Uncovering Fractional Monodromy

Abstract

Abstract:The uncovering of the role of monodromy in integrable Hamiltonian fibrations has been one of the major advances in the study of integrable Hamiltonian systems in the past few decades: on one hand monodromy turned out to be the most fundamental obstruction to the existence of global action-angle coordinates while, on the other hand, it provided the correct classical analogue for the interpretation of the structure of quantum joint spectra. Fractional monodromy is a generalization of the concept of monodromy: instead of restricting our attention to the toric part of the fibration we extend our scope to also consider singular fibres. In this paper we analyze fractional monodromy for n 1 :(−n 2 ) resonant Hamiltonian systems with n 1 , n 2 coprime natural numbers. We consider, in particular, systems that for n 1 , n 2 > 1 contain one-parameter families of singular fibres which are 'curled tori'. We simplify the geometry of the fibration by passing to an appropriate branched covering. In the branched covering the curled tori and their neighborhood become untwisted thus simplifying the geometry of the fibration: we essentially obtain the same type of generalized monodromy independently of n 1 , n 2 . Fractional monodromy is then recovered by pushing the results obtained in the branched covering back to the original system.

Parallel Transport Along Seifert Manifolds and Fractional Monodromy

“…Since the pioneering work [28], various proofs of Theorem 1.8 appeared; see [8,16,31,32] and [15]. Our proof, which is based on the singularities of the circle action, shows that…”

“…The definition of fractional monodromy (in the sense of [28] and [15]) reads as follows. Consider a singular Lagrangian fibration F : M → R over a n-dimensional manifold R, given by a proper integral map F .…”

“…Even though standard monodromy is related to singularities of the torus fibration F , it is an invariant of the regular, non-singular part F : F −1 (R) → R. An invariant that generalizes standard monodromy to singular torus fibrations is called fractional monodromy [28]. We note that fractional monodromy is not a complete invariant of such fibrations -it contains less information than the marked molecule in Fomenko-Zieschang theory [6,7,19] -but it is important for applications and appears, for instance, in the so-called m:(−n) resonant systems [15,27,28,30,31]; see Section 4.1 for details.…”

“…Cutting the manifold F −1 (γ)/Z 2 along any fiber F −1 (ξ 0 )/Z 2 , ξ 0 ∈ γ ∩ R, we get a manifold X with the boundary ∂X = X 0 X 1 consisting of the two tori X i . Following [15], we define the parallel transport using the connecting homomorphism of the long exact sequence of the pair (X, ∂X).…”

Bifurcations and monodromy of the axially symmetric 1:1:−2 resonance

“…The fundamental group π 1 (R ) is trivial and thus its image under the monodromy mapping is the identity. However, monodromy is also meaningful [12] for paths γ that do not lie completely in R or R , but pass from one connected component to the other through F • e . Then EM −1 (γ) is the disjoint union of a T 3 -bundle for which we can define monodromy and another manifold that can be ignored.…”