Compactness theorems for coupled Yang-Mills fields
The author considers the system of Young-Mills-Higgs equations (YMH equations) for the pair \((A,\Phi)\) where \(A \) is the potential of a connection in a vector bundle \(E\) over an \(n\)-dimensional manifold \(M\) with compact structure group \(G\) and \(\Phi \in \Gamma(E \otimes L)\) a Higgs field, where \(L = \text{vol}^{1/n}(M)\) is the line bundle of weight 1, which is the \(1/n\)-th power of the volume bundle. He gets two compactness theorems for solutions of this system which are generalizations of known results about pure Young-Mills equations. In particular, he proves the following theorem: Let \((A_i, \Phi_i)\) be a sequence of solutions of YMH equations. Then there exists a subsequence \((A_k, \Phi_k) \) and gauge transformations \(\sigma_k\) such that the sequence of solutions \(\sigma^*_k(A_k,\Phi_k)\) converges to a solution \((A,\Phi)\) outside a closed subset \(S\) of codimension at least 4.
scientific article; zbMATH DE number 2117740