zbMATH — the first resource for mathematics

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.

55P35 Loop spaces
54C35 Function spaces in general topology
81T13 Yang-Mills and other gauge theories in quantum field theory
Full Text: DOI Euclid
[1] R. Bott, A note on the Samelson product in the classical groups , Comment. Math. Helv. 34 (1960), 249-256. · Zbl 0094.01503
[2] C. P. Boyer, B. M. Mann, and D. Waggonner, On the homology of \(\operatorname{SU} (n)\) instantons , Trans. Amer. Math. Soc. 323 (1991), no. 2, 529-561. · Zbl 0725.53065
[3] D. H. Gottlieb, Applications of bundle map theory , Trans. Amer. Math. Soc. 171 (1972), 23-50. · Zbl 0251.55018
[4] H. Hamanaka and A. Kono, Unstable \(K^{1}\)-group and homotopy type of certain gauge groups , Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), 149-155. · Zbl 1103.55004
[5] B. Harris, On the homotopy groups of the classical groups , Ann. of Math. (2) 74 (1961), 407-413. · Zbl 0118.18501
[6] D. Kishimoto, Generating varieties, Bott periodicity and instantons , Topology Appl. 157 (2010), 657-668. · Zbl 1184.81070
[7] D. Kishimoto and A. Kono, Note on mod-\(p\) decompositions of gauge groups , Proc. Japan Acad. Ser. A. Math. Sci. 86 (2010), 15-17. · Zbl 1190.55007
[8] D. Kishimoto, A. Kono, and M. Tsutaya, Mod-\(p\) decompositions of gauge groups , Algebr. Geom. Topol. 13 (2013), 1757-1778. · Zbl 1276.57036
[9] A. Kono, A note on the homotopy type of certain gauge groups , Proc. Roy. Soc. Edinburgh Sect. A 117 (1991), 295-297. · Zbl 0722.55008
[10] G. E. Lang, The evaluation map and \(EHP\) sequences , Pacific J. Math. 44 (1973), 201-210. · Zbl 0217.20003
[11] M. Mimura, G. Nishida, and H. Toda, Mod-\(p\) decomposition of compact Lie groups , Publ. Res. Inst. Math. Sci. 13 (1977/78), 627-680. · Zbl 0383.22007
[12] S. D. Theriault, Odd primary decompositions of gauge groups , Algebr. Geom. Topol. 10 (2010), 535-564. · Zbl 1196.55009
[13] S.D. Theriault, The homotopy types of \(\operatorname{Sp} (2)\)-gauge groups , Kyoto J. Math. 50 2010, 591-605. · Zbl 1202.55004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.