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.

Geometrical semantics for linear logic (multiplicative fragment)

A standard question in the study of geometric quantization is whether symplectic reduction interacts nicely with the quantized theory, and in particular whether “quantization commutes with reduction.” Guillemin and Sternberg first proposed this question, and answered it in the affirmative for the case of a free action of a compact Lie group on a compact Kahler manifold. Subsequent work has focused mainly on extending their proof to non-free actions and non-Kahler manifolds. For realistic physical examples, however, it is desirable to have a proof which also applies to non-compact symplectic manifolds. In this thesis we give a proof of the quantization-reduction problem for general symplectic manifolds. This is accomplished by working in a particular wavefunction representation, associated with a polarization that is in some sense compatible with reduction. While the polarized sections described by Guillemin and Sternberg are nonzero on a dense subset of the Kahler manifold, the ones considered here are distributional, having support only on regions of the phase space associated with certain quantized, or admissible, values of momentum. We first propose a reduction procedure for the prequantum geometric structures that “covers” symplectic reduction, and demonstrate how both symplectic and prequantum reduction can be viewed as examples of foliation reduction. Consistency of prequantum reduction imposes the above-mentioned admissibility conditions on the quantized momenta, which can be seen as analogues of the Bohr-Wilson-Sommerfeld conditions for completely integrable systems. We then describe our reduction-compatible polarization, and demonstrate a one-to-one correspondence between polarized sections on the unreduced and reduced spaces. Finally, we describe a factorization of the reduced prequantum bundle, suggested by the structure of the underlying reduced symplectic manifold. This in turn induces a factorization of the space of polarized sections that agrees with its usual decomposition by irreducible representations, and so proves that quantization and reduction do indeed commute in this context. A significant omission from the proof is the construction of an inner product on the space of polarized sections, and a discussion of its behavior under reduction. In the concluding chapter of the thesis, we suggest some ideas for future work in this direction.

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:

We understand the symplectic geometry on manifolds not just as the geometry of the symplectic manifolds but as the study of various geometric structures on manifolds where a symplectic vector bundle plays an essential role. In the present chapter, we shall discuss about such structures. Of course, symplectic manifolds are the core of this theory. We shall also discuss the Lagrangian subbundles naturally related with the considered structures.KeywordsVector BundleSymplectic FormSymplectic ManifoldSymplectic StructureCotangent BundleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

Of the three organizers of the conference N. Hitchin is a mathematician, A. Kapustin and W. Nahm are physicists. Ideas from physics, in particular from string theory and related areas had a profound influence on algebraic and differential geometry. Conversely, many developments in string theory use very recent mathematical results. There were 19 talks at the workshop, 12 by mathematicians and 7 by physicists, but in many cases an outside observer would have had difficulties to sort them out. Quantum field theory often allows continuous interpolations between geometrical structures of topologically different manifolds. A particularly important example is mirror symmetry, where the complexified Kähler cone of one manifold corresponds to the moduli space of complex structures of the mirror manifold. Several talks explored geometrical and number theoretical aspects of this connection. Mirror symmetry can be understood in terms of torus fibrations and an isomorphism between the quantum field theories given by maps to a torus and to its dual (T-duality). The B-field of string theory can be described in terms of non-commutative geometry on the tori, and its effect on T-duality has been investigated. The quantum field theories which provide these dualities are supersymmetric conformally invariant quantum field theories in two dimensions. In mathematics, large parts of their structures have been axiomatised in the language of vertex operator algebras, but much remains to be done. A promising mathematical approach is provided by the chiral de Rham complex. This cohomology of this complex seems to be invariant under mirror symmetry, but its physical meaning is not yet clear. This led to many discussions between mathematicians and physicists. Indeed, the free time for discussions was at least as important as the lectures for learning the language of researchers with a different background and for making use of their ideas. An area in which the common understanding is well advance is topological field theory. Here one has good axioms and many of the analytic problems of conformally invariant quantum field theories can be ignored. One can solve rather complex problems and one can make contact with new geometrical ideas, in particular generalised complex structures. The latter allow to interpolate between complex and symplectic geometry, which should become important for both physics and mathematics. To understand all of conformal field theory at a similar mathematical depth will take more time, but already now relations to mixed Hodge structures have given much insight. Eventually many so far intractable problems should be solvable by the use of quantum field theoretical ideas, for example the construction of the non-singular Ricci flat metrics on K3. Perturbative string theory can be described in terms of conformal field theory, but the complete theory has more excitations than strings. Among them the best understood are the D-branes, many features of which can be understood in terms of boundary states in conformal field theory. Some of these branes are related to complex submanifolds, others to certain Lagrangian submanifolds, and their description involves twisted K-theory and derived categories. The development of a common language between mathematicians and physicists is well under way, and the workshop has made a contribution which all participants found very stimulating. In Europe such meetings between mathematicians and string theorists are rare, so the workshop was particularly useful. Much of the time was reserved for informal discussions. On one evening there was a discussion on the relation between classical and quantum geometry. Weather predictions for the free afternoon had been bad, but finally the weather was splendid and the supply of Black Forest cake matched the demand.

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.

Let (M,J,ω) be a manifold with an almost complex structure J tamed by a symplectic form ω. We suppose that M has the complex dimension two, is Levi-convex and with bounded geometry. We prove that a real two-sphere with two elliptic points, embedded into the boundary of M can be foliated by the boundaries of pseudoholomorphic discs.

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.

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden–Weinstein–Meyer reduction, Mikami–Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg–Kazhdan construction of Moore–Tachikawa varieties in topological quantum field theory. A key feature of our construction is a concrete and systematic association of a Hamiltonian $G$-space $\mathfrak {M}_{G, S}$ to each pair $(G,S)$, where $G$ is any Lie group and $S\subseteq \mathrm {Lie}(G)^{*}$ is any submanifold satisfying certain non-degeneracy conditions. The spaces $\mathfrak {M}_{G, S}$ satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. Although these Hamiltonian $G$-spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new.

In this article, we provide an exposition about symplectic toric manifolds, which are symplectic manifolds (M2n, ω) equipped with an effective Hamiltonian \(\mathbb {T}^n\cong (S^1)^n\)-action. We summarize the construction of M as a symplectic quotient of \(\mathbb {C}^d\), the \(\mathbb {T}^n\)-actions on M and their moment maps, and Guillemin’s Kähler potential on M. While the theories presented in this paper are for compact toric manifolds, they do carry over for some noncompact examples as well, such as the canonical line bundle KM, which is one of our main running examples, along with the complex projective space \(\mathbb {P}^n\) and its canonical bundle \(K_{\mathbb {P}^n}\). One main topic explored in this article is how to write the moment map in terms of the complex homogeneous coordinates \(z\in \mathbb {C}^d\), or equivalently, the relationship between the action-angle coordinates and the complex toric coordinates. We end with a brief review of homological mirror symmetry for toric geometries, where the main connection with the rest of the paper is that KM provides a prototypical class of examples of a Calabi-Yau toric manifold Y which serves as the total space of a symplectic fibration \(W: Y \to \mathbb {C}\) with a singular fiber above 0, known as a Landau-Ginzburg model in mirror symmetry. Here we write W in terms of the action-angle coordinates, which will prove to be useful in understanding the geometry of the fibration in our forthcoming work (Azam et al., Global homological mirror symmetry for genus two curves, in preparation).

This paper meticulously revisit and study the flux geometry of any compact connected oriented manifold (M, Ω). We generalize several well- known factorization results, exhibit some orbital conditions under which flux geometry can be studied, give a proof of the discreteness of the flux group for volume-preserving diffeomorphisms, derive that any smooth isotopy in the group of all vanishing-flux volume-preserving diffeomorphisms is a vanishing- flux path, and show that the kernel of flux for volume-preserving diffeomorphisms is C 1−closed inside the group of all volume-preserving diffeomorphisms isotopic to the identity map: We recover several well-known results from symplectic geometry. We use the above studies to construct a right-invariant metric on the group of all volume-preserving diffeomorphisms isotopic to the identity map and study the induced geometry. In the case of a symplectic volume form, the restriction of our metric to the group Ham(N, ω), of all Hamiltonian diffeomorphisms of a closed symplectic manifold (N, ω), is controlled from above by the usual Hofer metric in general, while the Hofer-like metric control our metric in the case where the Riemannian structure is compatible with the symplectic structure (in particular, our construction implies the non-degeneracy of the Hofer and Hofer-like norms).

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.