Equivalence of four-dimensional self-duality equations and the continuum analog of the principal chiral field problem

A connection is found between the self-dual equations of 4-dimensional space and the principal chiral field problem in $n$-dimensional space. It is shown that any solution of the principal chiral field equations in $n$-dimensional space with arbitrary 2-dimensional functions of definite linear combinations of 4 variables $y, \bar y, z, \bar z$ as independent arguments satisfies the system of self-dual equations of 4-dimensional space. General solution of self-dual equations depending on the suitable number of functions of three independent variables coincides with the general solution of the principal chiral field problem when the dimensionality of the space tends to the infinity.