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

L. N. Krivonosov ; V. A. Lukyanov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 23 (2019) no. 2, pp. 207-228

Résumé

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.

Détail

Keywords: manifold of conformal connection, curvature, Hodge operator, self-duality, anti-self-duality, Yang–Mills equations.
Mots-clés : torsion