Publication: Gauge forms on SU(2)-bundles
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Publication Date
1999-07
Authors
Muñoz Masqué, Jaime
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Elsevier
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Abstract
Let π:P→M be a principal SU(2)-bundle, let autP [resp. gauP⊂autP] be the Lie algebra [resp. the ideal] of all G-invariant [resp. G-invariant π-vertical] vector fields in X(P), and let p:C(P)→M be the bundle of connections of P. A differential form ωr on C(P) of arbitrary degree 0≤r≤4n, n=dimM, is said to be autP-invariant [resp. gauP-invariant] if it is invariant under the natural representation of autP [resp. gauP] on X(C(P)). The Z-graded algebra over Ω∙(M) of autP-invariant [resp. gauP-invariant] differential forms is denoted by IautP [resp. IgauP]. The basic results of this paper are the following: (1) The algebra of gauge invariant differential forms on p:C(P)→M is generated over the algebra of differential forms on M by a 4-form η4, i.e., IgauP(C(P))=(p∗Ω∙(M))[η4], where the form η4 is globally defined on C(P) by using the canonical su(s)-valued 1-form of the bundle T∗(M)⊗su(2) and the determinant function det:su(2)→R; its local expression is
η4=14S123(dA1i∧dxi∧dA1j∧dxj+2A2jA3kdxj∧dxk∧dA1i∧dxi),
(Aij,xj), 1≤i≤3, 1≤j≤n, being the coordinate system induced from (xj) and the standard basis (B1,B2,B3) of su(2) on C(P). (2) Assume M is connected. Then, IautP(C(P))=R[η4]. (3) The cohomology class of η4 in H4(C(P);R) coincides with −4π2p∗(c2(P)), where c2(P) stands for the second Chern class of P. Remark that p:C(P)→M is an affine bundle and hence one has a natural isomorphism p∗:H∙(M;R)→H∙(C(P);R). Another important remark is the following. If dimM≤3, then every principal SU(2)-bundle π:P→M is trivial and hence its Chern class vanishes, but the form η4 is not zero although its pull-back along every section of C(P) does vanish
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M.F Atiyah. Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc., 85 (1957), pp. 181–207
M.F Atiyah. Geometry of Yang-Mills Fields. Accademia Nazionale dei Lincei, Scuola Normale Superiore, Pisa (1979)
D Betounes. The geometry of gauge-particle field interaction: a generalization of Utiyama's theorem. J. Geom. Phys., 6 (1989), pp. 107–125
D Bleecker. Gauge Theory and Variational Principles. Addison-Wesley, Reading, MA (1981)
D.J Eck.Gauge-natural bundles and generalized gauge theories. Mem. Amer. Math. Soc., 247 (1981)
P.L García. Connections and 1-jet bundles. Rend. Sem. Mat. Univ. Padova, 47 (1972), pp. 227–242
P.L García. Gauge algebras, curvature and symplectic structure.J. Differential Geom., 12 (1977), pp. 209–227
V Guillemin, S Sternberg. Symplectic Techniques in Physics. Cambridge University Press, Cambridge, UK (1983)
L Hernández Encinas, J Muñoz Masqué. Symplectic structure and gauge invariance on the cotangent bundle
J. Math. Phys., 35 (1994), pp. 425–434
L Hernández Encinas, J Muñoz Masqué. Gauge invariance on the bundle of connections of a U (1)-principal bundle. C.R. Acad. Sci. Paris, t. 318, Série I (1994), pp. 1133–1138
D Husemoller. Fibre Bundles. (3rd ed.)Springer, New York (1994)
M Keyl. About the geometric structure of symmetry-breaking. J. Math. Phys., 32 (1991), pp. 1065–1071
S Kobayashi, K Nomizu. Foundations of Differential Geometry, vol. IWiley (Interscience Division), New York (1963)
S Kobayashi, K Nomizu. Foundations of Differential Geometry, vol. IIWiley (Interscience Division), New York (1969)
P.K Mitter, C.M Viallet. On the Bundle of Connections and the Gauge Orbit Manifold in Yang-Mills Theory. Commun. Math. Phys., 79 (1981), pp. 457–472
Ch Nash, S Sen. Topology and Geometry for Physicists. Academic Press, New York (1982)