Monopoles, vortices, and kinks in the framework of noncommutative geometry (original) (raw)

Generalized self-duality for the Yang-Mills-Higgs system

Physical Review D, 2021

Self-duality is a very important concept in the study and applications of topological solitons in many areas of Physics. The rich mathematical structures underlying it lead, in many cases, to the development of exact and non-perturbative methods. We present a generalization of the Yang-Mills-Higgs system by the introduction of scalar fields assembled in a symmetric and invertible matrix h of the same dimension as the gauge group. The coupling of such new fields to the gauge and Higgs fields is made by replacing the Killing form, in the contraction of the group indices, by the matrix h in the kinetic term for the gauge fields, and by its inverse in the Higgs field kinetic term. The theory is conformally invariant in the three dimensional space IR 3. An important aspect of the model is that for practically all configurations of the gauge and Higgs fields the new scalar fields adjust themselves to solve the modified self-duality equations. We construct solutions using a spherically symmetric ansätz and show that the 't Hooft-Polyakov monopole becomes a self-dual solution of such modified Yang-Mills-Higgs system. We use an ansätz based on the conformal symmetry to construct vacuum solutions presenting non-trivial toroidal magnetic fields.

Solitons in noncommutative gauge theory

Journal of High Energy Physics, 2001

We present a unified treatment of classical solutions of noncommutative gauge theories. We find all solutions of the noncommutative Yang-Mills equations of motion in 2 dimensions; and show that they are labeled by two integers-the rank of the gauge group and the magnetic charge. The magnetic vortex solutions are unstable in 2+1 dimensions, but correspond to the full, stable BPS solutions of N = 4 U (1) noncommutative gauge theory in 4 dimensions, that describes N infinite D1 strings that pierce a D3 brane at various points, in the presence of a background B-field in the Seiberg-Witten α ′ → 0 limit.

Galilean noncommutative gauge theory: symmetries and vortices

Nuclear Physics B, 2003

Noncommutative Chern-Simons gauge theory coupled to nonrelativistic scalars or spinors is shown to admit the "exotic" two-parameter-centrally extended Galilean symmetry, realized in a unique way consistent with the Seiberg-Witten map. Nontopological spinor vortices and topological external-field vortices are constructed by reducing the problem to previously solved self-dual equations.

Noncommutative Monopoles and Riemann-Hilbert Problems

Journal of High Energy Physics, 2004

The Bogomolny equations for Yang-Mills-Higgs monopoles follow from a system of linear equations which may be solved through a parametric Riemann-Hilbert problem. We extend this approach to noncommutative R 3 and use it to (re)construct noncommutative Dirac, Wu-Yang, and BPS monopole configurations in a unified manner. In all cases we write down the underlying matrix-valued functions for multi-monopoles and solve the corresponding Riemann-Hilbert problems for charge one. * On leave from Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia In modern string theory, solitonic solutions to the (effective) field equations are ubiquitous. A particular class of solitons are monopoles , which appear as D1-branes stretched between D3branes in type IIB superstring theory. When a constant NS B-field is turned on, gauge theory on a stack of D3-branes becomes noncommutative and supports noncommutative U(n) monopoles, including abelian ones [2] - .

A Monopole Solution From Noncommutative Multi-Instantons

Journal of High …, 2004

We extend the relation between instanton and monopole solutions of the selfduality equations in SU (2) gauge theory to noncommutative space-times. Using this approach and starting from a noncommutative multi-instanton solution we construct a U (2) monopole configuration which lives in 3 dimensional ordinary space. This configuration resembles the Wu-Yang monopole and satisfies the selfduality (Bogomol'nyi) equations for a U (2) Yang-Mills-Higgs system.

Global signatures of gauge invariance: Vortices and monopoles

Physical Review D - PHYS REV D, 1976

A comprehensive topological classification of vortices and their endpoint Dirac monopoles is formulated in gauge theories with an arbitrary compact Lie group. By way of homotopy theory, a simple analysis is presented for the global groups U (1)) O(3), SU(2) then SU(N)/ZN and SU(N). Finite vortices are achieved through the complementarity of Dirac strings and the nodal lines of the Higgs fields. In general, the varieties of topologically distinct vortices or monopoles are determined solely by the connectivity of the global group, specified by a discrete Abelian fundamental group. The close connection between our work and the Wu-Yang global formulation of gauge fields is pointed out.

Comments about higgs fields, noncommutative geometry, and the standard model

Lecture Notes in Physics, 2000

We make a short review of the formalism that describes Higgs and Yang Mills elds as two particular cases of an appropriate generalization of the notion of connection. We also comment about the several variants of this formalism, their interest, the relations with noncommutative geometry, the existence (or lack of existence) of phenomenological predictions, the relation with Lie super-algebras etc.

Higgs fields as Yang-Mills fields and discrete symmetries

Nuclear Physics B, 1991

We consider an extension of the formalism of gauge theories where the connection is no longer described by a Lie algebra valued one-form on space-time, but incorporates both the Yang-Mills fields and the Higgs fields. Higgs fields are associated here with the gauging of discrete directions. The formalism presented here is therefore not a Kaluza-Klein-like approach to Yang-Mills theory, but is, in a sense, a discrete analog of it. We first study a U(1) x U(1 } model, then a more realistic SUM x U(1) model. Algebraic and geometric aspects of those models in relation with the old and new ideas of "Non-Commutative Geometry" are also discussed.

Noncommutative Electromagnetism As A Large N Gauge Theory

Eur.Phys.J. C64 (2009) 445-457

We map noncommutative (NC) U(1) gauge theory on R^d_C X R^{2n}_{NC} to U(N -> \infty) Yang-Mills theory on R^d_C, where R^d_C is a d-dimensional commutative spacetime while R^{2n}_{NC} is a 2n-dimensional NC space. The resulting U(N) Yang-Mills theory on R^d_C is equivalent to that obtained by the dimensional reduction of (d+2n)-dimensional U(N) Yang-Mills theory onto R^d_C. We show that the gauge-Higgs system (A_\mu,\Phi^a) in the U(N -> \infty) Yang-Mills theory on R^d_C leads to an emergent geometry in the (d+2n)-dimensional spacetime whose metric was determined by Ward a long time ago. In particular, the 10-dimensional gravity for d=4 and n=3 corresponds to the emergent geometry arising from the 4-dimensional N=4 vector multiplet in the AdS/CFT duality. We further elucidate the emergent gravity by showing that the gauge-Higgs system (A_\mu,\Phi^a) in half-BPS configurations describes self-dual Einstein gravity.

Quiver Gauge Theory and Noncommutative Vortices

Progress of Theoretical Physics Supplement, 2007

We construct explicit BPS and non-BPS solutions of the Yang-Mills equations on noncommutative spaces R 2n θ × G/H which are manifestly G-symmetric. Given a G-representation, by twisting with a particular bundle over G/H, we obtain a Gequivariant U(k) bundle with a G-equivariant connection over R 2n θ × G/H. The U(k) Donaldson-Uhlenbeck-Yau equations on these spaces reduce to vortex-type equations in a particular quiver gauge theory on R 2n θ . Seiberg-Witten monopole equations are particular examples. The noncommutative BPS configurations are formulated with partial isometries, which are obtained from an equivariant Atiyah-Bott-Shapiro construction. They can be interpreted as D0-branes inside a space-filling brane-antibrane system.