Conjugacy classes of gauge groups
by
Renzo A. Piccinini
University of Milan

Let U={ U_i|i in J} be a ``good cover'' for a riemannian manifold or a CW-complex B and let G be a topological group; form the topological group

L : = Õ i in J  Map(Ui, G)

with the product topology. Then the gauge group G(\xi) of every principal G-bundle \xi over B is isomorphic to a subgroup of L. We wish to study the conjugacy classes of these gauge groups, viewed as subgroups of L. A first result in this direction is that two gauge groups G(\xi) and G(\xi') are conjugate if, and only if, \xi and \xi' differ by the action of a principal ZG-bundle over B, where ZG is the centre of G (for vector bundles, this means that \xi' is equivalent to the tensor product of \xi by a line bundle). The difficulty in classifying the conjugacy classes of gauge groups lies on the fact that the isotropy group of \xi might be non-trivial; for example, if \xi is a real n-bundle over the projective space RPⁿ, the isotropy group of \xi is Z₂ if, and only if, n is even and either \xi =~ \frac12n\gamma₁ⁿ or \xi =~ \frac12(n+2^f(n))\gamma₁ⁿ, where \gamma₁ⁿ is the canonical line bundle over RPⁿ and 2^f(n) is the order of the reduced KO-group of RPⁿ. For vector bundles we can obtain interesting results about the conjugacy of gauge groups by comparing the characteristic classes of the vector bundles. In general, we can transform the problem into a homotopy theory question via the classification theorem; to obtain significant results it is necessary to conduct a critical study into the Milgram-Steenrod construction of classifying spaces for topological groups.

Date received: May 26, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabo-08.