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Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries. (English) Zbl 1076.53102

The authors propose a reformulation of gauge theories based on some finite-dimensional “Lie algebroid”, that should include Chern-Simons gauge theories in three dimensions and the Poisson sigma model in two dimensions. This is principally made in order to define off-shell closed gauge symmetries. Furthermore, they state that their formulation “allows us to avoid complications arising in the infinite-dimension super-geometry”.
Reviewer’s remark: The supergeometry is a very muster method that allows us to eliminate renormalization problems in the quantization [see, e.g., the following paper: P. van Nieuvwenhuizen, Supergravity, Phys. Rep. 68, No. 4, 189–39 (1981)]. This aspect cannot be solved with the method proposed by these authors. Furthermore, from a more general point of view, any quantum field theory can be formulated today in the framework of a new noncommutative supergeometry. This utilizes infinite dimensional noncommutative topological algebras (quantum superalgebras), but in such a way to generate some non-commutative manifolds (quantum supermanifolds) that simulate in noncommutative way, the usual finite dimensional manifolds, i.e., there exist quantum coordinates (see recent works by the reviewer of this paper).
For these reasons, the classical formulation proposed in this paper, it appears inadequate to describe quantum phenomena like quantum Yang-Mills theories today.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
53C80 Applications of global differential geometry to the sciences
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T13 Yang-Mills and other gauge theories in quantum field theory
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References:

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