Grassmannian Sigma Models Research Articles

Description of surfaces associated with Grassmannian sigma models on Minkowski space

We construct and investigate smooth orientable surfaces in su(N) algebras. The structural equations of surfaces associated with Grassmannian sigma models on Minkowski space are studied using moving frames adapted to the surfaces. The first and second fundamental forms of these surfaces as well as the relations between them as expressed in the Gauss–Weingarten and Gauss–Codazzi–Ricci equations are found. The scalar curvature and the mean curvature vector expressed in terms of a solution of Grassmanian sigma model are obtained.

Sigma-Model Solitons in the Noncommutative Plane: Construction and Stability Analysis

Noncommutative multi-solitons are investigated in Euclidean two-dimensional U(n) and Grassmannian sigma models, using the auxiliary Fock-space formalism. Their construction and moduli spaces are reviewed in some detail, unifying abelian and nonabelian configurations. The analysis of linear perturbations around these backgrounds reveals an unstable mode for the U(n) models but shows stability for the Grassmannian case. For multi-solitons which are diagonal in the Fock-space basis we explicitly evaluate the spectrum of the Hessian and identify all zero modes. It is very suggestive but remains to be proven that our results qualitatively extend to the entire multi-soliton moduli space.

REGULAR AND BLACK HOLE SOLUTIONS TO HIGHER ORDER CURVATURE EINSTEIN–YANG–MILLS–GRASSMANNIAN SYSTEMS IN FIVE DIMENSIONS

Solutions to EYM systems in five space–time dimensions possessing no gravity decoupling limits, feature a peculiar critical behavior which is absent in their six-, seven- and eight-dimensional counterparts which do possess flat space limits. This critical behavior in five dimensions persists even when a scalar matter field is added, rendering the model nontrivial in the gravity decoupling limit. To this end, both regular and black hole spherically symmetric solutions to the higher curvature EYM–Grassmannian sigma model in d=5 space–time dimensions are constructed. A study of the solutions to the Grassmannian model in flat space is also carried out.

Vortices, instantons and branes

The purpose of this paper is to describe a relationship between the moduli space of vortices and the moduli space of instantons. We study charge k vortices in U(N) Yang-Mills-Higgs theories and show that the moduli space is isomorphic to a special Lagrangian submanifold of the moduli space of k instantons in non-commutative U(N) Yang-Mills theories. This submanifold is the fixed point set of a U(1) action on the instanton moduli space which rotates the instantons in a plane. To derive this relationship, we present a D-brane construction in which the dynamics of vortices is described by the Higgs branch of a U(k) gauge theory with 4 supercharges which is a truncation of the familiar ADHM gauge theory. We further describe a moduli space construction for semi-local vortices, lumps in the CP(N) and Grassmannian sigma-models, and vortices on the non-commutative plane. We argue that this relationship between vortices and instantons underlies many of the quantitative similarities shared by quantum field theories in two and four dimensions.

Solitons in a Grassmannian σ model coupled to a Chern-Simons term

We propose an exactly solvable Grassmannian sigma-model coupled to the Chern-Simons theory. In the presence of a novel topological term our model admits exact self-dual vortex solutions which are identical to those of pure Grassmannian model, but the topological charge has a physical meaning as a magnetic flux since the gauge field is no longer auxiliary. We also extend the theory to a noncommutative plane and analyze the BPS solutions.

SUBMODELS OF NONLINEAR GRASSMANN SIGMA MODELS IN ANY DIMENSION AND CONSERVED CURRENTS, EXACT SOLUTIONS

In the preceding paper,1 we constructed submodels of nonlinear Grassmann sigma models in any dimensions and, moreover, an infinite number of conserved currents and a wide class of exact solutions. In this letter, we first construct almost all conserved currents for the submodels and all those for CP1-model. We next review the Smirnov and Sobolev construction for the equations of CP1-submodel and extend the equations, the S-S construction and conserved currents to higher order ones.

Nonlinear Grassmann sigma models in any dimension and an infinite number of conserved currents

We first consider nonlinear Grassmann sigma models in any dimension and next construct their submodels. For these models we construct an infinite number of nontrivial conserved currents. Our result is independent of time-space dimensions and, therefore, is a full generalization of that of authors (Alvarez, Ferreira and Guillen). Our result also suggests that our method may be applied to other nonlinear sigma models such as chiral models, G/ H sigma models in any dimension.

Dynamical Gauge Boson and Strong-Weak Reciprocity

It is proposed that asymptotically nonfree gauge theories are consistently interpreted as theories of composite gauge bosons. It is argued that when hidden local symmetry is introduced, masslessness and coupling universality of dynamically generated gauge boson are ensured. To illustrate these ideas we take a four dimensional Grassmannian sigma model as an example and show that the model should be regarded as a cut-off theory and there is a critical coupling at which the hidden local symmetry is restored. Propagator and vertex functions of the gauge field are calculated explicitly and existence of the massless pole is shown. The beta function determined from the $ Z $ factor of the dynamically generated gauge boson coincides with that of an asymptotic nonfree elementary gauge theory. Using these theoretical machinery we construct a model in which asymptotic free and nonfree gauge bosons coexist and their running couplings are related by the reciprocally proportional relation.

NONLINEAR SIGMA MODELS ON A HALF PLANE

In the context of integrable field theory with boundary, the integrable nonlinear sigma models in two dimensions, for example the O(N), the principal chiral, the CPN−1 and the complex Grassmannian sigma models, are discussed on a half plane. In contrast to the well-known cases of sine-Gordon, nonlinear Schrödinger and affine Toda field theories, these nonlinear sigma models in two dimensions are not classically integrable if restricted on a half plane. It is shown that the infinite set of nonlocal charges characterizing the integrability on the whole plane is not conserved for the free (Neumann) boundary condition. If we require that these nonlocal charges be conserved, then the solutions become trivial.

QCD strings as a constrained Grassmannian σ model

We present calculations for the effective action of string world sheet in R3 and R4 utilizing its correspondence with the constrained Grassmannian sigma model. Minimal surfaces describe the dynamics of open strings while harmonic surfaces describe that of closed strings. The one-loop effective action for these are calculated with instanton and anti-instanton background, reprsenting N-string interactions at the tree level. The effective action is found to be the partition function of a classical modified Coulomb gas in the confining phase, with a dynamically generated mass gap.

FUSION RESIDUES

We discuss when and how the Verlinde dimensions of a rational conformal field theory can be expressed as correlation functions in a topological LG theory. It is seen that a necessary condition is that the RCFT fusion rules must exhibit an extra symmetry. We consider two particular perturbations of the Grassmannian superpotentials. The topological LG residues in one perturbation, introduced by Gepner are shown to be twisted version of the SU (N)k Verlinde dimensions. The residues in the other perturbation are the twisted Verlinde dimensions of another RCFT; these topological LG correlation functions are conjectured to be the correlation functions of the corresponding Grassmannian topological sigma model with a coupling in the action to instanton number.

Bound state solution of the grassmannian nonlinear sigma model with fermions

We construct the S-matrix for bound state (gauge-invariant) scattering for nonlinear sigma models defined on the manifold SU(n) S( U(p)⊗ U(n−p) ) with fermions. It is not possible to compute gauge non-singlet matrix elements. In the present language, constraints from higher conservation laws determine the bound state solution. An alternative derivation is also presented.

Fermionization of grassmannians and the quantized Hall effect

The rules of fermionization of grassmannian sigma models with instanton terms are derived. Their application to the quantized Hall effect is discussed.

Explicit forms of Bäcklund transformations for the Grassmannian and C P N−1 sigma models

A fairly wide class of classical solutions of the Euclidean two-dimensional Grassmannian and CPN−1 sigma models has been constructed explicitly and elementarily by the present author. Starting from these classical solutions we derive the explicit forms of the Bäcklund transformations following Harnad et al. Properties of the Bäcklund transformations for the Grassmannian models are discussed. In particular, a simple interpretation of the Bäcklund transformations for the CPN−1 model is obtained. Generalization of these results to the noncompact Grassmannian sigma models is straightforward.

Classical Solutions for the Supersymmetric Grassmannian Sigma Models in Two Dimensions. II

On etudie la version supersymetrique des modeles sigma complexes grassmanniens a 2 dimensions euclidiennes. On construit des solutions fermion classiques explicites pour les equations de Dirac supersymetriques linearisees

Classical Solutions for the Supersymmetric Grassmannian Sigma Models in Two Dimensions. I

The supersymmetric version of the complex Grassmannian sigma models (defined on the Grassmann manifold) in two euclidean dimensions is studied. By adopting the newly found solutions of the purely bosonic Grassmannian sigma model as the background fields, we construct explicit fermion classical solutions for the supersymmetric linearized Dirac eqations. These fermion solutions are obtained in an elementary way just like their bosonic partners.

Some properties of classical solutions in Grassmannian sigma models

We investigate the construction of a large class of classical solutions to the nonabelian Grassmannian sigma models in two Euclidean space-time dimensions and also discuss the problem of uniqueness. Some spectral types of solutions have interesting features when represented in the framework of the Riemann-Hilbert problem.

General classical solutions of the complex grassmannian and CP N−1 sigma models

General classical solutions are constructed for the complex grassmannian non-linear sigma models in two euclidean dimensions in terms of holomorphic functions. The grassmannian sigma models are a simple generalization of the well-known CP N−1 model in two dimensions and they share various interesting properties; existence of (anti-) instantons, an infinite number of conserved quantities and complete integrability.