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Laurent Charles

Publications and source records attributed to Laurent Charles.

At least 19 recordsLinked to original sources
arXiv2025

Quantum propagation for Berezin-Toeplitz operators

We describe the asymptotic behaviour of the quantum propagator generated by a Berezin-Toeplitz operator with real-valued principal symbol. We also give precise asymptotics for smoothed spectral projectors associated with the operator in the autonomous case; this leads us to introducting quantum states associated with immersed Lagrangian submanifolds. These descriptions involve geometric quantities of two origins, coming from lifts of the Hamiltonian flow to the prequantum bundle and the canonical bundle respectively. The latter are the main contribution of this article and are connected to the Maslov indices appearing in trace formulas, as will be explained in a forthcoming paper.

math.DG↗
arXiv2025

Semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces

On a closed hyperbolic surface, we investigate semiclassical defect measures associated with the magnetic Laplacian in the presence of a constant magnetic field. Depending on the energy level where the eigenfunctions concentrate, three distinct dynamical regimes emerge. In the low-energy regime, we show that any invariant measure of the magnetic flow in phase space can be obtained as a semiclassical measure. At the critical energy level, we establish Quantum Unique Ergodicity, together with a quantitative rate of convergence of eigenfunctions to the Liouville measure. In the high-energy regime, we prove a Shnirelman-type result: a density-one subsequence of eigenfunctions becomes equidistributed with respect to the Liouville measure.

math.AP↗
arXiv2025

Resolvents of Bochner Laplacians in the semiclassical limit

We introduce a new class of pseudodifferential operators, called Heisenberg semiclassical pseudodifferential operators, to study the space of sections of a power of a line bundle on a compact manifold, in the limit where the power is large. This class contains the Bochner Laplacian associated with a connection of the line bundle, and when the curvature is nondegenerate, its resolvent and some associated spectral projections, including generalized Bergman kernels.

math.DG↗
arXiv2023

Landau levels on a compact manifold

We consider a magnetic Laplacian on a compact manifold, with a constant non-degenerate magnetic field. In the large field limit, it is known that the eigenvalues are grouped in clusters, the corresponding sums of eigenspaces being called the Landau levels. The first level has been studied in-depth as a natural generalization of the Kaehler quantization. The current paper is devoted to the higher levels: we compute their dimensions as Riemann-Roch numbers, study the associated Toeplitz algebras and prove that each level is isomorphic with a quantization twisted by a convenient auxiliary bundle.

math.DG↗
arXiv2021

On the spectrum of non degenerate magnetic Laplacian

We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is non degenerate. Under a general condition, the Laplacian acting on high tensor powers of the bundle exhibits gaps and clusters of eigenvalues. We prove that for each cluster, the number of eigenvalues that it contains, is given by a Riemann-Roch number. We also give a pointwise description of the Schwartz kernel of the spectral projectors onto the eigenstates of each cluster, similar to the Bergman kernel asymptotics of positive line bundles. Another result is that gaps and clusters also appear in local Weyl laws.

math.DG↗
arXiv2020

Asymptotic representations of Hamiltonian diffeomorphisms and quantization

We show that for a special class of geometric quantizations with "small" quantum errors, the quantum classical correspondence gives rise to an asymptotic projective representation of the group of Hamiltonian diffeomorphisms. As an application, we get an obstruction to Hamiltonian actions of finitely presented groups.

math-ph↗
arXiv2019

Analytic Berezin-Toeplitz operators

We introduce new tools for analytic microlocal analysis on Kähler manifolds. As an application, we prove that the space of Berezin-Toeplitz operators with analytic contravariant symbol is an algebra. We also give a short proof of the Bergman kernel asymptotics up to an exponentially small error.

math.CV↗
arXiv2019

On a quasimorphism of Hamiltonian diffeomorphisms and quantization

In the setting of geometric quantization, we associate to any prequantum bundle automorphism a unitary map of the corresponding quantum space. These maps are controlled in the semiclassical limit by two invariants of symplectic topology: the Calabi morphism and a quasimorphism of the universal cover of the Hamiltonian diffeomorphism group introduced by Entov, Py, Shelukhin.

math.SG↗
arXiv2018

Quantum speed limit vs. classical displacement energy

We discuss a link between symplectic displacement energy, a fundamental notion of symplectic topology, and the quantum speed limit, a universal constraint on the speed of quantum-mechanical processes. The link is provided by the quantum-classical correspondence formalized within the framework of the Berezin-Toeplitz quantization.

math-ph↗
arXiv2017

Quantization of compact symplectic manifolds

We develop the theory of Berezin-Toeplitz operator on any compact symplectic prequantizable manifold from scratch. Our main inspiration is the Boutet de Monvel-Guillemin theory, that we simplify in several ways to obtain a concise exposition. A comparison with the spin-c Dirac quantization is also included.

math.DG↗
arXiv2016

On the Witten asymptotic conjecture for Seifert manifolds

We give a new proof of Witten asymptotic conjecture for Seifert manifolds with non vanishing Euler class and one exceptional fiber. Our method is based on semiclassical analysis on a two dimensional phase space torus. We prove that the Witten-Reshetikhin-Turaev invariant of a Seifert manifold is the scalar product of two Lagrangian states, and we estimate this scalar product in the large level limit. The leading order terms of the expansion are naturally given in terms of character varieties, the Chern-Simons invariants and some symplectic volumes. For the analytic part, we establish some singular stationary phase lemma for discrete oscillatory sums.

math.GT↗
arXiv2016

Subprincipal symbol for Toeplitz operators

We establish some subprincipal estimates for Berezin-Toeplitz operators on symplectic compact manifolds. From this, we construct a family of subprincipal symbol maps and we prove that these maps are the only ones satisfying some expected conditions.

math.DG↗
arXiv2015

Torsion and symplectic volume in Seifert manifolds

For any oriented Seifert manifold X and compact connected Lie group G with finite center, we relate the Reidemeister density of the moduli space of representations of the fundamental group of X into G to the Liouville measure of some moduli spaces of representations of surface groups into G.

math.GT↗
arXiv2011

Isospectrality for quantum toric integrable systems

We settle affirmatively the isospectral problem for quantum toric integrable systems: the semiclassical joint spectrum of such a system, given by a sequence of commuting Toeplitz operators on a sequence of Hilbert spaces, determines the classical integrable system given by the symplectic manifold and Poisson commuting functions, up to symplectomorphisms. We also give a full description of the semiclassical spectral theory of quantum toric integrable systems. This type of problem belongs to the realm of classical questions in spectral theory going back to pioneer works of Colin de Verdiere, Guillemin, Sternberg and others in the 1970s and 1980s.

math.SG↗
arXiv2011

Torus knot state asymptotics

The state of a knot is defined in the realm of Chern-Simons topological quantum field theory as a holomorphic section on the SU(2) character manifold of the peripheral torus. We compute the asymptotics of the torus knot states in terms of the Alexander polynomial, the Reidemeister torsion and the Chern-Simons invariant. We also prove that the microsupport of the torus knot state is included in the character manifold of the knot exterior. As a corollary we deduce the Witten asymptotics conjecture for the Dehn filling of the torus knots and asymptotic expansions for the colored Jones polynomials.

math.GT↗
arXiv2011

Knot state asymptotics I, AJ Conjecture and abelian representations

Consider the Chern-Simons topological quantum field theory with gauge group SU(2) and level k. Given a knot in the 3-sphere, this theory associates to the knot exterior an element in a vector space. We call this vector the knot state and study its asymptotic properties when the level is large. The latter vector space being isomorphic to the geometric quantization of the SU(2)-character variety of the peripheral torus, the knot state may be viewed as a section defined over this character variety. We first conjecture that the knot state concentrates in the large level limit to the character variety of the knot. This statement may be viewed as a real and smooth version of the AJ conjecture. Our second conjecture says that the knot state in the neighborhood of abelian representations is a Lagrangian state. Using microlocal techniques, we prove these conjectures for the figure eight and torus knots. The proof is based on q-difference relations for the colored Jones polynomial. We also provide a new proof for the asymptotics of the Witten-Reshetikhin-Turaev invariant of the lens spaces and a derivation of the Melvin-Morton-Rozansky theorem from the two conjectures.

math.GT↗
arXiv2011

Knot state asymptotics II, Witten conjecture and irreducible representations

This article pursues the study of the knot state asymptotics in the large level limit initiated in "Knot sate Asymptotics I". As a main result, we prove the Witten asymptotic expansion conjecture for the Dehn fillings of the figure eight knot. The state of a knot is defined in the realm of Chern-Simons topological quantum field theory as a holomorphic section on the SU(2)-character manifold of the peripheral torus. In the previous paper, we conjectured that the knot state concentrates on the character variety of the knot with a given asymptotic behavior on the neighborhood of the abelian representations. In the present paper we study the neighborhood of irreducible representations. We conjecture that the knot state is Lagrangian with a phase and a symbol given respectively by the Chern-Simons and Reidemeister torsion invariants. We show that under some mild assumptions, these conjectures imply the Witten conjecture on the asymptotic expansion of WRT invariants of the Dehn fillings of the knot. Using microlocal techniques, we show that the figure eight knot state satisfies our conjecture starting from q-differential relations verified by the colored Jones polynomials. The proof relies on a differential equation satisfied by the Reidemeister torsion along the branches of the character variety, a phenomenon which has not been observed previously as far as we know.

math.GT↗