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

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

Ann. Phys. 185, No. 1, 1-21 (1988).

[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

MSC:

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

Keywords:

analyticity preservation principle; Yang-Mills theories; Kähler gravities; superfield gauge theories; Kähler potential; harmonic analyticity; prepotential; hyper-Kähler geometry

Citations:

Zbl 0643.53967

References:

[1]	Penrose, R.; Ward, R. S., (Held, A., General Relativity and Gravitation, Vol. 2 (1980), Plenum: Plenum New York/London), 283
[2]	Belavin, A. A., Inverse Scattering Problem and the Algebra-Geometric Construction of Instantons (1977), Preprint ITPh, Chernogolovka
[3]	Ko, M.; Ludvigsen, M.; Newman, E. T.; Tod, K. P., Phys. Rep., 71, 51 (1981)
[4]	Ward, R. S., Nucl. Phys. B, 236, 381 (1984)
[5]	Newman, E. T., J. Math. Phys., 27, 2797 (1986)
[6]	Yang, C. N., Phys. Rev. Lett., 38, 1377 (1977)
[7]	Galperin, A.; Ivanov, E.; Ogievetsky, V., JETP Lett., 33, 176 (1981)
[8]	Galperin, A.; Ivanov, E.; Kalitzin, S.; Ogievetsky, V.; Sokatchev, E., Class. Quantum Grav., 1, 469 (1984)
[9]	Galperin, A.; Ivanov, E.; Ogievetsky, V.; Sokatchev, E., Class. Quantum Grav., 2, 617 (1985)
[12]	Galperin, A.; Ivanov, E.; Kalitzin, S.; Ogievetsky, V.; Sokatchev, E., Class. Quantum Grav., 2, 155 (1985)
[13]	Witten, E., Phys. Lett. B, 77, 394 (1978)
[14]	Rosly, A. A., (Proc. Intern. Seminar on Group Theoretical Methods in Physics, Vol. I (1983), Nauka: Nauka Moscow), 263
[15]	Galperin, A.; Ivanov, E.; Ogievetsky, V.; Sokatchev, E., Ann. Phys. (N.Y.), 185, 22 (1988)
[17]	Zupnik, B. M., Phys. Lett. B, 183, 175 (1987)
[18]	Zumino, B., Phys. Lett. B, 87, 203 (1979)
[19]	Ogievetsky, V. I.; Sokatchev, E. S., Phys. Lett., 79, 222 (1978)
[20]	Rosly, A. A., J. Phys. A, 16, L663 (1983)
[21]	Roček, M., Physica D, 15, 75 (1985)
[24]	Belavin, A. A.; Polyyakov, A. M.; Schwarz, A. S.; Tyupkin, Yu. S., Phys. Lett. B, 59, 85 (1975)
[25]	Atiyah, M. F.; Drinfeld, V. G.; Hitchin, N. J.; Manin, Yu. I., Phys. Lett. A, 65, 185 (1978)
[26]	Grimm, R.; Sohnius, M.; Wess, J., Nucl. Phys. B, 133, 275 (1978)