Noncommutative supergeometry, duality and deformations

We introduce a notion of $Q$-algebra that can be considered as a generalization of the notion of $Q$-manifold (a supermanifold equipped with an odd vector field obeying $\{Q,Q\} =0$). We develop the theory of connections on modules over $Q$-algebras and prove a general duality theorem for gauge theories on such modules. This theorem containing as a simplest case $SO(d,d,{\bf Z})$-duality of gauge theories on noncommutative tori can be applied also in more complicated situations. We show that $Q$-algebras appear naturally in Fedosov construction of formal deformation of commutative algebras of functions and that similar $Q$-algebras can be constructed also in the case when the deformation parameter is not formal.Comment: Extended version of hep-th/991221

Special Issue on Deformation Quantization

The need for quantization was felt already around 1900, but quantum mechanics proper started about 80 years ago. It then took half a century to express in a mathematically and physically precise way what was intuitively felt by many, that quantization is deformation. In the past 30 years what is now called “deformation quantization” developed and has proved seminal in a variety of domains, from abstract mathematics to modern theoretical physics. The aim of this special “anniversary” issue is to present some developments in that vast frontier area