scholarly journals DECONSTRUCTING MONOPOLES AND INSTANTONS

2000 ◽  
Vol 12 (10) ◽  
pp. 1367-1390 ◽  
Author(s):  
GIOVANNI LANDI
Keyword(s):  
Vector Bundle ◽  
Lie Groups ◽  
Finite Type ◽  
Topological Charge ◽  
Projective Modules ◽  
Dirac Monopole ◽  
Canonical Connection ◽  
Equivariant Maps ◽  
K Theory

We give a unifying description of the Dirac monopole on the 2-sphere S2, of a graded monopole on a (2, 2)-supersphere S2, 2 and of the BPST instanton on the 4-sphere S4, by constructing a suitable global projector p via equivariant maps. This projector determines the projective modules of finite type of sections of the corresponding vector bundle. The canonical connection ∇ = p ◦ d is used to compute the topological charge which is found to be equal to -1 for the three cases. The transposed projector q = pt gives the value +1 for the charges; this showing that transposition of projectors, although an isomorphism in K-theory, is not the identity map. We also study the invariance under the action of suitable Lie groups.

scholarly journals Twisted K-theory constructions in the case of a decomposable Dixmier-Douady class

2014 ◽  
Vol 14 (2) ◽  
pp. 247-272 ◽  
Author(s):  
Antti J. Harju ◽  
Jouko Mickelsson
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Lie Groups ◽  
Explicit Construction ◽  
Theory Model ◽  
Quantum Field ◽  
Compact Lie Groups ◽  
Field Theory Model ◽  
Cup Product ◽  
K Theory

AbstractTwisted K-theory on a manifold X, with twisting in the 3rd integral cohomology, is discussed in the case when X is a product of a circle and a manifold M. The twist is assumed to be decomposable as a cup product of the basic integral one form on and an integral class in H2(M,ℤ). This case was studied some time ago by V. Mathai, R. Melrose, and I.M. Singer. Our aim is to give an explicit construction for the twisted K-theory classes using a quantum field theory model, in the same spirit as the supersymmetric Wess-Zumino-Witten model is used for constructing (equivariant) twisted K-theory classes on compact Lie groups.

Some Results in the Connective K-Theory of Lie Groups

1988 ◽  
Vol 31 (2) ◽  
pp. 194-199
Author(s):  
L. Magalhães
Keyword(s):  
Hopf Algebra ◽  
Lie Groups ◽  
Lie Group ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
K Theory ◽  
Torsion Free ◽  
Connected Lie Group

AbstractIn this paper we give a description of:(1) the Hopf algebra structure of k*(G; L) when G is a compact, connected Lie group and L is a ring of type Q(P) so that H*(G; L) is torsion free;(2) the algebra structure of k*(G2; L) for L = Z2 or Z.

scholarly journals Chern Classes of Projective Modules

1963 ◽  
Vol 23 ◽  
pp. 121-152 ◽  
Author(s):  
Hideki Ozeki
Keyword(s):  
Vector Bundle ◽  
Cross Sections ◽  
Chern Class ◽  
Vector Bundles ◽  
Stein Manifold ◽  
Projective Modules ◽  
Singular Variety ◽  
Differentiable Functions ◽  
Holomorphic Vector Bundles ◽  
Differentiable Vector

In topology, one can define in several ways the Chern class of a vector bundle over a certain topological space (Chern [2], Hirzebruch [7], Milnor [9], Steenrod [15]). In algebraic geometry, Grothendieck has defined the Chern class of a vector bundle over a non-singular variety. Furthermore, in the case of differentiable vector bundles, one knows that the set of differentiable cross-sections to a bundle forms a finitely generated projective module over the ring of differentiable functions on the base manifold. This gives a one to one correspondence between the set of vector bundles and the set of f.g.-projective modules (Milnor [10]). Applying Grauert’s theorems (Grauert [5]), one can prove that the same statement holds for holomorphic vector bundles over a Stein manifold.

Transition of a spherical vortex to a Dirac monopole with fractional topological charge

2020 ◽  
Vol 35 (15) ◽  
pp. 2050118 ◽  
Author(s):  
Derar Altarawneh ◽  
Manfried Faber ◽  
Roman Höllwieser
Keyword(s):  
Dirac Operator ◽  
Boundary Conditions ◽  
Topological Charge ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Topological Properties ◽  
Dirac Monopole ◽  
Center Vortex ◽  
Spherical Vortex ◽  
Charge Formula

We study topological properties of classical spherical center vortices with the low-lying eigenmodes of the Dirac operator in the fundamental and adjoint representations using both the overlap and asqtad staggered fermion formulations. We find some evidence for fractional topological charge during cooling the spherical center vortex on a [Formula: see text] lattice. We identify the object with topological charge [Formula: see text] as a Dirac monopole with a gauge field fading away at large distances. Therefore, even for periodic boundary conditions, it does not need an anti-monopole.

scholarly journals On the K-theory of compact lie groups

Topology ◽  
1965 ◽  
Vol 4 (1) ◽  
pp. 95-99 ◽  
Author(s):  
M.F. Atiyah
Keyword(s):  
Lie Groups ◽  
Compact Lie Groups ◽  
K Theory

GEOMETRY OF QUANTUM HOMOGENEOUS VECTOR BUNDLES AND REPRESENTATION THEORY OF QUANTUM GROUPS I

1999 ◽  
Vol 11 (05) ◽  
pp. 533-552 ◽  
Author(s):  
A. R. GOVER ◽  
R. B. ZHANG
Keyword(s):  
Representation Theory ◽  
Finite Type ◽  
Quantum Groups ◽  
Vector Bundles ◽  
Homogeneous Spaces ◽  
Projective Modules ◽  
Geometrical Structures ◽  
Induced Representations ◽  
Compact Quantum Groups ◽  
Homogeneous Vector

Quantum homogeneous vector bundles are introduced in the context of Woronowicz type compact quantum groups. The bundles carry natural topologies, and their sections furnish finite type projective modules over algebras of functions on quantum homogeneous spaces. Further properties of the quantum homogeneous vector bundles are investigated, and applied to the study of the geometrical structures of induced representations of quantum groups.

scholarly journals Gravitating superconducting solitons in the (3+1)-dimensional Einstein gauged non-linear $$\sigma $$-model

2021 ◽  
Vol 81 (1) ◽  
Author(s):  
Fabrizio Canfora ◽  
Alex Giacomini ◽  
Marcela Lagos ◽  
Seung Hun Oh ◽  
Aldo Vera
Keyword(s):  
Gravitational Lensing ◽  
Topological Charge ◽  
Sigma Model ◽  
Persistent Current ◽  
Linear Sigma Model ◽  
Dirac Monopole ◽  
Topological Solitons ◽  
Abelian Field ◽  
Non Linear ◽  
Fully Coupled

AbstractIn this paper, we construct the first analytic examples of $$(3+1)$$ ( 3 + 1 ) -dimensional self-gravitating regular cosmic tube solutions which are superconducting, free of curvature singularities and with non-trivial topological charge in the Einstein-SU(2) non-linear $$\sigma $$ σ -model. These gravitating topological solitons at a large distance from the axis look like a (boosted) cosmic string with an angular defect given by the parameters of the theory, and near the axis, the parameters of the solutions can be chosen so that the metric is singularity free and without angular defect. The curvature is concentrated on a tube around the axis. These solutions are similar to the Cohen–Kaplan global string but regular everywhere, and the non-linear $$\sigma $$ σ -model regularizes the gravitating global string in a similar way as a non-Abelian field regularizes the Dirac monopole. Also, these solutions can be promoted to those of the fully coupled Einstein–Maxwell non-linear $$\sigma $$ σ -model in which the non-linear $$\sigma $$ σ -model is minimally coupled both to the U(1) gauge field and to General Relativity. The analysis shows that these solutions behave as superconductors as they carry a persistent current even when the U(1) field vanishes. Such persistent current cannot be continuously deformed to zero as it is tied to the topological charge of the solutions themselves. The peculiar features of the gravitational lensing of these gravitating solitons are shortly discussed.

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