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Refined gauge group decompositions. (English) Zbl 1321.55007
Let $$G$$ be a simple, simply connected, compact Lie group and let $$P_{k}\to S^4$$ be the principal $$G$$-bundle whose second Chern class has value $$k$$. In the paper under review, the authors give a $$p$$-local homotopy decomposition of the gauge group, $${\mathcal{G}}_{k}(G)$$, of $$P_{k}$$, when $$G$$ is a matrix group and $$p$$ is an odd prime. Their main result consists of the following homotopy equivalences, for $$n\geq 2$$ and any $$k\in {\mathbb Z}$$, ${\mathcal{G}}_{k}({\mathrm SU}(n)) \simeq \prod_{i=1,i\neq v-1,v}^{p-1} (B_{i}\times \Omega^4_{0}B_{i+2})\times X_{v-1}\times X_{v}.$ The spaces $$B_{\bullet}$$ come from a $$p$$-local homotopy equivalence, $$G\simeq \prod_{i=1}^{p-1} B_{i}$$, established by M. Mimura et al. [Publ. Res. Inst. Math. Sci. 13, 627–680 (1977; Zbl 0383.22007)]. Also, there is a homotopy fibration, $$\Omega^4_{0}B_{t+2}\to X_{t}\to B_{t}$$, for $$t\in\{v-1,v\}$$. A similar decomposition of $${\mathcal{G}}_{k}({\mathrm Sp}(n))$$ is also provided.

MSC:
 55P35 Loop spaces 54C35 Function spaces in general topology 81T13 Yang-Mills and other gauge theories in quantum field theory
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References:
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