Symmetry analysis of crystalline spin textures in dipolar spinor condensates

We study periodic crystalline spin textures in spinor condensates with dipolar interactions via a systematic symmetry analysis of the low-energy effective theory. By considering symmetry operations which combine real and spin space operations, we classify symmetry groups consistent with non-trivial experimental and theoretical constraints. Minimizing the energy within each symmetry class allows us to explore possible ground states.Comment: 19 pages, 4 figure

Surgery groups of the fundamental groups of hyperplane arrangement complements

Using a recent result of Bartels and Lueck (arXiv:0901.0442) we deduce that the Farrell-Jones Fibered Isomorphism conjecture in L-theory is true for any group which contains a finite index strongly poly-free normal subgroup, in particular, for the Artin full braid groups. As a consequence we explicitly compute the surgery groups of the Artin pure braid groups. This is obtained as a corollary to a computation of the surgery groups of a more general class of groups, namely for the fundamental group of the complement of any fiber-type hyperplane arrangement in the complex n-space.Comment: 11 pages, AMSLATEX file, revised following referee's comments and suggestions, to appear in Archiv der Mathemati

From Koszul duality to Poincar\'e duality

We discuss the notion of Poincar\'e duality for graded algebras and its connections with the Koszul duality for quadratic Koszul algebras. The relevance of the Poincar\'e duality is pointed out for the existence of twisted potentials associated to Koszul algebras as well as for the extraction of a good generalization of Lie algebras among the quadratic-linear algebras.Comment: Dedicated to Raymond Stora. 27 page

From double Lie groupoids to local Lie 2-groupoids

We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger's fundamental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated local Lie 2-groupoid, which leads us to propose a definition of a symplectic 2-groupoid.Comment: 23 pages, a few minor changes, including a correction to Lemma 6.

Obstructing extensions of the functor Spec to noncommutative rings

In this paper we study contravariant functors from the category of rings to the category of sets whose restriction to the full subcategory of commutative rings is isomorphic to the prime spectrum functor Spec. The main result reveals a common characteristic of these functors: every such functor assigns the empty set to M_n(C) for n >= 3. The proof relies, in part, on the Kochen-Specker Theorem of quantum mechanics. The analogous result for noncommutative extensions of the Gelfand spectrum functor for C*-algebras is also proved.Comment: 23 pages. To appear in Israel J. Math. Title was changed; introduction was rewritten; old Section 2 was removed to streamline the exposition; final section was rewritten to omit an error in the earlier proof of Theorem 1.

Generalized Euler Angle Paramterization for SU(N)

In a previous paper (math-ph/0202002) an Euler angle parameterization for SU(4) was given. Here we present the derivation of a generalized Euler angle parameterization for SU(N). The formula for the calculation of the Haar measure for SU(N) as well as its relation to Marinov's volume formula for SU(N) will also be derived. As an example of this parameterization's usefulness, the density matrix parameterization and invariant volume element for a qubit/qutrit, three qubit and two three-state systems, also known as two qutrit systems, will also be given.Comment: 36 pages, no figures; added qubit/qutrit work, corrected minor definition problems and clarified Haar measure derivation. To be published in J. Phys. A: Math. and Ge

Cuts and flows of cell complexes

We study the vector spaces and integer lattices of cuts and flows associated with an arbitrary finite CW complex, and their relationships to group invariants including the critical group of a complex. Our results extend to higher dimension the theory of cuts and flows in graphs, most notably the work of Bacher, de la Harpe and Nagnibeda. We construct explicit bases for the cut and flow spaces, interpret their coefficients topologically, and give sufficient conditions for them to be integral bases of the cut and flow lattices. Second, we determine the precise relationships between the discriminant groups of the cut and flow lattices and the higher critical and cocritical groups with error terms corresponding to torsion (co)homology. As an application, we generalize a result of Kotani and Sunada to give bounds for the complexity, girth, and connectivity of a complex in terms of Hermite's constant.Comment: 30 pages. Final version, to appear in Journal of Algebraic Combinatoric

On complex surfaces diffeomorphic to rational surfaces

In this paper we prove that no complex surface of general type is diffeomorphic to a rational surface, thereby completing the smooth classification of rational surfaces and the proof of the Van de Ven conjecture on the smooth invariance of Kodaira dimension.Comment: 34 pages, AMS-Te

Symmetry Decomposition of Chaotic Dynamics

Discrete symmetries of dynamical flows give rise to relations between periodic orbits, reduce the dynamics to a fundamental domain, and lead to factorizations of zeta functions. These factorizations in turn reduce the labor and improve the convergence of cycle expansions for classical and quantum spectra associated with the flow. In this paper the general formalism is developed, with the $N$-disk pinball model used as a concrete example and a series of physically interesting cases worked out in detail.Comment: CYCLER Paper 93mar01