All-order exact solutions for non-linear Yang–Mills theory

Abstract

We show that the so-called ’t Hooft–Polyakov monopole solution in D=(3,1) dimensions satisfies the field equations of a wide set of non-linear Yang–Mills (YM) actions to all orders in the parameter α that is inversely proportional to the string tension T. Here the expression D=(s,t) is for the (s+t)dimensional space–time with the signature ((+)s,(−)t), namely s positive and t negative signatures. We show that the so-called Born–Infeld (BI) action is only a special case of a general set of non-linear YM theories. We also consider BI-actions in D=(2,2) dimensions, and show that the self-dual SU(2) instanton (BPSTH) solution satisfies the field equations of BI-like actions perturbatively to all orders inα. In D=(2,2) dimensions, the usual action-principle problem for (anti)self-dual ((A)SD) YM field strengths is avoided by auxiliary-gravity lagrangians. The conditions on non-linear YM theories to have the ’t Hooft–Polyakov monopole solution to all orders in α in D=(3,1) dimensions, or similarly the BPSTH instanton solution in D=(2,2) dimensions, allow infinitely many non-linear YM actions in these dimensions.