|
SIGMA 10 (2014), 112, 6 pages arXiv:1407.8291
https://doi.org/10.3842/SIGMA.2014.112
Configurations of Points and the Symplectic Berry-Robbins Problem
Joseph Malkoun
Department of Mathematics and Statistics, Notre Dame University-Louaize, Lebanon
Received August 23, 2014, in final form December 17, 2014; Published online December 19, 2014
Abstract
We present a new problem on configurations of points, which is a new version of a similar problem by Atiyah and Sutcliffe, except it is related to the Lie group $\operatorname{Sp}(n)$, instead of the Lie group $\operatorname{U}(n)$. Denote by $\mathfrak{h}$ a Cartan algebra of $\operatorname{Sp}(n)$, and $\Delta$ the union of the zero sets of the roots of $\operatorname{Sp}(n)$ tensored with $\mathbb{R}^3$, each being a map from $\mathfrak{h} \otimes \mathbb{R}^3 \to \mathbb{R}^3$. We wish to construct a map $(\mathfrak{h} \otimes \mathbb{R}^3) \backslash \Delta \to \operatorname{Sp}(n)/T^n$ which is equivariant under the action of the Weyl group $W_n$ of $\operatorname{Sp}(n)$ (the symplectic Berry-Robbins problem). Here, the target space is the flag manifold of $\operatorname{Sp}(n)$, and $T^n$ is the diagonal $n$-torus. The existence of such a map was proved by Atiyah and Bielawski in a more general context. We present an explicit smooth candidate for such an equivariant map, which would be a genuine map provided a certain linear independence conjecture holds. We prove the linear independence conjecture for $n=2$.
Key words:
configurations of points; symplectic; Berry-Robbins problem; equivariant map; Atiyah-Sutcliffe problem.
pdf (251 kb)
tex (10 kb)
References
-
Atiyah M., The geometry of classical particles, in Surveys in Differential Geometry, Surv. Differ. Geom., Vol. 7, Int. Press, Somerville, MA, 2000, 1-15.
-
Atiyah M., Configurations of points, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 (2001), 1375-1387.
-
Atiyah M., Bielawski R., Nahm's equations, configuration spaces and flag manifolds, Bull. Braz. Math. Soc. (N.S.) 33 (2002), 157-176, math.RT/0110112.
-
Atiyah M., Sutcliffe P., The geometry of point particles, Proc. Roy. Soc. London Ser. A 458 (2002), 1089-1115, hep-th/0105179.
-
Berry M.V., Robbins J.M., Indistinguishability for quantum particles: spin, statistics and the geometric phase, Proc. Roy. Soc. London Ser. A 453 (1997), 1771-1790.
|
|