Vortices, strings, and pseudoinstantons

Vortices and strings in four-dimensional space are represented as a closed sector of solutions of the Yang–Mills equations in spaces of signatures $({+}{+}{+}{+})$ and $({+}{+}{-}{-})$. Pseudoinstantons are introduced as solutions of the self-duality equations for the gauge fields in the space of signature $({+}{+}{-}{-})$. Ansatzes that permit reduction of the Yang–Mills equation to the Liouville, sinh-Gordon, and sine-Gordon equations are given.