Finite orbits of the braid group action on sets of reflections

Special Symplectic Subgroup over Integers Arising as a Factor of The Braid Group

There is a natural action of the braid group on the symmetric matrices with units on the diagonal, appearing in various fields as Singularity Theory, Frobenius Manifolds or Isomonodromic deformations of certain classes of linear differential systems. The problem of finding the finite orbits of this action for the case of non-degenerate three-dimensional matrices was solved and it was conjectured that the finite orbits on any non-degenerate matrices correspond to finite Coxeter groups and conjugacy classes of quasicoxeter elements in them. In the present work it is considered another extremal c…

Finite orbits of the pure braid group on the monodromy of the $2$-variable Garnier system

In this paper we show that the $SL_{2}(\mathbb C)$ character variety of the Riemann sphere $Σ_5$ with five boundary components is a $5$-parameter family of affine varieties of dimension $4$. We endow this family of affine varieties with an action of the braid group and classify exceptional finite orbits. This action represents the nonlinear monodromy of the $2$ variable Garnier system and finite orbits correspond to algebraic solutions.

Braids with the symmetries of Platonic polyhedra in the Coulomb (N+1)-body problem

We take into account the Coulomb (N + 1)-body problem with N = 12, 24, 60. One of the particles has positive charge Q > 0, and the remaining N have all the same negative charge q < 0. These particles move under the Coulomb force, and the positive charge is assumed to be at rest at the center of mass. Imposing a symmetry constraint, given by the symmetry group of the Platonic polyhedra, we were able to compute periodic orbits, using a shooting method and continuation with respect to the value Q of the positive charge. In the setting of the classical N -body problem, the existence of such orbits…

牛顿系统中周期三体轨道的形成

The formation of periodic three-body orbits for Newtonian systems

研究通过模拟自引力系统，发现辫状结构可通过双体或三体相遇生成，且在浅势场中可能作为瞬态存在。

Borchers' Commutation Relations for Sectors with Braid Group Statistics in Low Dimensions

Borchers has shown that in a translation covariant vacuum representation of a theory of local observables with positive energy the following holds: The (Tomita) modular objects associated with the observable algebra of a fixed wedge region give rise to a representation of the subgroup of the Poincare group generated by the boosts and the reflection associated to the wedge, and the translations. We prove here that Borchers' theorem also holds in charged sectors with (possibly non-Abelian) braid group statistics in low space-time dimensions. Our result is a crucial step towards the Bisognano-Wic…