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Equivariant cohomology of the skyrmion bundle. (English) Zbl 0905.53052
Slovák, Jan (ed.), Proceedings of the 16th Winter School on geometry and physics, Srní, Czech Republic, January 13–20, 1996. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 46, 87-96 (1997).
The author constructs the gauged Skyrme model by introducing the skyrmion bundle as follows: instead of considering maps \(U: M\to \text{SU}_{N_F}\) he thinks of the meson fields as of global sections in a bundle \(B(M,\text{SU}_{N_F},G)=P(M,G)\times_G \text{SU}_{N_F}\). For calculations within the skyrmion bundle the author introduces by means of the so-called equivariant cohomology an analogue of the topological charge and the Wess-Zumino term. The final result of this paper is the following Theorem. For the skyrmion bundle with \(N_F\leq 6\), one has \[ H^{*}(EG\times_G \text{SU}_{N_F})\cong H^{*}(\text{SU}_{N_F})^G \cong \text{S}({\underline G}^{*})\otimes H^{*}(\text{SU}_{N_F}) \cong H^{*}(BG)\otimes H^{*}(\text{SU}_{N_F}), \] where \(EG(BG,G)\) is the universal bundle for the Lie group \(G\) and \(\underline G\) is the Lie algebra of \(G\).
For the entire collection see [Zbl 0866.00050].
53Z05 Applications of differential geometry to physics
55R91 Equivariant fiber spaces and bundles in algebraic topology
81V05 Strong interaction, including quantum chromodynamics
55N91 Equivariant homology and cohomology in algebraic topology