Bogomol’nyi Equations for Einstein–Yang–Mills–Dilaton theory

نویسنده

  • Mikhail S. Volkov
چکیده

A static, spherically symmetric and purely magnetic solution of the Einstein-YangMills-Dilaton theory, found previously by numerical integration is shown to obey a system of first order Bogomol’nyi equations. As common for such equations, there is a tight relation to supersymmetry, in the present case to the N=4 gauged SU(2)×SU(2) supergravity of Freedman and Schwarz. Specifically, the dilaton potential of the latter can be avoided by choosing one of the two gauge coupling constants to be imaginary. It is argued that this corresponds to a hitherto unknown N=4 gauged SU(2)×SU(1,1) supergravity in four Euclidean dimensions leading to Bogomol’nyi equations with asymptotically flat solutions. 1 Supported by the Deutsche Forschungsgemeinschaft, DFG-Wi 777/4-1 Preprint submitted to Elsevier Preprint 1 February 2008 Introduction.– Supergravity backgrounds play an important role in the analysis of string theory. Besides genuine fully supersymmetric string vacua, also particle like solutions with partial supersymmetry (p-branes, monopoles etc. [6]) are presently obtaining much consideration, in particular in view of their role in verifying various duality conjectures [8]. However, apart from stringy monopoles and the related solutions [6] obtained via the heterotic five-brane construction [11], most of the literature is devoted to solutions with Abelian gauge fields. This is easily understood, since such configurations can be obtained straightforwardly from the known solutions of the Einstein-Maxwell system. On the other hand, it is to be expected that also configurations with non-Abelian gauge fields will eventually play an important role. In addition, gauged supergravity models have recently regained considerable interest in view of the AdS/CFT correspondence [10], which also suggests studying classical solutions of supergravities with non-Abelian gauge fields. The incompleteness of the Abelian picture can be observed already in the (nonsupersymmetric) Einstein-Yang-Mills (EYM) theory. A number of interesting results have been obtained after the discovery in this theory of particle-like solutions by Bartnik and McKinnon [1] (see [12] for a recent review). In view of the complexity of the field equations even in the case of static, spherically symmetric solutions, our knowledge is largely based on numerical analysis. Nevertheless many interesting and partly surprising results are available as well on globally regular solutions as on black holes with ‘non-Abelian hair’. In particular they show that a number of the standard electrovacuum theorems do not apply in the non-Abelian domain: • The Birkhoff Theorem is not valid, i.e. there are time-dependent spherically symmetric solutions. • There exist globally regular, asymptotically flat static solutions. • Static black holes are not uniquely specified by their mass and their ‘gauge’ charges – the ‘No Hair’ Conjecture is violated. • Static black holes are not necessarily spherically symmetric – Israel’s theorems do not apply. • Non-rotating stationary black holes are not necessarily static – the Abelian staticity conjecture does not apply. In view of these results it becomes clear that experience gained in the Abelian domain is not universal, and the non-Abelian sector therefore should also be studied. We consider in this paper certain particle-like solutions of the coupled EinsteinYang-Mills-dilaton (EYMD) system. These solutions were obtained numerically some time ago [9,5,2]. They are static, spherically symmetric, globally regular, asymptotically flat and neutral – with the purely magnetic Yang-Mills (YM) field strength decaying as 1/r for r → ∞. In view of their instability

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تاریخ انتشار 1999