Energy machineries on a manifold; application to the construction of new energy representations of gauge groups
Let TX, \(T^*X\) respectively denote the tangent and cotangent bundles of a smooth n-dimensional Riemannian manifold X, and let \(D(T^*)\) denote the space of real-valued compactly supported smooth 1-forms on X endowed with the Schwartz topology of compactly supported smooth sections. The concept of an energy machinery on X is defined to be a quadruplet (Y,d\(\nu\),F,q), where Y is a connected submanifold of X endowed with a strictly positive measure \(d\nu\), and F is smooth sub-bundle of TX, while \(q=(q_ x)_{x\in Y}\) is a Euclidean structure on the restricted sub- bundle \(F^ Y\), it being assumed that, for all \(\omega\), \(\omega '\) in \(D(T^*X)\), the map \[ x\to q^*_ x(\omega^ F(x),\quad \omega^{'F}(x)),\quad x\in Y, \] is \(d\nu\)-integrable. This notion, together with that of an energy structure on X, gives rise to a systematic treatment of a large class of energy representations of gauge groups, including known cases. A sufficient condition for the irreducibility of these energy representations is obtained.
scientific article; zbMATH DE number 4173851