Gauge field geometry from complex and harmonic analyticities. II: Hyper- Kähler case. (English) Zbl 0643.53067

Galperin, A.; Ivanov, E.; Ogievetskij, V.; Sokachev, E.

Ann. Phys. 185, No. 1, 22-45 (1988).

[For part I, see the review above.]
The concept of preservation of harmonic analyticity is applied to find unconstrained prepotentials of hyper-Kähler geometry. We rewrite the \(R^{4n}\) \((n=1,2,...)\) hyper-Kähler constraints in the harmonic space \(R^{4n}\times S\) 2 and then solve them in terms of two prepotentials defined in an analytic subspace. The geometric meaning of prepotentials is revealed with introducing extra central charge coordinates. Working in the analytic basis permits one to define the metric by solving a linear differential equation on S 2. The procedure is illustrated by the Taub- NUT example. We also reproduce the well-known Ward ansatz for the metric and discuss the relation to twistor approaches. Finally, we establish the one-to-one correspondence between hyper-Kähler geometry and off-shell \(d=4\), \(N=2\) supersymmetric \(\sigma\) models. Their most general Lagrangian is shown to be uniquely composed of hyper-Kähler prepotentials, with the analytic space coordinates replaced by analytic hypermultiplet superfields defined on the same set of harmonic variables.

Reviewer: A.Galperin

MSC:

53C80	Applications of global differential geometry to the sciences
83E50	Supergravity
81T60	Supersymmetric field theories in quantum mechanics
53C55	Global differential geometry of Hermitian and Kählerian manifolds

Keywords:

harmonic analyticity; hyper-Kähler geometry; prepotentials; Ward ansatz; supersymmetric \(\sigma \) models

Citations:

Zbl 0643.53066

References:

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