Presymplectic Form Research Articles

Horizon phase spaces in general relativity

We derive a prescription for the phase space of general relativity on two intersecting null surfaces using the null initial value formulation. The phase space allows generic smooth initial data, and the corresponding boundary symmetry group is the semidirect product of the group of arbitrary diffeomorphisms of each null boundary which coincide at the corner, with a group of reparameterizations of the null generators. The phase space can be consistently extended by acting with half-sided boosts that generate Weyl shocks along the initial data surfaces. The extended phase space includes the relative boost angle between the null surfaces as part of the initial data.We then apply the Wald-Zoupas framework to compute gravitational charges and fluxes associated with the boundary symmetries. The non-uniqueness in the charges can be reduced to two free parameters by imposing covariance and invariance under rescalings of the null normals. We show that the Wald-Zoupas stationarity criterion cannot be used to eliminate the non-uniqueness. The different choices of parameters correspond to different choices of polarization on the phase space. We also derive the symmetry groups and charges for two subspaces of the phase space, the first obtained by fixing the direction of the normal vectors, and the second by fixing the direction and normalization of the normal vectors. The second symmetry group consists of Carrollian diffeomorphisms on the two boundaries.Finally we specialize to future event horizons by imposing the condition that the area element be non-decreasing and become constant at late times. For perturbations about stationary backgrounds we determine the independent dynamical degrees of freedom by solving the constraint equations along the horizons. We mod out by the degeneracy directions of the presymplectic form, and apply a similar procedure for weak non-degeneracies, to obtain the horizon edge modes and the Poisson structure. We show that the area operator of the black hole generates a shift in the relative boost angle under the Poisson bracket.

On the covariant formulation of gauge theories with boundaries

In the present article, we review the classical covariant formulation of Yang–Mills theory and general relativity in the presence of spacetime boundaries, focusing mainly on the derivation of the presymplectic forms and their properties. We further revisit the introduction of the edge modes and the conditions which justify them, in the context where only field-independent gauge transformations are considered. We particularly show that the presence of edge modes is not justified by gauge invariance of the presymplectic form, but rather by the condition that the presymplectic form is degenerate on the initial field space, which allows to relate this presymplectic form to the symplectic form on the gauge reduced field space via pullback.

Hamiltonian Lie algebroids

In previous work with M.C. Fernandes, we found a Lie algebroid symmetry for the Einstein evolution equations. The present work was motivated by the effort to combine this symmetry with the hamiltonian structure of the equations to explain the coisotropic structure of the constraint subset for the initial value problem. In this paper, we extend the notion of hamiltonian structure from Lie algebra actions to general Lie algebroids over presymplectic manifolds. Application of this construction to the problem in general relativity is still work in progress. After comparing a number of possible compatibility conditions between an anchor map A → T M A\to TM on a vector bundle A A and a presymplectic structure on the base M M , we choose the most natural of them, best formulated in terms of a suitably chosen connection on A A . We define a notion of momentum section of A ∗ A^* , and, when A A is a Lie algebroid, we specify a condition for compatibility with the Lie algebroid bracket. Compatibility conditions on an anchor, a Lie algebroid bracket, a momentum section, a connection, and a presymplectic structure are then the defining properties of a hamiltonian Lie algebroid. For an action Lie algebroid with the trivial connection, the conditions reduce to those for a hamiltonian action. We show that the clean zero locus of the momentum section of a hamiltonian Lie algebroid is a coisotropic submanifold. To define morphisms of hamiltonian Lie algebroids, we express the structure in terms of a bigraded algebra generated by Lie algebroid forms and de Rham forms on its base. We give an Atiyah-Bott type characterization of a bracket-compatible momentum map; it is equivalent to a closed basic extension of the presymplectic form, within the generalization of the BRST model of equivariant cohomology to Lie algebroids. We show how to construct a groupoid by reduction of an action Lie groupoid G × M G\times M by a subgroup H H of G G which is not necessarily normal, and we find conditions which imply that a hamiltonian structure descends to such a reduced Lie algebroid. We consider many examples and, in particular, find that the tangent Lie algebroid over a symplectic manifold is hamiltonian with respect to some connection if and only if the symplectic structure has a nowhere vanishing primitive. Recent results of Stratmann and Tang show that this is the case whenever the symplectic structure is exact.

Deformations of cluster mutations and invariant presymplectic forms

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A_2: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A_3 and A_4, we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions in the discrete sine-Gordon equation.

Existence of KAM tori for presymplectic vector fields

We prove the existence of a torus that is invariant with respect to the flow of a vector field that preserves the presymplectic form in an exact presymplectic manifold. The flow on this invariant torus is conjugate to a linear flow on a torus with a Diophantine velocity vector. The proof has an "a posteriori" format, the the invariant torus is constructed by using a Newton method in a space of functions, starting from a torus that is approximately invariant. The geometry of the problem plays a major role in the construction by allowing us to construct a special adapted basis in which the equations that need to be solved in each step of the iteration have a simple structure. In contrast to the classical methods of proof, this method does not assume that the system is close to integrable, and does not rely on using action-angle variables.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/126/abstr.html

Gauge equivalences for foliations and pre-symplectic structures

We consider the deformation theory of two kinds of geometric objects: foliations on one hand, pre-symplectic forms on the other. For each of them, we prove that the geometric notion of equivalence given by isotopies agrees with the algebraic notion of gauge equivalence obtained from the [Formula: see text]-algebras governing these deformation problems.

Integrable Evolutions of Twisted Polygons in Centro-Affine ℝm

Abstract We show that discrete $W_m$ lattices are bi-Hamiltonian, using geometric realizations of discretizations of the Adler–Gel’fand–Dikii flows as local evolutions of arc length-parametrized polygons in centro-affine space. We prove the compatibility of two known Hamiltonian structures defined on the space of geometric invariants by lifting them to a pair of pre-symplectic forms on the space of arc length parametrized polygons. The simplicity of the expressions of the pre-symplectic forms makes the proof of compatibility straightforward. We also study their kernels and possible integrable systems associated to the pair.

Geometry from divergence functions and complex structures

Motivated by the geometrical structures of quantum mechanics, we introduce an almost complex structure [Formula: see text] on the product [Formula: see text] of any parallelizable statistical manifold [Formula: see text]. Then, we use [Formula: see text] to extract a pre-symplectic form and a metric-like tensor on [Formula: see text] from a divergence function. These tensors may be pulled back to [Formula: see text], and we compute them in the case of an N-dimensional symplex with respect to the Kullback–Leibler relative entropy, and in the case of (a suitable unfolding space of) the manifold of faithful density operators with respect to the von Neumann–Umegaki relative entropy.

Whiskered Tori for Presymplectic Dynamical Systems

We prove persistence result of whiskered tori for the dynamical system which preserves an exact presymplectic form. The results are given in an a-posteriori format. Given an approximate solution of an invariance equation which satisfies some non-degeneracy assumptions, we conclude that there is a true solution close by. The proof is based on certain iterative procedure by which the accuracy of the approximate solutions of the invariance equation can be improved. The iterative procedure is not based on transformation theory, which is cumbersome for presymplectic systems, but on finding corrections to the solutions of the invariance equation. This iterative procedure takes advantage of identities that come from the preservation of the geometric structure and leads to a very efficient numerical method which has low storage requirements, low operator count per step and it is quadratically convergent. We note that a particular case of presymplectic systems is symplectic perturbed by quasi-periodic systems.

Holomorphic pre-symplectic form on the nested Hilbert scheme Hilb3,4 (C 2 )

Holomorphic pre-symplectic form on the nested Hilbert scheme Hilb3,4 (C 2 )

The infinitesimal counterpart of tangent presymplectic groupoids of higher order

Let $\left(G, \omega \right)$ be a presymplectic groupoid. In this paper we characterize the infinitesimal counter part of the tangent presymplectic groupoid of higher order, $(T^{r}G, \omega ^{\left(c\right)})$ where $T^{r}G$ is the tangent groupoid of higher order and $\omega ^{\left(c\right)}$ is the complete lift of higher order of presymplectic form $\omega $.

Moduli spaces of framed flags of sheaves on the projective plane

We study the moduli space of framed flags of sheaves on the projective plane via an adaptation of the ADHM construction of framed sheaves. In particular, we prove that, for certain values of the topological invariants, the moduli space of framed flags of sheaves is an irreducible, nonsingular variety carrying a holomorphic pre-symplectic form.

Removable Presymplectic Singularities and the Local Splitting of Dirac Structures

We call a singularity of a presymplectic form $\omega$ removable in its graph if its graph extends to a smooth Dirac structure over the singularity. An example for this is the symplectic form of a magnetic monopole. A criterion for the removability of singularities is given in terms of regularizing functions for pure spinors. All removable singularities are poles in the sense that the norm of $\omega$ is not locally bounded. The points at which removable singularities occur are the non-regular points of the Dirac structure for which we prove a general splitting theorem: Locally, every Dirac structure is the gauge transform of the product of a tangent bundle and the graph of a Poisson structure. This implies that in a neighborhood of a removable singularity $\omega$ can be split into a non-singular presymplectic form and a singular presymplectic form which is the partial inverse of a Poisson bivector that vanishes at the singularity. An interesting class of examples is given by log-Dirac structures which generalize log-symplectic structures. The analogous notion of removable singularities of Poisson structures is also studied.

Symplectic and Killing symmetries of AdS3 gravity: holographic vs boundary gravitons

The set of solutions to the AdS$_3$ Einstein gravity with Brown-Henneaux boundary conditions is known to be a family of metrics labeled by two arbitrary periodic functions, respectively left and right-moving. It turns out that there exists an appropriate presymplectic form which vanishes on-shell. This promotes this set of metrics to a phase space in which the Brown-Henneaux asymptotic symmetries become symplectic symmetries in the bulk of spacetime. Moreover, any element in the phase space admits two global Killing vectors. We show that the conserved charges associated with these Killing vectors commute with the Virasoro symplectic symmetry algebra, extending the Virasoro symmetry algebra with two $U(1)$ generators. We discuss that any element in the phase space falls into the coadjoint orbits of the Virasoro algebras and that each orbit is labeled by the $U(1)$ Killing charges. Upon setting the right-moving function to zero and restricting the choice of orbits, one can take a near-horizon decoupling limit which preserves a chiral half of the symplectic symmetries. Here we show two distinct but equivalent ways in which the chiral Virasoro symplectic symmetries in the near-horizon geometry can be obtained as a limit of the bulk symplectic symmetries.

Isometry group orbit quantization of spinning strings in AdS3 × S3

Describing the bosonic AdSS3 particle and string in SU(1,1) × SU(2) group variables, we provide a Hamiltonian treatment of the isometry group orbits of solutions via analysis of the pre-symplectic form. For the particle we obtain a one-parameter family of orbits parameterized by creation–annihilation variables, which leads to the Holstein–Primakoff realization of the isometry group generators. The scheme is then applied to spinning string solutions characterized by one winding number in AdS3 and two winding numbers in S3. We find a two-parameter family of orbits, where quantization again provides the Holstein–Primakoff realization of the symmetry generators with an oscillator-type energy spectrum. Analyzing the minimal energy at strong coupling, we verify the spectrum of short strings at special values of winding numbers.

Quantisation of presymplectic manifolds, $K$-theory and group representations

Let $G$ be a semisimple Lie group with finite component group, and let $K<G$ be a maximal compact subgroup. We obtain a quantisation commutes with reduction result for actions by $G$ on manifolds of the form $M = G\times_K N$, where $N$ is a compact prequantisable Hamiltonian $K$-manifold. The symplectic form on $N$ induces a closed two-form on $M$, which may be degenerate. We therefore work with presymplectic manifolds, where we take a presymplectic form to be a closed two-form. For complex semisimple groups and semisimple groups with discrete series, the main result reduces to results with a more direct representation theoretic interpretation. The result for the discrete series is a generalised version of an earlier result by the author. In addition, the generators of the $K$-theory of the $C^*$-algebra of a semisimple group are realised as quantisations of fibre bundles over suitable coadjoint orbits.

K-Symplectic Lie systems: theory and applications

A Lie system is a system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot–Guldberg Lie algebra. We suggest the definition of a particular class of Lie systems, the k-symplectic Lie systems, admitting a Vessiot–Guldberg Lie algebra of Hamiltonian vector fields with respect to the presymplectic forms of a k-symplectic structure. We devise new k-symplectic geometric methods to study their superposition rules, t-independent constants of motion and general properties. Our results are illustrated through examples of physical and mathematical interest. As a byproduct, we find a new interesting setting of application of the k-symplectic geometry: systems of first-order ordinary differential equations.

Presymplectic current and the inverse problem of the calculus of variations

The inverse problem of the calculus of variations asks whether a given system of partial differential equations (PDEs) admits a variational formulation. We show that the existence of a presymplectic form in the variational bicomplex, when horizontally closed on solutions, allows us to construct a variational formulation for a subsystem of the given PDE. No constraints on the differential order or number of dependent or independent variables are assumed. The proof follows a recent observation of Bridges, Hydon, and Lawson [Math. Proc. Cambridge Philos. Soc. 148(01), 159–178 (2010)] and generalizes an older result of Henneaux [Ann. Phys. 140(1), 45–64 (1982)] from ordinary differential equations (ODEs) to PDEs. Uniqueness of the variational formulation is also discussed.

Pontryagin forms on [formula omitted]-manifolds and symplectic structures on the spaces of Riemannian metrics

The Pontryagin forms on the 1-jet bundle of Riemannian metrics, are shown to provide in a natural way diffeomorphism-invariant pre-symplectic structures on the space of Riemannian metrics for the dimensions n≡2(mod4). The equivariant Pontryagin forms provide canonical moment maps for these structures. In dimension two, the symplectic reduction corresponding to the pre-symplectic form and its moment map attached to the first Pontryagin form, is proved to coincide with the Teichmüller space endowed with the Weil–Petersson symplectic form.

Chern–Simons pre-quantizations over four-manifolds

We endow the space of connections on an SU ( n ) -principal bundle over a four-manifold with a pre-symplectic structure and define a Hamiltonian action on it of the group of gauge transformations that are trivial on the boundary. Then we consider the trivial SU ( n ) -principal bundle for n ⩾ 3 over the four-manifold that is a submanifold of a null-cobordant four-manifold, and we construct on the moduli space of flat connections a hermitian line bundle with connection whose curvature is given by the pre-symplectic form. This is the Chern–Simons pre-quantization of moduli spaces. The group of gauge transformations on the boundary of the four-manifold acts on the moduli space of flat connections by an infinitesimally symplectic way. When the four-manifold is a 4-dimensional disc we show that this action is lifted to the pre-quantization by its Lie group extension. The geometric description of the latter is related to the 4-dimensional Wess–Zumino–Witten model.