Odd coset quantum mechanics

The standard quantum states of n complex Grassmann variables with a free-particle Lagrangian transform as a spinor of SO(2n). However, the same ‘free-fermion’ model has a non-linearly realized SU(n|1) symmetry; it can be viewed as the mechanics of a ‘particle’ on the Grassmann-odd coset space SU(n|1)/U(n). We implement a quantization of this model for which the states with non-zero norm transform as a representation of SU(n|1), the representation depending on the U(1) charge of the wave-function. For n=2 the wave-function can be interpreted as a BRST superfield.