David Sauzin

Bio: David Sauzin is an academic researcher from Centre national de la recherche scientifique. The author has contributed to research in topics: Series (mathematics) & Analytic continuation. The author has an hindex of 16, co-authored 62 publications receiving 866 citations. Previous affiliations of David Sauzin include Scuola Normale Superiore di Pisa & Institut de mécanique céleste et de calcul des éphémérides.

Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems

Abstract: . – We prove a theorem about the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems and construct in that context a system with an unstable orbit whose mean speed of drift allows us to check the optimality of the stability theorem.¶Our stability result generalizes those by Lochak-Neishtadt and Poschel, which give precise exponents of stability in the Nekhoroshev Theorem for the quasi-convex case, to the situation in which the Hamiltonian function is only assumed to belong to some Gevrey class instead of being real-analytic. For n degrees of freedom and Gevrey-α Hamiltonians, α ≥ 1, we prove that one can choose a = 1/2nα as an exponent for the time of stability and b = 1/2n as an exponent for the radius of confinement of the action variables, with refinements for the orbits which start close to a resonant surface (we thus recover the result for the real-analytic case by setting α = 1).¶On the other hand, for α > 1, the existence of compact-supported Gevrey functions allows us to exhibit for each n ≥ 3 a sequence of Hamiltonian systems with wandering points, whose limit is a quasi-convex integrable system, and where the speed of drift is characterized by the exponent 1/2(n−2)α. This exponent is optimal for the kind of wandering points we consider, inasmuch as the initial condition is located close to a doubly-resonant surface and the stability result holds with precisely that exponent for such an initial condition. We also discuss the relationship between our example of instability, which relies on a specific construction of a perturbation of a discrete integrable system, and Arnold’s mechanism of instability, whose main features (partially hyperbolic tori, heteroclinic connections) are indeed present in our system.

Borel summation and splitting of separatrices for the Hénon map

Abstract: We study two complex invariant manifolds associated with the parabolic fixed point of the area-preserving Henon map. A single formal power series corresponds to both of them. The Borel transform of the formal series defines an analytic germ. We explore the Riemann surface and singularities of its analytic continuation. In particular we give a complete description of the first singularity and prove that a constant, which describes the splitting of the invariant manifold, does not vanish. An interpretation in terms of Resurgence theory is also given.

Introduction to 1-summability and resurgence

Abstract: This text is about the mathematical use of certain divergent power series. The rst part is an introduction to 1-summability. The denitions rely on the formal Borel transform and the Laplace transform along an arbitrary direction of the complex plane. Given an arc of directions, if a power series is 1-summable in that arc, then one can attach to it a Borel-Laplace sum, i.e. a holomorphic function dened in a large enough sector and asymptotic to that power series in Gevrey sense. The second part is an introduction to Ecalle’s resurgence theory. A power series is said to be resurgent when its Borel transform is convergent and has good analytic continuation properties: there may be singularities but they must be isolated. The analysis of these singularities, through the so-called alien calculus, allows one to compare the various BorelLaplace sums attached to the same resurgent 1-summable series. In the context of analytic dierence-or-dier ential equations, this sheds light on the Stokes phenomenon. A few elementary or classical examples are given a thorough treatment (the Euler series, the Stirling series, a less known example by Poincar e). Special attention is devoted to non-linear operations: 1-summable series as well as resurgent series are shown to form algebras which are stable by composition. As an application, the resurgent approach to the classication of tangent-to-identity germs of holomorphic dieomorphisms in the simplest case is included. An example of a class of non-linear dierential equations giving rise to resurgent solutions is also presented. The exposition is as self-contained as can be, requiring only some familiarity with holomorphic functions of one complex variable.

A new method for measuring the splitting of invariant manifolds

Abstract: We study the so-called Generalized Arnol'd Model (a weakly hyperbolic near-integrable Hamiltonian system), with d+1 degrees of freedom (d⩾2), in the case where the perturbative term does not affect a fixed invariant d-dimensional torus. This torus is thus independent of the two perturbation parameters which are denoted e (e>0) and μ. We describe its stable and unstable manifolds by solutions of the Hamilton–Jacobi equation for which we obtain a large enough domain of analyticity. The splitting of the manifolds is measured by the partial derivatives of the difference ΔS of the solutions, for which we obtain upper bounds which are exponentially small with respect to e. A crucial tool of the method is a characteristic vector field, which is defined on a part of the configuration space, which acts by zero on the function ΔS and which has constant coefficients in well-chosen coordinates. It is in the case where |μ| is bounded by some positive power of e that the most precise results are obtained. In a particular case with three degrees of freedom, the method leads also to lower bounds for the splitting.

Resurgent functions and splitting problems

Abstract: The present text is an introduction to Ecalle's theory of resurgent functions and alien calculus, in connection with problems of exponentially small separatrix splitting. An outline of the resurgent treatment of Abel's equation for resonant dynamics in one complex variable is included. The emphasis is on examples of nonlinear difference equations, as a simple and natural way of introducing the concepts.

The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects

Abstract: We study the dynamics of four dimensional gauge theories with adjoint fermions for all gauge groups, both in perturbation theory and non-perturbatively, by using circle compactication with periodic boundary conditions for the fermions. There are new gauge phenomena. We show that, to all orders in perturbation theory, many gauge groups are Higgsed by the gauge holonomy around the circle to a product of both abelian and nonabelian gauge group factors. Non-perturbatively there are monopole-instantons with fermion zero modes and two types of monopole-anti-monopole molecules, called bions. One type are magnetic bions which carry net magnetic charge and induce a mass gap for gauge uctuations. Another type are neutral bions which are magnetically neutral, and their un- derstanding requires a generalization of multi-instanton techniques in quantum mechanics | which we refer to as the Bogomolny-Zinn-Justin (BZJ) prescription | to compactied eld theory. The BZJ prescription applied to bion-anti-bion topological molecules predicts a singularity on the positive real axis of the Borel plane (i.e., a divergence from summing large orders in peturbation theory) which is of order N times closer to the origin than the leading 4-d BPST instanton-anti-instanton singularity, where N is the rank of the gauge group. The position of the bion-anti-bion singularity is thus qualitatively similar to that of the 4-d IR renormalon singularity, and we conjecture that they are continuously re- lated as the compactication radius is changed. By making use of transseries and Ecalle's resurgence theory we argue that a non-perturbative continuum denition of a class of eld theories which admit semi-classical expansions may be possible.

Lectures on non‐perturbative effects in large N gauge theories, matrix models and strings

Abstract: In these lectures I present a review of non-perturbative instanton effects in quantum theories, with a focus on large N gauge theories and matrix models. I first consider the structure of these effects in the case of ordinary differential equations, which provide a model for more complicated theories, and I introduce in a pedagogical way some technology from resurgent analysis, like trans-series and the resurgent version of the Stokes phenomenon. After reviewing instanton effects in quantum mechanics and quantum field theory, I address general aspects of large N instantons, and then present a detailed review of non-perturbative effects in matrix models. Finally, I consider two applications of these techniques in string theory.

Resurgence and trans-series in Quantum Field Theory: the $\mathbb{C}{{\mathbb{P}}^{N-1 }}$ model

Abstract: This work is a step towards a non-perturbative continuum definition of quantum field theory (QFT), beginning with asymptotically free two dimensional non-linear sigma-models, using recent ideas from mathematics and QFT. The ideas from mathematics are resurgence theory, the trans-series framework, and Borel-Ecalle resummation. The ideas from QFT use continuity on ${{\mathbb{R}}^1}\times \mathbb{S}_L^1$ , i.e., the absence of any phase transition as N → ∞ or rapid-crossovers for finite-N, and the small-L weak coupling limit to render the semi-classical sector well-defined and calculable. We classify semi-classical configurations with actions 1/N (kink-instantons), 2/N (bions and bi-kinks), in units where the 2d instanton action is normalized to one. Perturbation theory possesses the IR-renormalon ambiguity that arises due to non-Borel summability of the large-orders perturbation series (of Gevrey-1 type), for which a microscopic cancellation mechanism was unknown. This divergence must be present because the corresponding expansion is on a singular Stokes ray in the complexified coupling constant plane, and the sum exhibits the Stokes phenomenon crossing the ray. We show that there is also a non-perturbative ambiguity inherent to certain neutral topological molecules (neutral bions and bion-anti-bions) in the semiclassical expansion. We find a set of “confluence equations” that encode the exact cancellation of the two different type of ambiguities. There exists a resurgent behavior in the semi-classical trans-series analysis of the QFT, whereby subleading orders of exponential terms mix in a systematic way, canceling all ambiguities. We show that a new notion of “graded resurgence triangle” is necessary to capture the path integral approach to resurgence, and that graded resurgence underlies a potentially rigorous definition of general QFTs. The mass gap and the Θ angle dependence of vacuum energy are calculated from first principles, and are in accord with large-N and lattice results.