Topological classification of unknotted ring defects

Unknotted ring defects in ordered media are classified in terms of the homotopy theory. It is also investigated what type of point defects will appear when a radius of the ring defect tends to zero.     

APPLYING THE HOMOTOPY EQUIVALENCE TRANSFORMATION OF TOPOLOGICAL SPACE SETS TO THE TOPOLOGICAL CLASSIFICATION OF STATES AND DEFECTS IN ORDERED MEDIA

In this paper the application of homotopy equivalence transformation (HET) of topological space sets to the topological classification of states and defects in ordered media is discussed. Firstly, an argument is pres-ented about the idea that for simplifying and even working out the classification and constructing homotopy class sets into groups, it is crucial to utilize the HET. As the theoretical basis for doing this we sum up the relevant results in homotopy theory into a theorem, called the "invariance theorem for HET". Secondly, in order to favor the utilization of this theorem, several propositions on homotopy equivalence between space sets are given. Finally, the absolute and relative topological dassification of states and defects is systemtically studied. The main results obtained are embodied in eight theorems.     

同伦群理论在液晶缺陷中的应用

本文介绍了液晶缺陷研究的进展,并用同伦群理论讨论了各种类型液晶中不同维数的缺陷的分类,还简单讨论了液晶中缺陷强度所遵循的拓扑性质。     

有序介质中缺陷的拓扑理论

本文评述介绍了有序介质中缺陷的拓扑理论,着重于表述同伦论的数学框架并讨论了几个理想的可解的例。     

The homology of defective crystal lattices and the continuum limit

The problem of extending fields that are defined on lattices to fields defined on the continua that they become in the continuum limit is basically one of continuous extension from the 0‐skeleton of a simplicial complex to its higher‐dimensional skeletons. If the lattice in question has defects, as well as the order parameter space of the field, then this process might be obstructed by characteristic cohomology classes on the lattice with values in the homotopy groups of the order parameter space. The examples from solid‐state physics that are discussed are quantum spin fields on planar lattices with point defects or orientable space lattices, vorticial flows or director fields on lattices with dislocations or disclinations, and monopole fields on lattices with point defects.     

Differentials on graph complexes II: hairy graphs

We study the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory. We provide spectral sequences converging to zero whose first pages contain the hairy graph cohomology. Our results yield a way to construct many nonzero hairy graph cohomology classes out of (known) non-hairy classes by studying the cancellations in those sequences. This provide a first glimpse at the tentative global structure of the hairy graph cohomology.     

On the topological equivalence of gauge and Higgs fields in the dyon sectors

We prove by topological methods methods that in the context of SU(2) gauge theory the integers labelling the homotopy classes of Yang-Mills and Higgs fields are equal for periodic instanton and dyon configurations. A similar statement is true for the group SO(3) and, in the periodic instanton case, whenever a simply connected group is broken down to an abelian subgroup. We briefly discuss how the result goes over to the canonically quantized theory.     

Classifying spaces for quantum principal bundles

A construction of a quantum analogue of principal bundles is discussed. Deformations of quantum groups in the sense of Woronowicz allow to relax the condition of local triviality of a principal bundle; the fibres need not be all identical any longer. This leads to deformations of structure group and bundles. There is still a classifying space in the sense that homotopy classes of bundles are classified by homotopy classes of maps from the base space into the classifying space.     

Tensor constructions of open string theories (I) foundations

The possible tensor constructions of open string theories are analyzed from first principles. To this end the algebraic framework of open string field theory is clarified, including the role of the homotopy associative A_∞ algebra, the odd symplectic structure, cyclicity, star conjugation, and twist. It is also shown that two string theories are off-shell equivalent if the corresponding homotopy associative algebras are homotopy equivalent in a strict sense.It is demonstrated that a homotopy associative star algebra with a compatible even bilinear form can be attached to an open string theory. If this algebra does not have a space-time interpretation, positivity and the existence of a conserved ghost number require that its cohomology is at degree zero, and that it has the structure of a direct sum of full matrix algebras. The resulting string theory is shown to be physically equivalent to a string theory with a familiar open string gauge group.     

BFV-Complex and Higher Homotopy Structures

We present a connection between the BFV-complex (abbreviation for Batalin-Fradkin-Vilkovisky complex) and the strong homotopy Lie algebroid associated to a coisotropic submanifold of a Poisson manifold. We prove that the latter structure can be derived from the BFV-complex by means of homotopy transfer along contractions. Consequently the BFV-complex and the strong homotopy Lie algebroid structure are L _∞ quasi-isomorphic and control the same formal deformation problem. However there is a gap between the non-formal information encoded in the BFV-complex and in the strong homotopy Lie algebroid respectively. We prove that there is a one-to-one correspondence between coisotropic submanifolds given by graphs of sections and equivalence classes of normalized Maurer-Cartan elemens of the BFV-complex. This does not hold if one uses the strong homotopy Lie algebroid instead.     

Quantum Open-Closed Homotopy Algebra and String Field Theory

We reformulate the algebraic structure of Zwiebach’s quantum open-closed string field theory in terms of homotopy algebras. We call it the quantum open-closed homotopy algebra (QOCHA) which is the generalization of the open-closed homotopy algebra (OCHA) of Kajiura and Stasheff. The homotopy formulation reveals new insights about deformations of open string field theory by closed string backgrounds. In particular, deformations by Maurer Cartan elements of the quantum closed homotopy algebra define consistent quantum open string field theories.     

Homotopy Classification of Bosonic String Field Theory

We prove the decomposition theorem for the loop homotopy Lie algebra of quantum closed string field theory and use it to show that closed string field theory is unique up to gauge transformations on a given string background and given S-matrix. For the theory of open and closed strings we use results in open-closed homotopy algebra to show that the space of inequivalent open string field theories is isomorphic to the space of classical closed string backgrounds. As a further application of the open-closed homotopy algebra, we show that string field theory is background independent and locally unique in a very precise sense. Finally, we discuss topological string theory in the framework of homotopy algebras and find a generalized correspondence between closed strings and open string field theories.     

管状和闭合磁畴壁的拓扑分类

本文从拓扑学角度,对管状磁畴壁和闭合磁畴壁静态结构的分类问题,做了统一处理。这两种磁畴壁的同伦类集合G_W⁽ⁿ⁾和G_W⁽ⁿ⁾,分别与S²→S²和S³→S²的n+1基点连续映射的相对同化类集合是一一对应的,因而构成同构于整数加群Z的群,分别称为n式管壁群和n式闭壁群。重新讨论了Slonczewski等人定义的“绕数”,找出了用它表征管壁 关键词：     

Time-delay in potential scattering theory

We show that a gauge field uniquely determines its potential if and only if its holonomy group coincides with the gauge group on every open set in spacetime, provided that the field is not degenerate as a 2-form over spacetime. In other words, there is no potential ambiguity whenever such a field is irreducible everywhere in spacetime. We then show that the ambiguous potentials for those gauge fields are partitioned into gauge-equivalence classes (modulo certain homotopy classes) as a consequence of the nontrivial connectivity of spacetime. These homotopy classes depend on the gauge group, on the holonomy group and on this last group's centralizer in the gauge group.To the Memory of Jorge André SwiecaResearch supported by C.N.Pq. and M.E.C. (Brazil)     

多基点同伦群理论

本文建立了一个定理:当拓扑空间V单连通时,N(≥1)基点n阶同伦类集合π_n(V;v₁,v₂,…,v_N)可被构造成同构于单基点n阶同伦群π_n(V)的群,叫做N基点n阶同伦群;并且V的单连通性一般是不可省略的条件。给出了一些推论。简要地触及了这个定理及其推论在铁磁状态拓扑分类中的应用问题。 关键词：     

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.     

Changing topology and nontrivial homotopy

A method is proposed for extending the path integral to a sum over paths which change topology. This is used to show that topological contributions to a field theory path integral arising from nontrivial homotopy on the background space are completely determined in the free theory. The method is applied to extend (“third quantize”) field theory to backgrounds which change topology (e.g. string theory) and to show that topological contributions from nontrivial homotopy of field histories in the target space are completely determined by the field theory on a fixed background.     

The correlation of length spectra of two hyperbolic surfaces

We derive an asymptotic formula for the number of free homotopy classes on a closed surface which have (approximately) the same length with respect to two different hyperbolic structures on the surface. The growth rate in the asymptotic formula is described in terms of the thermodynamic formalism for the geodesic flow.     

Topological Classification of the Magnetization States in the Hole-Including Ferromagnets

In this paper the problem of topological classification of the magnetization states in the hole-including ferromagnets is comprehensively studied. The main result is that the set of homotopy classes of magnetization states, in a ferromagnet including a family of noninteracting holes of cylinder, sphere and annulus types with numbers m, k and l, respectively, can be constructed into a group isomorphic to z^k+l, the (k + l)-dimensional discrete vector group.