By Lefort G.

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**Extra info for Algebra and analysis: problems and solutions**

**Example text**

A , b can be viewed as the holonomies of a Yang-Mills connection on an SO(2n + 1)-bundle Q0 → Σ0 . Also, c = = diag(Hn , (−1)n I1 ) can be viewed as the holonomy of a ﬂat connection on an SO(2n + 1)-bundle Q1 over Σ01 = RP2 , and c = , d = I2n+1 can be viewed as the holonomies of a ﬂat connection on an SO(2n + 1)-bundle Q2 over Σ02 (a Klein bottle). Let Σ be obtained by gluing Σ0 and Σ0i at a point, and let P → Σ be the (topological) principal SO(2n + 1)bundle over Σ such that P |Σ0 = Q0 and P |Σ0i = Qi .

Then VYM identiﬁed with the representation variety Vss (Pµ ) of central Yang-Mills connections on Pµ . We have homeomorphisms ,0 ∼ Nµ (P )/G(P ) ∼ = XYM (GR )P µ /GR = Vss (Pµ )/(GR )µ and homotopy equivalences of homotopic orbit spaces: hG(P ) Nµ (P ) ,0 ∼ XYM (GR )P µ hGR ∼ Vss (Pµ )h(GR )µ . hG(ξ ) hG(P ) 0 Combined with the homotopy equivalence Cµ (ξ0 ) ∼ Nµ (P ) , we conclude that (GR )µ G(ξ ) ,0 Pt 0 (Cµ (ξ0 ); Q) = PtGR (XYM (GR )P (Vss (Pµ ); Q). 3. The connectedness of Nµ (P ) implies the connectedness of Vss (Pµ ), but not vise versa, because G0 (P ) is not connected in general.

11. Let GR = SO(2n). Then √ C 0 = { −1diag(t1 J, . . , tn J) | t1 ≥ · · · ≥| tn |≥ 0}, √ −C 0 = { −1diag(v1 J, . . , vn J) | v1 ≤ · · · ≤ − | vn |≤ 0}. The unique w in W ∼ = SG(n), the subgroup of G(n) consisting of even permutations, that maps C 0 to −C 0 , belongs to the Z2 part of SG(n), and √ √ w · −1diag(t1 J, . . , tn J) = −1diag(−t1 J, . . , −tn−1 J, (−1)n−1 tn J). Thus τ √ √ −1diag(t1 J, . . , tn J) = −1diag(t1 J, . . , tn−1 J, (−1)n tn J) of itself. If n is odd, If n is even, then any Y ∈ C 0 in conjugate to the negative √ then Y ∈ C 0 is conjugate to −Y iﬀ Y is of the form −1diag(t1 J, .