Dynamical Contents of Unconventional Supersymmetry

TL;DR

This paper analyzes a (2+1)-dimensional system with a Dirac field coupled to Chern-Simons connections, revealing the dynamical content and gauge structure of an unconventional supersymmetric theory relevant to graphene's low-energy physics.

Contribution

It explicitly performs the Hamiltonian analysis of an unconventional supersymmetric Chern-Simons theory, identifying its degrees of freedom and gauge symmetries, with applications to graphene and boundary conditions.

Findings

Propagating fermionic modes depend on background geometry and electromagnetic fields.

Explicit separation of first- and second-class constraints in the theory.

Analysis of boundary effects and extension to SU(2) gauge group.

Abstract

The Dirac Hamiltonian formalism is applied to a system in $(2+1)$-dimensions consisting of a Dirac field $\psi$ minimally coupled to Chern-Simons $U(1)$ and $SO(2,1)$ connections, $A$ and $\omega$, respectively. This theory is connected to a supersymmetric Chern-Simons form in which the gravitino has been projected out (unconventional supersymmetry) and, in the case of a flat background, corresponds to the low energy limit of graphene. The separation between first-class and second-class constraints is performed explicitly, and both the field equations and gauge symmetries of the Lagrangian formalism are fully recovered. The degrees of freedom of the theory in generic sectors shows that the propagating states correspond to fermionic modes in the background determined by the geometry of the graphene sheet and the nondynamical electromagnetic field. This is shown for the following canonical sectors: i) a conformally invariant generic description where the spinor field and the dreibein are locally rescaled; ii) a specific configuration for the Dirac fermion consistent with its spin, where Weyl symmetry is exchanged by time reparametrizations; iii) the vacuum sector $\psi=0$, which is of interest for perturbation theory. For the latter the analysis is adapted to the case of manifolds with boundary, and the corresponding Dirac brackets together with the centrally extended charge algebra are found. Finally, the $SU(2)$ generalization of the gauge group is briefly treated, yielding analogous conclusions for the degrees of freedom.