spo(2|2)-Equivariant Quantizations on the Supercircle S¹|²

Abstract

We consider the space of differential operators Dλμ acting between λ- and μ-densities defined on S¹|² endowed with its standard contact structure. This contact structure allows one to define a filtration on Dλμ which is finer than the classical one, obtained by writting a differential operator in terms of the partial derivatives with respect to the different coordinates. The space Dλμ and the associated graded space of symbols Sδ (δ=μ−λ) can be considered as spo(2|2)-modules, where spo(2|2) is the Lie superalgebra of contact projective vector fields on S¹|². We show in this paper that there is a unique isomorphism of spo(2|2)-modules between Sδ and Dλμ that preserves the principal symbol (i.e. an spo(2|2)-equivariant quantization) for some values of δ called non-critical values. Moreover, we give an explicit formula for this isomorphism, extending in this way the results of [Mellouli N., SIGMA 5 (2009), 111, 11 pages] which were established for second-order differential operators. The method used here to build the spo(2|2)-equivariant quantization is the same as the one used in [Mathonet P., Radoux F., Lett. Math. Phys. 98 (2011), 311-331] to prove the existence of a pgl(p+1|q)-equivariant quantization on Rp|q.