Direct test of the integral Yang-Mills equations through SU(2) monopoles

We use the SU(2) 't Hooft-Polyakov monopole configuration, and its Bogomolny-Prasad-Sommerfield (BPS) version, to test the integral equations of the Yang-Mills theory. Those integral equations involve two (complex) parameters which do not appear in the differential Yang-Mills equations, and if they are considered to be arbitrary, it then implies that non-Abelian gauge theories (but not Abelian ones) possess an infinity of integral equations. For static monopole configurations, only one of those parameters is relevant. We expand the integral Yang-Mills equation in a power series of that parameter and show that the 't Hooft-Polyakov monopole and its BPS version satisfy the integral equations obtained in first and second order of that expansion. Our results point to the importance of exploring the physical consequences of such an infinity of integral equations on the global properties of the Yang-Mills theory.