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Let $Y$ be Poincaré's integral homology 3-sphere. Let $\pi:P\to Y$ be a (necessarily trivial) $\mathrm{SU}(2)$-principal bundle over $Y$. Fix $x_0\in Y$ and $p_0\in P_x:=\pi^{-1}(x_0)$. The fundamental group $\pi_1(Y,x_0)$ of $Y$ is isomorphic to the binary icosahedral group $2I$ which can be seen as a subgroup of $\mathrm{SU}(2)$. Let $A$ be a flat connection on $P$ such that its holonomy group $H<\mathrm{Aut}(P_x)\cong \mathrm{SU}(2)$ based at $x_0$ is isomorphic to the binary icosahedral group $2I$ inside $\mathrm{SU}(2)$. Let $Q\subset P$ be the set of points in $P$ that are horizontally reachable from $p_0$, i.e. : $$ Q := \left\{p\in P \;|\; \exists \gamma\in \mathcal{C}^1_{pw}([0,1];P) \;\;\text{s.t.}\;\; \gamma(0)=p_0,\gamma(1)=p\;\; \text{and $\gamma$ horizontal for $A$}\right\} \subset P $$ where $\mathcal{C}^1_{pw}$ means "piecewise differentiable". From theorem 7.1 at p.83-84 of [KN-I], it happens that $Q$ is a structural reduction from the bundle $\mathrm{SU}(2)\hookrightarrow P\to Y$ to the bundle $H\hookrightarrow Q\to Y$ such that $A$ reduces to a connection on $Q\to Y$. So here, according to [KN-I] and to this (at middle of p.15), $A$ is a reducible connection (in fact, $Q\to Y$ has discrete fibers so the restriction of $A$ to $Q$ vanish because, here, $\mathrm{Lie}(H)=\{0\}$).

On the other side, the centralizer of $H$ in $\mathrm{SU}(2)$ is minimal : $C_{\mathrm{SU}(2)}(H)=C_{\mathrm{SU}(2)}(2I)=\{-1,1\}$. So here, according to the mainstream gauge theory's definition of reducibility, $A$ is an irreducible connection.

Questions :

  1. Am I right that there is a conflict between [KN-I]'s definition of reducible connections and the definition found in gauge theory litterature in the last forty or fifty years ?
  2. If I'm wrong, please point where I missed something.
  3. If I'm right, do you know around which paper was the first instance of the definition of an irreducible connection being that $C_G(H)$ is minimal ?

Maybe it's just that the notion of "reducibility of $A$ in the context of structural reduction" is not the same thing as "reductibility of $A$ in the context of reducibility representations of the fundamental group".


p.s. this question is a sequel to this previous question.

[KN-I] : Foundations of Geometry, Vol. I (Kobayashi, Nomizu)


Conclusion : The two notions cited above of reducible connections are mostly the same, but aren't the same.

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I wouldn't call it a conflict. Words are often overused in mathematics. But they are different notions.

In gauge theory the fundamental notion is that of a (unitary, orthogonal) connection on a vector bundle. A reducible connection is one that is induced from a vector bundle of smaller dimension. To Kobayashi and Nomizu, the fundamental notion is a connection on a principal $G$-bundle, where $G$ is a Lie group. Then the right notion of irreducible is whether or not the connection can be induced from a smaller (Lie) structure group. This is precisely whether or not $H_A$ is a proper subgroup of $G$. If you further think the primitive notion is that of connections on principal $G$-bundles where $G$ is compact Lie, then you care about whether or not $\overline{H_A}$ is a proper subgroup of $G$.

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  • $\begingroup$ Yep it seems to me "irreducible/reducible" is used in two close but different notions. Though, gauge theory isn't always about connexions on a vector bundle. I mostly see connections on principal bundles. Then, this induces connections on associated bundles. e.g., for the canonical representation of $\mathrm{SU}(2)$ on $\mathbb{C}^2$ we have a vector bundle $E$ with typical fiber $\mathbb{C}^2$ where $A$ on $P$ induces, lets say, $\nabla$ on $E$. But yeah, one can go the opposite way also, from $E$ to $P$ considering the special unitary frame bundle of $E$ which is our original $P$. $\endgroup$ – Noé AC Jan 20 '18 at 22:58
  • $\begingroup$ @NAC I know :) But at least for $SU(2)$, $SO(3)$, $C(H_A)$ is the center if and only if the connection is not induced from connections on smaller rank bundles. I'd guess that's true for arbitary $SO(N)$ but I haven't thought about it. $\endgroup$ – user98602 Jan 20 '18 at 23:03
  • $\begingroup$ @NAC Similar but not precisely the same for positive dimensional subgroups; e.g. $\text{Pin}(2) \subset SU(2)$ is a positive-dimensional subgroup not contained in $U(1)$ (which is what I would say means it's induced by a connection on a bundle of lower rank). A careful author should be precise about what they mean in their context (most usually that the stabilizer of the gauge group action is as small as possible). $\endgroup$ – user98602 Jan 20 '18 at 23:21
  • $\begingroup$ Ok, thanks. I think that concludes my series of questions/doubts about irreducible connections $\mathrm{SU}(2)$. $\endgroup$ – Noé AC Jan 20 '18 at 23:23

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