Gauge Equivalence Research Articles

Equivalence of Projections as Gauge Equivalence¶on Noncommutative Space

Projections play crucial roles in the ADHM construction on noncommutative $\R^4$. In this article a framework for the description of equivalence relations between projections is proposed. We treat the equivalence of projections as ``gauge equivalence'' on noncommutative space. We find an interesting application of this framework to the study of U(2) instanton on noncommutative $\R^4$: A zero winding number configuration with a hole at the origin is ``gauge equivalent'' to the noncommutative analog of the BPST instanton. Thus the ``gauge transformation'' in this case can be understood as a noncommutative resolution of the singular gauge transformation in ordinary $\R^4$.

Triangular de Rham cohomology of compact Kahler manifolds

The de Rham of a smooth manifold with values in a group Lie is studied. By definition, this is the quotient of the set of flat connections in the trivial principal bundle by the so-called gauge equivalence. The case under consideration is the one when is a compact Kahler manifold and is a soluble complex linear algebraic group in a special class containing the Borel subgroups of all complex classical groups and, in particular, the group of all triangular matrices. In this case a description of the set in terms of the cohomology of with values in the (Abelian) sheaves of flat sections of certain flat Lie algebra bundles with fibre (the tangent Lie algebra of ) or, equivalently, in terms of the harmonic forms on representing this cohomology is obtained.

Time evolution as a gauge transformation: x5-dependent cosmological solution in 5d Kaluza–Klein theory

We discuss a new feature of the 5d Kaluza–Klein cosmology. For that purpose, we obtain the simplest x 5-dependent solution which, in the reduced description, is associated with a radiation-dominated Robertson–Walker universe, and also can be regarded as an extension of the Schwarzschild solution. This solution enables us to deduce an important result that an evolving universe is related with a static universe by the gauge transformation, i.e., they are gauge equivalent. This means that having a different universe simply corresponds to choosing a different gauge.

Dynamical twists in group algebras

We classify twists in group algebras of finite groups. Namely, we set up a bijective correspondence between gauge equivalence classes of twists (which are solutions of a certain non-linear functional equation) and isomorphism classes of dynamical data described in purely group theoretical terms. This generalizes the classification of usual twists obtained by Movshev and Etingof-Gelaki.

Iso-Huygens Deformations of Homogeneous Differential Operators Related to a Special Cone of Rank \({\text{3}}\)

We consider iso-Huygens deformations of homogeneous hyperbolic Gindikin operators related to a special cone of rank \({\text{3}}\). The deformations are carried out with the use of Stellmacher--Lagnese and Calogero--Moser potentials. Using the notion of gauge equivalence of operators and the algebraic method of intertwining operators, we write out the fundamental solutions of the deformed operators in closed form and give sufficient conditions for the Huygens principle to hold for these operators in the strengthened or ordinary form.

Noncommutative gauge theories from deformation quantization

We construct noncommutative gauge theories based on the notion of the Weyl bundle, which appears in Fedosov's construction of deformation quantization on an arbitrary symplectic manifold. These correspond to D-brane worldvolume theories in non-constant B-field and curved backgrounds in string theory. All such theories are embedded into a “universal” gauge theory of the Weyl bundle. This shows that the combination of a background field and a noncommutative field strength has universal meaning as a field strength of the Weyl bundle. We also show that the gauge equivalence relation is a part of such a “universal” gauge symmetry.

A discretization of the matrix nonlinear Schrödinger equation

In this paper, we shall show that a class of solutions to the discrete coupled matrix nonlinear Schrodinger equation (DCMNLSE) is gauge equivalent to the discrete equation of the Schrodinger flow of maps into the Grassmannian and the realizing gauge transformation is only the discretization of a classical gauge transformation between the matrix nonlinear Schrodinger equation (MNLSE) and the Schrodinger flow of maps into the Grassmannian. In other words, from the viewpoint of gauge equivalence, a class of solutions of the DCMNLSE is a correct discretization of the MNLSE.

The NLS-equation and itsSL(2,R) structure

The relationship of the group SL(2,R) and the NLS- equation is presented. As a consequence, the SL(2,R) gauge equivalence between the NLS- and the M-HF model is proved, which provides a new example in geometrically explaining dynamical properties of soliton equations by the SL(2,R) structure.

Pohlmeyer's transformation and the (2+0) integrable equations of statistical physics

Using the Pohlmeyer transformation in Euclidean space, we obtain an equation which provides the possibility of constructing a wide class of two-dimensional elliptic integrable systems. The problem of the gauge equivalence of these systems with the nonlinear sigma-model and the Getmanov and Bitsadze models is investigated. Exact solutions of some equations are constructed. Bibliography: 34 titles.

Topological dynamics on moduli spaces, I

Let M be an orientable genus g > 0 surface with boundary O3M. Let 17 be the mapping class group of M fixing OM. The group 17 acts on Mc = Homc(7r1(M),SU(2))/SU(2), the space of SU(2)-gauge equivalence classes of flat SU(2)-connections on M with fixed holonomy on O3M. We study the topological dynamics of the 1-action and give conditions for the individual F-orbits to be dense in Mc.

Gauge equivalence and l-reductions of the differential–difference KP equation

The gauge equivalence of the differential–difference KP hierarchy is studied in the frame work of Sato theory and the l-reductions are investigated. The infinitely many conservation laws and symmetries of the resulting systems from this analysis is also been presented explicitly.

Comments on gauge equivalence in noncommutative geometry

We investigate the transformation from ordinary gauge field to noncommutative one which was introduced by N.Seiberg and E.Witten (hep-th/9908142). It is shown that the general transformation which is determined only by gauge equivalence has a path dependence in `\theta-space'. This ambiguity is negligible when we compare the ordinary Dirac-Born-Infeld action with the noncommutative one in the U(1) case, because of the U(1) nature and slowly varying field approximation. However, in general, in the higher derivative approximation or in the U(N) case, the ambiguity cannot be neglected due to its noncommutative structure. This ambiguity corresponds to the degrees of freedom of field redefinition.

A note on the Lax pairs for Painlevéequations

For the classical Painleveequations, besides the method of similarity reduction of Lax pairs for integrable partial differential equations, two ways are known for Lax pair generation. The first is based on the confluence procedure in Fuchs' linear ODE with four regular singularities isomonodromy deformation which is governed by the sixth Painleveequation. The second method treats the hypergeometric equation and confluent hypergeometric equations as the isomonodromy deformation equations for the triangular systems of ODEs, in whose non-triangular extensions give rise to the Lax pairs for the Painleveequations. The theory of integrable integral operators suggests a new way of Lax pair generation for the classical Painleveequations. This method involves a special kind of gauge transformation that is applied to linear systems which are exactly solvable in terms of the classical special functions. Some of the Lax pairs we introduce are known, others are new. The question of gauge equivalence of different Lax pairs for the Painleveequations is considered as well.

Inverse scattering method and vector higher order non-linear Schrödinger equation

A generalized inverse scattering method has been developed for arbitrary n-dimensional Lax equations. Subsequently, the method has been used to obtain N-soliton solutions of a vector higher order non-linear Schrödinger equation, proposed by us. It has been shown that under a suitable reduction, the vector higher order non-linear Schrödinger equation reduces to the higher order non-linear Schrödinger equation. An infinite number of conserved quantities have been obtained by solving a set of coupled Riccati equations. Gauge equivalence is shown between the vector higher order non-linear Schrödinger equation and the generalized Landau–Lifshitz equation and the Lax pair for the latter equation has also been constructed in terms of the spin field, establishing direct integrability of the spin system.

Further comments on the background field method and gauge-invariant effective actions

The aim of this paper is to give a firm and clear proof of the existence in the background field framework of a gauge-invariant effective action for any gauge theory (background gauge equivalence). Here by effective action we mean a functional whose Legendre transform restricted to the physical shell generates the matrix elements of the connected S-matrix. We resume and clarify a former argument due to Abbott, Grisaru and Schaefer based on the gauge-artifact nature of the background fields and on the identification of the gauge-invariant effective action with the generator of the proper, background field, vertices.

On the relation between 2+1 Einstein gravity and Chern-Simons theory

A simple example is given to show that the gauge equivalence classes of physical states in Chern-Simons theory are not in one-to-one correspondence with those of Einstein gravity in three spacetime dimensions. The two theories are therefore not equivalent. It is shown that including singular metrics into general relativity has more, and in fact a quite counter-intuitive, impact on the theory than one naively expects.

Examples of vector bundles admitting unique ASD connections on quaternion-Kähler manifolds

We prove a series of rigidity theorems. Examples of higher rank ASD vector bundles are given on quaternion-Kähler manifolds, which admit the one and only ASD connections modulo the gauge equivalence.

Soliton solutions, Liouville integrability and gauge equivalence of Sasa Satsuma equation

Exact integrability of the Sasa Satsuma equation (SSE) in the Liouville sense is established by showing the existence of an infinite set of conservation laws. The explicit form of the conserved quantities in terms of the fields are obtained by solving the Riccati equation for the associated 3×3 Lax operator. The soliton solutions, in particular, one and two soliton solutions, are constructed by the Hirota’s bilinear method. The one soliton solution is also compared with that found through the inverse scattering method. The gauge equivalence of the SSE with a generalized Landau Lifshitz equation is established with the explicit construction of the new equivalent Lax pair.

Complex Anti-Self-Dual Connections on a Product of Calabi-Yau Surfaces and Triholomorphic Curves

We study the adiabatic limit of a sequence of Ω-anti-self-dual connections on unitary bundles over a product of two compact Calabi–Yau surfaces M×N by scaling metrics to shrink N to a point. We show that after fixing gauge transformations, a subsequence of the N-components of these connections converges to a triholomorphic curve from M away from a Cayley cycle in M×N to the moduli space \({\cal M}_N\) of instantons on M×N modulo gauge equivalence in the Hausdorff topology, and converges on the blow-up locus to a family, which is parameterized by the Cayley cycle, of triholomorphic curves from C 2 to \({\cal M}_N\).

The gauge equivalence of the NLS and the Schrödinger flow of maps in 2 + 1 dimensions

The gauge equivalence of the analogues of the nonlinear Schrodinger equation NLS- and those of the Schrodinger flow of maps into H2 (the Minkowski HF model) in 2 + 1 dimensions are proved, respectively. Combining these with the already-known results, we obtain a complete understanding of the gauge equivalence of the analogous models of the nonlinear Schrodinger equation (for = 1 or -1) and those of the Heisenberg ferromagnet model (for Euclidean or Minkowski) in 2 + 1 dimensions.