Flat space holography in spin-2 extended dilaton-gravity

We present an interacting spin-2 gauge theory coupled to the two-dimensional dilaton-gravity in flat spacetime. The asymptotic symmetry group is enhanced to the central extension of Diff(S1)⋉C∞(S1)⋉Vec(S1) when the central element of the Heisenberg subgroup is zero (vanishing U(1) level). Using the BF-formulation of the model we derive the corresponding boundary coadjoint action which is the spin-2 extension of the warped Schwarzian theory at vanishing U(1) level. We also discuss the thermodynamics of black holes in this model.

New boundary conditions for AdS2

We describe new boundary conditions for AdS2 in Jackiw-Teitelboim gravity. The asymptotic symmetry group is enhanced to Diff(S1) ⋉ C∞(S1) whose breaking to SL(2, ℝ) × U(1) controls the near-AdS2 dynamics. The action reduces to a boundary term which is a generalization of the Schwarzian theory and can be interpreted as the coadjoint action of the warped Virasoro group. This theory reproduces the low-energy effective action of the complex SYK model. We compute the Euclidean path integral and derive its relation to the random matrix ensemble of Saad, Shenker and Stanford. We study the flat space version of this action, and show that the corresponding path integral also gives an ensemble average, but of a much simpler nature. We explore some applications to near-extremal black holes.

Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation

In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their “space of evolution” associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (e.g., Galileo’s group in classical physics). Souriau Lie Group Thermodynamics was also addressed 30 years after Souriau by R.F. Streater in the framework of Quantum Physics by Information Geometry for some Lie algebras, but only in the case of null cohomology. Souriau method could then be applied on Lie groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map. For this case, the coadjoint action on moment map is equivariant. For non-null cohomology, we give the case of Lie group SE(2). Finally, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping.

A Basis of Casimirs in 3D Magnetohydrodynamics

We prove that any regular Casimir in 3D magnetohydrodynamics (MHD) is a function of the magnetic helicity and cross-helicity. In other words, these two helicities are the only independent regular integral invariants of the coadjoint action of the MHD group $\textrm{SDiff}(M)\ltimes \mathfrak X^*(M)$, which is the semidirect product of the group of volume-preserving diffeomorphisms and the dual space of its Lie algebra.

Displacing (Lagrangian) submanifolds in the manifolds of full flags

In symplectic geometry a question of great importance is whether a (Lagrangian) submanifold is displaceable, that is, if it can be made disjoint from itself by a Hamiltonian isotopy.We analyze the coadjoint orbits of SU(n) and their Lagrangian submanifolds that are the fibers of the Gelfand-Tsetlin map.We use the coadjoint action to displace a large collection of these fibers. Thenwe concentrate on the case n = 3 and apply McDuff’s method of probes to show that “most” of the generic Gelfand-Tsetlin fibers are displaceable. “Most” means “all but one” in the non-monotone case, and it means “all but a 1-parameter family” in the monotone case. In the case of a non-monotone manifold of full flags we present explicitly a unique non-displaceable Lagrangian fiber (S