Affine Symmetry Research Articles

Common structures of quantum field theories and lattice systems through boundary symmetry

The sine-Gordon model and affine Toda field theories on the half-line, on the one hand, the XXZ spin chain with nondiagoual boundary terms, and interacting many-body lattice systems with a flow, on the other, have a common characteristic. They possess nonlocal conserved boundary charges, generating the Askey-Wilson algebra, a coideal subalgebra of the bulk quantized affine symmetry. We argue that the boundary Askey-Wilson symmetry is the deep algebraic property allowing for integrability of the physical system in consideration.

SPACE–TIME STRUCTURE IN THE ULTIMATE THEORY

After criticizing the various existing attempts at extending the concept of the Minkowski space, the following problem is considered: If there is the ultimate theory at all, how should the space–time in it be formulated? The principle of "quantum priority" is proposed, and under this principle, it is argued that the ultimate theory should have the affine symmetry in the framework of quantum gravity. It is shown that the Poincaré symmetry of particle physics is realized as a result of spontaneous breakdown of the affine symmetry.

A note on proper affine vector fields in non-static plane symmetric space-times

Themost general form of non-static plane symmetric space-times is considered to study proper affine vector fields by using holonomy and decomposability, the rank of the 6 × 6 Riemannmatrix and direct integration techniques. Studying proper affine vector fields in each nonstatic case of the above space-times it is shown that very special classes of the above space-times admit proper affine vector fields. We also discuss the Lie algebra in each case.

An Affine Symmetric Image Model and Its Applications

Natural images contain considerable redundancy, some of which is successfully captured using recently developed directional wavelets. In this paper, an affine symmetric image model is considered. It provides a flexible scheme to exploit geometric redundancy. A patch of texture from an image is rotated, scaled and sheared to approximate other similar parts in the image, revealing the self-similarity relation. The general scheme is derived as follows. A texture model is required that identifies structural patterns. Then the affine symmetry is exploited between structural textures at a local level, the objective being to find the minimum residual error by estimating the affine transform relating two patches of texture. Having developed a local model, the methodology is extended to the whole image to estimate the global affine relation. This global model is further developed in a multiresolution framework for multiscale analysis, by which the self similarity of the image is exploited across space and scale. The multiresolution model can be applied to a series of practical problems. Experimental evaluation demonstrates the effectiveness of the approach in affine invariant texture segmentation and image approximation.

Defect loops in gauged Wess-Zumino-Witten models

We consider loop observables in gauged Wess-Zumino-Witten models, and study the action of renormalization group flows on them. In the WZW model based on a compact Lie group G, we analyze at the classical level how the space of renormalizable defects is reduced upon the imposition of global and affine symmetries. We identify families of loop observables which are invariant with respect to an affine symmetry corresponding to a subgroup H of G, and show that they descend to gauge-invariant defects in the gauged model based on G/H. We study the flows acting on these families perturbatively, and quantize the fixed points of the flows exactly. From their action on boundary states, we present a derivation of the "generalized Affleck-Ludwig rule, which describes a large class of boundary renormalization group flows in rational conformal field theories.

Tensor generalizations of affine symmetry vectors

A definition is suggested for affine symmetry tensors, which generalize the notion of affine vectors in the same way that (conformal) Killing tensors generalize (conformal) Killing vectors. An identity for these tensors is proven, which gives the second derivative of the tensor in terms of the curvature tensor, generalizing a well-known identity for affine vectors. Additionally, the definition leads to a good definition of homothetic tensors. The inclusion relations between these types of tensors are exhibited. The relationship between affine symmetry tensors and solutions to the equation of geodesic deviation is clarified, again extending known results about Killing tensors.

Switching Rule Design for Switched Dynamic Systems With Affine Vector Fields

In this paper, we propose a method for designing switching rules that drive the state of the switched dynamic system to a desired equilibrium point. The method deals with the class of switched systems where each mode of operation is represented by a dynamical system with an affine vector field. The results are given in terms of linear matrix inequalities and they guarantee global asymptotic stability of the tracking error dynamics. Switching rules based on complete and partial state measurements are proposed. The case of uncertain polytopic systems is also considered and a numerical example illustrates the approach.

Equivalence of control systems with linear systems on Lie groups and homogeneous spaces

The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only if the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear if its flow is a one parameter group of automorphisms. An affine vector field is obtained by adding a left invariant one. Its projection on a homogeneous space, whenever it exists, is still called affine. Affine vector fields on homogeneous spaces can be characterized by their Lie brackets with the projections of right invariant vector fields. A linear system on a homogeneous space is a system whose drift part is affine and whose controlled part is invariant. The main result is based on a general theorem on finite dimensional algebras generated by complete vector fields, closely related to a theorem of Palais, and which has its own interest. The present proof makes use of geometric control theory arguments.

Higher genus partition functions of meromorphic conformal field theories

It is shown that the higher genus vacuum amplitudes of a meromorphic conformal field theory determine the affine symmetry of the theory uniquely, and we give arguments that suggest that also the representation content with respect to this affine symmetry is specified, up to automorphisms of the finite Lie algebra. We illustrate our findings with the self-dual theories at c = 16 and c = 24; in particular, we give an elementary argument that shows that the vacuum amplitudes of the E8 × E8 theory and the Spin(32)/2 theory differ at genus g = 5. The fact that the discrepancy only arises at rather high genus is a consequence of the modular properties of higher genus amplitudes at small central charges. In fact, we show that for c ≤ 24 the genus one partition function specifies already the partition functions up to g ≤ 4 uniquely. Finally we explain how our results generalise to non-meromorphic conformal field theories.

Transformations of locally conformally Kähler manifolds

We consider several transformation groups of a locally conformally K\"ahler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperk\"ahler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally K\"ahler manifold which is neither Weyl-reducible nor locally conformally hyperk\"ahler are holomorphic and conformal

ON SIX-DIMENSIONAL FUNDAMENTAL SUPERSTRINGS AS HOLOGRAMS

In this paper we discuss a possible holographic dual of the two-dimensional conformal field theory associated with the world-sheet of a macroscopic superstring in a compactification on a four-torus. We assume that the near-horizon geometry of the black string has symmetries of AdS 3×S3×T4 and construct a sigma model in the bulk. Analyzing the symmetries of the bulk theory and comparing them with those of the CFT in a special light-cone gauge, we find agreement between global symmetries. Due to nonstandard gauge realization it is not clear how affine symmetries can be realized.

Form factors of integrable higher-spin XXZ chains and the affine quantum-group symmetry

We derive exactly scalar products and form factors for integrable higher-spin XXZ chains through the algebraic Bethe-ansatz method. Here spin values are arbitrary and different spins can be mixed. Transforming the symmetric R-matrix into the asymmetric one that is compatible with subalgebra U q ( sl 2 ) of U q ( sl 2 ˆ ) , we calculate the exact expressions by the representation theory of U q ( sl 2 ) . The results should be fundamental for calculating form factors and correlation functions systematically for various solvable models associated with the integrable XXZ spin chains.

Affine symmetries of the equivariant quantum cohomology ring of rational homogeneous spaces

Let $X$ be a rational homogeneous space and let $QH^*(X)_{loc}^\times$ be the group of invertible elements in the small quantum cohomology ring of $X$ localised in the quantum parameters. We generalise results of \cite{cmp2} and realise explicitly the map $\pi_1({\rm Aut}(X))\to QH^*(X)_{loc}^\times$ described in \cite{seidel}. We even prove that this map is an embedding and realise it in the equivariant quantum cohomology ring $QH^*_T(X)_{loc}^\times$. We give explicit formulas for the product by these elements. The proof relies on a generalisation, to a quotient of the equivariant homology ring of the affine Grassmannian, of a formula proved by Peter Magyar \cite{Magyar}. It also uses Peterson's unpublished result \cite{Peterson} --- recently proved by Lam and Shimozono in \cite{Lam-Shi} --- on the comparison between the equivariant homology ring of the affine Grassmannian and the equivariant quantum cohomology ring.

Affine symmetry groups in 2D-quasicrystals

This paper is devoted to the mathematical description of possible finite affine symmetry groups in two-dimensional quasicrystals.

Baxter's Q-operator for the W-algebra WN

The q-oscillator representation for the Borel subalgebra of the affine symmetry is presented. By means of this q-oscillator representation, we give the free field realizations of Baxter's Q-operator for the W-algebra WN. We give functional relations of the T–Q operators, including the higher-rank generalization of Baxter's T–Q relation.

On lie algebras of affine vector fields of ideal realizations of holomorphic linear connections

We study the properties of real realizations of holomorphic linear connections over associative commutative algebras $$ \mathbb{A} $$ m with unity. The following statements are proved. If a holomorphic linear connection ∇ on M n over $$ \mathbb{A} $$ m (m ≥ 2) is torsion-free and R ≠ 0, then the dimension over ℝ of the Lie algebra of all affine vector fields of the space (M ℝ , ∇ℝ) is no greater than (mn)2 − 2mn + 5, where m = dimℝ $$ \mathbb{A} $$ , $$ n = dim_\mathbb{A} $$ M n , and ∇ℝ is the real realization of the connection ∇. Let ∇ℝ =1 ∇ ×2 ∇ be the real realization of a holomorphic linear connection ∇ over the algebra of double numbers. If the Weyl tensor W = 0 and the components of the curvature tensor 1 R ≠ 0, 2 R ≠ 0, then the Lie algebra of infinitesimal affine transformations of the space (M 2 ℝ , ∇ℝ) is isomorphic to the direct sum of the Lie algebras of infinitesimal affine transformations of the spaces ( a M n , a ∇) (a = 1, 2).

From quantum affine symmetry to the boundary Askey–Wilson algebra and the reflection equation

Within the quantum affine algebra representation theory, we construct linear covariant operators that generate the Askey–Wilson algebra. It has the property of a coideal subalgebra, which can be interpreted as the boundary symmetry algebra of a model with quantum affine symmetry in the bulk. The generators of the Askey–Wilson algebra are implemented to construct an operator-valued K-matrix, a solution of a spectral-dependent reflection equation. We consider the open driven diffusive system where the Askey–Wilson algebra arises as a boundary symmetry and can be used for an exact solution of the model in the stationary state. We discuss the possibility of a solution beyond the stationary state on the basis of the proposed relation of the Askey–Wilson algebra to the reflection equation.

ON SOME INFINITESIMAL AUTOMORPHISMS OF RIEMANNIAN FOLIATION

In Riemannian foliation, a transverse affine vector field preserves the curvature and its covariant derivatives. In this paper we solve the converse problem. Actually, we show that an infinitesimal automorphism of a Riemannian foliation which preserves the curvature and its covariant derivatives induces a transverse almost homothetic vector field. If in addition the manifold is closed and the foliation is irreducible harmonic, then a such infinitesimal automorphism induces a transverse killing vector field.

On real dimensions of lie algebras of holomorphic affine vector fields

On real dimensions of lie algebras of holomorphic affine vector fields

Affine Subdivision, Steerable Semigroups, and Sphere Coverings

Let ∆ be a Euclidean n-simplex and let {∆j} denote a finite union of simplices which partition ∆. We assume that the partition is invariant under the affine symmetry group of ∆. A classical example of such a partition is the one obtained from barycentric subdivision, but there are plenty of other possibilities. (See §4.1, or else [Sp, p. 123], for a definition of barycentric subdivision.) Our partition gives rise to an affine subdivision rule for ∆, which may be iterated. To subdivide each ∆j , we choose an affine map Aj with ∆j = Aj(∆), and then partition ∆j into the collection {Aj(∆i)}. The affine invariance of the partition translates into the fact that our partition of ∆j is independent of the (n+ 1)! different choices for Aj . Now we iterate. A basic question one can ask is Does the iteration of the subdivision rule produce a dense set of shapes of simplices? By shape of a simplex, we mean a simplex considered mod similarities. In [BBC] this question was raised and answered affirmatively for the case of 2-dimensional barycentric subdivision. In [S] we got the same result in 3 dimensions. In general, a first step for the kind of density results just mentioned is as follows: Let Cn be the collection of all n-dimensional simplices obtained by iteratively applying the subdivision rule. Let Ωn be the collection of matrices of the form ± L | det(L)|1/n ,