Duality equations on a 4-manifold of conformal torsion-free connection and some of their solutions for the zero signature

• Main Authors: L. N. Krivonosov, V. A. Lukyanov
• Format: Article
• Language: English
• Published: Samara State Technical University 2019-06-01
• Series: Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki
• Subjects: manifold of conformal connection; curvature; torsion; hodge operator; self-duality; anti-self-duality; yang–mills equations
• Online Access: http://mi.mathnet.ru/eng/vsgtu1674

On a 4-manifold of conformal torsion-free connection with zero signature (−−++) we found conditions under which the conformal curvature matrix is dual (self-dual or anti-self-dual). These conditions are 5 partial differential equations of the 2nd order on 10 coefficients of the angular metric and 4 partial differential equations of the 1st order, containing also 3 coefficients of external 2-form of charge. (External 2-form of charge is one of the components of the conformal curvature matrix.) Duality equations for a metric of a diagonal type are composed. They form a system of five second-order differential equations on three unknown functions of all four variables. We found several series of solutions for this system. In particular, we obtained all solutions for a logarithmically polynomial diagonal metric, that is, for a metric whose coefficients are exponents of polynomials of four variables.