Energy representations of gauge groups associated with Riemannian flags
Let (X,g) be a smooth Riemannian n-manifold and \(Y_ i\) be k submanifolds with C 1 Riemannian structures \(g_ i\), and let \(Y_ i\subset Y_{i-1}\) and dim \(Y_ i<\dim Y_{i-1}\) \((Y_ 0=X)\). This defines a Riemannian flag. If dim \(Y_ i=n-i\), then the Riemannian flag is regular. Let G be a semisimple Lie group and \({\mathfrak g}\) its Lie algebra with the Euclidean structure given by the negative of the Killing form. The Riemannian structures g and \(g_ i\) enable one to define the energy function on the space \(\Omega\) (X,g) of all smooth \({\mathfrak g}\)- valued 1-forms on X. The completion of complexified \(\Omega\) (X,g) defines the Hilbert space \(H_ Y\) identified with the state space of a 1-particle system and its symmetric tensor algebra leads to the Fock space. The so-called energy representation \(U_ Y\) of the gauge group G X of all smooth compactly supported G-valued maps on X is then defined. Two main theorems are proved: I. If \(n\geq 3\) and the Riemannian flag Y is regular, then \(U_ Y\) is irreducible; if \(n=2\), Y is regular and the length of each root of \({\mathfrak g}\) is bounded from below by a positive constant \(K_ Y\), then \(U_ Y\) is irreducible. II. Let Y and Y' be two distinct regular Riemannian flags and let \(n\geq 3\), or \(n=2\) and the length of each root of \({\mathfrak g}\) be greater than or equal to \(Max(K_ Y,K_{Y'})\), where the positive constants \(K_ Y\) and \(K_{Y'}\) are as in I. Then \(U_ Y\) and \(U_{Y'}\) are not unitarily equivalent. The paper is completely self-contained in all its mathematical aspects. The energy representations discussed are on the Fock space and hence there must be some important physical implications (which are not discussed here) for the quantization of gauge fields.
scientific article; zbMATH DE number 4063981