Gauge Equivalence Research Articles

Reciprocal diffusions and symmetries

We prove that the partition of Markov diffusions into reciprocal classes is equivalent to the partition of the one parameter symmetry groups of second order linear partial differential operators under gauge equivalence. In order to do so we explicit the system of Determining Equations which characterize the symmetries of a second order linear p.d.e. and we prove that the set of compatibility conditions for this system only depends on the reciprocal class. We then show that the reciprocal class of a Markov diffusion is preserved under the symmetries of the associated partial differential operator

Gauge equivalence between -dimensional continuous Heisenberg ferromagnetic models and nonlinear Schrödinger-type equations

The gauge equivalence between the.2C 1/-dimensional Zakharov equation and the .2C 1/-dimensional integrable continuous Heisenberg ferromagnetic model is established. Their integrable reductions are also shown explicitly.

Canonical gauge equivalences of the sAKNS and sTB hierarchies

We study the gauge transformations between the supersymmetric AKNS (sAKNS) and supersymmetric two-boson (sTB) hierarchies. The Hamiltonian nature of these gauge transformations is investigated, which turns out to be canonical. We also obtain the Darboux-Backlund transformations for the sAKNS hierarchy from these gauge transformations.

Nonlinear σ-model in a curved space, gauge equivalence, and exact solutions of (2+0)-dimensional integrable equations

We propose a nonlinear σ-model in a curved space as a general integrable elliptic model. We construct its exact solutions and obtain energy estimates near the critical point. We consider the Pohlmeyer transformation in Euclidean space and investigate the gauge equivalence conditions for a broad class of elliptic equations. We develop the inverse scattering transform method for the sinh-Gordon equation and evaluate its exact and asymptotic solutions.

A note on the gauge equivalence between the Manin-Radul and Laberge-Mathieu super KdV hierarchies

The gauge equivalence between the Manin-Radul and Laberge-Mathieu super KdV hierarchies is revisited. Apart from the Inami-Kanno transformation, we show that there is another gauge transformation which also possess the canonical property. We explore the relationship of these two gauge transformations from the Kupershmidt-Wilson theorem viewpoint and, as a by-product, obtain the Darboux-Backlund transformation for the Manin-Radul super KdV hierarchy. The geometrical intepretation of these transformations is also briefly discussed.

On the simplest (2+1) dimensional integrable spin systems and their equivalent nonlinear Schrödinger equations

Using a moving space curve formalism, geometrical as well as gauge equivalence between a (2+1) dimensional spin equation (M-I equation) and the (2+1) dimensional nonlinear Schrödinger equation (NLSE) originally discovered by Calogero, discussed then by Zakharov and recently rederived by Strachan, have been estabilished. A compatible set of three linear equations are obtained and integrals of motion are discussed. Through stereographic projection, the M-I equation has been bilinearized and different types of solutions such as line and curved solitons, breaking solitons, induced dromions, and domain wall type solutions are presented. Breaking soliton solutions of (2+1) dimensional NLSE have also been reported. Generalizations of the above spin equation are discussed.

Uniqueness of the Functional Determinant

The functional determinant of the conformal laplacian and the square of the Dirac operator are known to be extremized at the standard round metric of the four-sphere among all conformal metrics (up to gauge equivalence). In this article we show that this is the unique critical point, thus extending the work of Onofri and Osgood, Phillips and Sarnak for the functional determinant on S 2 which characterized the constant curvature metric as the unique critical point of the determinant. In addition, we introduce a new symmetric two-tensor field which is defined on any conformally flat four-manifold and can be viewed as a fourth order generalization of the Einstein gravitational tensor. As a consequence we prove a Pohozaev identity for manifolds with boundary which admit conformal Killing vector fields.

Some geometric aspects of the nonlinear O(3) sigma-model in the dimension (2+0)

On the basis of the gauge equivalence between an elliptic analog of the O(3) sigma-model and thesinh-Gordon equation proved by the author, connection formulas are obtained. Their interpretation in terms of differential geometry is given. Bibliography: 9 titles.

Manifestly supersymmetric Lax integrable hierarchies

A systematic method of constructing manifestly supersymmetric 1 + 1-dimensional KP Lax hierarchies is presented. Closed expressions for the Lax operators in terms of superfield eigenfunctions are obtained. All hierarchy equations being eigenfunction equations are shown to be automatically invariant under the (extended) supersymmetry. The supersymmetric Lax models existing in the literature are found to be contained (up to a gauge equivalence) in our formalism.

Drinfeld-Sokolov gravity

A lagrangian euclidean model of Drinfeld-Sokolov (DS) reduction leading to generalW-algebras on a Riemann surface of any genus is presented. The background geometry is given by the DS principal bundleK associated to a complex Lie groupG and anSL(2,ℂ) subgroupS. The basic fields are a hermitian fiber metricH ofK and a (0, 1) Koszul gauge fieldA * ofK valued in a certain negative graded subalgebrar ofg related tos. The action governing theH andA * dynamics is the effective action of a DS field theory in the geometric background specified byH andA *. Quantization ofH andA * implements on one hand the DS reduction and on the other defines a novel model of 2d gravity, DS gravity. The gauge fixing of the DS gauge symmetry yields an integration on a moduli space of DS gauge equivalence classes ofA * configurations, the DS moduli space. The model has a residual gauge symmetry associated to the DS gauge transformations leaving a given fieldA * invariant. This is the DS counterpart of conformal symmetry. Conformal invariance and certain non-perturbative features of the model are discussed in detail.

Resolution of constraints and gauge equivalence in algebraic Schrödinger representation of quantum electrodynamics

Abstract The algebraic formalism of QED is expounded in order to demonstrate both the resolution of constraints and to verify gauge equivalence between temporal gauge and Coulomb gauge on the quantum level. In the algebraic approach energy eigenstates of QED in temporal gauge are represented in an algebraic GNS basis. The corresponding Hilbert space is mapped into a functional space of generating functional states. The image of the QED-Heisenberg dynamics becomes a functional energy equation for these states. In the same manner the Gauß constraint is mapped into functional space. By suitable transformations the functional image of the Coulomb forces is recovered in temporal gauge. The equivalence of this result with the functional version of QED in Coulomb gauge is demonstrated. The meaning of the various transformations and their relations are illustrated for the case of harmonic oscillators. If applied to QCD this method allows an exact derivation of effective color "Coulomb" forces, in addition it implies a clear conception for the incorporation of various algebraic representations into the formal Heisenberg dynamics and establishes the algebraic "Schrödinger" equation for quantum fields in functional space.

More on generalized Heisenberg ferromagnet models

We generalize the integrable Heisenberg ferromagnet model according to each Hermitian symmetric spaces and address various new aspects of the generalized model. Using the first order formalism of generalized spins which are defined on the coadjoint orbits of arbitrary groups, we construct a Lagrangian of the generalized model from which we obtain the Hamiltonian structure explicitly in the case of CP( N − 1) orbit. The gauge equivalence between the generalized Heisenberg ferromagnet and the nonlinear Schrödinger models is given. Using the equivalence, we find infinitely many conserved integrals of both models.

On YM2 measures and area-preserving diffeomorphisms

For a given gauge group and compact Riemannian two-manifold, it is known that the associated Yang-Mills measure can be defined directly as a finitely additive measure on the space of connections, and this finitely additive measure is invariant with respect to SDiff, the group of all area-preserving diffeomorphisms of the surface. The first question we address is whether this symmetry essentially characterizes the projection of the Yang-Mills measure to the space of gauge equivalence classes. The proper formulation of this question entails the construction of an SDiff-equivariant equivariant completion of the space of continuous connections, such that the projection of the Yang-Mills measure to the space of gauge equivalence classes has a countably additive extension. We also consider the coupling of the Yang-Mills measure to determinants of Dirac operators. The basic problems are to prove that the coupled measure is absolutely continuous with respect to the background Yang-Mills measure, to find a reasonable formula for the Radon-Nikodym derivative, and to analyze the action of SDiff.

A Note on Supersymmetric Two-Boson Equation11This project supported by National Natural Science Foundation of China

We show that the supersymmetric version of two-boson or dispersive water wave system is nothing but one of the systems. We also consider a constrained , which is a gauge equivalence of it.

Soliton solutions and gauge equivalence for the problem of Zakharov–Shabat and its generalizations

In a paper by Takhtadjan and Faddeev [Hamiltonian approach to the soliton theory (in Russian) (Nauka, Moskov, 1986)] the N-soliton solutions, related to the nonlinear Schrödinger equation (NSE), are given. A generalization of this approach allows us to apply it not only to the NSE, but to the whole hierarchy of the Zakharov–Shabat problem, to the quadratic bundle problem, and to the ones gauge equivalent to them [where one can find, for example, the Heisenberg ferromagnet equation, the relativistic Mikhailov model (which, in appropriate reduction, is equivalent to the massive Thirring model), the derivative nonlinear Schrödinger equation (which is equivalent to the derivative Landau–Lifshitz equation), etc.]. We have used an appropriate reduction, giving us interesting, from a physical point of view, results. Thus we manage to obtain the soliton solutions for the whole hierarchy of the quadratic bundle problem (free and under reduction) and for the ones gauge equivalent to it. This result was first announced in previous articles by Vaklev, but here we manage to solve the determinants given there and to present the searched result as simple fractions of products of numerical differences.

Real instantons, Dirac operators and quaternionic classifying spaces

Let M ( k , S O ( n ) ) M(k,SO(n)) be the moduli space of based gauge equivalence classes of S O ( n ) SO(n) instantons on principal S O ( n ) SO(n) bundles over S 4 S^4 with first Pontryagin class p 1 = 2 k p_1=2k . In this paper, we use a monad description (Y. Tian, The Atiyah-Jones conjecture for classical groups, preprint, S. K. Donaldson, Comm. Math. Phys. 93 (1984), 453–460) of these moduli spaces to show that in the limit over n n , the moduli space is homotopy equivalent to the classifying space B S p ( k ) BSp(k) . Finally, we use Dirac operators coupled to such connections to exhibit a particular and quite natural homotopy equivalence.

Classifying Spaces and Dirac Operators Coupled to Instantons

Let $M(k,SU(l))$ denote the moduli space of based gauge equivalence classes of $SU(l)$ instantons on principal bundles over ${S^4}$ with second Chern class equal to $k$. In this paper we use Dirac operators coupled to such connections to study the topology of these moduli spaces as $l$ increases relative to $k$. This "coupling" procedure produces maps ${\partial _u}:M(k,SU(l)) \to BU(k)$, and we prove that in the limit over $l$ such maps recover Kirwan’s $[\text {K}]$ homotopy equivalence $M(k,SU) \simeq BU(k)$. We also compute, for any $k$ and $l$, the image of the homology map ${({\partial _u})_ * }:{H_ * }(M(k,SU(l));Z) \to {H_ * }(BU(k);Z)$. Finally, we prove all the analogous results for $Sp(l)$ instantons.

Boundary-value problem for the two-dimensional stationary heisenberg magnet with nontrivial background. II

The investigation of the boundary-value problem on a half-plane for the two-dimensional stationary Heisenberg magnet is continued. The asymptotic behavior of “N-soliton” solutions is discussed. The asymptotic contribution of the continuous spectrum is calculated. The gauge equivalence of the boundary-value problems for the models of a magnet and the elliptic equation represented by the sinh—Gordon equation is considered.

ELECTROWEAK VORTICES AND GAUGE EQUIVALENCE

Vortex configurations in the electroweak gauge theory are investigated. Two gauge-inequivalent solutions of the field equations, the Z and W vortices, have previously been found. They correspond to embeddings of the Abelian Nielsen-Olesen vortex solution into a U(1) subgroup of SU(2)×U(1). It is shown here that any electroweak vortex solution can be mapped into a solution of the same energy with a vanishing upper component of the Higgs field. The correspondence is a gauge equivalence for all vortex solutions except those for which the winding numbers of the upper and lower Higgs components add to zero. This class of solutions, which includes the W vortex, corresponds to a singular solution in the one-component gauge. The results, combined with numerical investigations, provide an argument against the existence of other vortex solutions in the gauge-Higgs sector of the Standard Model.

Classifying spaces and Dirac operators coupled to instantons

LetM(k,SU(l))M(k,SU(l))denote the moduli space of based gauge equivalence classes ofSU(l)SU(l)instantons on principal bundles overS4{S^4}with second Chern class equal tokk. In this paper we use Dirac operators coupled to such connections to study the topology of these moduli spaces asllincreases relative tokk. This "coupling" procedure produces maps∂u:M(k,SU(l))→BU(k){\partial _u}:M(k,SU(l)) \to BU(k), and we prove that in the limit overllsuch maps recover Kirwan’s[K][\text {K}]homotopy equivalenceM(k,SU)≃BU(k)M(k,SU) \simeq BU(k). We also compute, for anykkandll, the image of the homology map(∂u)∗:H∗(M(k,SU(l));Z)→H∗(BU(k);Z){({\partial _u})_ * }:{H_ * }(M(k,SU(l));Z) \to {H_ * }(BU(k);Z). Finally, we prove all the analogous results forSp(l)Sp(l)instantons.