The goal of the talk is proving a conjecture of Claude Roger about the universal central extension of the Lie algebra of volume-preserving vector fields. In the beginning we will briefly review the notion of central extensions of Lie algebras and their link to Chevalley-Eilenberg-cohomology. We will then proceed to Roger's conjecture, which lies in the (continuous) infinite-dimensional...
Multisymplectic manifolds provide a natural generalization of symplectic manifolds by considering closed, non-degenerate k-forms in place of 2-forms. A central theme in the study of (multi)symplectic structures is the investigation of the relationship between symmetries, encoded by group actions that preserve the differential form, and reduction procedures. These reduction schemes yield a...
In relativistic field theory, fields are sections of a vector bundle over spacetime and observables are functionals of the fields generated by distributional sections of the bundle. Observables form a well-known Poisson algebra (induced by a given Lagrangian), which is then quantized by deformation. The algebra structure breaks the covariant dependence on the bundle, because there is no nice...
In relativistic field theory, fields are sections of a vector bundle over spacetime and observables are functionals of the fields generated by distributional sections of the bundle. Observables form a well-known Poisson algebra (induced by a given Lagrangian), which is then quantized by deformation. The algebra structure breaks the covariant dependence on the bundle, because there is no nice...
Using K-theory and cyclic cohomology to describe topological phases for solids gives us the possibility to describe various topological relations between physical systems by means of the boundary map in K-theory. The perhaps best known example of that is the bulk edge correspondence which equates the quantised Hall conductivity with the edge conductivity. I will try to give an overview of that.
I will report on work in very slow progress with Ferdinand-Lin-To where we try to implement rigorously the Wick rotation on curved space time. Our ultimate goal would be the construction of a nonperturbative Phi43 theory on de Sitter space which gives a non trivial example of analytic state. I will emphasize the difficulties that we have not been able to overcome and stress the central role...
In this talk I will give a brief overview of the computation of the form factors of integrable spin chains from the Algebraic Bethe Ansatz in the thermodynamic limit. As an illustration I’ll show how to compute boundary overlaps for the open spin chains after a change of one boundary parameter.
Steinberg symbol is a function of two elements of a ring satisfying very simple conditions. It's main application so far is in number theory and algebraic K-theory. We will show that it is a very transparent construction and it gives the central extension giving just infinite Heisenberg group. The same symbol can be used to define an explicit formula for central extension of affine Lie...
In this talk, I introduce new q-deformations of Lie algebras linked to the modular group and the q-rational numbers of Morier-Genoud and Ovsienko. In particular, a deformation of the Lie algebra sl(2), the Witt algebra and a glimpse on the Virasoro algebra will be presented. These deformations are realized through concrete differential operators and lead to a new understanding of q-rationals.
There is a plethora of classical integrable systems that are named after Toda, Calogero, Moser, Sutherland, Ruijsenaars, and many other researchers. These are toy models of interacting particles which have appeared since the late 1960s, and which have been related to many areas in mathematics and theoretical physics. My aim is to provide a historical account of these systems (biased towards...
I will introduce a correspondence between the language of quantum error correcting codes and that of gauge theory. I will focus more specifically on the well-studied family of \emph{stabilizer codes}, which can be interpreted as Abelian gauge theories with gauge group a product of $\mathbb{Z}_2$. This class of codes includes repetition codes such as the elementary three-qubit code, and...
Motivated by classical mechanics and classical field theory, I will give an elementary introduction to two different approaches
to the inverse problem of variational calculus (for systems of differential equations
and systems of partial differential equations).
In doing so, I will try to indicate the state of the art and mention some other approaches that have been considered.
After some brief reminders on the octonions and the exceptional Lie group G_2, we will be interested in the 2-plectic structure on the 6-sphere induced by the octonions and the links with the almost complex structure. In a second part of the talk, we will discuss the dynamic in the two possible cases : vector field and bivector field, and we shall study in more detail the multisymplectic structure.
I will present a family of models, based on variational problems defined on fields on manifolds, the classical solutions of which lead to an at least locally principal bundle structure on the given manifold. Moreover each critical point of such models allows us to recover solutions of gauge fields theories such as Yang-Mills or Einstein gravity equations. The 'discovery' of these models...