A quick note on integrable systems on orbifolds

The aim of this note is to present some relatively simple remarks and observations about integrable Hamiltonian and non-Hamiltonian dynamical systems on orbifolds.

Motivation and definition

Orbifolds appear very naturally in many situations, in particular in the reduction theory of symmetric dynamical systems. So it is natural to develop a theory of integrable systems on orbifolds. But we have not seen any paper on this topic, except for the case of toric systems (toric orbifolds).

Our observation is that the theory of integrable systems admits a rather straightforward generalization to the case of orbifolds, and many theorems remain valid in this case.

First, we must define precisely what is an integrable system on an orbifold?

Consider first the general non-Hamiltonian case.

On a manifold on dimension n, an integrable system consists of p vector fields X1, …, Xp and q functions F1,…, Fq, which are independent and commute pairwise, and p+q=n.

An orbifold is locally the quotient of a manifold by the action of a finite group. On the orbifold, we still want to have vector field, so on the local model D^n/G (where G is a finite group acting linearly on a n-dimensional disk D^n) we will require the existence of G-invariant vector fields X1,…,X_p on D^n (so that they can project to the quotient D^n/G) .

A priori, the map F=(F_1,…,F_q) : D^n -> D^q needs not be G-invariant, just G-equivariant w.r.t. some linear action of G on D^q. But we can turn equivariant functions into invariant functions (by Schwarz lemma: on R^q there are q independent polynomial G-invariant functions if G acts on R^q linearly) , so without loss of generality, we can require that F is also G-invariant.

Thus, in the local model D/G, we have an integral system on D which is G-invariant, and so it projects to an integrable system on D/G.

The symplectic (or Poisson, etc.): the underlying symplectic (Poisson etc.) structure should be a G-invariant structure on D^n in the local model. The momentum map is invariant under G, then the Hamiltonian vector fields are also invariant under G, and one can take the quotient of everything w.r.t. G to get a projection to the orbifold.

Liouville theorem for orbifolds

Question: Orbifold version of regular Liouville theorem? Things are regular except for the action of G, then how can G act in this case?

Observation: If G fixes a point, then it fixes the whole orbit through that point. If the orbit is regular, then we get a regular orbit fixed by G. So the local model can be extended to a semi-local model, and G acts on the base space of orbits.  In the Hamiltonian case, this action must preserve the integral affine structure. What are elements of GL(n,Z) of finite order? Rotations by pi/2 or by pi or reflections, what else?

Singularities of the system

Observation: Orbifold blow-up of singularities (like what Crainic-Fernandes-Martinez did with Poisson manifolds of compact type), can turn singularities of compact group actions into orbifolds?!

Observation: Differential Galois theory should work the same on orbifolds.

Question: What are nondegenerate singularities of integrable systems on orbifolds?

Theorem: By a small integrable perturbation, can turn any “not too degenerate” integrable system into an integrable system such that all orbifold points are either regular or elliptic (no hyperbolic singularities at orbifold points)

Local normal forms?

Miranda-Zung has symmetry group for Eliasson nondegenerate normal forms -> orbifold group is (Z2)^k (for hyperbolic part) times cyclic for elliptic  and focus-focus part.

So if the model is Hamiltonian nondegenerate (before the action of G) then the choice of G is limited: only products Z2 for hyperbolic components, and cyclic for elliptic or focus components, similarly to the case of toric orbifolds (toric orbifolds have nondegenerate singularities in this sense)

If G is given then it’s a different story. The question the becomes: what are the most nondegenerate singularities which admit a given symmetry group G with a fixed point ?

Refs: Schwarz, Wassermann, my paper on corank 1 …

Example: the case D^2/Z_3 hyperbolic (snowflake): degenerate if not taking into the account the Z3 group, but among unstable singuarities with this symmetry group the “snowflake” is “the least degenerate” one. Still, by a small perturbation, the orbifold point becomes an elliptic point and there will be a nearby nondegenerate hyperbolic point. So this “snowflake” situation is unstable under small perturbation, not “robust/rigid”.

Subproblem: Rn-actions on n-dimensional orbifolds

Local symmetry groups = reflections

Orbifold structure = manifold corner.

Complexification and Galoisian theory on orbifolds

 

Various examples

  • Serfert fibrations. Base space = orbifoldon which we may have an integrable system
  • Toric orbifolds -> simple rational polytopes

OK, so finally what are the results of this small note?

  1. Definition of integrable systems of orbifolds
  2. Examples & restrictions of orbifold groups in each situation
  3. Theorem: Liouville theorem and natural associated torus actions for orbifolds
  4. Theorem: Under some conditions, perturbation of the system such that all singular orbifold points become elliptic (generic local stability at orbifold points)
  5. Extension of differential Galois theory to orbifolds

 

 

 

 

 

 

 

 

 

 

 

 

 

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