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Noncommutative supergeometry, duality and deformations. (English) Zbl 1006.81083

Summary: Following Lett. Math. Phys. 50, 309-321 (1999; Zbl 0967.58004), 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\((d,d,\mathbb{Z})\)-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.

MSC:

81T75 Noncommutative geometry methods in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
81T13 Yang-Mills and other gauge theories in quantum field theory

Citations:

Zbl 0967.58004
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References:

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