On complex and symplectic toric stacks

Chapter 3 Symplectic geometry

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.

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.

Moment maps, strict linear precision, and maximum likelihood degree one

Del Pezzo surfaces with irregularity and intersection numbers on quotients in geometric invariant theory Zachary Alexander Maddock This thesis comprises two parts covering distinct topics in algebraic geometry. In Part I, we construct the first examples of regular del Pezzo surfaces for which the first cohomology group of the structure sheaf is nonzero. Such surfaces, which only exist over imperfect fields, arise as generic fibres of fibrations of singular del Pezzo surfaces in positive characteristic whose total spaces are smooth, and their study is motivated by the minimal model program. We also find a restriction on the integer pairs that are possible as the irregularity (that is, the dimension of the first cohomology group of the structure sheaf) and anti-canonical degree of regular del Pezzo surfaces with positive irregularity. In Part II, we consider a connected reductive group acting linearly on a projective variety over an arbitrary field. We prove a formula that compares intersection numbers on the geometric invariant theory quotient of the variety by the reductive group with intersection numbers on the geometric invariant theory quotient of the variety by a maximal torus, in the case where all semi-stable points are properly stable. These latter intersection numbers involve the top equivariant Chern class of the maximal torus representation given by the quotient of the adjoint representation on the Lie algebra of the reductive group by that of the maximal torus. We provide a purely algebraic proof of the formula when the root system decomposes into irreducible root systems of type A. We are able to remove this restriction on root systems by applying a related result of Shaun Martin from symplectic geometry.

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:

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.

Simple polytopes are a classical object of convex geometry. They play a key role in many modern fields of research, such as algebraic and symplectic geometry, toric topology, enumerative combinatorics, and mathematical physics. In this paper, the results of a new approach based on a differential ring of simple polytopes are described. This approach allows one to apply the theory of differential equations to the study of combinatorial invariants of simple polytopes.

In this paper we discuss three distance functions on the set of convex bodies. In particular we study the convergence of Delzant polytopes, which are fundamental objects in symplectic toric geometry. By using these observations, we derive some convergence theorems for symplectic toric manifolds with respect to the Gromov–Hausdorff distance.

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.

The purpose of this workshop was to discuss recent results in Several Complex Variables, Complex Geometry and Symplectic/Contact Geometry with a special focus on the interplay and exchange of ideas among these areas, which proved to be very fruitful in the last years. The main topics of the workshop included Symplectic and Contact Geometry, Almost Complex Geometry, Pluripotential Theory and the Monge-Ampère equation, Complex Dynamics, Geometric Questions of Complex Analysis (including Theory of Foliations) and Applications, the \bar{\partial} -equation and Geometry.

The tree amplituhedra are mathematical objects generalising the notion of polytopes into the Grassmannian. Proposed for as a geometric construction encoding tree-level scattering amplitudes in planar super Yang–Mills theory, they are mathematically interesting for any . In this paper we strengthen the relation between scattering amplitudes and geometry by linking the amplituhedron to the Jeffrey–Kirwan residue, a powerful concept in symplectic and algebraic geometry. We focus on a particular class of amplituhedra in any dimension, namely cyclic polytopes, and their even-dimensional conjugates. We show how the Jeffrey–Kirwan residue prescription allows to extract the correct amplituhedron volume functions in all these cases. Notably, this also naturally exposes the rich combinatorial and geometric structures of amplituhedra, such as their regular triangulations.