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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 21st 2016
   • (edited Sep 21st 2016)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexIf you have seen my recent postings either on G+ [1,2,4,5] or on MO [3] then you may have wondered what all this is in aid of. It's an attempt to show that the moduli space of Yang-Mills instantons on $S^5$ on the unique nontrivial $SU(2)$ bundle there is non-empty (or rather, on a non-compact Riemannian manifold that compactifies to $S^5$). This is parallel with simply trying to find solutions to the relevant PDEs, since it's possible there are no solutions and I'm wasting my time. The story is roughly as follows. * Anti-self-dual contact instantons are also Yang-Mills instantons, and in fact the Yang-Mills action is bounded below by the action for contact instantons. * Contact instantons on a quasiregular K-contact Sasaki-Einstein 5-manifold $M$ can arise by pullback of ordinary (A)SD instantons from the quotient by the Reeb foliation, which is generically a Kähler 4-orbifold, but in the case at hand it is $\mathbb{CP}^2$ ($S^5$ satisfies all these hypotheses). This is work of [Baraglia and Hekmati](https://arxiv.org/abs/1401.5140) * In particular, ASD instantons on $\mathbb{CP}^2$ pull back to give ASD contact instantons on $S^5$. * However, all $SU(2)$-bundles on $\mathbb{CP}^2$, classified by $c_2 \in \mathbb{Z}$ pull back to give a trivial bundle on $S^5$ (Tyler Lawson told me this, in discussion on [1]) * There are of course _more_ ASD instantons on $\mathbb{CP}^2$ with structure group $SO(3)$, classified by $w_2\in \{0,1\}$ and $p_1\in \mathbb{Z}$ (such that $p_1 \cong w_2 (mod 4)$), with the $SU(2)$ ones arising from the cases $w_2=0$, and _subject to the requirement that $p_1 \leq -3$_, otherwise the moduli space of ASD instantons is empty. This is where serious analytic theorems come into play. * Note that there is no difference at the level of Lie algebras, so local data for an $SO(3)$-connection is identical to the local data for an $SU(2)$-connection on a lifted bundle. So if we can get an $SO(3)$ instanton on $S^5$, we get an $SU(2)$ instanton (and the unique nontrivial $SU(2)$-bundle corresponds to a unique nontrivial $SO(3)$-bundle. * So if there is an $SO(3)$-bundle on $\mathbb{CP}^2$ with $w_2=1$ and $p_1 \leq -3$ that pulls back to the nontrivial bundle on $S^5$ we know that we have an $SO(3)$, hence $SU(2)$, ASD contact instanton, hence a Yang-Mills instanton. The current most concrete problem I'm trying to solve is at [5], where I have set up a rather classical geometric situation with concrete models for (skeleta of) classifying spaces in order to see if I can detect whether $S^5 \to \mathbb{CP}^2 \to BSO(3)$ is nontrivial from other easier-to-handle data, in particular an element of $[S^1,\mathbb{RP}^2] = \mathbb{Z}/2$. [1] [Here's a bit of a puzzle...](https://plus.google.com/+DavidRoberts/posts/KHeDwabxDXS) [2] [Looking at the Postnikov tower of BSO(3)](https://plus.google.com/+DavidRoberts/posts/NuVp7zEnQYT) [3] [The fifth k-invariant of BSO(3)](http://mathoverflow.net/questions/250303/) [4] [...what happens to SO(3) bundles when pulled back along S^5 --> CP^2](https://plus.google.com/+DavidRoberts/posts/8hrkFb2NNfa) [5] [Real and quaternionic projective spaces and Grassmannians, and a fibre sequence of classifying spaces](https://plus.google.com/+DavidRoberts/posts/T6yeav82GZG)

   If you have seen my recent postings either on G+ [1,2,4,5] or on MO [3] then you may have wondered what all this is in aid of. It’s an attempt to show that the moduli space of Yang-Mills instantons on $S^5$ on the unique nontrivial $SU(2)$ bundle there is non-empty (or rather, on a non-compact Riemannian manifold that compactifies to $S^5$). This is parallel with simply trying to find solutions to the relevant PDEs, since it’s possible there are no solutions and I’m wasting my time.

   The story is roughly as follows.

   • Anti-self-dual contact instantons are also Yang-Mills instantons, and in fact the Yang-Mills action is bounded below by the action for contact instantons.
   • Contact instantons on a quasiregular K-contact Sasaki-Einstein 5-manifold $M$ can arise by pullback of ordinary (A)SD instantons from the quotient by the Reeb foliation, which is generically a Kähler 4-orbifold, but in the case at hand it is $\mathbb{CP}^2$ ($S^5$ satisfies all these hypotheses). This is work of Baraglia and Hekmati
   • In particular, ASD instantons on $\mathbb{CP}^2$ pull back to give ASD contact instantons on $S^5$.
   • However, all $SU(2)$-bundles on $\mathbb{CP}^2$, classified by $c_2 \in \mathbb{Z}$ pull back to give a trivial bundle on $S^5$ (Tyler Lawson told me this, in discussion on [1])
   • There are of course more ASD instantons on $\mathbb{CP}^2$ with structure group $SO(3)$, classified by $w_2\in \{0,1\}$ and $p_1\in \mathbb{Z}$ (such that $p_1 \cong w_2 (mod 4)$), with the $SU(2)$ ones arising from the cases $w_2=0$, and subject to the requirement that $p_1 \leq -3$, otherwise the moduli space of ASD instantons is empty. This is where serious analytic theorems come into play.
     • Note that there is no difference at the level of Lie algebras, so local data for an $SO(3)$-connection is identical to the local data for an $SU(2)$-connection on a lifted bundle. So if we can get an $SO(3)$ instanton on $S^5$, we get an $SU(2)$ instanton (and the unique nontrivial $SU(2)$-bundle corresponds to a unique nontrivial $SO(3)$-bundle.
   • So if there is an $SO(3)$-bundle on $\mathbb{CP}^2$ with $w_2=1$ and $p_1 \leq -3$ that pulls back to the nontrivial bundle on $S^5$ we know that we have an $SO(3)$, hence $SU(2)$, ASD contact instanton, hence a Yang-Mills instanton.

   The current most concrete problem I’m trying to solve is at [5], where I have set up a rather classical geometric situation with concrete models for (skeleta of) classifying spaces in order to see if I can detect whether $S^5 \to \mathbb{CP}^2 \to BSO(3)$ is nontrivial from other easier-to-handle data, in particular an element of $[S^1,\mathbb{RP}^2] = \mathbb{Z}/2$.

   [1] Here’s a bit of a puzzle…

   [2] Looking at the Postnikov tower of BSO(3)

   [3] The fifth k-invariant of BSO(3)

   [4] …what happens to SO(3) bundles when pulled back along S^5 –> CP^2

   [5] Real and quaternionic projective spaces and Grassmannians, and a fibre sequence of classifying spaces

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 21st 2016
   • (edited Sep 21st 2016)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI've had to roll back a couple of my claims at [5], since they were completely wrong. But I feel the map $$ \mathbb{HP}^2 \to \widetilde{Gr}_3(\mathfrak{g}), $$ for some Lie algebra $\mathfrak{g}$, should be well known. Alternatively, I would be just as happy to know what the corresponding $\mathbb{RP}^2$ bundle $E$ on an appropriately large $\widetilde{Gr}_3(V)$ is, where $V$ has large enough dimension to be classifying for 4-manifolds. Because then I would pull $E$ back along a classifying map $\mathbb{CP}^2\to \widetilde{Gr}_3(V)$ and check that the obstruction to $S^5\to E$ descending to a section of $E\to \mathbb{CP}^2$ was precisely the homotopy class of the map $S^1\to \mathbb{RP}^2$ on fibres.

   I’ve had to roll back a couple of my claims at [5], since they were completely wrong. But I feel the map

   $$\mathbb{HP}^2 \to \widetilde{Gr}_3(\mathfrak{g}),$$

   for some Lie algebra $\mathfrak{g}$, should be well known.

   Alternatively, I would be just as happy to know what the corresponding $\mathbb{RP}^2$ bundle $E$ on an appropriately large $\widetilde{Gr}_3(V)$ is, where $V$ has large enough dimension to be classifying for 4-manifolds. Because then I would pull $E$ back along a classifying map $\mathbb{CP}^2\to \widetilde{Gr}_3(V)$ and check that the obstruction to $S^5\to E$ descending to a section of $E\to \mathbb{CP}^2$ was precisely the homotopy class of the map $S^1\to \mathbb{RP}^2$ on fibres.