Complex of twistor operators in symplectic spin geometry

Chapter 3 Symplectic geometry

<p style='text-indent:20px;'>In this paper we look at the question of integrability, or not, of the two natural almost complex structures <inline-formula><tex-math id="M1">\begin{document}$ J^{\pm}_\nabla $\end{document}</tex-math></inline-formula> defined on the twistor space <inline-formula><tex-math id="M2">\begin{document}$ J(M, g) $\end{document}</tex-math></inline-formula> of an even-dimensional manifold <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> with additional structures <inline-formula><tex-math id="M4">\begin{document}$ g $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \nabla $\end{document}</tex-math></inline-formula> a <inline-formula><tex-math id="M6">\begin{document}$ g $\end{document}</tex-math></inline-formula>-connection. We measure their non-integrability by the dimension of the span of the values of <inline-formula><tex-math id="M7">\begin{document}$ N^{J^\pm_\nabla} $\end{document}</tex-math></inline-formula>. We also look at the question of the compatibility of <inline-formula><tex-math id="M8">\begin{document}$ J^{\pm}_\nabla $\end{document}</tex-math></inline-formula> with a natural closed <inline-formula><tex-math id="M9">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-form <inline-formula><tex-math id="M10">\begin{document}$ \omega^{J(M, g, \nabla)} $\end{document}</tex-math></inline-formula> defined on <inline-formula><tex-math id="M11">\begin{document}$ J(M, g) $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M12">\begin{document}$ (M, g) $\end{document}</tex-math></inline-formula> we consider either a pseudo-Riemannian manifold, orientable or not, with the Levi Civita connection or a symplectic manifold with a given symplectic connection <inline-formula><tex-math id="M13">\begin{document}$ \nabla $\end{document}</tex-math></inline-formula>. In all cases <inline-formula><tex-math id="M14">\begin{document}$ J(M, g) $\end{document}</tex-math></inline-formula> is a bundle of complex structures on the tangent spaces of <inline-formula><tex-math id="M15">\begin{document}$ M $\end{document}</tex-math></inline-formula> compatible with <inline-formula><tex-math id="M16">\begin{document}$ g $\end{document}</tex-math></inline-formula>. In the case <inline-formula><tex-math id="M17">\begin{document}$ M $\end{document}</tex-math></inline-formula> is oriented we require the orientation of the complex structures to be the given one. In the symplectic case the complex structures are positive.</p>

The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold ℳ is presented as a second class constrained surface in the fibre bundle ?* ρℳ which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure. The second class system is converted into the first class one by continuation of the constraints into the extended manifold, being a direct sum of ?* ρℳ and the tangent bundle Tℳ. This extended manifold is equipped with a nontrivial Poisson bracket which naturally involves two basic ingredients of Fedosov geometry: the symplectic structure and the symplectic connection. The constructed first class constrained theory, being equivalent to the original symplectic manifold, is quantized through the BFV-BRST procedure. The existence theorem is proven for the quantum BRST charge and the quantum BRST invariant observables. The adjoint action of the quantum BRST charge is identified with the Abelian Fedosov connection while any observable, being proven to be a unique BRST invariant continuation for the values defined in the original symplectic manifold, is identified with the Fedosov flat section of the Weyl bundle. The Fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum BRST invariant observables.

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.

Let $M$ be a holomorphic symplectic K\"ahler manifold equipped with a Lagrangian fibration $\pi$ with compact fibers. The base of this manifold is equipped with a special K\"ahler structure, that is, a K\"ahler structure $(I, g, \omega)$ and a symplectic flat connection $\nabla$ such that the metric $g$ is locally the Hessian of a function. We prove that any Lagrangian subvariety $Z\subset M$ which intersects smooth fibers of $\pi$ and smoothly projects to $\pi(Z)$ is a toric fibration over its image $\pi(Z)$ in $B$, and this image is also special K\"ahler. This answers a question of N. Hitchin related to Kapustin-Witten BBB/BAA duality.

The aim of our article is the study of solution space of the symplectic twistor operator $$T_s$$ in symplectic spin geometry on standard symplectic space $$(\mathbb R ^{2n},\omega )$$ , which is the symplectic analogue of the twistor operator in (pseudo) Riemannian spin geometry. In particular, we observe a substantial difference between the case $$n=1$$ of real dimension $$2$$ and the case of $$\mathbb R ^{2n}, n>1$$ . For $$n>1$$ , the solution space of $$T_s$$ is isomorphic to the Segal–Shale–Weil representation.

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.

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.

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.

The Marsden-Weinstein reduction procedure fashions new symplectic manifolds from Hamiltonian actions on old symplectic manifolds. It is of considerable interest to determine the way in which quantization of the reduced phase spaces relates to that of the original phase spaces. In particular, it is natural to ask for conditions under which prequantization data may be passed through the operation of reduction. Note that reduced phase spaces can fail to be metaplectic, so in general it cannot be expected that traditional prequantization data (Kostant-Souriau line bundle plus metaplectic structure: see [2, 3] for example) will pass to reduced phase spaces; moreover, even when reduced phase spaces admit metaplectic structures, these do not generally arise from metaplectic structures on the original phase spaces in a natural manner. Following Hess, we take the view that prequantization data consists of a prequantized Mpc structure: see [6] for full details. In this note, we describe when and how such prequantization data may be passed to reduced phase spaces . En route, we encounter a generalized version of the Bohr-Wilson-Sommerfeld rule as an obstruction to this passage. We illustrate our account by reference to coadjoint orbits as reduced phase spaces.

An affine connection is one of the central objects in differential geometry. Long ago, Berger presented a very restricted list of possible irreducibly acting holonomies of torsion-free affine connections. This chapter explores that any torsion-free holomorphic affine connection with irreducibly acting holonomy group can, in principle, be constructed by twistor methods. It explains the key elements of the twistor constructions without indulging in rather lengthy proofs. The chapter focuses on the category of complex manifolds, holomorphic affine connections, and so on, though many results can be easily adapted to the real analytic case along the lines. The class of affine connections with very little torsion is a sibling of the class of torsion-free affine connections in the sense that both these classes have involutive holonomy bundles. This means in particular that all connections of both types can be constructed by twistor methods on appropriate Legendre moduli spaces.

Let Q: C ∞ (M)[[h]] → W D be a quantization map considered in the author’s papers [2, 3, 4]. Here M is a symplectic manifold of dimension 2n, W D the algebra of flat sections of the Weyl algebras bundle W with respect to an Abelian connection D on W. The coefficient bundle of W is supposed to be Hom(E,E) for some m-dimensional complex vector bundle E. The Abelian connection D depends thus on the symplectic connection ∂s on M and the connection ∂ E on E.

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.