Multi-instantons in seven dimensions.pdfVIP

a r X i v : h e p - t h / 0 5 0 3 2 3 0 v 1 2 9 M a r 2 0 0 5 Multi-instantons in seven dimensions E.K. Loginov? Physics Department, Ivanovo State University Ermaka St. 39, Ivanovo, 153025, Russia February 19, 2005 Abstract We consider the self-dual Yang-Mills equations in seven dimen- sions. Modifying the t’Hooft construction of instantons in d = 4, we find N -instanton 7d solutions which depend on 8N effective parame- ters and are E6-invariant. 1 Introduction The pure Yang-Mills (YM) theory defined in the four-dimensional Euclidean space has a rich and interesting structure even at the classical level. The discovery of regular solutions to the YM field equations, which correspond to absolute minimum of the action (Belavin et al.) [1], has led to an intensive study of such a classical theory. One hopes that a deep understanding of the classical theory will be invaluable when one tries to quantize such a theory. In the past few years, increased attention has been paid to gauge field equations in space-time of dimension greater than four, with a view to ob- taining physically interesting theories via dimensional reduction [2]. Such equations appear in the many-dimensional theory of supergravity, in the low-energy effective theory of d-branes, and in M-theory [3]. Using solutions of the YM equations in d 4 makes possible to obtain soliton solutions in these theories [4]. It is known also that the YM theory in d dimensions may ?Research supported by RFBR Grant 04-02-16324; E-mail: loginov@ivanovo.ac.ru 1 be reduced to the Yang-Mills-Higgs (YMH) theory in k d dimensions [5]. Hence, solutions of the YMH equations in d = 4 may be obtained from solutions of the YM equations in d 4 dimensions. In Ref. [6], the 4d self-dual Yang-Mills equations was generalized to the higher-dimensional linear relations (CDFN equations) cmnpsF ps = λFmn, (1) where the numerical tensor cmnps is completely antisymmetric and λ = const is a non-zero eigenvalue. It is obviously that these