Carroll contractions of Lorentz-invariant theories

We consider Carroll-invariant limits of Lorentz-invariant field theories. We show that just as in the case of electromagnetism, there are two inequivalent limits, one “electric” and the other “magnetic”. Each can be obtained from the corresponding Lorentz-invariant theory written in Hamiltonian form through the same “contraction” procedure of taking the ultrarelativistic limit c → 0 where c is the speed of light, but with two different consistent rescalings of the canonical variables. This procedure can be applied to general Lorentz-invariant theories (p-form gauge fields, higher spin free theories etc) and has the advantage of providing explicitly an action principle from which the electrically-contracted or magnetically-contracted dynamics follow (and not just the equations of motion). Even though not manifestly so, this Hamiltonian action principle is shown to be Carroll invariant. In the case of p-forms, we construct explicitly an equivalent manifestly Carroll-invariant action principle for each Carroll contraction. While the manifestly covariant variational description of the electric contraction is rather direct, the one for the magnetic contraction is more subtle and involves an additional pure gauge field, whose elimination modifies the Carroll transformations of the fields. We also treat gravity, which constitutes one of the main motivations of our study, and for which we provide the two different contractions in Hamiltonian form.

The nondegenerate generalized Kähler Calabi–Yau problem

We formulate a Calabi–Yau-type conjecture in generalized Kähler geometry, focusing on the case of nondegenerate Poisson structure. After defining natural Hamiltonian deformation spaces for generalized Kähler structures generalizing the notion of Kähler class, we conjecture unique solvability of Gualtieri’s Calabi–Yau equation within this class. We establish the uniqueness, and moreover show that all such solutions are actually hyper-Kähler metrics. We furthermore establish a GIT framework for this problem, interpreting solutions of this equation as zeroes of a moment map associated to a Hamiltonian action and finding a Kempf–Ness functional. Lastly we indicate the naturality of generalized Kähler–Ricci flow in this setting, showing that it evolves within the given Hamiltonian deformation class, and that the Kempf–Ness functional is monotone, so that the only possible fixed points for the flow are hyper-Kähler metrics. On a hyper-Kähler background, we establish global existence and weak convergence of the flow.

Stacky Hamiltonian Actions and Symplectic Reduction

We introduce the notion of a Hamiltonian action of an étale Lie group stack on an étale symplectic stack and establish versions of the Kirwan convexity theorem, the Meyer–Marsden–Weinstein symplectic reduction theorem, and the Duistermaat–Heckman theorem in this context.

Examples of Gibbs States of Mechanical Systems with Symmetries

In a previous paper, the notion of Gibbs state for the Hamiltonian action of a Lie group on a symplectic manifold was given, together with its applications in Statistical Mechanics, and the works in this field of the French mathematician and physicist Jean-Marie Souriau were presented. Using an adaptation of the cross product for pseudo-Euclidean three-dimensional vector spaces, we present several examples of such Gibbs states, together with the associated thermodynamic functions, for various two-dimensional symplectic manifolds, including the pseudo-spheres, the Poincar{\'e} disk and the Poincar{\'e} half-plane.

On Gibbs States of Mechanical Systems with Symmetries

The French mathematician and physicist Jean-Marie Souriau studied Gibbs states for the Hamiltonian action of a Lie group on a symplectic manifold and considered their possible applications in Physics and Cosmology. These Gibbs states are presented here with detailed proofs of all the stated results. A companion paper to appear will present examples of Gibbs states on various symplectic manifolds on which a Lie group of symmetries acts by a Hamiltonian action, including the Poincar\'e disk and the Poincaré half-plane.

Toric Geometry of Spin(7)-Manifolds

We study $ \operatorname{Spin}(7) $-manifolds with an effective multi-Hamiltonian action of a four-torus. On an open dense set, we provide a Gibbons–Hawking type ansatz that describes such geometries in terms of a symmetric $ 4\times 4 $-matrix of functions. This description leads to the 1st known $ \operatorname{Spin}(7) $-manifolds with a rank $ 4 $ symmetry group and full holonomy. We also show that the multi-moment map exhibits the full orbit space topologically as a smooth four-manifold, containing a trivalent graph in $ \mathbb{R}^4 $ as the image of the set of the special orbits.

On the classic geometrodynamics of a spherically-symmetric configuration of gravitational and electromagnetic fields

Analytical aspects of the classical geometrodynamics for the spherically-symmetric configuration of the electromagnetic and gravitational fields in GR are considered. The feature of such configurations is that they admit two motion integrals – the total mass and charge. The Einstein-Hilbert action for the configuration, after dimensional reduction, by means of the Legendre transformation is reduced to the Hamiltonian action. Using the conservation laws for the mass and charge, as well as the Hamiltonian constraint, the momenta are found as functions of configuration variables. The set of equations, which associate momenta and functional derivatives of the action in the configuration space (CS) is integrable. This allows us to obtain the action functional as a solution of the Einstein-Hamilton-Jacobi equation in functional derivatives. Variations of the action functional with respect to mass M and charge Q of the configuration lead to the motion trajectories in the CS. We note that the minisuperspace metric, which is induced by the kinetic part of the Lagrangian, does not coincide with the CS metric that arises when the function of lapse N is excluded from the action. The space-time metric for which the indicated metrics coincide in the T-region up to a coefficient are considered. The metric of CS is constructed and its geometry is studied. Under the trivial embedding of hypersurfaces of the foliation into a dynamical T-region, the CS is flat. It allows introducing pseudo-Cartesian coordinates in which the CS metric takes the Lorentz form.

Hyperfunctions, the Duistermaat–Heckman Theorem and Loop Groups

This chapter investigates the Duistermaat–Heckman theorem using the theory of hyperfunctions. In applications involving Hamiltonian torus actions on infinite-dimensional manifolds, the more general theory seems to be necessary in order to accommodate the existence of the infinite-order differential operators which arise from the isotropy representations on the tangent spaces to fixed points. The chapter quickly reviews the theory of hyperfunctions and their Fourier transforms. It then applies this theory to construct a hyperfunction analogue of the Duistermaat–Heckman distribution. The main goal will be to study the Duistermaat–Heckman distribution of the loop space of SU(2) but it will also characterize the singular locus of the moment map for the Hamiltonian action of T×S 1 on the loop space of G. The main goal of this chapter is to present a Duistermaat–Heckman hyperfunction arising from a Hamiltonian group action on an infinite-dimensional manifold.