Journées du réseau Dijon-Lyon-Metz de physique mathématique

10 oct. 2024, 14:00 → 11 oct. 2024, 12:15 Europe/Paris

Salle A318 (IMB)

This is the third edition of the research network workshop of Dijon-Lyon-Metz on mathematical physics. Topics of the workshop include (but are not restricted to):

• Classical and Quantum Field Theory (multisymplectic geometry, Higher Structures, Gauge theories, CFT, TQFT QFT, GR)
• Non commutative geometry and spectral theory (K-theory, index, topological insulators)
• Integrable systems (classical and quantum, integrable hierarchies, spin models, AdS/CFT correspondence)

Previous events

• 2nd edition (Lyon, September 2024): https://indico.math.cnrs.fr/event/12743/
• 1st edition (Lyon, December 2023): https://indico.math.cnrs.fr/event/10922/

 

jeudi 10 octobre

1 Laplace coupling quantization

Quantization can be seen as a well-known deformation of the product of a cocommutative Hopf algebra (called twisted product or circle product). After a short introduction to Hopf algebras, the principle of this deformation is presented and illustrated by the operator product and the time-ordered product in quantum field theory. The first step of the renormalization of time-ordered products will be sketched.

Orateur: Christian Brouder (IMPMC, Sorbonne Université)

2 Fredholm Determinants and Random Matrices

First, random matrix theory will be briefly introduced, summarizing the main results necessary to discuss the cumulative distribution for the largest eigenvalue of the Gaussian Unitary Ensemble. When the size of the matrix becomes infinite, this limiting distribution becomes a Fredholm determinant which can be evaluated by a functional formula, yielding the Tracy-Widom distribution. Then, a generalized setup will be described and motivated. Finally, our recent result about these generalized considerations will be presented as a deformation of the Tracy-Widom distribution.

Orateur: Xavier Navand (Université de Bourgogne)

16:00 Coffee break

3 $\hbar$-expansion of Wightman distributions

Twisted $\hbar$-deformations by classical wave operators are introduced for the $\lambda \Phi^4$-theory in Minkowski spacetime. These deformations are non-perturbative in the coupling constant $\lambda$. The corresponding Wightman $n$-functions are defined as evaluations at $0$ of the $n$-fold deformed products of classical solutions of the $\lambda \Phi^4$ wave equation. We show that, in this setting, the $2$-point function is well-defined as a formal series in $\hbar$ of tempered distributions. Interestingly, these twisted deformations appear to possess an inherent renormalization scheme.

Orateur: Giuseppe Dito (Université de Bourgogne)

vendredi 11 octobre

4 Introduction to $n$-Poisson (and $n$-Dirac) manifolds (2) : examples and Lie $n$-algebras.

Reminders on what are $n$-Poisson (and $n$-Dirac) manifolds, without using groupoids (only in terms of some structures in the generalised tangent space). Examples of such structures. Natural structure(s) of Lie $n$-algebra coming from these manifolds.

Orateur: Philippe Bonneau (Université de Lorraine)

10:00 Coffee break

5 Existence of crystallographic-invariant exponentially localized Wannier functions in topological insulators

The problem of investigating the presence of topological obstructions to the existence of crystallographic invariant exponentially localized Wannier functions in topological insulators is addressed. When crystallographic groups are considered, twists in ordinary K-theory must be taken into account. The case of the Haldane model is used to demonstrate how an explicit algorithmic procedure can be set up to effectively construct invariant exponentially localized Wannier functions whenever no topological obstructions are present.

Orateur: Marco D'Agostino (INSA, Lyon)

6 Cohomology, deformations and structure of local homogeneous Poisson brackets of arbitrary degree

Dubrovin and Novikov initiated the study of local homogeneous differential-geometric Poisson brackets of arbitrary degree $k$ in their seminal 1984 paper. Despite many efforts, and several results in low degree, very little is known about their structure for arbitrary $k$. After an introduction to the topic we report on our recent results on the structure of DN brackets of degree $k$. By applying homological algebra methods to the computation of their Poisson cohomology (or rather of an associated differential complex) we show that certain linear combinations of the coefficients of a degree $k$ DN bracket define $k$ flat connections. Moreover the Poisson cohomology of such brackets is related with the Chevalley-Eilenberg cohomology of an associate finite-dimensional Lie algebra. Work in progress with M. Casati.

Orateur: Guido Carlet (Université de Bourgogne)