Born-Infeld Kinematics

We encode dynamical symmetries of Born-Infeld theory in a geometry on the tangent bundle of generally curved spacetime manifolds. The resulting covariant formulation of a maximal acceleration extension of special and general relativity is put to use in the discussion of particular point particle dynamics and the transition to a first quantized theory.     

Quantization of Poisson Families and of Twisted Poisson Structures

We describe a deformation quantization of a modification of Poisson geometry by a closed 3-form. Under suitable conditions, it gives rise to a stack of algebras. The basic object used for this aim is a kind of families of Poisson structures given by a Maurer–Cartan equation; they are easily quantized using Kontsevich's formality theorem. We conclude with a section on quantization of complex manifolds.     

Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries

Chern–Simons (CS) gauge theories in three dimensions and the Poisson sigma model (PSM) in two dimensions are examples of the same theory, if their field equations are interpreted as morphisms of Lie algebroids and their symmetries (on-shell) as homotopies of such morphisms. We point out that the (off-shell) gauge symmetries of the PSM in the literature are not globally well defined for non-parallelizable Poisson manifolds and propose a covariant definition of the off-shell gauge symmetries as left action of some finite-dimensional Lie algebroid.

Our approach allows us to avoid complications arising in the infinite-dimensional super-geometry of the BV- and AKSZ-formalism. This preprint is a starting point in a series of papers meant to introduce Yang–Mills type gauge theories of Lie algebroids, which include the standard YM theory, gerbes, and the PSM.     

Deformation Quantization of Algebraic Varieties

The paper is devoted to peculiarities of the deformation quantization in the algebro-geometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic geometry of an actual deformation (versus a formal one). Deformed algebras obtained by semiformal deformations are Noetherian and have polynomial growth. We propose constructions of semiformal quantizations of projective and affine algebraic Poisson manifolds satisfying certain natural geometric conditions. Projective symplectic manifolds (e.g. K3 surfaces and Abelian varieties) do not satisfy our conditions, but projective spaces with quadratic Poisson brackets and Poisson–Lie groups can be semiformally quantized.     

Symmetry Reduction in Symplectic and Poisson Geometry

We present a quick review of several reduction techniques for symplectic and Poisson manifolds using local and global symmetries compatible with these structures. Reduction based on the standard momentum map (symplectic or Marsden–Weinstein reduction) and on generalized distributions (the optimal momentum map and optimal reduction) is emphasized. Reduction of Poisson brackets is also discussed and it is shown how it defines induced Poisson structures on cosymplectic and coisotropic submanifolds.     

Symmetry Reduction in Symplectic and Poisson Geometry

We present a quick review of several reduction techniques for symplectic and Poisson manifolds using local and global symmetries compatible with these structures. Reduction based on the standard momentum map (symplectic or Marsden–Weinstein reduction) and on generalized distributions (the optimal momentum map and optimal reduction) is emphasized. Reduction of Poisson brackets is also discussed and it is shown how it defines induced Poisson structures on cosymplectic and coisotropic submanifolds.     

Wheeled Pro(p)file of Batalin-Vilkovisky Formalism

Using a technique of wheeled props we establish a correspondence between the homotopy theory of unimodular Lie 1-bialgebras and the famous Batalin-Vilkovisky formalism. Solutions of the so-called quantum master equation satisfying certain boundary conditions are proven to be in 1-1 correspondence with representations of a wheeled dg prop which, on the one hand, is isomorphic to the cobar construction of the prop of unimodular Lie 1-bialgebras and, on the other hand, is quasi-isomorphic to the dg wheeled prop of unimodular Poisson structures. These results allow us to apply properadic methods for computing formulae for a homotopy transfer of a unimodular Lie 1-bialgebra structure on an arbitrary complex to the associated quantum master function on its cohomology. It is proven that in the category of quantum BV manifolds associated with the homotopy theory of unimodular Lie 1-bialgebras quasi-isomorphisms are equivalence relations.     

Strong homotopy Lie algebras,homotopy Poisson manifolds and Courant algebroids

We study Maurer–Cartan elements on homotopy Poisson manifolds of degree n. They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson \(\mathfrak g\)-manifolds, and twisted Courant algebroids. Using the fact that the dual of an n-term \(L_\infty \)-algebra is a homotopy Poisson manifold of degree \(n-1\), we obtain a Courant algebroid from a 2-term \(L_\infty \)-algebra \(\mathfrak g\) via the degree 2 symplectic NQ-manifold \(T^*[2]\mathfrak g^*[1]\). By integrating the Lie quasi-bialgebroid associated to the Courant algebroid, we obtain a Lie-quasi-Poisson groupoid from a 2-term \(L_\infty \)-algebra, which is proposed to be the geometric structure on the dual of a Lie 2-algebra. These results lead to a construction of a new 2-term \(L_\infty \)-algebra from a given one, which could produce many interesting examples.     

Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model

General boundary conditions (branes') for the Poisson sigma model are studied. They turn out to be labeled by coisotropic submanifolds of the given Poisson manifold. The role played by these boundary conditions both at the classical and at the perturbative quantum level is discussed. It turns out to be related at the classical level to the category of Poisson manifolds with dual pairs as morphisms and at the perturbative quantum level to the category of associative algebras (deforming algebras of functions on Poisson manifolds) with bimodules as morphisms. Possibly singular Poisson manifolds arising from reduction enter naturally into the picture and, in particular, the construction yields (under certain assumptions) their deformation quantization.     

A Counterexample to the Quantizability of Modules

Let a Poisson structure on a manifold M be given. If it vanishes at a point m, the evaluation at m defines a one dimensional representation of the Poisson algebra of functions on M. We show that this representation can, in general, not be quantized. Precisely, we give a counterexample for , such that: (i) The evaluation map at zero can not be quantized to a representation of the algebra of functions with product the Kontsevich product associated to the Poisson structure. (ii) For any formal Poisson structure extending the given one and still vanishing at zero up to second order in epsilon, (i) still holds. We do not know whether the second claim remains true if one allows the higher order terms in epsilon to attain nonzero values at zero.      

Conformal Hamiltonian systems

Vector fields whose flow preserves a symplectic form up to a constant, such as simple mechanical systems with friction, are called “conformal”. We develop a reduction theory for symmetric conformal Hamiltonian systems, analogous to symplectic reduction theory. This entire theory extends naturally to Poisson systems: given a symmetric conformal Poisson vector field, we show that it induces two reduced conformal Poisson vector fields, again analogous to the dual pair construction for symplectic manifolds. Conformal Poisson systems form an interesting infinite-dimensional Lie algebra of foliate vector fields. Manifolds supporting such conformal vector fields include cotangent bundles, Lie–Poisson manifolds, and their natural quotients.     

Homotopy of rational maps and the quantization of Skyrmions

The Skyrme model is a classical field theory which models the strong interaction between atomic nuclei. It has to be quantized in order to compare it to nuclear physics. When the Skyrme model is semi-classically quantized it is important to take the Finkelstein-Rubinstein constraints into account. The aim of this paper is to show how to calculate these FR constraints directly from the rational map ansatz using basic homotopy theory. We then apply this construction in order to quantize the Skyrme model in the simplest approximation, the zero mode quantization. This is carried out for up to 22 nucleons and the results are compared to experiment.     

Canonical quantization of the Parisi-Sourlas model

The Parisi-Sourlas model is canonically quantized. The superalgebra of the symmetries present in this model is given. The Noether currents and charges for these symmetries are calculated.     

Morita Invariant Properties of Twisted Poisson Manifolds

In this letter, we study some invariants of integrable twisted Poisson manifolds under Morita equivalence. We show that two twisted symplectic manifolds are Morita equivalent if and only if their fundamental groups are isomorphic. Moreover, the first cohomology group of twisted Poisson manifolds is invariant under Morita equivalence. Finally, we prove that if two twisted Poisson manifolds are Morita equivalent, then their representation categories are equivalent.      

Poisson Quasi-Nijenhuis Manifolds

We introduce the notion of Poisson quasi-Nijenhuis manifolds generalizing Poisson-Nijenhuis manifolds of Magri-Morosi. We also investigate the integration problem of Poisson quasi-Nijenhuis manifolds. In particular, we prove that, under some topological assumption, Poisson (quasi)-Nijenhuis manifolds are in one-one correspondence with symplectic (quasi)-Nijenhuis groupoids. As an application, we study generalized complex structures in terms of Poisson quasi-Nijenhuis manifolds. We prove that a generalized complex manifold corresponds to a special class of Poisson quasi-Nijenhuis structures. As a consequence, we show that a generalized complex structure integrates to a symplectic quasi-Nijenhuis groupoid, recovering a theorem of Crainic. Francqui fellow of the Belgian American Educational Foundation. Research supported by NSF grant DMS03-06665 and NSA grant 03G-142.     

Symmetry Analysis and Conservation Laws for the Hunter-Saxton Equation

In this paper,the problem of determining the most general Lie point symmetries group and conservation laws of a well known nonlinear hyperbolic PDE in mathematical physics called the Hunter-Saxton equation(HSE) is analyzed.By applying the basic Lie symmetry method for the HSE,the classical Lie point symmetry operators are obtained.Also,the algebraic structure of the Lie algebra of symmetries is discussed and an optimal system of onedimensional subalgebras of the HSE symmetry algebra which creates the preliminary classification of group invariant solutions is constructed.Particularly,the Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained.Mainly,the conservation laws of the HSE are computed via three different methods including Boyer's generalization of Noether's theorem,first homotopy method and second homotopy method.     

Geometry and topology of chiral anomalies in gauge theories

Geometrical and topological aspects of chiral anomalies in gauge theories are reviewed. Geometrical and topological concepts and results for chiral anomalies in gauge theories are considered, including differential forms, Lie groups, homotopy, homology, cohomology, Riemannian manifolds, fibre bundles, characteristic classes, index theorems and spectral flow. Gauge theories and their formulation in terms of differential forms and fibre bundles are described. The quantisation of gauge theories is performed using path integrals, and the orbit space, BRST symmetries and ? vacuum are discussed. Gauge theories with fermions are formulated, including realistic models of strong and weak interactions. Chiral anomalies and related issues such as the existence of Schwinger terms, their origins in terms of differential forms, cohomology, the orbit space, BRST identities, Hamiltonian systems and relations to index theorems are analysed. Constraints on models for particle physics from chiral anomalies and theories involving spontaneously broken chiral symmetry described by effective Lagrangians are also mentioned.     

相对论Birkhoff系统动力学研究

给出相对论系统的Birkhoff函数和Birkhoff函数组、Pfaff作用量、PfaffBirkhoff原理、Birkhoff方程;研究相对论动力学系统的Birkhoff表示方法;根据在无限小变换下相对论Pfaff作用量的不变性和相对论Birkhoff方程的不变性,得到相对论Birkhoff系统的Noether对称性理论和Lie对称性理论;研究相对论Birkhoff系统的代数结构和Poisson积分方法. 关键词： 相对论 Birkhoff系统 Noether对称性 Lie对称性 代数结构 Poisson积分     

On Poisson geometries related to noncommutative emergent gravity

We study metric-compatible Poisson structures in the semi-classical limit of noncommutative emergent gravity. Space–time is realized as quantized symplectic submanifold embedded in R^D${\mathbb{R}}^D$, whose effective metric depends on the embedding as well as on the Poisson structure. We study solutions of the equations of motion for the Poisson structure, focusing on a natural class of solutions such that the effective metric coincides with the embedding metric. This leads to i$i$-(anti-) self-dual complexified Poisson structures in four space–time dimensions with Lorentzian signature. Solutions on manifolds with conformally flat metric are obtained and tools are developed which allow to systematically re-derive previous results, e.g. for the Schwarzschild metric. It turns out that the effective gauge coupling is related to the symplectic volume density, and may vary significantly over space–time. To avoid this problem, we consider in a second part space–time manifolds with compactified extra dimensions and split noncommutativity, where solutions with constant gauge coupling are obtained for several physically relevant geometries.