Supersymmetric Quantum Mechanics on a noncommutative plane through the lens of deformation quantization

A gauge invariant mathematical formalism based on deformation quantization is outlined to model an $\mathcal{N}=2$ supersymmetric system of a spin $1\mathrm{/}2$ charged particle placed in a nocommutative plane under the influence of a vertical uniform magnetic field. The noncommutative involutive algebra $(C^{\mathrm{\infty }}({\mathbb{R}}^2)[[\vartheta ]],{\ast }^r)$ of formal power series in $\vartheta$ with coefficients in the commutative ring $C^{\mathrm{\infty }}({\mathbb{R}}^2)$ was employed to construct the relevant observables, viz., SUSY Hamiltonian $H$, supercharge operator $Q$ and its adjoint $Q^{\text{†}}$ all belonging to the $2\times 2$ matrix algebra ${\mathcal{M}}_2(C^{\mathrm{\infty }}({\mathbb{R}}^2)[[\vartheta ]],{\ast }^r)$ with the help of a family of gauge-equivalent star products ${\ast }^r$. The energy eigenvalues of the SUSY Hamiltonian all turned out to be independent of not only the gauge parameter $r$ but also the noncommutativity parameter $\vartheta$. The nontrivial Fermionic ground state was subsequently computed associated with the zero energy which indicates that supersymmetry remains unbroken in all orders of $\vartheta$. The Witten index for the noncommutative SUSY Landau problem turns out to be $-1$ corroborating the fact that there is no broken supersymmetry for the model we are considering.