General Solution of the non-abelian Gauss law and non-abelian analogs of the Hodge decomposition

摘要

General solution of the non-abelian Gauss law in terms of covariant curls andgradients is presented. Also two non-abelian analogs of the Hodge decompositionin three dimensions are addressed. i) Decomposition of an isotriplet vectorfield $V_{i}^{a}(x)$ as sum of covariant curl and gradient with respect to anarbitrary background Yang-Mills potential is obtained. ii) A decomposition ofthe form $V_{i}^{a}=B_{i}^{a}(C)+D_{i}(C) \phi^{a} $ which involves non-abelianmagnetic field of a new Yang-Mills potential C is also presented. These resultsare relevant for duality transformation for non-abelian gauge fields.