On isomorphism classes of gauge groups
Let \(G\) be a compact connected Lie group with center \(Z\). Let \(\pi :P\rightarrow X\) be a principal \(G\) bundle over a closed connected smooth manifold \(X\) and \(\mathcal{G}\) its gauge group. Then \(\mathcal{G}\) is naturally identified with the space \(\Gamma (\text{Ad}P)\) of all smooth sections of the bundle Ad\(P=P\times _{\text{Ad}}G\). Let \(P'\) be another \(G\) bundle over \(X\) and \(\mathcal{G}'\) its gauge group. Theorem 1. If \(G\) is a unitary group or a simple compact connected Lie group (except semi-spinor groups) then \(\mathcal{G}\cong \mathcal{G}'\) as Fréchet Lie groups if and only if \(\text{Ad } P'\) is isomorphic to \(\text{Ad } P\). In [Pac. J. Math. 68, 331-338 (1977; Zbl 0357.55022)] \textit{W. D. Curtis} and \textit{F. R. Miller} obtained similar results in the case that \(G\) has trivial center or \(G\) is \(SO(2n)\), \(n\geq 3\), or \(U(n)\), \(n\geq 2\). Let \(\mathcal{D}\) denote the equivariant diffeomorphism group of \(P\times_{G}\Aut(G)\) and \(\mathcal{G}_{0}\) the identity component of the gauge group \(\mathcal{G}\). Theorem. If \(G\) is a compact connected simple Lie group (except semi-spinor groups) then \(\Aut \mathcal{G}_0\cong \mathcal{D}\). The group \(G/Z\) splits into a product of simple Lie groups \(G/Z\cong G_{1}\times \ldots \times G_{k}\). Let \(\text{Ad }P_{i}\) denote the associated bundle \(P\times _{\text{Ad}}G_{i}\). Let \(G'\) be a another compact Lie group and \(\pi :P'\rightarrow Y\) a principal \(G'\) bundle over a closed connected smooth manifold \(Y\), \(\mathcal{G}'\) its gauge group and \(\mathcal{Z}\), \(\mathcal{Z}'\) the centres of \(\mathcal{G}\), \(\mathcal{G}'\), respectively. Theorem 3. \((\mathcal{G}/ \mathcal{Z})_{0} \cong (\mathcal{G}'/ \mathcal{Z}')_{0}\) if and only if \(G/Z\cong G'/Z'\) and \(\text{Ad }P_{i}\) is isomorphic to \(\text{Ad }P_{\sigma(i)}'\) for each \(i\) where \(\sigma \) is an element of the \(k\)th symmetry group. In particular, if \(\mathcal{G}\cong \mathcal{G}'\) then \(X\) and \(Y\) are diffeomorphic.
scientific article; zbMATH DE number 1307953
scientific article