be the CFT$_1$ dual to
AdS$_2$. The $N$-point correlation function that satisfy conformal constraints
have been constructed from a non-conformal vacuum and the insertion of a
non-primary operator. The main goal of this paper is to find an interpretation
of this oddness. For this purpouse, we study possible gravitational dual models
and propose a two-dimensional dilaton gravity with a massless fermion for the
description of conformal quantum mechanics. We find a universal correspondence
between states in the conformal quantum mechanics model and two-dimensional
spacetimes. Moreover, the solutions of the Dirac equation can be interpreted as
zero modes of a Floquet-Dirac system. Within this system, the oddness of the
non-conformal vacuum and non-primary operator is elucidated. As a possible
application, we interpret the gauge symmetries of the Floquet-Dirac system as
the corresponding infinite symmetries of the Schr\"odinger equation which are
conjectured to be related to higher spin symmetries.
