Space-Time Defects and Group Momentum Space

We study massive and massless conical defects in Minkowski and de Sitter spaces in various space-time dimensions. The energy momentum of a defect, considered as an (extended) relativistic object, is completely characterized by the holonomy of the connection associated with its space-time metric. The possible holonomies are given by Lorentz group elements, which are rotations and null rotations for massive and massless defects, respectively. In particular, if we fix the direction of propagation of a massless defect in n+1-dimensional Minkowski space, then its space of holonomies is a maximal Abelian subgroup of the AN(n-1) group, which corresponds to the well known momentum space associated with the n-dimensional κ-Minkowski noncommutative space-time and κ-deformed Poincaré algebra. We also conjecture that massless defects in n-dimensional de Sitter space can be analogously characterized by holonomies belonging to the same subgroup. This shows how group-valued momenta related to four-dimensional deformations of relativistic symmetries can arise in the description of motion of space-time defects.

CommutativeC⁎-Algebras of Toeplitz Operators via the Moment Map on the Polydisk

We found that in the polydiskDnthere exist(n+1)(n+2)/2different classes of commutativeC⁎-algebras generated by Toeplitz operators whose symbols are invariant under the action of maximal Abelian subgroups of biholomorphisms. On the other hand, using the moment map associated with each (not necessary maximal) Abelian subgroup of biholomorphism we introduced a family of symbols given by the moment map such that theC⁎-algebra generated by Toeplitz operators with this kind of symbol is commutative. Thus we relate to each Abelian subgroup of biholomorphisms a commutativeC⁎-algebra of Toeplitz operators.

Confinement on R3 × S1: Continuum and lattice

There has been substantial progress in understanding confinement in a class of four-dimensional SU(N) gauge theories using semiclassical methods. These models have one or more compact directions, and much of the analysis is based on the physics of finite temperature gauge theories. The topology R3 × S1 has been most often studied using a small compactification circumference L such that the running coupling g2(L) is small. The gauge action is modified by a double-trace Polyakov loop deformation term, or by the addition of periodic adjoint fermions. The additional terms act to preserve Z(N) symmetry and thus confinement. An area law for Wilson loops is induced by a monopole condensate. In the continuum, the string tension can be computed analytically from topological effects. Lattice models display similar behavior, but the theoretical analysis of topological effects is based on Abelian lattice duality rather than on semiclassical arguments. In both cases, the key step is reducing the low-energy symmetry group from SU(N) to the maximal Abelian subgroup U(1)N-1 while maintaining Z(N) symmetry.

ANATOMY OF EINSTEIN'S THEORY: ABELIAN DECOMPOSITION OF GENERAL RELATIVITY

Treating Einstein's theory as a gauge theory of Lorentz group, we decompose the gravitational connection (the gauge potential of Lorentz group) Γμ into the restricted connection of the maximal Abelian subgroup of Lorentz group and the valence connection which transforms covariantly under Lorentz gauge transformation. With this decomposition we show that the Einstein's theory can be decomposed into the restricted part made of the restricted connection which has the full Lorentz gauge invariance and the valence part made of the valence connection which plays the role of gravitational source of the restricted gravity. We show that there are two different Abelian decomposition of Einstein's theory, the light-like (or null) decomposition and the non light-like (or non-null) decomposition. In this decomposition the role of the metric gμν is replaced by a four-index metric tensor gμν which transforms covariantly under the Lorentz group, and the metric-compatibility condition ∇αgμν = 0 of the connection is replaced by the gauge and generally covariant condition [Formula: see text]. The decomposition shows the existence of a restricted theory of gravitation which has the full general invariance but is much simpler and has less physical degrees of freedom than Einstein's theory. Moreover, it tells that the restricted gravity can be written as an Abelian gauge theory, which implies that the graviton can be described by a massless spin-one field.

ON HOLOMORPHIC EFFECTIVE ACTIONS OF HYPERMULTIPLETS COUPLED TO EXTERNAL GAUGE SUPERFIELDS

We study the structure of holomorphic effective action generated by hypermultiplet models interacting with background super-Yang–Mills fields. A general form of holomorphic effective action depending on background fields is found for hypermultiplet belonging to arbitrary representation of any semisimple compact Lie group spontaneously broken to its maximal Abelian subgroup. Resulting effective action is defined by the superfield strengths lying in Cartan subalgebra of the gauge algebra. The applications of the obtained results to hypermultiplets in fundamental and adjoint representations of the SU (n), SO (n), Sp (n) groups are considered.

A REMARK ON WITTEN EFFECT FOR QCD MONOPOLES IN MATRIX QUANTUM MECHANICS

In a recent work3 we argued that a certain matrix quantum mechanics may describe 't Hooft's monopoles which emerge in QCD when the theory is projected to its maximal Abelian subgroup. In this letter we find further evidence which supports this interpretation. We study the theory with a nonzero theta-term. In this case, 't Hooft's QCD monopoles become dyons since they acquire electric charges due to the Witten effect. We calculate a potential between a dyon and an anti-dyon in the matrix quantum mechanics, and find that the attractive force between them grows as the theta angle increases.

CSA-groups and separated free constructions

A group G is called a CSA-group if all its maximal Abelian subgroups are malnormal; that is, Mx ∩ M = 1 for every maximal Abelian subgroup M and x ∈ G − M. The class of CSA-groups contains all torsion-free hyperbolic groups and all groups freely acting on λ-trees. We describe conditions under which HNN-extensions and amalgamated products of CSA-groups are again CSA. One-relator CSA-groups are characterised as follows: a torsion-free one-relator group is CSA if and only if it does not contain F2 × Z or one of the nonabelian metabelian Baumslag-Solitar groups B1, n = 〈x, y | yxy−1 = xn〉, n ∈ Z ∂ {0, 1}; a one-relator group with torsion is CSA if and only if it does not contain the infinite dihedral group.

Maximal abelian subgroups of free profinite groups

The aim of this note is to answer in the negative a question of W. -D. Geyer, asked at the 1983 Group Theory Meeting in Oberwolfach: Is a maximal abelian subgroup A of a free profinite group F necessarily isomorphic to , the profinite completion of

Maximal Abelian Subgroups of the Symmetric Groups

Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires.Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdös and Turán deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turán who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn, what kind of properties hold for almost all subgroups of the class?