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Structure of gauge-invariant Lagrangians. (English) Zbl 1433.53034

Summary: The theory of gauge fields in Theoretical Physics poses several mathematical problems of interest in Differential Geometry and in Field Theory. Below, we tackle one of these problems: the existence of a finite system of generators of gauge-invariant Lagrangians and how to compute them. More precisely, if \(p:C\rightarrow M\) is the bundle of connections on a principal \(G\)-bundle \(\pi :P\rightarrow M\), and \(G\) is a semisimple connected Lie group, then a finite number \(L_1,\dotsc,L_{N^\prime}\) of gauge-invariant Lagrangians defined on \(J^1C\) is proved to exist such that for any other gauge-invariant Lagrangian \(L\in C^\infty (J^1C)\), there exists a function \(F\in C^\infty (\mathbb{R}^{N^\prime})\), such that \(L=F(L_1,\dotsc,L_{N^\prime})\). Several examples are dealt with explicitly.

MSC:

53C05 Connections (general theory)
35F20 Nonlinear first-order PDEs
58A20 Jets in global analysis
58D19 Group actions and symmetry properties
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
58E30 Variational principles in infinite-dimensional spaces
81T13 Yang-Mills and other gauge theories in quantum field theory
53C80 Applications of global differential geometry to the sciences
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References:

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