Representation of a gauge group as motions of a Hilbert space
Annales Mathématiques Blaise Pascal, Volume 11 (2004) no. 2, pp. 131-153.

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, ${𝔊}_{P}$, 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 $P$.

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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] R. Abraham; J.E. Marsden; J Ratiu Manifolds, Tensor Analysis and Applications, Springer-Verlag, New York, 1988 | MR: 960687 | Zbl: 0875.58002

[2] S. Albeverio; S. R. Høegh-Krohn; D. Testard 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 | Article | MR: 619959 | Zbl: 0488.22038

[3] M. F. Atiyah; R. Bott The Yang-Mills Equations over Riemann Surfaces, Phil. Trans. R. Soc. Lond. A, Volume 308 (1982), pp. 523-615 | MR: 702806 | Zbl: 0509.14014

[4] E. Binz; J. Sniatycki; H. Fischer Geometry of Classical Fields, North-Holland, Amsterdam, 1988 | MR: 972499 | Zbl: 0675.53065

[5] J. Dieudonné Treatise on Analysis Volume IV, Academic Press, New York, 1974 | MR: 362066 | Zbl: 0292.58001

[6] J. Dieudonné Treatise on Analysis Volume V, Academic Press, New York, 1977 | Zbl: 0418.22007

[7] D. S. Freed; K. K. Uhlenbeck Instantons and Four-Manifolds, Springer-Verlag, New York, 1984 | MR: 757358 | Zbl: 0559.57001

[8] I. M. Gelfand; M. I. Graev; A. Vershik 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: 0368.53034

[9] I. M. Gelfand; M. I. Graev; A. Vershik 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: 0449.22019

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

[11] A. Knapp Lie Groups Beyond an Introduction, Birkhäuser, New York, 1996 | MR: 1399083 | Zbl: 0862.22006

[12] S. Kobayashi; K. Nomizu Foundations of Differential Geometry Volume 1, John Wiley & Sons, Inc, United States of America, 1996

[13] A. Kriegl; P. W. Michor The Convenient Setting of Global Analysis, American Mathematical Society, United States of America, 1997 | MR: 1471480 | Zbl: 0889.58001

[14] A. Kriegl; P. W. Michor Regular infinite dimensional Lie groups, Journal of Lie Theory, Volume 7 (1997), pp. 61-99 | MR: 1450745 | Zbl: 0893.22012

[15] S. Lang Differential and Riemannian Manifolds, Springer-Verlag, New York, 1995 | MR: 1335233 | Zbl: 0824.58003

[16] J. A. Leslie On a Differential Structure for the Group of Diffeomorphism, Topology, Volume 6 (1967), pp. 263-271 | Article | MR: 210147 | Zbl: 0147.23601

[17] P. Libermann; C. Marle Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Company, Dordrecht, Holland, 1987 | MR: 882548 | Zbl: 0643.53002

[18] K. B. Marate; G. Martucci The geometry of gauge fields, Journal of Geometry and Physics, Volume Vol. 6, N.3 (1989) | Zbl: 0679.53023

[19] J. Mickelsson Current Algebras and Groups, Plenum Monographs in Nonlinear Physics, New York, 1989 | MR: 1032521 | Zbl: 0726.22015

[20] H. Omori 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: 1421572 | Zbl: 0871.58007

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

[22] A. L. Onishchik; E. B. Vinberg Lie Groups and Algebraic Groups, Springer-Verlag, Berlin, 1990 | MR: 1064110 | Zbl: 0722.22004

[23] R. Palais Seminar on the Atiyah-Singer Index Theorem, Princeton University Press, 1965 | MR: 198494 | Zbl: 0137.17002

[24] A. Pressley; G. Segal Loop Groups, Oxford University Press, New York, 1986 | MR: 900587 | Zbl: 0618.22011

[25] W. Rudin Functional Analysis, McGraw-Hill, Singapore, 1991 | MR: 1157815 | Zbl: 0867.46001

[26] F. Treves Topological Vector Spaces, Distributions and Kernels, Academis Press, INC, New York, 1973 | MR: 225131

[27] N. R. Wallach On the irreducibility and inequivalence of unitary representations of gauge groups, Compositio Mathematica, Volume 64 (1987), pp. 3-29 | Numdam | MR: 911356 | Zbl: 0632.22014

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