Searcharxiv⌕ Search

arXiv subjects

David Sauzin

Publications and source records attributed to David Sauzin.

At least 19 recordsLinked to original sources
arXiv2025

A proof of Witten's asymptotic expansion conjecture for WRT invariants of Seifert fibered homology spheres

Let $X$ be a general Seifert fibered integral homology $3$-sphere with $r\ge3$ exceptional fibers. For every root of unity $\zeta\not=1$, we show that the SU(2) WRT invariant of $X$ evaluated at $\zeta$ is (up to an elementary factor) the non-tangential limit at $\zeta$ of the GPPV invariant of $X$, thereby generalizing a result from [Andersen-Mistegard 2022]. Based on this result, we apply the quantum modularity results developed in [Han-Li-Sauzin-Sun 2023] to the GPPV invariant of $X$ to prove Witten's asymptotic expansion conjecture [Witten 1989] for the WRT invariant of $X$. We also prove that the GPPV invariant of $X$ induces a higher depth strong quantum modular form. Moreover, when suitably normalized, the GPPV invariant provides an ``analytic incarnation'' of the Habiro invariant.

math.CV↗
arXiv2025

Explicit linearization of multi-dimensional germs and vector fields through Ecalle's tree expansions

We provide explicit formulas of non-recursive type for the linearizing transformations of a non-resonant analytic germ of diffeomorphism at a fixed point or a non-resonant analytic germ of vector field at a singular point, in any complex dimension. The formal expressions we obtain rely on a part of Ecalle's tree-based combinatorics called ``armould calculus" and they have the same shape for dynamical systems with discrete or continuous time. They allow us to recover in a straightforward manner, under Bruno's arithmetical condition, the best known estimates for the domains of convergence of the analytic linearizing changes of variables in terms of the value of the Bruno series, including a new precise dependence with respect to the dimension of the problem.

math.DS↗
arXiv2025

Geometric normalization

For a local analytic diffeomorphism of the plane with an irrational elliptic fixed point at 0, we introduce the notion of ``geometric normalization'', which includes the classical formal normalizations as a special case: it is a formal conjugacy to a formal diffeomorphism which preserves the foliation by circles centered at 0. We show that geometric normalizations, despite of non-uniqueness, correspond in a natural way to a unique formal invariant foliation. We show, in various contexts, generic results of divergence for the geometric normalizations, which amount to the generic non-existence of any analytic invariant foliation.

math.DS↗
arXiv2025

Resurgence in the Universal Structures in B-model Topological String Theory

We propose a systematic analysis of Alim-Yau-Zhou's double scaling limit and Couso-Santamar\'{i}a's large radius limit for the perturbative free energies in B-model topological string theory based on \'Ecalle's Resurgence Theory. Taking advantage of the known resurgent properties of the formal solutions to the Airy equation and of the stability of resurgent series under exponential/logarithm and nonlinear changes of variable, we show how to rigorously derive the non-perturbative information from the perturbative one by means of alien calculus in this context, spelling out the notions of formal integral and Bridge Equation, typical of the resurgent approach to ordinary differential equations. We also discuss the Borel-Laplace summation of the obtained resurgent transseries, including a study of real analyticity based on the connection formulas stemming from the resummation of the Bridge Equation.

math-ph↗
arXiv2023

Resurgence and Mould Calculus

Resurgence Theory and Mould Calculus were invented by J. Ecalle around 1980 in the context of analytic dynamical systems and are increasingly more used in the mathematical physics community, especially since the 2010s. We review the mathematical formalism and touch on the applications. This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics.

math-ph↗
arXiv2022

Resurgence and Partial Theta Series

We consider partial theta series associated with periodic sequences of coefficients, of the form $Θ(τ) := \sum_{n>0} n^νf(n) e^{iπn^2τ/M}$, with $ν$ non-negative integer and an $M$-periodic function $f : \mathbb{Z} \rightarrow \mathbb{C}$. Such a function is analytic in the half-plane $\{Im(τ)>0\}$ and as $τ$ tends non-tangentially to any $α\in\mathbb{Q}$, a formal power series appears in the asymptotic behaviour of $Θ(τ)$, depending on the parity of $ν$ and $f$. We discuss the summability and resurgence properties of these series by means of explicit formulas for their formal Borel transforms, and the consequences for the modularity properties of $Θ$, or its ``quantum modularity'' properties in the sense of Zagier's recent theory. The Discrete Fourier Transform of $f$ plays an unexpected role and leads to a number-theoretic analogue of Écalle's ``Bridge Equations''. The motto is: (quantum) modularity = Stokes phenomenon + Discrete Fourier Transform.

math.CV↗
arXiv2021

Variations on the Resurgence of the Gamma Function

We review Écalle's formalism of minors, natural-majors and real-majors, and provide explicit formulas in the Borel plane that show the resurgence of the exponential of the Stirling series. We also discuss its Stokes phenomena in the framework of alien calculus.

math.CV↗
arXiv2021

Elliptic fixed points with an invariant foliation: Some facts and more questions

We address the following question: let F:(R^2,0)->(R^2,0) be an analytic local diffeomorphism defined in the neighborhood of the non resonant elliptic fixed point 0 and let Φbe a formal conjugacy to a normal form N. Supposing F leaves invariant the foliation by circles centered at 0, what is the analytic nature of Φand N?

math.DS↗
arXiv2020

On the Moyal Star Product of Resurgent Series

We analyze the Moyal star product in deformation quantization from the resurgence theory perspective. By putting algebraic conditions on Borel transforms, one can define the space of ``algebro-resurgent series'' (a subspace of $1$-Gevrey formal series in $i\hbar/2$ with coefficients in $C\{q,p\}$), which we show is stable under Moyal star product.

math-ph↗
arXiv2020

Hadamard Product and Resurgence Theory

We discuss the analytic continuation of the Hadamard product of two holomorphic functions under assumptions pertaining to Ecalle's Resurgence Theory, proving that if both factors are endlessly continuable with prescribed sets of singular points $A$ and $B$, then so is their Hadamard product with respect to the set $\{0\}\cup A \cdot B$. In this generalization of the classical Hadamard Theorem, all the branches of the multivalued analytic continuation of the Hadamard product are considered.

math.CV↗
arXiv2020

On the regularity of Mather's $β$-function for standard-like twist maps

We consider the minimal average action (Mather's $β$ function) for area preserving twist maps of the annulus. The regularity properties of this function share interesting relations with the dynamics of the system. We prove that the $β$-function associated to a standard-like twist map admits a unique $C^1$-holomorphic complex extension, which coincides with this function on the set of real diophantine frequencies.

math.DS↗
arXiv2018

KAM tori are no more than sticky

When a Gevrey smooth perturbation is applied to a quasi-convex integrable Hamiltonian, it is known that the KAM invariant tori that survive are sticky, that is, doubly exponentially stable. We show by examples the optimality of this effective stability.

math.DS↗
arXiv2018

The Baker-Campbell-Hausdorff formula via mould calculus

The well-known Baker-Campbell-Hausdorff theorem in Lie theory says that the logarithm of a noncommutative product e X e Y can be expressed in terms of iterated commutators of X and Y. This paper provides a gentle introduction t{ó} Ecalle's mould calculus and shows how it allows for a short proof of the above result, together with the classical Dynkin explicit formula [Dy47] for the logarithm, as well as another formula recently obtained by T. Kimura [Ki17] for the product of exponentials itself. We also analyse the relation between the two formulas and indicate their mould calculus generalization to a product of more exponentials.

math.RA↗
arXiv2017

Iterated convolutions and endless Riemann surfaces

We discuss a version of Écalle's definition of resurgence, based on the notion of endless continuability in the Borel plane. We relate this with the notion of Ω-continuability, where Ω is a discrete filtered set, and show how to construct a universal Riemann surface X_Ω whose holomorphic functions are in one-to-one correspondence with Ω-continuable functions. We then discuss the Ω-continuability of convolution products and give estimates for iterated convolutions of the form \hatϕ_1*\cdots *\hatϕ_n. This allows us to handle nonlinear operations with resurgent series, e.g. substitution into a convergent power series.

math.DS↗
arXiv2017

Normalization in Banach scale Lie algebras via mould calculus and applications

We study a perturbative scheme for normalization problems involving resonances of the unperturbed situation, and therefore the necessity of a non-trivial normal form, in the general framework of Banach scale Lie algebras (this notion is defined in the article). This situation covers the case of classical and quantum normal forms in a unified way which allows a direct comparison. In particular we prove a precise estimate for the difference between quantum and classical normal forms, proven to be of order of the square of the Planck constant. Our method uses mould calculus (recalled in the article) and properties of the solution of a universal mould equation studied in a preceding paper.

math.AP↗
arXiv2017

Rayleigh-Schrödinger series and Birkhoff decomposition

We derive new expressions for the Rayleigh-Schrödinger seriesdescribing the perturbation of eigenvalues of quantumHamiltonians. The method, somehow close to the so-called dimensionalrenormalization in quantum field theory, involves the Birkhoffdecomposition of some Laurent series built up out of explicit fullynon-resonant terms present in the usual expression of theRayleigh-Schrödinger series. Our results provide new combinational formulae and a new way \ff{of deriving} perturbation series in Quantum Mechanics. More generally we prove that such a decomposition provides solutions of general normal form problems in Lie algebras.

math.AP↗