This is a survey article based on the author’s Master thesis on affine representations of a gauge group. Most of the results presented here are well-known to differential geometers and physicists familiar with gauge theory. However, we hope this short systematic presentation offers a useful self-contained introduction to the subject.

In the first part we present the construction of the group of motions of a Hilbert space and we explain the way in which it can be considered as a Lie group. The second part is about the definition of the gauge group, ${\U0001d50a}_{P}$, associated to a principal bundle, $P$. In the third part we present the construction of the Hilbert space where the representation takes place. Finally, in the fourth part, we show the construction of the representation and the way in which this representation goes to the set of connections associated to $P$.

@article{AMBP_2004__11_2_131_0, author = {Clara Luc{\'\i}a Aldana Dom{\'\i}nguez}, title = {Representation of a gauge group as motions of a {Hilbert} space}, journal = {Annales math\'ematiques Blaise Pascal}, pages = {131--153}, publisher = {Annales math\'ematiques Blaise Pascal}, volume = {11}, number = {2}, year = {2004}, doi = {10.5802/ambp.189}, mrnumber = {2109604}, zbl = {1077.58006}, language = {en}, url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.189/} }

TY - JOUR AU - Clara Lucía Aldana Domínguez TI - Representation of a gauge group as motions of a Hilbert space JO - Annales mathématiques Blaise Pascal PY - 2004 SP - 131 EP - 153 VL - 11 IS - 2 PB - Annales mathématiques Blaise Pascal UR - https://ambp.centre-mersenne.org/articles/10.5802/ambp.189/ DO - 10.5802/ambp.189 LA - en ID - AMBP_2004__11_2_131_0 ER -

%0 Journal Article %A Clara Lucía Aldana Domínguez %T Representation of a gauge group as motions of a Hilbert space %J Annales mathématiques Blaise Pascal %D 2004 %P 131-153 %V 11 %N 2 %I Annales mathématiques Blaise Pascal %U https://ambp.centre-mersenne.org/articles/10.5802/ambp.189/ %R 10.5802/ambp.189 %G en %F AMBP_2004__11_2_131_0

Clara Lucía Aldana Domínguez. Representation of a gauge group as motions of a Hilbert space. Annales mathématiques Blaise Pascal, Volume 11 (2004) no. 2, pp. 131-153. doi : 10.5802/ambp.189. https://ambp.centre-mersenne.org/articles/10.5802/ambp.189/

[1] Manifolds, Tensor Analysis and Applications, Springer-Verlag, New York, 1988 | MR | Zbl

[2] Irreducibility and reducibility for the energy representation of the group of mappings of a riemannian manifold into a compact semisimple Lie group, Journal of Functional Analysis, Volume 41 (1981), pp. 378-396 | DOI | MR | Zbl

[3] The Yang-Mills Equations over Riemann Surfaces, Phil. Trans. R. Soc. Lond. A, Volume 308 (1982), pp. 523-615 | MR | Zbl

[4] Geometry of Classical Fields, North-Holland, Amsterdam, 1988 | MR | Zbl

[5] Treatise on Analysis, IV, Academic Press, New York, 1974 | MR | Zbl

[6] Treatise on Analysis, V, Academic Press, New York, 1977 | Zbl

[7] Instantons and Four-Manifolds, Springer-Verlag, New York, 1984 | MR | Zbl

[8] Representation of the group of smooth mappings of a manifold $X$ into a compact Lie group, Compositio Math, Volume 35, Fasc. 3 (1977), pp. 299-334 | Numdam | Zbl

[9] Representation of the group of functions taking values in a compact Lie group, Compositio Math, Volume 42, Fasc. 2 (1981), pp. 217-243 | Numdam | Zbl

[10] Unitary Representations of Gauge Groups (1996) (Ph.D. thesis, University of Massachusetts)

[11] Lie Groups Beyond an Introduction, Birkhäuser, New York, 1996 | MR | Zbl

[12] Foundations of Differential Geometry, 1, John Wiley & Sons, Inc, United States of America, 1996

[13] The Convenient Setting of Global Analysis, American Mathematical Society, United States of America, 1997 | MR | Zbl

[14] Regular infinite dimensional Lie groups, Journal of Lie Theory, Volume 7 (1997), pp. 61-99 | MR | Zbl

[15] Differential and Riemannian Manifolds, Springer-Verlag, New York, 1995 | MR | Zbl

[16] On a Differential Structure for the Group of Diffeomorphism, Topology, Volume 6 (1967), pp. 263-271 | DOI | MR | Zbl

[17] Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Company, Dordrecht, Holland, 1987 | MR | Zbl

[18] The geometry of gauge fields, Journal of Geometry and Physics, Volume Vol. 6, N.3 (1989) | Zbl

[19] Current Algebras and Groups, Plenum Monographs in Nonlinear Physics, New York, 1989 | MR | Zbl

[20] Infinite-Dimensional Lie Groups, Translations of Mathematical Monographs. American Mathematical Society, United States of America, 1984 (Original published in Japanese by Kinokuniya Co., Ltd., Tokyo, 1979) | MR | Zbl

[21] On Regular Fréchet-Lie Groups IV, Tokyo J. Math, Volume 5 No. 2 (1981)

[22] Lie Groups and Algebraic Groups, Springer-Verlag, Berlin, 1990 | MR | Zbl

[23] Seminar on the Atiyah-Singer Index Theorem, Princeton University Press, 1965 | MR | Zbl

[24] Loop Groups, Oxford University Press, New York, 1986 | MR | Zbl

[25] Functional Analysis, McGraw-Hill, Singapore, 1991 | MR | Zbl

[26] Topological Vector Spaces, Distributions and Kernels, Academis Press, INC, New York, 1973 | MR

[27] On the irreducibility and inequivalence of unitary representations of gauge groups, Compositio Mathematica, Volume 64 (1987), pp. 3-29 | Numdam | MR | Zbl

*Cited by Sources: *