Let $C_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in Euclidean $3$-space and let $U(n)/T^n$ denote the flag manifold associated to the unitary group $U(n)$, i.e. the homogeneous space obtained by quotienting $U(n)$ by the action of the $n$-torus $T^n$, where $T^n$ acts on $U(n)$ by multiplication from the right.
The group $SU(2)$ acts on $\mathbb{R}^3$ via its natural $2$-to-$1$ homomorphism $SU(2) \to SO(3)$ and this induces an action of $SU(2)$ on $C_n(\mathbb{R}^3)$. Fix a regular homomorphism $\rho: SU(2) \to SU(n) \subset U(n)$. We then have an action of $SU(2)$ on $U(n)$ by acting by mutliplication from the left via $\rho$ and this descends to an action of $SU(2)$ on $U(n)/T^n$, since the action of $SU(2)$ and of $T^n$ on $U(n)$ commute.
Let $S_n$ denote the symmetric group on $n$ elements. It naturally acts on $C_n(\mathbb{R}^3)$ by permuting the points comprising each configuration $\mathbf{x} \in C_n(\mathbb{R}^3)$. Moreover, $S_n$ acts on $U(n)$ by permuting the columns of every $g \in U(n)$ and this descends to an action of $S_n$ on $U(n)/T^n$, as can be checked.
For the sake of this post, a smooth map $f: C_n(\mathbb{R}^3) \to U(n)/T^n$ is said to be a solution of the Berry-Robbins problem (BR problem, for short), if it is both $SU(2)$ and $S_n$ equivariant.
It is known, from the work of Atiyah and Bielawski, that there is at least one solution of the Berry-Robbins problem.
Let $H$ be the subgroup of $U(n)$ consisting of all unitary matrices $h \in U(n)$ such that $h \rho(k) = \rho(k) h$ for any $k \in SU(2)$. I think that $H$ is nothing but the center $Z$ of $U(n)$, which is isomorphic to $U(1)$.
Let $\mathcal{G}$ denote the group of all smooth maps $g: C_n(\mathbb{R}^3) \to H$ which are invariant under the action of $S_n$ on the domain $C_n(\mathbb{R}^3)$.
Given any solution $f$ of the BR problem, the pointwise product $g.f$ is another solution of the BR problem, which we think of as being "gauge"-equivalent to $f$.
Question 1: how many "gauge" equivalent solutions of the BR problem are there? Question 2: can one perhaps use the moduli space of solutions of the BR problem, up to equivalence, in order to construct interesting invariants of configurations of points in Euclidean $3$-space?
Remark: is the moduli space a single point? I wonder if some kind of uniqueness result was proved in the article by Atiyah and Bielawski. I should probably have a look at that article again.
Edit 1 (containing some interesting but tough conjectures): I am not sure if the following statements are true, but if they are, then they would have some interesting consequences.
Conjecture 1: there is only a single solution of the Berry-Robbins problem, as formulated above, up to "gauge" equivalence (in the sense I have explained above).
Conjecture 2 (probably related to conjecture 1): if we assume that our configuration space consists precisely of configurations of $n$ distinct points whose center of mass is the origin in $\mathbb{R}^3$ and if we replace $U(n)/T^n$ with $SU(n)/T^n$, then the modified BR problem has a unique solution up to "gauge" equivalence.
If conjecture 2 is true and assuming the Atiyah problem on configurations of points has a positive solution (i.e. the linear independence conjecture holds), it would then follow that the Atiyah construction of a candidate map in his problem on configurations of points (which he has verified numerically up to some somewhat large $n$ with the impressive scientific programming skills of P. Sutcliffe) is "gauge" equivalent to the solution contained in the article by Atiyah and Bielawski.
What this would mean, is that for the corresponding boundary conditions of Nahm's equations (a pole of type $\rho$ at $t = 0$ and $(T_1, T_2, T_3)$ approaches a regular commuting triple as $t \to \infty$; see the article by Atiyah and Bielawski for more details), we would know explicitly the output of the Nahm's equations/flow, for it would be given by the elementary construction of Sir Michael in his problem of configurations of points.
I have no idea where to start to investigate these conjectures, but I thought I would put them out there.
Note that Sir Michael had already asked whether these 2 works (the problem on configurations of points and his work with Bielawski) were related (see his Edinburgh lectures on Geometry and Physics, which are on arxiv. I am making a bold conjectural statement that they are essentially the same...
