Symplectic Form Research Articles

A note on “Perturbation bounds for Williamson's symplectic normal form”

A perturbation bound for Williamson's symplectic normal form under any unitarily invariant norm has been derived in Idel et al. (2017) [3]. We improve this bound to a tight one, and derive a few alternative forms of our new bound.

Block perturbation of symplectic matrices in Williamson’s theorem

AbstractWilliamson’s theorem states that for any $2n \times 2n$ real positive definite matrix A, there exists a $2n \times 2n$ real symplectic matrix S such that $S^TAS=D \oplus D$ , where D is an $n\times n$ diagonal matrix with positive diagonal entries known as the symplectic eigenvalues of A. Let H be any $2n \times 2n$ real symmetric matrix such that the perturbed matrix $A+H$ is also positive definite. In this paper, we show that any symplectic matrix $\tilde {S}$ diagonalizing $A+H$ in Williamson’s theorem is of the form $\tilde {S}=S Q+\mathcal {O}(\|H\|)$ , where Q is a $2n \times 2n$ real symplectic as well as orthogonal matrix. Moreover, Q is in symplectic block diagonal form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of A. Consequently, we show that $\tilde {S}$ and S can be chosen so that $\|\tilde {S}-S\|=\mathcal {O}(\|H\|)$ . Our results hold even if A has repeated symplectic eigenvalues. This generalizes the stability result of symplectic matrices for non-repeated symplectic eigenvalues given by Idel, Gaona, and Wolf [Linear Algebra Appl., 525:45–58, 2017].

Paracausal deformations of Lorentzian metrics and Møller isomorphisms in algebraic quantum field theory

Given a pair of normally hyperbolic operators over (possibnly different) globally hyperbolic spacetimes on a given smooth manifold, the existence of a geometric isomorphism, called Møller operator, between the space of solutions is studied. This is achieved by exploiting a new equivalence relation in the space of globally hyperbolic metrics, called paracausal relation. In particular, it is shown that the Møller operator associated to a pair of paracausally related metrics and normally hyperbolic operators also intertwines the respective causal propagators of the normally hyperbolic operators and it preserves the natural symplectic forms on the space of (smooth) initial data. Finally, the Møller map is lifted to a *-isomorphism between (generally off-shell) CCR-algebras. It is shown that the Wave Front set of a Hadamard bidistribution (and of a Hadamard state in particular) is preserved by the pull-back action of this *-isomorphism.

Hydro & thermo dynamics at causal boundaries, examples in 3d gravity

We study 3-dimensional gravity on a spacetime bounded by a generic 2-dimensional causal surface. We review the solution phase space specified by 4 generic functions over the causal boundary, construct the symplectic form over the solution space and the 4 boundary charges and their algebra. The boundary charges label boundary degrees of freedom. Three of these charges extend and generalize the Brown-York charges to the generic causal boundary, are canonical conjugates of boundary metric components and naturally give rise to a fluid description at the causal boundary. Moreover, we show that the boundary charges besides the causal boundary hydrodynamic description, also admit a thermodynamic description with a natural (geometric) causal boundary temperature and angular velocity. When the causal boundary is the asymptotic boundary of the 3d AdS or flat space, the hydrodynamic description respectively recovers an extension of the known conformal or conformal-Carrollian asymptotic hydrodynamics. When the causal boundary is a generic null surface, we recover the null surface thermodynamics of [1] which is an extension of the usual black hole thermodynamics description.

Commuting symplectomorphisms on a surface and the flux homomorphism

Let $$(S,\omega )$$ be a closed connected oriented surface whose genus l is at least two equipped with a symplectic form. Then we show the vanishing of the cup product of the fluxes of commuting symplectomorphisms. This result may be regarded as an obstruction for commuting symplectomorphisms. In particular, the image of an abelian subgroup of $$\textrm{Symp}^c_0(S, \omega )$$ under the flux homomorphism is isotropic with respect to the natural intersection form on $$H^1(S;{\mathbb {R}})$$ . The key to the proof is a refinement of the non-extendability result, previously given by the first-named and second-named authors, for Py’s Calabi quasimorphism $$\mu _P$$ on $$\textrm{Ham}^c(S, \omega )$$ .

Motion of a spinning particle under the conservative piece of the self-force is Hamiltonian to first order in mass and spin

We consider the motion of a point particle with spin in a stationary spacetime. We define, following Witzany et al. [Classical Quantum Gravity 36, 075003 (2019)] and later Ramond [arXiv:2210.03866], a 12-dimensional Hamiltonian dynamical system whose orbits coincide with the solutions of the Mathisson-Papapetrou-Dixon equations of motion with the Tulczyjew-Dixon spin supplementary condition, to linear order in spin. We then perturb this system by adding the conservative pieces of the leading order gravitational self-force and self-torque sourced by the particle's mass and spin. We show that this perturbed system is Hamiltonian and derive expressions for the Hamiltonian function and symplectic form. This result extends a previous result for spinless point particles [F. M. Blanco and E. E. Flanagan, Phys. Rev. Lett. 130, 051201 (2023)].

Breakdown of rotational tori in 2D and 4D conservative and dissipative standard maps

We study the breakdown of rotational invariant tori in 2D and 4D standard maps by implementing three different methods. First, we analyze the domains of analyticity of a torus with given frequency through the computation of the Lindstedt series expansions of the embedding of the torus and the drift term. The Padé and log-Padé approximants provide the shape of the analyticity domains by plotting the poles of the polynomial at the denominator of the Padé approximants. Secondly, we implement a Newton method to construct the embedding of the torus; the breakdown threshold is then estimated by looking at the blow-up of the Sobolev norms of the embedding. Finally, we implement an extension of Greene method to get information on the breakdown threshold of an invariant torus with irrational frequency by looking at the stability of the periodic orbits with periods approximating the frequency of the torus.We apply these methods to 2D and 4D standard maps. The 2D maps can either be conservative (symplectic) or dissipative (more precisely, conformally symplectic, namely a dissipative map with the geometric property to transform the symplectic form into a multiple of itself). The conformally symplectic maps depend on a dissipative parameter and a drift term, which is needed to get the existence of invariant attractors. The 4D maps are obtained coupling (i) two symplectic standard maps, or (ii) two conformally symplectic standard maps, or (iii) a symplectic and a conformally symplectic standard map.Concerning the results, Padé and Newton methods perform well and provide reliable and consistent results (although we implemented Newton method only for symplectic and conformally symplectic maps). Our implementation of the extension of Greene method is inconclusive, since it is computationally expensive and delicate, especially in 4D non-symplectic maps, also due to the existence of Arnold tongues.

Geometry of symmetric spaces of type EVI

We generalize Atsuyama's result on the geometry of symmetric spaces of type EVI to the case of fields of characteristic zero. We relate the possible mutual positions of two points with the classification of balanced symplectic ternary algebras (also known as Freudenthal triple systems).

Note on the action with the Schwarzian at the stretched horizon

In this paper, we discuss the quantization of an interesting model of Carlip which appeared recently. It shows a way to associate boundary degrees of freedom to the stretched horizon of a stationary non-extremal black hole, as has been done in JT gravity for near-extremal black holes. The path integral now contains an integral over the boundary degrees of freedom, which are time reparametrizations of the stretched horizon keeping its length fixed. These boundary degrees of freedom can be viewed as elements of $Diff(S^1)/S^1$, which is the coadjoint orbit of an ordinary coadjoint vector under the action of the Virasoro group. From the symplectic form on this manifold, we obtain the measure in the boundary path integral. Doing a one-loop computation about the classical solution, we find that the one-loop answer is not finite, signalling that either the classical solution is unstable or there is an indefiniteness problem with this action, similar to the conformal mode problem in quantum gravity. Upon analytically continuing the field, the boundary partition function we get is independent of the inverse temperature and does not contribute to the thermodynamics at least at one-loop. This is in contrast to the study of near-extremal black holes in JT gravity, where the entire contribution to thermodynamics is from boundary degrees of freedom.

Pseudoholomorphic curves on the LCS-fication of contact manifolds

Abstract For each contact diffeomorphism ϕ : (Q, ξ) → (Q, ξ) of (Q, ξ), we equip its mapping torus Mϕ with a locally conformal symplectic form of Banyaga’s type, which we call the lcs mapping torus of the contact diffeomorphism ϕ. In the present paper, we consider the product Q × S 1 = M id (corresponding to ϕ = id) and develop basic analysis of the associated J-holomorphic curve equation, which has the form ∂ ˉ π w = 0 , w ∗ λ ∘ j = f ∗ d θ $$\bar{\partial}^{\pi} w=0, \quad w^{*} \lambda \circ j=f^{*} d \theta$$ for the map u = (w, f) : Σ ˙ → Q × S 1 $\dot{\Sigma} \rightarrow Q \times S^{1}$ for a λ-compatible almost complex structure J and a punctured Riemann surface ( Σ ˙ , j ) . $(\dot{\Sigma}, j).$ In particular, w is a contact instanton in the sense of [31], [32].We develop a scheme of treating the non-vanishing charge by introducing the notion of charge class in H 1 ( Σ ˙ , Z ) $H^{1}(\dot{\Sigma}, \mathbb{Z})$ and develop the geometric framework for the study of pseudoholomorphic curves, a correct choice of energy and the definition of moduli spaces towards the construction of a compactification of the moduli space on the lcs-fication of (Q, λ) (more generally on arbitrary locally conformal symplectic manifolds).

B-Contact structures on tentacular hyperboloids

This paper connects two different approaches to the analysis of Hamiltonian dynamics on non-compact energy hypersurfaces - b-symplectic geometry with its singular symplectic form and Floer techniques for tentacular Hamiltonians. More precisely, we show how to equip tentacular hyperboloids with a b-contact structure. We construct a b3-symplectic manifold (X,Z,ωb), such that each connected component of X∖Z is symplectomorphic to the standard symplectic space (T⁎Rn,ω0). For a tentacular hyperboloid S⊆T⁎Rn we look at its copies in X∖Z and show that their completion in (X,Z,ωb) is a smooth hypersurface of b-contact type.

Hamiltonian flows for pseudo-Anosov mapping classes

For a given pseudo-Anosov homeomorphism \varphi of a closed surface S , the action of \varphi on the Teichmüller space \mathcal{T}(S) preserves the Weil–Petersson symplectic form. We give explicit formulae for two invariant functions \mathcal{T}(S)\to \mathbb{R} whose symplectic gradients generate autonomous Hamiltonian flows that coincide with the action of \varphi at time one. We compute the Poisson bracket between these two functions. This amounts to computing the variation of length of a Hölder cocycle on one lamination along a shear vector field defined by another. For a measurably generic set of laminations, we prove that the variation of length is expressed as the cosine of the angle between the two laminations integrated against the product Hölder distribution, generalizing a result of Kerckhoff. We also obtain rates of convergence for the supports of germs of differentiable paths of measured laminations in the Hausdorff metric on a hyperbolic surface, which may be of independent interest.

Multi-image encryption scheme with quaternion discrete fractional Tchebyshev moment transform and cross-coupling operation

With quaternion theory, the traditional discrete fractional Tchebyshev transform is extended to the quaternion algebra domain for multi-image processing. A new multi-image encryption scheme based on quaternion discrete fractional Tchebyshev moment transform (QDFrTMT) and the cross-coupling chaotic system is suggested. The original images are first confused by fractal sorting square matrix and sine-logistic exponential chaotic map, and then the resulting image is divided into four matrices. With the quaternion symplectic form representation, a quaternion signal can be formed by the divided matrices, which can reduce the number of the discrete fractional Tchebyshev transforms used and enhance the computational efficiency. Subsequently, the quaternion array is encrypted with the proposed QDFrTMT. To overcome the weakness of the permutation-diffusion operation based on a single chaotic map, a Logistic-sine exponential chaotic map and a piece-wise linear chaotic map are cross-coupled. The final ciphertext images can be acquired by carrying out the dual-layer encryption processes in horizontal and vertical directions. Numerical simulations and security analyses verify the effectiveness of the multi-image encryption algorithm and its strong ability to counteract common attacks.

Topological Manin pairs and (n,s)-type series

Lie subalgebras of L = {mathfrak {g}}(!(x)!) times {mathfrak {g}}[x]/x^n{mathfrak {g}}[x] , complementary to the diagonal embedding Delta of {mathfrak {g}}[![x]!] and Lagrangian with respect to some particular form, are in bijection with formal classical r-matrices and topological Lie bialgebra structures on the Lie algebra of formal power series {mathfrak {g}}[![x]!] . In this work we consider arbitrary subspaces of L complementary to Delta and associate them with so-called series of type (n, s) . We prove that Lagrangian subspaces are in bijection with skew-symmetric (n, s) -type series and topological quasi-Lie bialgebra structures on {mathfrak {g}}[![x]!] . Using the classificaiton of Manin pairs we classify up to twisting and coordinate transformations all quasi-Lie bialgebra structures. Series of type (n, s) , solving the generalized classical Yang-Baxter equation, correspond to subalgebras of L. We discuss their possible utility in the theory of integrable systems.

Flat symplectic Lie algebras

Let be a symplectic Lie group, i.e., a Lie group endowed with a left invariant symplectic form. If is the Lie algebra of G then we call a symplectic Lie algebra. The product on defined by extends to a left invariant connection on G which is torsion free and symplectic (. When has vanishing curvature, we call a flat symplectic Lie group and a flat symplectic Lie algebra. In this paper, we study flat symplectic Lie groups. We start by showing that the derived ideal of a flat symplectic Lie algebra is degenerate with respect to ω. We show that a flat symplectic Lie group must be nilpotent with degenerate center. This implies that the connection of a flat symplectic Lie group is always complete. We prove that the flat symplectic double extension process can be applied to characterize all flat symplectic Lie algebras. More precisely, we show that every flat symplectic Lie algebra is obtained by a sequence of flat symplectic double extension of flat symplectic Lie algebras starting from {0}. As examples in low dimensions, we classify all flat symplectic Lie algebras of dimension .

A duality between vertex superalgebras [formula omitted] and [formula omitted] and generalizations to logarithmic vertex algebras

We introduce a certain subalgebra F‾ of the Clifford vertex superalgebra (bc system), which is completely reducible (as a LVir(−2,0)–module) and C2–cofinite, but not conformal (and not isomorphic to the symplectic fermion algebra SF(1)). We show that there is an interesting duality between SF(1) and F‾, given by the fact that F‾ can be equipped with the structure of a SF(1)–module and vice versa.Using the decomposition of F‾ and a free-field realization from [3], we decompose Lk(osp(1|2)) at the critical level k=−3/2 as a module for Lk(sl(2)). The decomposition of Lk(osp(1|2)) is exactly the same as that of the N=4 superconformal vertex algebra with central charge c=−9, denoted by V(2). Using the duality between F‾ and SF(1), we prove that Lk(osp(1|2)) and V(2) are in a duality of the same type. As an application, we construct and classify all irreducible Lk(osp(1|2))–modules in the category O and the category R, which includes relaxed highest weight modules. We also describe the structure of the parafermion algebra N−3/2(osp(1|2)) as a N−3/2(sl(2))–module.Extending this example, we introduce for each p≥2 a non-conformal vertex algebra Anew(p) and show that Anew(p) is isomorphic to the doublet vertex algebra as a module for the Virasoro algebra. We also construct the vertex algebra Vnew(p), which is isomorphic to the logarithmic vertex algebra V(p) as a module for slˆ(2).We also conjecture the existence of the conformal vertex algebra V(sp(2n)) which is in a duality with the affine vertex algebra L−n−1/2(osp(1|2n)).

Poisson Trace Orders

Abstract The two main approaches to the study of irreducible representations of orders (via traces and Poisson orders) have so far been applied in a completely independent fashion. We define and study a natural compatibility relation between the two approaches leading to the notion of Poisson trace orders. It is proved that all regular and reduced traces are always compatible with any Poisson order structure. The modified discriminant ideals of all Poisson trace orders are proved to be Poisson ideals and the zero loci of discriminant ideals are shown to be unions of symplectic cores, under natural assumptions (maximal orders and Cayley–Hamilton algebras). A base change theorem for Poisson trace orders is proved. A broad range of Poisson trace orders are constructed based on the proved theorems: quantized universal enveloping algebras, quantum Schubert cell algebras and quantum function algebras at roots of unity, symplectic reflection algebras, 3D and 4D Sklyanin algebras, Drinfeld doubles of pre-Nichols algebras of diagonal type, and root of unity quantum cluster algebras.

Characteristic Foliation on Hypersurfaces With Positive Beauville–Bogomolov–Fujiki Square

Abstract Let $Y$ be a smooth hypersurface in a projective irreducible holomorphic symplectic manifold X of dimension 2n. The characteristic foliation $F$ is the kernel of the symplectic form restricted to Y. In this article, we prove that a generic leaf of the characteristic foliation is dense in Y if Y has positive Beauville–Bogomolov–Fujiki square.

Asymptotically holomorphic theory for symplectic orbifolds

We extend Donaldson’s asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a compact symplectic orbifold such that the symplectic form defines an integer cohomology class, we prove that there exist sections of large tensor powers of the prequantizable line bundle such that their zero sets are symplectic suborbifolds. We then derive a Lefschetz hyperplane theorem for these suborbifolds, that computes their real cohomology up to middle dimension. We also get the hard Lefschetz and formality properties for them, when the ambient orbifold satisfies those properties.

Moduli Space of Factorized Ramified Connections and Generalized Isomonodromic Deformation

We introduce the notion of factorized ramified structure on a generic ramified irregular singular connection on a smooth projective curve. By using the deformation theory of connections with factorized ramified structure, we construct a canonical 2-form on the moduli space of ramified connections. Since the factorized ramified structure provides a duality on the tangent space of the moduli space, the 2-form becomes nondegenerate. We prove that the 2-form on the moduli space of ramified connections is d -closed via constructing an unfolding of the moduli space. Based on the Stokes data, we introduce the notion of local generalized isomonodromic deformation for generic unramified irregular singular connections on a unit disk. Applying the Jimbo-Miwa-Ueno theory to generic unramified connections, the local generalized isomonodromic deformationis equivalent to the extendability of the family of connections to an integrable connection. We give the same statement for ramified connections. Based on this principle of Jimbo-Miwa-Ueno theory, we construct a global generalized isomonodromic deformation on the moduli space of generic ramified connections by constructing a horizontal lift of a universal family of connections. As a consequence of the global generalized isomonodromic deformation, we can lift the relative symplectic form on the moduli space to a total closed form, which is called a generalized isomonodromic 2-form.