Equivariant cohomology of the skyrmion bundle

Gross, Christian

  • Proceedings of the 16th Winter School "Geometry and Physics", Publisher: Circolo Matematico di Palermo(Palermo), page [87]-96

Abstract

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The author constructs the gauged Skyrme model by introducing the skyrmion bundle as follows: instead of considering maps U : M → SU N F he thinks of the meson fields as of global sections in a bundle B ( M , SU N F , G ) = P ( M , G ) × G 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 ≤ 6 , one has H * ( E G × G SU N F ) ≅ H * ( SU N F ) G ≅ S ( G ̲ * ) ⊗ H * ( SU N F ) ≅ H * ( B G ) ⊗ H * ( SU N F ) , where E G ( B G , G ) is the universal bundle for the Lie group G and G ̲ is the Lie algebra of G .

How to cite

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Gross, Christian. "Equivariant cohomology of the skyrmion bundle." Proceedings of the 16th Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 1997. [87]-96. <http://eudml.org/doc/221614>.

@inProceedings{Gross1997,
abstract = {The author constructs the gauged Skyrme model by introducing the skyrmion bundle as follows: instead of considering maps $U: M\rightarrow \text\{SU\}_\{N_F\}$ 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 $N_F\le 6$, 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 $G$.},
author = {Gross, Christian},
booktitle = {Proceedings of the 16th Winter School "Geometry and Physics"},
keywords = {Proceedings; Winter school; Srní (Czech Republic); Geometry; Physics},
location = {Palermo},
pages = {[87]-96},
publisher = {Circolo Matematico di Palermo},
title = {Equivariant cohomology of the skyrmion bundle},
url = {http://eudml.org/doc/221614},
year = {1997},
}

TY - CLSWK
AU - Gross, Christian
TI - Equivariant cohomology of the skyrmion bundle
T2 - Proceedings of the 16th Winter School "Geometry and Physics"
PY - 1997
CY - Palermo
PB - Circolo Matematico di Palermo
SP - [87]
EP - 96
AB - The author constructs the gauged Skyrme model by introducing the skyrmion bundle as follows: instead of considering maps $U: M\rightarrow \text{SU}_{N_F}$ 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 $N_F\le 6$, 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 $G$.
KW - Proceedings; Winter school; Srní (Czech Republic); Geometry; Physics
UR - http://eudml.org/doc/221614
ER -

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