A Brief Introduction to Quantum Geometry

Supergeometry

From the sheaf-theoretic point of view, a supermanifold is defined as a ringed space consisting of a smooth manifold M and a sheaf of Z/2-graded algebras , which is locally isomorphic to the coordinate sheaf

In terms of "local coordinates" supermanifolds are described by local coordinate systems containing, besides standard local coordinates for M, also some mutually anticommuting coordinates . The picture is completed by assuming that that coordinates and mutually commute.

From the theoretical physics prospective, supergeometry plays a central role in the foundations of supersymmetry. A principal goal of supersymmetry was to provide a unifying view of bosonic and fermionic fields, and to establish a framework for a mathematically consistent formulation of quantum theory of gravity. The space-time is viewed as a supermanifold, and the symmetry is described by supergroups, which are the supergeometric counterparts of Lie groups. Elementary particles are grouped into supermultiplets, that generally contain both bosons and fermions. For more informations about supergeometry, we refer to [SV1,SV2,R].

Many constructions of supergeometry are naturally generalizable to the level of braided structures. In this context, the *-algebra is equipped with an additional structure, given by the appropriate operators satisfying the braid equation:

The operator replaces the standard transposition, and is generally not involutive. In the context Z/2-graded supercommutative algebras used in supergeometry, the braiding reduces to the involution

In classical geometry, all the braidings are the standard transpositions. In quantum geometry, an interesting phenomena appears. It turns out that quantum objects with a sufficiently "rich" geometrical structure (as quantum groups and quantum bundles) are always intrinsically braided, in the sense that the geometrical structure allows us to construct very interesting braid operators!