Symplectic Geometry on Manifolds

Chapter 3 Symplectic geometry

Toric varieties play an important role both in symplectic and complex geometry. In symplectic geometry, the construction of a symplectic toric manifold from a smooth polytope is due to Delzant. In algebraic geometry, there is a more general construction using fans rather than polytopes. However, in case the fan is induced by a smooth polytope Audin showed both constructions to give isomorphic projective varieties. For rational but not necessarily smooth polytopes the Delzant construction was refined by Lerman and Tolman, leading to symplectic toric orbifolds or more generally, symplectic toric DM stacks (Lerman and Malkin). We show that the stacks resulting from the Lerman-Tolman construction are isomorphic to the stacks obtained by Borisov et al. in case the stacky fan is induced by a polytope. No originality is claimed (cf. also an article by Sakai). Rather we hope that this text serves as an example driven introduction to symplectic toric geometry for the algebraically minded reader.

We study the mirror operation of the Atiyah flop in symplectic geometry. We formulate the operation for a symplectic manifold with a Lagrangian fibration. Furthermore we construct geometric stability conditions on the derived Fukaya category of the deformed conifold and study the action of the mirror Atiyah flop on these stability conditions.

This paper discusses the local and global geometry of coisotropic foliations and complex polarizations on symplectic manifolds and draws attention to an analogy between coisotropic foliations and Hermitian vector bundles, in which connections and characteristic classes are modelled by objects in symplectic geometry.

Multisymplectic geometry is a generalization of symplectic geometry suitable for n-dimensional field theories, in which the nondegenerate 2-form of symplectic geometry is replaced by a nondegenerate (n + 1)-form. The case n = 2 is relevant to string theory: we call this '2-plectic geometry.' Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, the observables associated to a 2-plectic manifold form a 'Lie 2-algebra,' which is a categorified version of a Lie algebra. Any compact simple Lie group G has a canonical 2-plectic structure, so it is natural to wonder what Lie 2-algebra this example yields. This Lie 2-algebra is infinite-dimensional, but we show here that the sub-Lie-2-algebra of left-invariant observables is finite-dimensional, and isomorphic to the already known 'string Lie 2-algebra' associated to G. So, categorified symplectic geometry gives a geometric construction of the string Lie 2-algebra. © 2010, John C. Baez and Christopher L. Rogers.

The aim of this chapter is to introduce the basic problems and (soft!) techniques in symplectic geometry by presenting examples—more exactly series of examples— of almost complex and symplectic manifolds: it is obviously easier to understand the classification of symplectic ruled surfaces if you have already heard of Hirzebruch surfaces for instance.

Preface * 1. Symplectic Manifolds * 2. Principal S1-bundles * 3. Contact Manifolds * 4. Associated Metrics * 5. Integral Submanifolds and Contact Transformations * 6. Sasakian and Cosymplectic Manifolds * 7. Curvature of Contact Metric Manifolds * 8. Submanifolds of Kahler and Sasakian Manifolds * 9. Tangent Bundles and Tangent Sphere Bundles * 10. Curvature Functionals and Spaces of Associated Metrics * 11. Negative Xi-sectional Curvature * 12. Complex Contact Manifolds * 13. Additional Topics in Complex Geometry * 14. 3-Sasakian Manifolds * Bibliography * Subject Index * Author Index

This study examines the intricate interplay between Hamiltonian dynamics and spectral analysis within the context of algebraic structures and symmetry groups. By integrating perspectives from symplectic geometry, Lie theory, and nonlinear dynamics, we introduce a framework that aims to unify classical conservation laws with advanced spectral decomposition techniques. Our approach synthesizes geometric mechanics with adaptive numerical strategies, revealing novel relationships between phase-space invariants and multi-scale Fourier analysis. Applications in quantum mechanics, signal processing, and machine learning illustrate the potential of Hamiltonian-preserving algorithms to maintain spectral integrity while ensuring long-term stability. These findings assist in detecting chaotic transitions and demonstrate compatibility with established FFT methodologies, thereby offering new insights into the synergy between geometric structures and spectral methods.

It is shown that the geometry of quantum theory can be derived from geometrical structure that may be considered more fundamental. The basic elements of this reconstruction of quantum theory are the natural metric on the space of probabilities (information geometry), the description of dynamics using a Hamiltonian formalism (symplectic geometry), and requirements of consistency (Kähler geometry). The theory that results is standard quantum mechanics, but in a geometrical formulation that includes also a particular case of a family of nonlinear gauge transformations introduced by Doebner and Goldin. The analysis is carried out for the case of discrete quantum mechanics. The work presented here relies heavily on, and extends, previous work done in collaboration with M. J. W. Hall.

The Schläfli identity, which is important in Regge calculus and loop quantum gravity, is examined from a symplectic and semiclassical standpoint in the special case of flat, three-dimensional space. In this case a proof is given, based on symplectic geometry. A series of symplectic and Lagrangian manifolds related to the Schläfli identity, including several versions of a Lagrangian manifold of tetrahedra, are discussed. Semiclassical interpretations of the various steps are provided. Possible generalizations to three-dimensional spaces of constant (nonzero) curvature, involving Poisson–Lie groups and q-deformed spin networks, are discussed.

For a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure), we construct a sequence consisting of differential operators using a symplectic torsion-free affine connection. All but one of these operators are of first order. The first order ones are symplectic analogues of the twistor operators known from Riemannian spin geometry. We prove that under the condition the symplectic Weyl curvature tensor field of the symplectic connection vanishes, the mentioned sequence forms a complex. This gives rise to a new complex for the so called Ricci type symplectic manifolds, which admit a metaplectic structure.

This is a book on symplectic topology, a rapidly developing field of mathematics which originated as a geometric tool for problems of classical mechanics. Since the 1980s, powerful methods such as Gromov's pseudo-holomorphic curves and Morse-Floer theory on loop spaces gave rise to the discovery of unexpected symplectic phenomena. The present book focuses on function spaces associated with a symplectic manifold. A number of recent advances show that these spaces exhibit intriguing properties and structures, giving rise to an alternative intuition and new tools in symplectic topology. The book provides an essentially self-contained introduction into these developments along with applications to symplectic topology, algebra and geometry of symplectomorphism groups, Hamiltonian dynamics and quantum mechanics. It will appeal to researchers and students from the graduate level onwards.

Let G G be a compact connected Lie group, and ( M , ω ) (M,\omega ) a Hamiltonian G G -space with proper moment map μ \mu . We give a surjectivity result which expresses the K K -theory of the symplectic quotient M / / G M /\!\!/G in terms of the equivariant K K -theory of the original manifold M M , under certain technical conditions on μ \mu . This result is a natural K K -theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the K K -theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian G G -spaces. We discuss this lemma in detail and highlight the differences between the K K -theory and rational cohomology versions of this lemma. We also introduce a K K -theoretic version of equivariant formality and prove that when the fundamental group of G G is torsion-free, every compact Hamiltonian G G -space is equivariantly formal. Under these conditions, the forgetful map K G ∗ ( M ) → K ∗ ( M ) K_{G}^{*}(M)\to K^{*}(M) is surjective, and thus every complex vector bundle admits a stable equivariant structure. Furthermore, by considering complex line bundles, we show that every integral cohomology class in H 2 ( M ; Z ) H^{2}(M;\mathbb {Z}) admits an equivariant extension in H G 2 ( M ; Z ) H_{G}^{2}(M;\mathbb {Z}) .

A real Bott manifold is the total space of an iterated R P 1 \mathbb {R}P^1 -bundle over a point, where each R P 1 \mathbb {R}P^1 -bundle is the projectivization of a Whitney sum of two real line bundles. In this paper, we characterize real Bott manifolds which admit a symplectic form. In particular, it turns out that a real Bott manifold admits a symplectic form if and only if it is cohomologically symplectic. In this case, it admits even a Kähler structure. We also prove that any symplectic cohomology class of a real Bott manifold can be represented by a symplectic form. Finally, we study the flux of a symplectic real Bott manifold.

The Heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum-mechanical particle. This can be rephrased as follows: the smallest subsets of classical phase space in which the presence of a quantum-mechanical particle can be detected are its Lagrangian submanifolds. For this reason it makes sense to regard the Lagrangian submanifolds of phase space as being its true "points"; see Weinstein [17]. Now let G be a compact Lie group and G x X ~ X a Hamiltonian action of G on X (see w for definitions). It is well-known that the fixed points of this action form a symplectic submanifold of X. (See for instance Guillemin and Sternberg [5].) However, what can one say about the fixed "points" of G? We will show that they are also the "points'" of a symplectic manifold, Xc. This manifold is the Marsden-Weinstein reduction of X with respect to the zero orbit in g*, and will be described in Sect. 2. (It was introduced in a completely different context fl'om ours by Marsden and Weinstein [12].) Problems in classical mechanics can often be reduced to the study of Hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on Hilbert space. This fact has inspired a number of efforts to "quantize" symplectic geometry by devising schemes for associating Hilbert space to symplectic manifolds. The "no-go" theorems of Groenwald and Van Hove impose some embarrassing limitations on all such schemes; however, it seems to be a useful idea heuristically to think of every symplectic manifold, X~a~ic, 1, as being symbiotically associated with a Hilbert space, Xq,a, t ..... in such a way that the classical observables on the first space correspond to quantum observables on the second space. The heuristics further suggests that if G is a group of symmetries of X~la~,i~, j, it should also be a group of symmetries of Xq,a,,tum. In this heuristic spirit, we will state the main conjecture of this paper: