En collaboration avec Claude Roger:
The Schrödinger-Virasoro algebra. Mathematical structure and dynamical Schrödinger symmetries.

Le livre est disponible (en attendant la publication) sur cette page (cliquer
sur le lien).

L'algèbre de Schrödinger-Virasoro a été introduite à l'origine par Malte Henkel (Laboratoire de Physique des Matériaux, Nancy) comme
algèbre d'invariance d'échelle locale potentielle en physique statistique hors-équilibre, au lieu et à la place de la célèbre algèbre de
Virasoro (ou algèbre de transformations conformes locales) qui joue un rôle clé dans la compréhension de la physique statistique au point
critique en dimension 2. Cette nouvelle algèbre de dimension infinie est effectivement une sorte d'analogue newtonien de l'algèbre conforme.
On peut espérer qu'elle joue un rôle dans la compréhension des modèles statistiques dynamiques avec exposant dynamique z=2; c'est
le sujet des recherches à venir.

Le livre se concentre sur des aspects plus mathématiques: géométrie de Newton-Cartan, structure algébrique (cohomologie, représentations),
supersymétrisations (en lien avec les algèbres de super-contact...). Une large place est faite à l'étude de l'action de cette algèbre sur des espaces
d'opérateurs de Schrödinger dépendant périodiquement du temps, en combinant les outils géométriques classiques provenant de l'étude des
algèbres de Lie de dimension infinie et des systèmes intégrables, et des outils plus analytiques provenant de la mécanique quantique. Une
classification des orbites, avec un système complet de formes normales, permet notamment de déterminer la monodromie de ces opérateurs.
On montre également que cette action est hamiltonienne pour une structure de Poisson à la Kirillov-Kostant-Souriau provenant d'une algèbre
lacée sur l'algèbre des symboles pseudo-différentiels.

Nous développons dans cet article les principaux arguments constructifs utilisés en théorie quantique des champs,
en nous cantonnant aux théories bosoniques, pour lesquelles il n'existe pas de présentation générale récente.

L'article s'adresse d'abord et avant tout à des mathématiciens ou physiciens mathématiciens connaissant les
arguments de base de la théorie perturbative des champs, et souhaitant connaître un cadre général dans lequel
ils peuvent être rendus rigoureux. Il fournit également un apercu d'une série d'articles récents en collaboration
avec J. Magnen (cf. ci-dessous), visant à donner une définition constructive des chemins rugueux et du calcul
stochastique fractionnaire.

C'est en quelque sorte un résumé de l'article (II) "Constructive proof of convergence" avec J. Magnen (cf. ci-dessous),
mais mettant plus l'emphase sur les liens entre théorie perturbative et théorie constructive.

A renormalized rough path over fractional Brownian motion. http://arxiv.org/abs/1006.5604

We construct in this article a rough path over fractional Brownian motion with arbitrary Hurst index by (i)
using the Fourier normal ordering algorithm introduced in \cite{Unt-Holder} to reduce the problem to that of
regularizing tree iterated integrals and (ii) applying the Bogolioubov-Parasiuk-Hepp-Zimmermann (BPHZ)
renormalization algorithm to Feynman diagrams representing tree iterated integrals.

en collaboration avec Jacques Magnen: From constructive field theory to fractional stochastic calculus.

(I) An introduction:
rough path theory and perturbative heuristics. A paraître à: Annales Henri Poincaré. http://arxiv.org/abs/1012.3873

Let $B=(B_1(t),\ldots,B_d(t))$ be a $d$-dimensional fractional Brownian motion
with Hurst index $\alpha\le 1/4$, or more generally a Gaussian process whose paths have the same local regularity.
Defining properly iterated integrals of $B$ is a difficult
task because of the low H\"older regularity index of its paths. Yet rough path theory shows it is
the key to the construction of a stochastic calculus with respect to $B$, or to solving differential
equations driven by $B$.

We intend to show in a forthcoming series of papers how to desingularize iterated integrals by a weak singular
non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved
by using "standard" tools of constructive field theory, in particular cluster expansions and renormalization.
These powerful tools allow optimal estimates of the moments and call for an extension of the Gaussian tools
such as for instance the Malliavin calculus.

This first paper aims to be both a presentation of the basics of rough path theory to physicists, and of perturbative field theory to probabilists;
it is only heuristic, in particular because the desingularization of iterated integrals is really a {\em non-perturbative} effect. It is also meant to be
a general motivating introduction to the subject, with some insights into quantum field theory and stochastic calculus. The interested reader
should read in a second time the companion article \cite{MagUnt2} for the constructive proofs.

(II) Constructive proof of convergence for the Lévy area of fractional Brownian motion with Hurst index $\alpha\in(1/8,1/4)$.
http://arxiv.org/abs/1103.1750

After this first introductory paper, this one concentrates on the details of the constructive proof of convergence for second-order
iterated integrals, also known as Lévy area.

En collaboration avec Loic Foissy: Ordered forests, permutations and iterated integrals. http://arxiv.org/abs/1004.5208

We construct an explicit Hopf algebra isomorphism from the algebra of heap-ordered trees to that of quasi-symmetric
functions, generated by formal permutations, which is a lift of the natural projection of the Connes-Kreimer algebra of decorated rooted trees onto
the shuffle algebra. This isomorphism gives a universal way of lifting measure-indexed characters of the Connes-Kreimer algebra into
measure-indexed characters of the shuffle algebra, already introduced in \cite{Unterberger} in the framework of rough path theory as the so-called
Fourier normal ordering algorithm.

A Levy area by Fourier normal ordering for multidimensional fractional Brownian motion with small Hurst index.
http://arxiv.org/abs/0906.1416

A central limit theorem for the rescaled L\'evy area of two-dimensional fractional
Brownian motion with Hurst index H<1/4.
http://arxiv.org/abs/0808.3458

19. avec Samy Tindel,
The rough path associated to the multidimensional analytic fBm with any
Hurst parameter. Collectanea Mathematica 62 (2), 197- (2011). http://arxiv.org/abs/0810.1408

In this paper, we consider a complex-valued d-dimensional fractional Brownian motion defined on the closure of the complex
upper half-plane, called analytic fractional Brownian motion and denoted by Gamma. This process has been introduced in
\cite{Un}, and both its real and imaginary parts, restricted to the real axis, are usual fractional Brownian motions. The current
note is devoted to prove that a rough path based on Gamma can be constructed for any value of the Hurst parameter in (0,1/2).
We also show how to solve differential equations driven by Gamma in a neighborhood of 0 of the complex upper half-plane, by means
of elementary arguments.

18. avec Claude Roger,
      A Hamiltonian action of the Schr\"odinger-Virasoro algebra on a space of
      periodic time-dependent Schr\"odinger operators in (1+1)-dimensions. Journal of
      Nonlinear Mathematical Physics 17 (3), 257--279 (2010). http://arxiv.org/abs/0810.0902

Let ${\cal S}^{lin}:=\{a(t)(-2\II \partial_t-\partial_r^2+V(t,r)\ |\ a\in C^{\infty}(\R/2\pi\Z),
V\in C^{\infty}(\R/2\pi\Z\times\R)\}$ be the space of Schr\"odinger operators in $(1+1)$-dimensions with periodic
time-dependent potential. The action on ${\cal S}^{lin}$ of a large infinite-dimensional reparametrization group $SV$
with Lie algebra $\sv$ \cite{RogUnt06,Unt08}, called the Schrödinger-Virasoro group and containing the Virasoro group,
is proved to be Hamiltonian for a certain Poisson structure on ${\cal S}^{lin}$. More precisely, the infinitesimal action
of $\sv$ appears to be part of a coadjoint action of a Lie algebra of pseudo-differential symbols, $\g$, of which $\sv$ is
a quotient, while the Poisson structure is inherited from the corresponding Kirillov-Kostant-Souriau form.

17. avec Andreas Neuenkirch et Samy Tindel, Discretizing the fractional Levy area. Stochastic
Processes and their Applications 120 (2), 223-254 (2010). http://arxiv.org/abs/0902.0497

16. avec Albrecht Boettcher, Sacha Grudsky, Egor A. Maksimenko,
The first order asymptotics of the extreme eigenvectors of certain Hermitian
Toeplitz matrices. Integral Equations and Operator Theory 63, 165-180 (2009).

15. Stochastic calculus for fractional Brownian motion with Hurst exponent H larger
than 1/4: a rough path method by analytic extension. Annals of Probability 37 (2), 565-614 (2009).
http://arxiv.org/abs/math/0703697

14. On vertex algebra representations of the Schr\"odinger-Virasoro Lie algebra.
Nuclear Physics B823 (3), 320-371 (2009). http://arxiv.org/abs/cond-mat/0703214

The Schrödinger-Virasoro Lie algebra $\sv$ is an extension of the Virasoro Lie algebra by a nilpotent Lie algebra
formed with a bosonic current of weight 3/2 and a bosonic current of weight 1. It is also a natural
infinite-dimensional extension of the Schrödinger Lie algebra, which -- leaving aside the invariance under
time-translation -- has been proved to be a symmetry algebra for many statistical physics models undergoing a
dynamics with dynamical exponent z=2.

We define in this article general Schrödinger-Virasoro primary fields by analogy with conformal field theory,
characterized by a 'spin' index and a (non-relativistic) mass, and construct vertex algebra representations of $\sv$
out of a charged symplectic boson and a free boson and its associated vertex operators. We also compute two- and
three-point functions of still conjectural massive fields that are defined by an analytic continuation with respect to a
formal parameter.

13. avec Albrecht Boettcher et Sacha Grudsky,
Asymptotic pseudomodes of Toeplitz matrices. Operators and Matrices, 2 (4),
525--531 (2008).

12. The Schrödinger-Virasoro Lie algebra: a mathematical structure between conformal field theory
and non-equilibrium dynamics, Journal of Physics, Conference Series 40, 156 (2006).

11. avec Claude Roger,
The Schrödinger-Virasoro Lie group and algebra: from geometry to representation
theory, Ann. Henri Poincaré 7 (2006), 1477--1529. http://arxiv.org/abs/math-ph/0601050

This article is devoted to an extensive study of a infinite-dimensional Lie algebra $\sv$, introduced in \cite{Henk94} in the context
of non-equilibrium statistical physics, containing as subalgebras both the Lie algebra ofinvariance of the free
Schrödinger equation and the central charge-free Virasoro algebra $\Vect(S^1).$ We call $\sv$ the
Schrödinger-Virasoro Lie algebra. We study its representation theory: realizations as Lie symmetries of field equations,
coadjoint representation, coinduced representations in connection with Cartan's prolongation method (yielding analogues
of the tensor density modules for $\Vect(S^1)$). We also present a detailed cohomogical study, providing
in particular a classification of deformations and central extensions; there appears
a non-local cocycle.

10. avec Malte Henkel, René Schott et Stoimen Stoimenov,
The Poincaré algebra in the context of ageing systems: Lie structure, representations, Appell systems and coherent states.
http://arxiv.org/abs/math-ph/0601028 ,

9. avec Malte Henkel,
Supersymmetric extensions of Schrödinger-invariance, Nucl. Phys. B746, 155--201 (2006).
http://arxiv.org/abs/math-ph/0512024

The set of dynamic symmetries of the scalar free Schrödinger equation in d space dimensions gives a realization
of the Schrödinger algebra that may be extended into a representation of the conformal algebra in d+2
dimensions, which yields the set of dynamic symmetries of the same equation where the mass is not viewed as a constant,
but as an additional coordinate. An analogous construction also holds for the spin-1/2 L\'evy-Leblond equation.
An (N=2) supersymmetric extension of these equations leads, respectively, to a `super-Schrödinger' model
and to the (3|2)-supersymmetric model. Their dynamic supersymmetries form the Lie superalgebras $\osp(2|2)\ltimes\sh(2|2)$
and osp(2|4), respectively. The Schrödinger algebra and its supersymmetric counterparts are found to be the largest
finite-dimensional Lie subalgebras of a family of infinite-dimensional Lie superalgebras that are systematically
constructed in a Poisson algebra setting, including the Schrödinger-Neveu-Schwarz algebra $\sns^{(N)}$ with $N$ supercharges.

Covariant two-point functions of quasiprimary superfields are calculated for several subalgebras of osp(2|4). If one includes both
(N=2) supercharges and time-inversions, then the sum of the scaling dimensions is restricted to a finite set of possible values.

8. avec Malte Henkel, Rene Schott et Stoimen Stoimenov,
On the dynamical symmetric algebra of ageing: Lie structure, representations and
Appell systems, Quantum Probab. White Noise Anal. 20 (2007), 233-240. http://arxiv.org/abs/math/0510096

7. avec Malte Henkel, Alan Picone et Michel Pleimling, http://arxiv.org/abs/cond-mat/0307649
Local scale invariance and its applications to strongly anisotropic critical phenomena (2003).

5. avec Nils B. Andersen,
    An application of shift operators to ordered symmetric spaces,
    publié dans: Ann. Inst. Fourier 52 (2002).

4. Hypergeometric functions of second kind and spherical functions on an ordered symmetric space,
    J. Funct. Anal. 188 (2002).

3. avec Fulvio Ricci,
Solvability of invariant sublaplacians on spheres and group contractions,
Rend. Mat. Acc. Lincei 9-12 (2001).

2. avec Nils B. Andersen,
Harmonic analysis on $SU(n,n)/SL(n,\C)\times\R_+^*$,
J. Lie Theory 10 (2000).