Singular Monopoles and Gravitational Instantons.pdf

a r X i v : h e p - t h / 9 8 0 3 1 6 0 v 1 2 0 M a r 1 9 9 8 IASSNS-HEP-98/26 CALT-68-2166 Singular Monopoles and Gravitational Instantons Sergey A. Cherkis? California Institute of Technology Pasadena, CA 91125 Anton Kapustin? School of Natural Sciences, Institute for Advanced Study Olden Lane, Princeton, NJ 08540 Abstract We model Ak and Dk asymptotically locally flat gravitational in- stantons on the moduli spaces of solutions of U(2) Bogomolny equa- tions with prescribed singularities. We study these moduli spaces using Ward correspondence and find their twistor description. This enables us to write down the Ka?hler potential for Ak and Dk gravita- tional instantons in a relatively explicit form. ?Research supported in part by DOE grant DE-FG03-92-ER40701 ?Research supported in part by DOE grant DE-FG02-90-ER40542 1 Introduction A gravitational instanton is a smooth four-dimensional manifold with a Rie- mannian metric satisfying Einstein equations. A particularly interesting class of gravitational instantons is that of four-dimensional hyperka?hler manifolds, i.e. manifolds with holonomy group contained in SU(2). A hyperka?hler man- ifold can be alternatively characterized as a Riemannian manifold admitting three covariantly constant complex structures I, J,K satisfying the quater- nion relations IJ = ?JI = K, etc. (1) such that the metric is Hermitian with respect to I, J,K. Covariant con- stancy of I, J,K implies that three 2-forms ω1 = g(I·, ·), ω2 = g(J ·, ·), ω3 = g(K·, ·) are closed. If we pick one of the complex structures, say I, we may regard a hyperka?hler manifold as a complex manifold equipped with Ka?hler metric (with Ka?hler form ω1) and a complex symplectic form ω = ω2 + iω3. Hyperka?hler four-manifolds arise in several physical problems. For exam- ple, compactification of string and M-theory on hyperka?hler four-manifolds preserves one half of supersymmetries and provides exact solutions of stringy equations of motion. The only compact hype