Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1996-12-19
Commun.Math.Phys. 193 (1998) 527-594
Physics
High Energy Physics
High Energy Physics - Theory
125 pages, Plain TeX file
Scientific paper
10.1007/s002200050339
We reconsider differential geometry from the point of view of the quantum theory of non-relativistic spinning particles, which provides examples of supersymmetric quantum mechanics. This enables us to encode geometrical structure in algebraic data consisting of an algebra of functions on a manifold and a family of supersymmetry generators represented on a Hilbert space. We show that known types of differential geometry can be classified in terms of the supersymmetries they exhibit. Replacing commutative algebras of functions by non-commutative *-algebras of operators, while retaining supersymmetry, we arrive at a formulation of non-commutative geometry encompassing and extending Connes' original approach. We explore different types of non-commutative geometry and introduce notions of non-commutative manifolds and non-commutative phase spaces. One of the main motivations underlying our work is to construct mathematical tools for novel formulations of quantum gravity, in particular for the investigation of superstring vacua.
Froehlich Juerg
Grandjean Olivier
Recknagel Andreas
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