Zero Modes of the Dirac Operator in Vortex-Type Gauge Fields

We consider the $D4$ Euclidean Dirac equation in a $SU(2)$ gauge field with the form of intersecting central vortices and obtain its solutions corresponding to normalizable zero modes. In the context of the problem of constructing the mechanism of chiral symmetry breaking based on the model with randomly distributed central vortices, we study a possibility for splitting fermionic zero modes in the case of a superposition of many vortices with Pontryagin indices of different signs.