Abelian Structure Research Articles

Calculating the parabolic Chern character of a locally abelian parabolic bundle

We calculate the parabolic Chern character of a bundle with locally abelian parabolic structure on a smooth strict normal crossings divisor, using the definition in terms of Deligne–Mumford stacks. We obtain explicit formulas for ch 1, ch 2 and ch 3, and verify that these correspond to the formulas given by Borne for ch 1 and Mochizuki for ch 2.

General Heart Construction on a Triangulated Category (I): Unifying t-Structures and Cluster Tilting Subcategories

In the paper of Keller and Reiten, it was shown that the quotient of a triangulated category (with some conditions) by a cluster tilting subcategory becomes an abelian category. After that, Koenig and Zhu showed in detail, how the abelian structure is given on this quotient category, in a more abstract setting. On the other hand, as is well known since 1980s, the heart of any t-structure is abelian. We unify these two constructions by using the notion of a cotorsion pair. To any cotorsion pair in a triangulated category, we can naturally associate an abelian category, which gives back each of the above two abelian categories, when the cotorsion pair comes from a cluster tilting subcategory, or a t-structure, respectively.

Right side of TeV scale spontaneousR-parity violation

We study a simple extension of the minimal supersymmetric standard model in which the Abelian sector of the theory consists of $B\ensuremath{-}L$ and right-handed isospin. In the minimal model this Abelian gauge structure is broken to the standard model hypercharge gauge group by nonvanishing vacuum expectation values of the right-handed sneutrinos, resulting in spontaneous $R$-parity violation. This theory can emerge as a low energy effective theory of a left-right symmetric theory realized at a high scale. We determine the mass spectrum of the theory, and discuss the generation of neutrino masses and $R$-parity violating interactions. The possibility of distinguishing between $R$-parity violating models with a gauged $U(1{)}_{B\ensuremath{-}L}$ broken at the TeV scale at the Large Hadron Collider is discussed.

Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review

A novel algebraic topology approach to supersymmetry (SUSY) and symmet- ry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconduc- tors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes ex- tended symmetries and quantum groupoid, algebroid and functorial representations of non- Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier- Stieltjes transforms, and duality relations link, respectively, the quantum groups and quan- tum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasi-triangular, quasi-Hopf algebras, bialgebroids, Grassmann-Hopf algebras and higher dimensional algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang-Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant general relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin) models with the extended quantum symmetry of entangled, 'string-net condensed' (ground) states.

An analogue of Feller's theorem for logarithmic combinatorial assemblies

We continue our investigations on iterated logarithm laws for additive functions defined on random combinatorial structures called assemblies or abelian partitional structures. Exploiting Feller's theorem, we obtain sharp upper bounds for a sequence of truncated additive functions. The results imply bounds for the sequence of sizes of components. The main ideas originated from the first author's number-theoretical papers.

From triangulated categories to abelian categories: cluster tilting in a general framework

A general framework for cluster tilting is set up by showing that any quotient of a triangulated category modulo a tilting subcategory (i.e., a maximal 1-orthogonal subcategory) carries an induced abelian structure. These abelian quotients turn out to be module categories of Gorenstein algebras of dimension at most one.

COBORDISMS OF FOLD MAPS AND MAPS WITH A PRESCRIBED NUMBER OF CUSPS

A generic smooth map of a closed $2k$-manifold into $(3k-1)$-space has a finite number of cusps ($\Sigma^{1,1}$-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities ($\Sigma^{1,0}$-singularities). Two fold maps are fold bordant if there are cobordisms between their source- and target manifolds with a fold map extending the two maps between the boundaries, if the two targets agree and the target cobordism can be taken as a product with a unit interval then the maps are fold cobordant. We compute the cobordism groups of fold maps of $(2k-1)$-manifolds into $(3k-2)$-space. Analogous cobordism semi-groups for arbitrary closed $(3k-2)$-dimensional target manifolds are endowed with Abelian group structures and described. Fold bordism groups in the same dimensions are described as well.

Counting abelian group structures

A bijective proof is given of a recurrence for the function counting the number of binary operations which endow a finite set with the structure of an abelian group. The proof depends on a lemma in “labelled homological algebra” and provides a simple route to a “curious result” of Philip Hall.

STABILITY OF ABELIAN COMPLEX STRUCTURES

Let M = Γ\G be a nilmanifold endowed with an invariant complex structure. We prove that Kuranishi deformations of abelian complex structures are all invariant complex structures, generalizing a result in [7] for 2-step nilmanifolds. We characterize small deformations that remain abelian. As an application, we observe that at real dimension six, the deformation process of abelian complex structures is stable within the class of nilpotent complex structures. We give an example to show that this property does not hold in higher dimension.

DEFORMATION OF 2-STEP NILMANIFOLDS WITH ABELIAN COMPLEX STRUCTURES

We develop deformation theory for abelian invariant complex structures on a nilmanifold, and prove that in this case the invariance property is preserved by the Kuranishi process. A purely algebraic condition characterizes the deformations leading again to abelian structures, and we prove that such deformations are unobstructed. Various examples illustrate the resulting theory, and the behavior possible in three complex dimensions.

On pairs of free modules over a Dedekind domain

The study of pairs of modules (over a Dedekind domain) arises from two different perspectives, as a starting step in the analysis of tuples of submodules of a given module, or also as a particular case in the analysis of Abelian structures made by two modules and a morphism between them. We discuss how these two perspectives converge to pairs of modules, and we follow the latter one to obtain an alternative approach to the classification of pairs of torsionfree objects. Then we restrict our attention to pairs of free modules. Our main results are that the theory of pairs of free Abelian groups is co-recursively enumerable, and that a few remarkable extensions of this theory are decidable.

Abelian subgroup structure of square complex groups and arithmetic of quaternions

A square complex is a 2-complex formed by gluing squares together. This article is concerned with the fundamental group Γ of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup structure of Γ is studied in some detail. Before outlining the results, it is necessary to describe the construction of Γ. In [Moz, Section 3], there is constructed a lattice subgroup Γ = Γp,l of G = PGL2(Qp)×PGL2(Ql), where p, l ≡ 1 (mod 4) are two distinct primes. This restriction was made because −1 has a square root in Qp if and only if p ≡ 1 (mod 4), but the construction of Γ is generalized in [Rat, Chapter 3] to all pairs (p, l) of distinct odd primes. The affine building ∆ of G is a product of two homogeneous trees of degrees (p + 1) and (l + 1) respectively. The group Γ is a finitely presented torsion free group which acts freely and transitively on the vertices of ∆, with a finite square complex as quotient ∆/Γ. Here is how Γ is constructed. Let

Two- and three-body color flux tubes in the chromodielectric model

Using the framework of the Chromo Dielectric Model we perform an analysis of color electric flux tubes in meson-like $q\bar{q}$ and baryon-like $qqq$ quark configurations. We discuss the Abelian color structure of the model and point out a symmetry in color space as a remnant of the SU(3) symmetry of QCD. The generic features of the model are discussed by varying the model parameters. We fix these parameters by reproducing the string tension $\tau=980 $MeV/fm and the transverse width $\rho=0.35 $fm of the $q\bar{q}$ flux tube obtained in lattice calculations. We use a bag constant $B^{1/4}=(240-260) $MeV, a glueball mass $m_g = (1000-1700) $MeV and a strong coupling constant $C_F \alpha_s = 0.2-0.3$. We show that the asymptotic string profile of an infinitely long flux tube is already reached for $q\bar{q}$ separations $R\ge1.0 $fm. A connection to the Dual Color Superconductor is made by extracting a magnetic current from the model equations and a qualitative agreement between the two descriptions of confinement is shown. In the study of the $qqq$ system we observe a $\mathsf{\Delta}$-like geometry for the color electric fields and a \textsf{Y}-like geometry in the scalar fields both in the energy density distribution and in the corresponding potentials. The resulting total $qqq$ potential is described neither by the $\mathsf{\Delta}$-picture nor by the \textsf{Y}-picture alone.

Elementary abelian Hopf Galois structures and polynomial formal groups

We find a relationship between regular embeddings of G, an elementary abelian p-group of order p n , into unipotent upper triangular matrices with entries in F p and commutative dimension n degree 2 polynomial formal groups with nilpotent upper triangular structure matrices. We classify the latter when n = 3 up to linear isomorphism, and use that classification to determine the number of Hopf Galois structures on a Galois extension L / K of fields with Galois group G elementary abelian of order p 3 . We also obtain a lower bound on the number of Hopf Galois structures on a Galois extension L / K when the Galois group is elementary abelian of order p n .

Phases of bosonic strings and two dimensional gauge theories

We suggest that the extrinsic curvature and torsion of a bosonic string can be employed as variables in a two dimensional Landau-Ginzburg gauge field theory. Their interpretation in terms of the abelian Higgs multiplet leads to two different phases. In the phase with unbroken gauge symmetry the ground state describes open strings while in the phase with broken gauge symmetry the ground state involves closed strings. When we allow for an additional abelian gauge structure along the string, we arrive at an interpretation in terms of the two dimensional SU(2) Yang-Mills theory.

Non Abelian Phases, Charge Pumping and Holonomic Computation with Josephson Junctions

In 1984 Berry pointed out that abelian gauge structures exist naturally in slowly varying quantum system. Since then, a great deal of interest both theoretically and experimentally has been devoted to the interpretation, generalization and detection of geometric phases. In particular, relevant for the present work, it has been shown that a non-Abelian gauge structure emerges if a set of quantum states remains degenerate as the Hamiltonian varies. Non-Abelian phases have been originally investigated in Nuclear Quadrupole Resonance. More recently, they have found a renewed interest within the field of Quantum Computation, since it has been proved that quantum gates can be also implemented by geometrical means using both Abelian and non-Abelian phases. In this novel geometric approach, known as Holonomic Quantum Computation (HQC), information is encoded in a degenerate eigenspace of a parametric family of Hamiltonians and the computational network of unitary quantum gates is realized by driving adiabatically the Hamiltonian parameters along loops in a control manifold. By properly designing such loops, the non trivial curvature of the geometry of the control manifold gives rise to unitary transformations, i.e. holonomies, that implement desired quantum gates. Until now only one scheme has been proposed to realize HQC in trapped ions by means of pulsed coherent excitations. On the other hand, it would be of great interest to have an implementation for solid-state systems, in view of their scalability and integrability. In particular, the impressing development of nanotechnologies with the consequent control of artificial two-level system in superconducting devices stimulates a more detailed investigation of geometric interference in mesoscopic systems. Here we show that non-Abelian phases can appear in the quantum charge dynamics of Josephson networks. Besides their possible detection, we shall see how they may be employed both in adiabatic charge pumping and as an implementation of HQC with Josephson junctions. 2. Josephson network for non abelian phases

Yang-Mills measure on compact surfaces

Discrete Yang-Mills measure Continuous Yang-Mills measure Abelian gauge theory Small scale structure in the semi-simple case Surgery of the Yang-Mills measure Bibliography.

HyperKähler torsion structures invariant by nilpotent Lie groups

We study hyperKähler torsion (HKT) structures on nilpotent Lie groups and on associated nilmanifolds. We show three weak HKT structures on ℝ8 which are homogeneous with respect to extensions of Heisenberg-type Lie groups. The corresponding hypercomplex structures are of a special kind called Abelian. We prove that on any 2-step nilpotent Lie group all invariant HKT structures arise from Abelian hypercomplex structures. Furthermore, we use a correspondence between Abelian hypercomplex structures and subspaces of \U0001d530p(n) to produce continuous families of compact and noncompact manifolds carrying non-isometric HKT structures. Finally, geometrical properties of invariant HKT structures on 2-step nilpotent Lie groups are obtained.

Splittings and Cr-structures for manifolds with nonpositive sectional curvature

Let M n denote the universal covering space of a compact Riemannian manifold, M n , with sectional curvature, −1≤K Mn ≤0. We show that a collection of deck transformations of M n , satisfying certain (metric dependent) conditions, determines an open dense subset of M n , at every point of which, there exists a local isometric splitting with nontrivial flat factor. Such a collection, which we call an abelian structure, also gives rise to an essentially canonical Cr-structure in the sense of Buyalo, i.e an atlas for an injective F-structure, for which additional conditions hold. It follows in particular that the minimal volume of M n vanishes. We show that an abelian structure exists if the injectivity radius at all points of M n is less than e(n)>0. This yields a conjecture of Buyalo as well as a strengthened version of the conclusion of Gromov’s “gap conjecture” in our special situation. In addition, we observe that abelian structures on nonpositively curved manifolds have certain stability properties under suitably controlled changes of metric.

Abelian hypercomplex structures on central extensions of H-type Lie algebras

It is the aim of this work to give a characterization of the two-step nilpotent Lie algebras carrying abelian hypercomplex structures. In the special case of trivial extensions of irreducible H-type Lie algebras this characterization is given in terms of the dimension of the commutator subalgebra. As a consequence, we obtain the corresponding theorem for arbitrary H-type Lie algebras, extending a result in Barberis and Dotti, J. Math. Oxford (2) 47 (1996) 389–404.