Gauge Equivalence Research Articles

Broken gauge equivalence of Hamiltonians due to time-evolution and the Berry phase

It is shown that the Berry phase is invariant under both time-dependent unitary and gauge transformations. The initial condition of the quantum system plays a role in uniquely determining the Berry phase. The quantum states transported separately by two Hamiltonians, which are initially related with a static gauge transformation, differ by a Berry phase. The phase in this case cannot be removed by a valid unitary transformation.

NON-LINEAR GENERALIZATION OF THE GAUGE AMBIGUITIES OF THE LAGRANGIAN FORMALISM

A generalization of the gauge equivalence of Lagrangians is developed. In this approach the different Lagrangians determine not only the same set of solutions but also the same energies. Some general properties are studied and then the case of velocity-free forces is considered. In particular we show the non-existence for n=2 (two degrees of freedom) of such equivalent Lagrangians and that when n=3 the only forces admitting such equivalent Lagrangians correspond to those deriving from central potentials.

On the initial condition for instanton solutions

To each gauge equivalence class of both local and global framed (in the sense of Donaldson) self-dual solutions with the gauge group U(r) there is related the unique canonical initial condition (in the sense of Takasaki) and in this way the gauge freedom is eliminated. A geometric interpretation is given and consequently the complete transcription of the ADHM construction into the inverse scattering formalism is derived. As an application, an injection holomorphic mapping of the instanton moduli space into a finite-dimensional complex vector space is described and the loop group action on the transition functions is discussed. The results suggest the possibility of a new description of the framed instanton moduli spaces directly as algebraic sets.

On the Homology of SU(n) Instantons

In this paper we study the homology of the moduli spaces of instantons associated to principal SU(n) bundles over the four-sphere. This is accomplished by exploiting an loop structure implicit in the disjoint union of all moduli spaces associated to a fixed SU(n) with arbitrary instanton number and relating these spaces to the known homology structure of the four-fold loop space on BSU(n) . Moduli spaces of instantons (self-dual connections with respect to a conformal class of metrics) associated to principal G-bundles Pk(G) = P over S4 have proven to be basic objects in modern geometry. Here G is any simple compact Lie group and k is the integer that classifies the bundle P and is referred to as the instanton number. We denote these moduli spaces by At (G). There is a natural inclusion (0. 1) ik (G): Z, (G) Qk BG induced by forgetting the self-duality condition where we have identified the moduli space of based gauge equivalence classes of all connections on P with the kth component of the four-fold loop space Q?4BG [5]. In a fundamental paper, Atiyah and Jones [5] studied the inclusion (0.1) for G = SU(2) (when S4 has its standard conformally flat metric) and posed several fundamental questions. In a series of papers Taubes (cf. [24-26]) proved several basic existence theorems, stability theorems in terms of k and provided a basis framework to describe how the topology of Ak changes as k increases. Taubes' work is much more general than we describe here in that he not only studied general Lie groups G but also replaced 54 by an arbitrary compact closed Riemannian four-manifold (with arbitrary conformal class of metric). In [7] it was observed that, over the four-sphere but with G = Sp(n), the disjoint union of .k over all k form a homotopy C4 space and that iterated loop space techniques may be profitably used to study H. (xk) for individual k. Certain computational results may be obtained immediately from these Received by the editors October 17, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C55, 58E05; Secondary 53C57, 55P35.

Boundary values of hyperbolic monopoles

The authors show that a magnetic monopole on hyperbolic space is determined, up to gauge equivalence, by its asymptotic boundary value. Their basic tool is the ADHM construction which specializes to a discrete Nahm equation. They also construct a complete metric on the moduli spaces.

Planar radially symmetric Heisenberg spin system and generalized nonlinear Schrödinger equation: Gauge equivalence, Bäcklund transformations and explicit solutions

Under suitable transformations, we show that the planar radially symmetric continuum Heisenberg ferromagnetic spin system and the geometrically equivalent generalized nonlinear Schrödinger equation (NLSE) are equivalent to a linearly x-dependent inhomogeneous spin system and generalized inhomogeneous NLSE respectively on a half-line. We construct appropriate Bäcklund transformations and indicate the construction of explicit solutions in the radially symmetric case.

The gauge equivalence of the Davey-Stewartson equation and (2+1)-dimensional continuous Heisenberg ferromagnetic model

The gauge equivalence between the Davey-Stewartson (DS) equation and the (2+1)-dimensional continuous Heisenberg ferromagnetic model is shown explicitly. Through the gauge transformation, solutions of the DSII are obtained from the vortex type solutions of the former equation.

Structure of the space of reducible connections for Yang-Mills theories

The geometrical structure of the gauge equivalence classes of reducible connections are investigated. The general procedure to determine the set of orbit types (strata) generated by the action of the gauge group on the space of gauge potentials is given. In the so obtained classification, a stratum, containing generically certain reducible connections, corresponds to a class of isomorphic subbundles given by an orbit of the structure and gauge group. The structure of every stratum is completely clarified. A nonmain stratum can be understood in terms of the main stratum corresponding to a stratification at the level of a subbundle.

Leading terms in the heat invariants

Let D D be a second-order differential operator with leading symbol given by the metric tensor on a compact Riemannian manifold. The asymptotics of the heat kernel based on D D are given by homogeneous, invariant, local formulas. Within the set of allowable expressions of a given homogeneity there is a filtration by degree, in which elements of the smallest class have the highest degree. Modulo quadratic terms, the linear terms integrate to zero, and thus do not contribute to the asymptotics of the L 2 {L^2} trace of the heat operator; that is, to the asymptotics of the spectrum. We give relations between the linear and quadratic terms, and use these to compute the heat invariants modulo cubic terms. In the case of the scalar Laplacian, qualitative aspects of this formula have been crucial in the work of Osgood, Phillips, and Sarnak and of Brooks, Chang, Perry, and Yang on compactness problems for isospectral sets of metrics modulo gauge equivalence in dimensions 2 and 3.

Linear integral equations and multicomponent nonlinear integrable systems II

In a previous paper (I) an extension of the direct linearization method was developed for obtaining solutions of multicomponent generalizations of integrable nonlinear partial differential equations (PDE's). The method is based on a general type of linear integral equations containing integrations over an arbitrary contour with an arbitrary measure in the complex plane of the spectral parameter. In I the general framework has been presented, with as immediate application the direct linearization of multicomponent versions of the nonlinear Schrödinger equation and the (complex) modified Korteweg-de Vries equation. In the present paper we treat a varìety of examples of other multicomponent PDE's, and we also discuss Miura transformations and gauge equivalences. The examples include the direct linearization of multicomponent generalizations of the isotropic Heisenberg spin chain equation, the complex sine-Gordon equation, the Getmanov equation, the derivative non-linear Schrödinger equation and the massive Thirring model equations.

Gauge transformations for the quadratic bundle

Gauge transformations constructed by means of the Jost solutions related to two similar cases of the quadratic bundle of general form are studied. Those transformations are taken at arbitrary fixed point λ=λ0 of the continuous spectrum of the problem. The entire class of gauge-equivalent nonlinear evolution equations (NLEE) and their Hamiltonian structures are obtained. An interesting case of reduction of the potential is examined. It was shown that the entire class of NLEE splits into three different (in terms of coefficient functions) subclasses of gauge-equivalent NLEE. The gauge equivalence between the derivative nonlinear Schrödinger equation and some modifications of the derivative Landau–Lifshitz equation (DLLE) is demonstrated. The simplest soliton solutions of the DLLE and its higher analogs are obtained.

Monopoles, non-linear ? models, and two-fold loop spaces

In this paper we study the topology of\(\hat M_k\), the moduli spaces ofSU(2) monopoles associated with the Yang-Mills-Higgs and Bogomol'nyi equations, and ℜ(m)k, non-linear σ models from quantum field theory. Beautiful work of Donaldson [18, 19], Hitchin [24, 25] and Taubes [37, 39, 40] shows that gauge equivalence classes of monopoles correspond to based rational self-maps of the Riemann sphere. Similarly, the non-linear σ models we consider here are based harmonic maps from the Riemann sphere to complex projectivem space. In seminal work, Segal [35] studied ℛ(m)k, the space of based rational maps from the Riemann sphere to complex projectivem space of a fixed degreek. Any element of ℛ(m)k is clearly an element of Ωk2CP(m), the space of all based continuous maps from the Riemann sphere to complex projectivem space of a fixed degreek, and this assignment gives rise to the natural inclusion of ℛ(m)k in Ωk2CP(m). Segal showed that these natural inclusions are homotopy equivalences through dimensionk(2m − 1). As the topology of the two-fold loop space Ω2CP(m) is well understood, Segal's result gives a very efficient way to explicitly determine the low dimensional topology of ℛ(m)k. Thus iterated loop spaces have much to say about the topology of monopoles and non-linear σ models.

Holomorphic curves in loop groups

It was observed by Atiyah that there is a correspondence between based gauge equivalence classes ofSU n -instantons overS 4 of charged on the one hand, and based holomorphic curves of genus zero inΩSU n of degreed on the other hand. In this paper we study the parameter space of such holomorphic curves which have the additional property that they lie entirely in the subgroupΩ alg SU n of algebraic loops. We describe a cell decomposition of this parameter space, and compute its complex dimension to be (2n−1)d.

Gauge equivalence, supersymmetry and classical solutions of the ospu (1, 1/1) Heisenberg model and the nonlinear Schr�dinger equation

An integrable generalization of the continuous classical O(2, 1) pseudospin Heisenberg model to the case of the ospu(1, 1/1) superalgebra is constructed. The gauge equivalence of the constructed model and the related NLSE is established. We indicate a method of generating classical solutions using the global ospu(1, 1/1) supersymmetry. The relationship between solutions of O(2, 1) HM and ‘superpartners’ of NLSE is obtained.

Generalized Hidden Local Symmetry and the A1 Meson

We present a generalized formalism of the hidden local symmetry in which any nonlinear sigma model based on the manifold C/H is gauge equivalent to a model possessing GglobalX Goeal symmetry, which is a natural extension of the well-known gauge equivalence between a C/H nonlinear sigma model and a CglObal X H'oea' linear model. As an application of this formalism, we reexamine a possibility that the Al mesons as well as the p mesons and their U(3) partners are dynamical gauge bosons of the hidden local symmetry, [U(3)L x U(3)RJ.oeal, in the U(3h x U(3)R/U(3)v nonlinear sigma model.

Gauge equivalence of two different IAnsaaumlItze Rfor non-Abelian charged vortices.

Recently the existence of non-Abelian charged vortices has been established by taking two different Ansa$uml: tze in SU(2) gauge theories. We point out that these two Ansa$uml: tze are in two topologically equivalent prescriptions. We show that they are gauge equivalent only at infinity. We also show that this gauge equivalence is not possible for Z/sub N/ vortices in SU(N) gauge theories for Ngreater than or equal to3.

Algebraic structures of degenerate systems and the indefinite metric

It is shown that the indefinite metric structures of degenerate systems as given by Strocchi and Wightman [F. Strocchi and A. S. Wightman, J. Math. Phys. 15, 2198 (1974); 17, 1930 (1976)] arise in a natural fashion from the algebraic structure of such systems, where the latter has been developed in a C*-context by Grundling and Hurst [H. B. G. S Grundling and C. A. Hurst, Commun. Math. Phys. 98, 369 (1985)]. Auxiliary concepts like gauge equivalence are examined, and the preceding general theory is specialized to the situation of linear boson fields with linear Hermitian constraints. Two examples of this situation are given—a one-dimensional scalar boson in a periodic universe and Landau gauge electromagnetism.

The renormalization group and string field theory

The field theoretic equation of motion of string is interpreted as the exact renormalization group equation. The generators of string gauge invariance are the redundant operators of the RG; gauge equivalence is related to universality. We give a geometric interpretation of the theory and show in principle how to derive the S-matrix.

Linear integral equations and multicomponent nonlinear integrable systems I

A systematic method for obtaining multicomponent generalizations of integrable nonlinear partial differential equations (PDE's) is developed. The method starts from a general type of linear integral equations, containing integrations over an arbitrary contour with an arbitrary measure in the complex plane of k (the spectral parameter). Special k-dependent factors in the integrand are shown to induce an extra coupling between solutions of these integral equations with a different source term. In this way the direct linearization is obtained of multicomponent generalizations of various nonlinear PDE's, such as the nonlinear Schrödinger equation, the derivative nonlinear Schrödinger equations, the isotropic Heisenberg spin chain equation, the (complex) sine-Gordon equation, and the massive Thirring model equations. In the present paper (I) we present the general framework to derive finite-matrix PDE's, and we also discuss Bäcklund transformations and multicomponent lattice versions. In a subsequent paper (II) we treat a variety of examples of multicomponent PDE's, and discuss Miura transformations and gauge equivalences.

Gauge equivalence of sigma models with non-compact Grassmannian manifolds

The gauge equivalence (GE) of sigma models associated with non-compact Grassmannian manifolds is investigated with emphasis on the necessary restrictions for the choice of gauge elements in such cases. The importance of GE in solving a non-linear system with the help of inverse scattering data of its gauge related counterpart is demonstrated. The gauge relations between generalised Landau-Lifshitz (LL) and non-linear Schrodinger (NLS) type equations and also between non-linear sigma models and generalised 'sine-sinh-Gordon' equations for non-compact SU(p,q)/S(U(u,v)*U(s,t)) manifolds are established. Using H-gauge invariance of LL, the GE is extended to some higher-order specific non-linear systems. The gauge connection among various LL and NLS equations are schematically represented. Along with the recovery of earlier results important new results, some with significant non-compact structures, are discovered.