A geometric approach to scalar field theories on the supersphere

Following a strictly geometric approach we construct globally supersymmetric scalar field theories on the supersphere, defined as the quotient space $S^{2\mathrm{\mid }2}=UOSp(1\mathrm{\mid }2)\mathrm{/}\mathcal{U}(1)$. We analyze the superspace geometry of the supersphere, in particular deriving the invariant vielbein and spin connection from a generalization of the left-invariant Maurer-Cartan form for Lie groups. Using this information we proceed to construct a superscalar field action on $S^{2\mathrm{\mid }2}$, which can be decomposed in terms of the component fields, yielding a supersymmetric action on the ordinary two-sphere. We are able to derive Lagrange equations and Noether’s theorem for the superscalar field itself.