Matrix Cartan Superdomains, Super Toeplitz Operators, and Quantization

We present a general theory of non-perturbative quantization of a class of hermitian symmetric supermanifolds. The quantization scheme is based on the notion of a super Toeplitz operator on a suitable Z2-graded Hilbert space of superholomorphic functions. The quantized supermanifold arises as the C-algebra generated by all such operators. We prove that our quantization framework reproduces the invariant super Poisson structure on the classical supermanifold as Planck’s constant tends to zero. 1 Supported in part by the National Science Foundation under grant DMS–9206936 2 Supported in part by the Department of Energy under grant DE–FG02–88ER25065 3 Supported in part by the Consiglio Nazionale delle Ricerche (CNR)