Trivial Structure Research Articles

Localizing gravity on exotic thick 3-branes

We consider localization of gravity on thick branes with a non trivial structure. Double walls that generalize the thick Randall-Sundrum solution, and asymmetric walls that arise from a Z_2-symmetric scalar potential, are considered. We present a new asymmetric solution: a thick brane interpolating between two AdS_5 spacetimes with different cosmological constants, which can be derived from a ``fake supergravity'' superpotential, and show that it is possible to confine gravity on such branes.

Classification of Subsystems for Graded-Local Nets with Trivial Superselection Structure

We classify Haag-dual Poincaré covariant subsystems of a graded-local net on 4D Minkowski spacetime which satisfies standard assumptions and has trivial superselection structure. The result applies to the canonical field net of a net of local observables satisfying natural assumptions. As a consequence, provided that it has no nontrivial internal symmetries, such an observable net is generated by (the abstract versions of) the local energy-momentum tensor density and the observable local gauge currents which appear in the algebraic formulation of the quantum Noether theorem. Moreover, for a net of local observables as above, we also classify the Poincaré covariant local extensions which preserve the dynamics.

Second Quantized Frobenius Algebras

We show that given a Frobenius algebra there is a unique notion of its second quantization, which is the sum over all symmetric group quotients of nth tensor powers, where the quotients are given by symmetric group twisted Frobenius algebras. To this end, we consider the setting of Frobenius algebras given by functors from geometric categories whose objects are endowed with geometric group actions and prove structural results, which in turn yield a constructive realization in the case of nth tensor powers and the natural permutation action. We also show that naturally graded symmetric group twisted Frobenius algebras have a unique algebra structure already determined by their underlying additive data together with a choice of super–grading. Furthermore we discuss several notions of discrete torsion and show that indeed a non–trivial discrete torsion leads to a non–trivial super structure on the second quantization.

Injective Hopf bimodules, cohomologies of infinite dimensional Hopf algebras and graded-commutativity of the Yoneda product

We prove that the category of Hopf bimodules over any Hopf algebra has enough injectives, which enables us to extend some results on the unification of Hopf bimodule cohomologies of [R. Taillefer, PhD thesis, 2001; arXiv preprint math.QA/0005019] to the infinite dimensional case. We also prove that the cup-product defined on these cohomologies is graded-commutative. Unlike the algebra case (see [S. Schwede, J. Reine Angew. Math. 498 (1998) 153–172]), these methods do not give a non-trivial Gerstenhaber algebra structure on the cohomology we consider. We also comment that the other approach to finding such a structure that we know of (see [M. Farinati, A. Solotar, arXiv preprint math.KT/0207243]) also gives a trivial Gerstenhaber algebra structure.

Nonextensive statistical mechanics: A brief introduction

Boltzmann-Gibbs statistical mechanics is based on the entropy $S_{BG}=-k \sum_{i=1}^W p_i \ln p_i$. It enables a successful thermal approach of ubiquitous systems, such as those involving short-range interactions, markovian processes, and, generally speaking, those systems whose dynamical occupancy of phase space tends to be ergodic. For systems whose microscopic dynamics is more complex, it is natural to expect that the dynamical occupancy of phase space will have a less trivial structure, for example a (multi)fractal or hierarchical geometry. The question naturally arises whether it is possible to study such systems with concepts and methods similar to those of standard statistical mechanics. The answer appears to be {\it yes} for ubiquitous systems, but the concept of entropy needs to be adequately generalized. Some classes of such systems can be satisfactorily approached with the entropy $S_q=k\frac{1-\sum_{i=1}^W p_i^q}{q-1}$ (with $q \in \cal R$, and $S_1 =S_{BG}$). This theory is sometimes referred in the literature as {\it nonextensive statistical mechanics}. We provide here a brief introduction to the formalism, its dynamical foundations, and some illustrative applications. In addition to these, we illustrate with a few examples the concept of {\it stability} (or {\it experimental robustness}) introduced by B. Lesche in 1982 and recently revisited by S. Abe.

Two-Dimensional Gauge Theories of the Symmetric Group S n in the Large-n Limit

We study the two-dimensional gauge theory of the symmetric group S n describing the statistics of branched n-coverings of Riemann surfaces. We consider the theory defined on the disc and on the sphere in the large-n limit. A non trivial phase structure emerges, with various phases corresponding to different connectivity properties of the covering surface. We show that any gauge theory on a two-dimensional surface of genus zero is equivalent to a random walk on the gauge group manifold: in the case of S n , one of the phase transitions we find can be interpreted as a cutoff phenomenon in the corresponding random walk. A connection with the theory of phase transitions in random graphs is also pointed out. Finally we discuss how our results may be related to the known phase transitions in Yang-Mills theory. We discover that a cutoff transition occurs also in two dimensional Yang-Mills theory on a sphere, in a large N limit where the coupling constant is scaled with N with an extra log N compared to the standard ‘t Hooft scaling.

Asymptotic estimates of Dirac and Laplace eigenvalues on warped products over S 1

In this paper we show that the space of spinors over a warped product over S1 has a certain splitting ΓΣ=⊕W(k,n) in spaces of spinors of weight k and winding number n which is respected by the Dirac operator. The same holds for the space of functions and the Laplace operator. We give eigenvalue estimates for the eigenvalues λk,n of the Dirac operator with eigenspinors of weight k and winding number n and eigenvalues μm,n of the Laplace operator with eigenfunctions of weight m and winding number n. In particular, we show that μ2k,n≥λ2k,n holds for large n on S1×fTl where Tl is a flat torus with the trivial spin structure.

The Bundle Structure of Noncommutative Tori over $UHF$-Algebras

The noncommutative torus $C^*(\mathbb Z^n,\,\omega)$ of rank $n$ is realized as the $C^*$-algebra of sections of a locally trivial continuous $C^*$-algebra bundle over $\widehat{S_{\omega}}$ with fibres $C^*(\mathbb Z^n/S_{\omega},\, \omega_1)$ for some totally skew multiplier $\omega_1$ on $\mathbb Z^n/S_{\omega}$. It is shown that $C^*(\mathbb Z^n/S_{\omega},\,\omega_1)$ is isomorphic to $A_{\varphi}\otimes M_k(\mathbb C)$ for some completely irrational noncommutative torus $A_{\varphi}$ and some positive integer $k$, and that $A_{\omega} \otimes M_{l^{\infty}}$ has the trivial bundle structure if and only if the set of prime factors of $k$ is a subset of the set of prime factors of $l$. This is applied to understand the bundle structure of the tensor products of Cuntz algebras with noncommutative tori.

Towards classification of semigraphoids

Semigraphoids are special sets of triples (I,J,K), I,J,K disjoint subsets of a finite set, that mimic conditional independences. New constructions on semigraphoids are introduced, the most crucial being factors and expansions. They are aimed at study of new classes of semigraphoids, that are constructed from semigraphoids of a trivial structure, e.g. from uniform semigraphoids, and at bringing each semigraphoid to a canonical form. Canonical semigraphoids are defined and each semigraphoid is constructed from a canonical one by means of a pure minor and an expansion. Semigraphoid closure and generators are investigated. The case of two generators is analysed in detail. Invariants for semigraphoids based on relations among generators are introduced and the corresponding classes of semigraphoids are related to classes built from uniform semigraphoids. Representability of semigraphoids by linear spaces and random variables is reexamined. The semigraphoids with at most two generators are proved to be linear and hence, by a simple lemma, probabilistic.

Quasiequatorial gravitational lensing by spinning black holes in the strong field limit

Spherically symmetric black holes produce, by strong field lensing, two infinite series of relativistic images, formed by light rays winding around the black hole at distances comparable to the gravitational radius. In this paper, we address the relevance of the black hole spin for the strong field lensing phenomenology, focusing on trajectories close to the equatorial plane for simplicity. In this approximation, we derive a two-dimensional lens equation and formulae for the position and the magnification of the relativistic images in the strong field limit. The most outstanding effect is the generation of a non trivial caustic structure. Caustics drift away from the optical axis and acquire finite extension. For a high enough black hole spin, depending on the source extension, we can practically observe only one image rather than two infinite series of relativistic images. In this regime, additional non equatorial images may play an important role in the phenomenology.

Nontrivial Galois module structure of cyclotomic fields

We say a game Galois field extension L/K with Galois group G has trivial Galois module structure if the rings of integers have the property that OL is a free OK[G]-module. The work of Greither, Rep...

On a special kind of homomorphism defined on the multiplicative group of an algebraic number field

For a Galois extension L|K of algebraic number fields with the group G and any abelian group A, we consider the group \( \Hom_{G}(L^{*}, A) \) of all G-homomorphisms \( L^{*} \rightarrow A \), where \( L^* \) and A are endowed with the natural and the trivial G-module structure, respectively. If G itself is abelian, then \( \Hom_{G}(L^{*}, G) \) consists of homomorphic couplings on L, which are the main tools for the construction of near fields.

Asymptotic Behaviour of Tame Nilpotent Harmonic Bundles with Trivial Parabolic Structure

Let E be a holomorphic vector bundle. Let θ be a Higgs field, that is a holomorphic section of End (E) ⊗ Ω1,0X satisfying θ2 = 0. Let h be a pluriharmonic metric of the Higgs bundle (E, θ). The tuple (E, θ, h) is called a harmonic bundle. Let X be a complex manifold, and D be a normal crossing divisor of X. In this paper, we study the harmonic bundle (E, θ, h) over X − D. We regard D as the singularity of (E, θ, h), and we are particularly interested in the asymptotic behaviour of the harmonic bundle around D. We will see that it is similar to the asymptotic behaviour of complex variation of polarized Hodge structures, when the harmonic bundle is tame and nilpotent with the trivial parabolic structure. For example, we prove constantness of general monodromy weight filtrations, compatibility of the filtrations, norm estimates, and the purity theorem. For that purpose, we will obtain a limiting mixed twistor structure from a tame nilpotent harmonic bundle with trivial parabolic structure, on a punctured disc. It is a solution of a conjecture of Simpson.

Hochschild homology criteria for trivial algebra structures

We prove two similar results by quite different methods. The first one deals with augmented artinian algebras over a field: we characterize the trivial algebra structure on the augmentation ideal in terms of the maximality of the dimensions of the Hochschild homology (or cyclic homology) groups. For the second result, let X X be a 1-connected finite CW-complex. We characterize the trivial algebra structure on the cohomology algebra of X X with coefficients in a fixed field in terms of the maximality of the Betti numbers of the free loop space.

Nontrivial Galois module structure of cyclotomic fields

We say a tame Galois field extension L / K L/K with Galois group G G has trivial Galois module structure if the rings of integers have the property that O L \mathcal {O}_{L} is a free O K [ G ] \mathcal {O}_{K}[G] -module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes l l so that for each there is a tame Galois field extension of degree l l so that L / K L/K has nontrivial Galois module structure. However, the proof does not directly yield specific primes l l for a given algebraic number field K . K. For K K any cyclotomic field we find an explicit l l so that there is a tame degree l l extension L / K L/K with nontrivial Galois module structure.

Nil R-Mod Abelian Groups

We study mixed abelian groups such that End(M), the endomorphism ring of M, has no nilpotent elements other than 0. These are the abelian groups M that only admit the trivial R-module structure for any zero-ring R.

Site Selective Processes: A Combined Theoretical and Experimental Investigation of Thermally Activated Tautomerization Processes in 2(2,4-Dinitrobenzyl) Pyridine Derivatives

The thermally activated NH→CH tautomerization process in crystals of 2(2,4-dinitrobenzyl) pyridine derivatives was investigated. The results indicate the absence of any trivial molecular structure–reactivity correlation, in contrast to what is found for similar systems in solutions. The absence of molecular structure–reactivity correlation in the crystalline state is attributed to the participation of neighboring molecules in the proton transfer process. A combined experimental and theoretical approach is presented and applied to the study of the tautomerization reaction in crystalline environments.

Some remarks on Buchsbaum bundles

Let E be a vector bundle on Pn . We say E is Buchsbaum if Hp(EM(∗)) has trivial module structure for each linear space M and for all 1≤p≤dimM−1. In this note, we show that a Buchsbaum bundle is either an Ω-bundle (i.e. a direct sum of Ωp(h)'s, where Ωp is the pth exterior power of the contangent bundle of Pn) or the quotient of an Ω-bundle by a direct sum of line bundles. We also give a strategy to characterize/construct Buchsbaum bundles, give conditions for an Ω-bundle to be extendable.

Webs, Lenard schemes, and the local geometry of bi-Hamiltonian Toda and Lax structures

We introduce a criterion that a given bi-Hamiltonian structure admits a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bi-Hamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with constant coefficients. This shows that the open Toda lattice cannot be locally represented as a product of two bi-Hamiltonian structures. Near, a generic point, the bi-Hamiltonian periodic Toda lattice is shown to be isomorphic to a product of two open Toda lattices (one of which is a (trivial) structure of dimension 1). While the above results might be obtained by more traditional methods, we use an approach based on general results on geometry of webs. This demonstrates the possibility of applying a geometric language to problems on bi-Hamiltonian integrable systems; such a possibility may be no less important than the particular results proved in this paper. Based on these geometric approaches, we conjecture that decompositions similar to the decomposition of the periodic Toda lattice exist in local geometry of the Volterra system, the complete Toda lattice, the multidimensional Euler top, and a regular bi-Hamiltonian Lie coalgebra. We also state general conjectures about the geometry of more general finite-dimensional bi-Hamiltonian structures. The class of homogeneous structures is shown to coincide with the class of systems integrable by Lenard scheme. The bi-Hamiltonian structures which admit a non-degenerate Lax structure are shown to be locally isomorphic to the open Toda lattice.

Classification of Subsystems for Local Nets¶with Trivial Superselection Structure

Let F be a local net of von Neumann algebras in four spacetime dimensions satisfying certain natural structural assumptions. We prove that if F has trivial superselection structure then every covariant, Haag-dual subsystem B is the fixed point net under a compact group action on one component in a suitable tensor product decomposition of F. Then we discuss some application of our result, including free field models and certain theories with at most countably many sectors.