Fractional Monodromy Research Articles

Qualitative dynamics and inflationary attractors in loop cosmology

Qualitative dynamics of three different loop quantizations of spatially flat isotropic and homogeneous models is studied using effective spacetime description of the underlying quantum geometry. These include the standard loop quantum cosmology (LQC), its recently revived modification (referred to as mLQC-I), and another related modification of LQC (mLQC-II) whose dynamics is studied in detail for the first time. Various features of LQC, including quantum bounce and pre-inflationary dynamics, are found to be shared with the mLQC-I and mLQC-II models. We study universal properties of dynamics for chaotic inflation, fractional monodromy inflation, Starobinsky potential, non-minimal Higgs inflation, and an exponential potential. We find various critical points and study their stability, which reveal various qualitative similarities in the post-bounce phase for all these models. The pre-bounce qualitative dynamics of LQC and mLQC-II turns out to be very similar, but is strikingly different from that of mLQC-I. In the dynamical analysis, some of the fixed points turn out to be degenerate for which center manifold theory is used. For all these potentials, non-perturbative quantum gravitational effects always result in a non-singular inflationary scenario with a phase of super-inflation succeeded by the conventional inflation. We show the existence of inflationary attractors, and obtain scaling solutions in the case of the exponential potential. Since all of the models agree with general relativity at late times, our results are also of use in classical theory where qualitative dynamics of some of the potentials has not been studied earlier.

Parallel Transport Along Seifert Manifolds and Fractional Monodromy

The notion of fractional monodromy was introduced by Nekhoroshev, Sadovskií and Zhilinskií as a generalization of standard (‘integer’) monodromy in the sense of Duistermaat from torus bundles to singular torus fibrations. In the present paper we prove a general result that allows one to compute fractional monodromy in various integrable Hamiltonian systems. In particular, we show that the non-triviality of fractional monodromy in 2 degrees of freedom systems with a Hamiltonian circle action is related only to the fixed points of the circle action. Our approach is based on the study of a specific notion of parallel transport along Seifert manifolds.

Uncovering Fractional Monodromy

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 n1:(−n2) resonant Hamiltonian systems with n1, n2 coprime natural numbers. We consider, in particular, systems that for n1, n2 > 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 n1, n2. Fractional monodromy is then recovered by pushing the results obtained in the branched covering back to the original system.

A simple proof of the geometric fractional monodromy theorem

A simple proof of the “geometric fractional monodromy theorem” (Broer-Efstathiou-Lukina 2010) is presented. The fractional monodromy of a Liouville integrable Hamiltonian system over a loop γ ⊂ ℝ2 is a generalization of the classic monodromy to the case when the Liouville foliation has singularities over γ. The “geometric fractional monodromy theorem” finds, up to an integral parameter, the fractional monodromy of systems similar to the 1: (−2) resonance system. A handy equivalent definition of fractional monodromy is presented in terms of homology groups for our proof.

Generating functions for effective hamiltonians. Symmetry, topology, combinatorics

Recent developments associated with old technique of generating functions and invariant theory which I have started to apply to molecular problems due to my collaboration with Yu.F. Smirnov about 25 years ago are discussed. Three aspects are presented: the construction of diagonal in polyad quantum number effective resonant vibrational Hamiltonians using the symmetrized Hadamard product; the decomposition of the number of state generating function into regular and oscillatory contributions and its relation with Todd polynomials and topological characterization of energy bands; qualitative aspects of resonant oscillators and fractional monodromy as one of new generalizations of Hamiltonian monodromy.

Dynamics near the [formula omitted] resonance

We study the dynamics near the truncated p : ± q resonant Hamiltonian equilibrium for p , q coprime. The critical values of the momentum map of the Liouville integrable system are found. The three basic objects reduced period, rotation number, and non-trivial action for the leading order dynamics are computed in terms of complete hyperelliptic integrals. A relation between the three functions that can be interpreted as a decomposition of the rotation number into geometric and dynamic phase is found. Using this relation we show that the p : − q resonance has fractional monodromy. Finally we prove that near the origin of the 1 : − q resonance the twist vanishes.

Fractional Bidromy in the Vibrational Spectrum of HOCl

We introduce the notion of fractional bidromy which is the combination of fractional monodromy and bidromy, two recent generalizations of Hamiltonian monodromy. We consider the vibrational spectrum of the HOCl molecule which is used as an illustrative example to show the presence of nontrivial fractional bidromy. To our knowledge, this is the first example of a molecular system where such a generalized monodromy is exhibited.

A geometric fractional monodromy theorem

We prove the existence of fractional monodromy for two degree of freedom integrable Hamiltonian systems with one-parameter families of curled tori under certain general conditions. We describe the action coordinates of such systems near curled tori and we show how to compute fractional monodromy using the notion of the rotation number.

Fuzzy Fractional Monodromy and the Section–Hyperboloid

A theorem about the matrix of fractional monodromy will be formulated. The monodromy corresponds to going around a fiber with a singular point of oscillator type with arbitrary resonance. The reason of fractional monodromy and fuzziness of such a monodromy is explained. Some ideas for the proof of the theorem are given. A few remarks about the semi-global structure of singular lagrangian fibration are made.

Fractional monodromy: parallel transport of homology cycles

A 2 n-dimensional completely integrable system gives rise to a singular fibration whose generic fiber is the n-torus T n . In the classical setting, it is possible to define a parallel transport of elements of the fundamental group of a fiber along a path, when the path describes a loop around a singular fiber, it defines an automorphism of π 1 ( T n ) called monodromy transformation [J.J. Duistermaat, On global action-angle coordinates, Communications on Pure and Applied Mathematics 33 (6) (1980) 687–706]. Some systems give rise to a non-classical setting, in which the path can wind around a singular fiber only by crossing a codimension 1 submanifold of special singular fibers (a wall), in this case a non-classical parallel transport can be defined on a subgroup of the fundamental group. This gives rise to what is known as monodromy with fractional coefficients [N. Nekhoroshev, D. Sadovskiì, B. Zhilinskiì, Fractional monodromy of resonant classical and quantum oscillators, Comptes Rendus Mathematique 335 (11) (2002) 985–988]. In this article, we give a precise meaning to the non-classical parallel transport. In particular we show that it is a homologic process and not a homotopic one. We justify this statement by describing the type of singular fibers that generate a wall that can be crossed, by describing the parallel transport in a semi-local neighbourhood of the wall of singularities, and by producing a family of 4-dimensional examples.

Infinitesimally stable and unstable singularities of 2-degrees of freedom completely integrable systems

In this article we give a list of 10 rank zero and 6 rank one singularities of 2-degrees of freedom completely integrable systems. Among such singularities, 14 are the singularities that satisfy a non-vanishing condition on the quadratic part, the remaining 2 are rank 1 singularities that play a role in the geometry of completely integrable systems with fractional monodromy. We describe which of them are stable and which are unstable under infinitesimal completely integrable deformations of the system.

The 1:±2 resonance

On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Non-linear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio 1/2 and its unfolding. In particular we show that for the indefinite case (1:-2) the frequency ratio map in a neighbourhood of the origin has a critical point, i.e. the twist condition is violated for one torus on every energy surface near the energy of the equilibrium. In contrast, we show that the frequency map itself is non-degenerate (i.e. the Kolmogorov non-degeneracy condition holds) for every torus in a neighbourhood of the equilibrium point. As a byproduct we are able to obtain another proof of fractional monodromy in the 1:-2 resonance.

Fractional monodromy in systems with coupled angular momenta

We present a one-parameter family of systems with fractional monodromy, which arises from a 1:2 diagonal action of a dynamical symmetry SO(2). In a regime of adiabatic separation of slow and fast motions, we relate the presence of fractional monodromy to a redistribution of states both in the quantum and in the semi-quantum spectra.

Fractional monodromy in the case of arbitrary resonances

The existence of fractional monodromy is proved for the compact Lagrangian fibration on a symplectic 4-manifold that corresponds to two oscillators with arbitrary non-trivial resonant frequencies. Here one means by the monodromy corresponding to a loop in the total space of the fibration the transformation of the fundamental group of a regular fibre, which is diffeomorphic to the 2-torus. In the example under consideration the fibration is defined by a pair of functions in involution, one of which is the Hamiltonian of the system of two oscillators with frequency ratio , where , are arbitrary coprime positive integers distinct from the trivial pair . This is a generalization of the result on the existence of fractional monodromy in the case , published before. Bibliography: 39 titles.

Fractional Hamiltonian Monodromy

We introduce fractional monodromy in order to characterize certain non-isolated critical values of the energy–momentum map of integrable Hamiltonian dynamical systems represented by nonlinear resonant two-dimensional oscillators. We give the formal mathematical definition of fractional monodromy, which is a generalization of the definition of monodromy used by other authors before. We prove that the 1:( − 2) resonant oscillator system has monodromy matrix with half-integer coefficients and discuss manifestations of this monodromy in quantum systems.

Fractional monodromy in the [formula omitted] resonance

We give an analytic proof of the fractional monodromy theorem for the 1 : − 2 oscillator system with S 1 symmetry formulated by N.N. Nekhoroshev, D.A. Sadovskií, and B.I. Zhilinskií in C. R. Acad. Sci. Paris, Ser. I 335 (2002) 985–988. Our proof is based on an analytic description of the Hamiltonian flow on the fibers of the integral map of this system.

Fractional monodromy of resonant classical and quantum oscillators

We introduce fractional monodromy for a class of integrable fibrations which naturally arise for classical nonlinear oscillator systems with resonance. We show that the same fractional monodromy characterizes the lattice of quantum states in the joint spectrum of the corresponding quantum systems. Results are presented on the example of a two-dimensional oscillator with resonance 1:(−1) and 1:(−2). To cite this article: N.N. Nekhoroshev et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 985–988.