Gauge field geometry from complex and harmonic analyticities. I: Kähler and self-dual Yang-Mills cases. (English) Zbl 0643.53066

[For part II, see the review Zbl 0643.53967 below.]
The analyticity preservation principle is employed to demonstrate an impressive affinity between the field theories with intrinsic analytic structure (Yang and self-dual Yang-Mills theories, Kähler and hyper- Kähler gravities) and superfield gauge theories \((N=1\) and \(N=2\) Yang- Mills, \(N=1\) and \(N=2\) supergravities). The defining constraints of the former theories are interpreted as the integrability conditions for the existence of appropriate analytic subspaces and are solved by passing to the basis with manifest analyticity. We prefer to work within the analytic basis. This allows one, e.g., to replace the nonlinear splitting problem of the twistor approach by solving a linear equation.
We begin with implications of familiar complex analyticity in Yang theory and Kähler gravity. A new development is the geometric interpretation of Kähler potential in an extended space with the central charge coordinate. Next, the analyticity of a different kind is introduced, the harmonic analyticity. It governs the geometry of self-dual Yang-Mills fields, in \(R^{4n}\) \((n=1,2,...)\), specifying it in terms of an unconstrained prepotential which lives in the analytic harmonic space involving the sphere S 2. The harmonic analyticity determines also hyper- Kähler geometry (see part II) and has deep parallels in the twistor approach. Detailed comparison with the latter is made.
Reviewer: A.Galperin


53C80 Applications of global differential geometry to the sciences
81T08 Constructive quantum field theory
83E50 Supergravity
53C55 Global differential geometry of Hermitian and Kählerian manifolds


Zbl 0643.53967
Full Text: DOI


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