The Yang-Mills Equations over Riemann Surfaces

Counting Homotopy Types of Gauge Groups

“…Recall from [3] that GP may be viewed as the group GP´A d G of sections of P´A d G. In the light of this, it is clear that if P 1 and P 2 are principal Gbundles over K such that the bundles of groups P 1´Ad G and P 2´Ad G are equivalent, then GP 1 and GP 2 are isomorphic as topological groups. (In this connection, we note that Tsukuda [26] has not only pointed out a similar result, but also established a converse for suitable G.) Likewise if the bundles of groups P 1´Ad G and P 2´Ad G are ®bre homotopy equivalent then GP 1 and GP 2 are homotopy equivalent.…”

“…Let D be a closed ball-neighbourhood of x 0 in M, and let D ± be its interior. As in [3], G 0 P may be identi®ed with the group of sections of P´A d G which are trivial at x 0 . This is topologically isomorphic to the group G D P of sections which are trivial over D (by the usual device of choosing a larger ballneighbourhood D 1 É D and using a relative homeomorphism of M; D with M; x 0 which is the identity outside D ± 1 ).…”

Around Polygons in ℝ3 and S3

“…A measure ν on S 2 is said to be stable (respectively, semi-stable) if the mass of any atom is less than (respectively, less than or equal to) 1 2 ν(S 2 ).…”

“…However, the equivalence relation used in the algebro-geometric construction of the moduli space of semi-stable parabolic bundles is exactly the isomorphism of associated graded bundles (associated to the Harder-Narasimhan filtration of a holomorphic bundle by successive maximally destabilizing subbundles; see [1]). Hence every semi-stable bundle is equivalent to one which is polystable and we can extend ξ to a bijection …”

“…But rather than approaching the intricacies of the algebraic geometry of these spaces, we choose to represent them as infinite-dimensional affine spaces and the resulting quotients by infinite-dimensional Lie groups. This is quite standard, since the work of Atiyah and Bott [1] and before, so our main work here will be in incorporating the parabolic structures of our bundles. First, since all holomorphic vector bundles of degree zero are isomorphic as smooth bundles, we shall work always in our trivial bundle E = C 2 × CP 1 .…”