Gauge Equivalence Research Articles

The geometry of fourth-order differential operator

In the present paper, the equivalence problem for fourth-order differential operators with one variable under general fiber-preserving transformation using the Cartan method of equivalence is applied. Two versions of equivalence problems are considered. First, the direct equivalence problem and second equivalence problem is to determine the sufficient and necessary conditions on two fourth-order differential operators such that there exists a fiber-preserving transformation mapping one to the other according to gauge equivalence.

Combinatorial descent data for gerbes

We consider descent data in cosimplicial crossed groupoids. This is a combinatorial abstraction of the descent data for gerbes in algebraic geometry. The main result is this: a weak equivalence between cosimplicial crossed groupoids induces a bijection on gauge equivalence classes of descent data.

Novel ambiguities in the Seiberg-Witten map and the emergent gravity

A homogeneous part of the Seiberg-Witten gauge equivalence relation for gauge fields can lead to solutions that involve matter fields in such a way that the gauge equivalence and the dimensional and index structures are preserved. In particular, we consider scalar fields coupled to U(1) gauge fields. The matter fields appear non-linearly in the map. As an application, we analyze the implication of this ambiguity to emergent gravity at the first order in noncommutative parameter and show that the new ambiguity restores the possibility of conformal coupling of real scalar density field that is coupled non-minimally to the emergent gravity induced by gauge fields -- a possibility that is strictly not allowed if we consider only the already known ambiguity in the Seiberg-Witten map.

Loop differential K-theory

In this paper we introduce an equivariant extension of the Chern-Simons form, associated to a path of connections on a bundle over a manifold M, to the free loop space LM, and show it determines an equivalence relation on the set of connections on a bundle. We use this to define a ring, loop differential K-theory of M, in much the same way that differential K-theory can be defined using the Chern-Simons form [14]. We show loop differential K-theory yields a refinement of differential K-theory, and in particular incorporates holonomy information into its classes. Additionally, loop differential K-theory is shown to be strictly coarser than the Grothendieck group of bundles with connection up to gauge equivalence. Finally, we calculate loop differential K-theory of the circle.

A Gauge Transformation between Ragnisco-Tu Hierarchy and a Related Lattice Hierarchy

A new lattice hierarchy related to Ragnisco-Tu equation is proposed and its gauge equivalence to Ragnisco-Tu equation is proven. As an application of gauge transformation, we construct Darboux transformation (DT) of this new equation through DT of Ragnisco-Tu equation. An explicit exact solution is presented as an example.

The Global Symmetry Group of Quantum Spectral Beams and Geometric Phase Factors

We propose a cohomological modelling schema of quantum state spaces and their connectivity structures in relation to the formulation of global geometric phase phenomena. In the course of this schema, we introduce the notion of Hermitian differential line sheaves or unitary rays and classify their gauge equivalence classes in terms of a global differential invariant given by the de Rham cohomology class of the curvature. Furthermore, we formulate and interpret physically the curvature recognition integrality theorem for unitary rays. Using this recognition theorem, we define the notion of a quantum spectral beam and show that it has an affine space structure with structure group given by the characters of the fundamental group.

Belavin-Drinfeld cohomologies and introduction to classification of quantum groups

In the present article we discuss the classification of quantum groups whose quasiclassical limit is a given simple complex Lie algebra g. This problem reduces to the classification of all Lie bialgebra structures on g(K), where K = C((ℏ)). The associated classical double is of the form g(K)⊗K A, where A is one of the following: K[ε], where ε2 =0, K ⊕ K or K[j], where j2 = ℏ. The first case relates to quasi-Frobenius Lie algebras. In the second and third cases we introduce a theory of Belavin-Drinfeld cohomology associated to any non-skewsymmetric r-matrix from the Belavin-Drinfeld list [1]. We prove a one-to-one correspondence between gauge equivalence classes of Lie bialgebra structures on g(K) and cohomology classes (in case II) and twisted cohomology classes (in case III) associated to any non-skewsymmetric r-matrix.

Twisted deformation quantization of algebraic varieties

Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf OX. These are stack-like versions of usual deformations. We prove that there is a twisted quantization operation from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on gauge equivalence classes. This result extends work of Kontsevich, and our own earlier work, on deformation quantization of algebraic varieties.

On quantum groups and Lie bialgebras related to sl(n)

Given an arbitrary field of characteristic 0, we study Lie bialgebra structures on sl(n, ), based on the description of the corresponding classical double. For any Lie bialgebra structure δ, the classical double D(sl(n, ),δ) is isomorphic to sl(n, ) ⊗A, where A is either [ε], with ε2 = 0, or ⊗ or a quadratic field extension of . In the first case, the classification leads to quasi-Frobenius Lie subalgebras of sl(n, ). In the second and third cases, a Belavin-Drinfeld cohomology can be introduced which enables one to classify Lie bialgebras on sl(n, ), up to gauge equivalence. The Belavin-Drinfeld untwisted and twisted cohomology sets associated to an r-matrix are computed.

Nonintegrable semidiscrete Hirota equation: gauge-equivalent structures and dynamical properties.

In this paper, we investigate nonintegrable semidiscrete Hirota equations, including the nonintegrable semidiscrete Hirota(-) equation and the nonintegrable semidiscrete Hirota(+) equation. We focus on the topics on gauge-equivalent structures and dynamical behaviors for the two nonintegrable semidiscrete equations. By using the concept of the prescribed discrete curvature, we show that, under the discrete gauge transformations, the nonintegrable semidiscrete Hirota(-) equation and the nonintegrable semidiscrete Hirota(+) equation are, respectively, gauge equivalent to the nonintegrable generalized semidiscrete modified Heisenberg ferromagnet equation and the nonintegrable generalized semidiscrete Heisenberg ferromagnet equation. We prove that the two discrete gauge transformations are reversible. We study the dynamical properties for the two nonintegrable semidiscrete Hirota equations. The exact spatial period solutions of the two nonintegrable semidiscrete Hirota equations are obtained through the constructions of period orbits of the stationary discrete Hirota equations. We discuss the topic regarding whether the spatial period property of the solution to the nonintegrable semidiscrete Hirota equation is preserved to that of the corresponding gauge-equivalent nonintegrable semidiscrete equations under the action of discrete gauge transformation. By using the gauge equivalent, we obtain the exact solutions to the nonintegrable generalized semidiscrete modified Heisenberg ferromagnet equation and the nonintegrable generalized semidiscrete Heisenberg ferromagnet equation. We also give the numerical simulations for the stationary discrete Hirota equations. We find that their dynamics are much richer than the ones of stationary discrete nonlinear Schrödinger equations.

A C*-Algebra for Quantized Principal U(1)-Connections on Globally Hyperbolic Lorentzian Manifolds

The aim of this work is to complete our program on the quantization of connections on arbitrary principal U(1)-bundles over globally hyperbolic Lorentzian manifolds. In particular, we show that one can assign via a covariant functor to any such bundle an algebra of observables which separates gauge equivalence classes of connections. The C*-algebra we construct generalizes the usual CCR-algebras, since, contrary to the standard field-theoretic models, it is based on a presymplectic Abelian group instead of a symplectic vector space. We prove a no-go theorem according to which neither this functor, nor any of its quotients, satisfies the strict axioms of general local covariance. As a byproduct, we prove that a morphism violates the locality axiom if and only if a certain induced morphism of cohomology groups is non-injective. We show then that, fixing any principal U(1)-bundle, there exists a suitable category of subbundles for which a quotient of our functor yields a quantum field theory in the sense of Haag and Kastler. We shall provide a physical interpretation of this feature and we obtain some new insights concerning electric charges in locally covariant quantum field theory.

Comments on observables for identity-based marginal solutions in Berkovits’ superstring field theory

We construct an analytic solution for tachyon condensation around identity-based marginal solutions in Berkovits' WZW-like open superstring field theory. Using this, which is a kind of wedge-based solution, the gauge invariant overlaps for the identity-based marginal solutions can be calculated analytically. This is a straightforward extension of a method in bosonic string field theory, which has been elaborated by the authors, to superstring. We also comment on a gauge equivalence relation between the tachyon vacuum solution and its marginally deformed one. From this viewpoint, we can find the vacuum energy of the identity-based marginal solutions to be zero, which agrees with the previous result as a consequence of $\xi$ zero mode counting.

Electromagnetism, Local Covariance, the Aharonov–Bohm Effect and Gauss’ Law

We quantise the massless vector potential A of electromagnetism in the presence of a classical electromagnetic (background) current, j, in a generally covariant way on arbitrary globally hyperbolic spacetimes M. By carefully following general principles and procedures we clarify a number of topological issues. First we combine the interpretation of A as a connection on a principal U(1)-bundle with the perspective of general covariance to deduce a physical gauge equivalence relation, which is intimately related to the Aharonov-Bohm effect. By Peierls' method we subsequently find a Poisson bracket on the space of local, affine observables of the theory. This Poisson bracket is in general degenerate, leading to a quantum theory with non-local behaviour. We show that this non-local behaviour can be fully explained in terms of Gauss' law. Thus our analysis establishes a relationship, via the Poisson bracket, between the Aharonov-Bohm effect and Gauss' law (a relationship which seems to have gone unnoticed so far). Furthermore, we find a formula for the space of electric monopole charges in terms of the topology of the underlying spacetime. Because it costs little extra effort, we emphasise the cohomological perspective and derive our results for general p-form fields A (p < dim(M)), modulo exact fields. In conclusion we note that the theory is not locally covariant, in the sense of Brunetti-Fredenhagen-Verch. It is not possible to obtain such a theory by dividing out the centre of the algebras, nor is it physically desirable to do so. Instead we argue that electromagnetism forces us to weaken the axioms of the framework of local covariance, because the failure of locality is physically well-understood and should be accommodated.

Module parallel transports in fuzzy gauge theory

In this paper, we define and investigate a notion of parallel transport on finite projective modules over finite matrix algebras. Given a derivation-based differential calculus on the algebra and a connection on the module, we construct for every derivation X a module parallel transport, which is a lift to the module of the one-parameter group of algebra automorphisms generated by X. This parallel transport morphism is determined uniquely by an ordinary differential equation depending on the covariant derivative along X. Based on these parallel transport morphisms, we define a basic set of gauge invariant observables, i.e. functions from the space of connections to the complex numbers. For modules equipped with a Hermitian structure, we prove that this set of observables is separating on the space of gauge equivalence classes of Hermitian connections. This solves the gauge copy problem for fuzzy gauge theories.

Quantized Abelian Principal Connections on Lorentzian Manifolds

We construct a covariant functor from a category of Abelian principal bundles over globally hyperbolic spacetimes to a category of *-algebras that describes quantized principal connections. We work within an appropriate differential geometric setting by using the bundle of connections and we study the full gauge group, namely the group of vertical principal bundle automorphisms. Properties of our functor are investigated in detail and, similar to earlier works, it is found that due to topological obstructions the locality property of locally covariant quantum field theory is violated. Furthermore, we prove that, for Abelian structure groups containing a nontrivial compact factor, the gauge invariant Borchers-Uhlmann algebra of the vector dual of the bundle of connections is not separating on gauge equivalence classes of principal connections. We introduce a topological generalization of the concept of locally covariant quantum fields. As examples, we construct for the category of principal U(1)-bundles two natural transformations from singular homology functors to the quantum field theory functor that can be interpreted as the Chern class and the electric charge. In this case we also prove that the electric charges can be consistently set to zero, which yields another quantum field theory functor that satisfies all axioms of locally covariant quantum field theory.

Removal of singularities and Gromov compactness for symplectic vortices

We prove that the moduli space of gauge equivalence classes of symplectic vortices with uniformly bounded energy in a compact Hamiltonian manifold admits a Gromov compact- ication by polystable vortices. This extends results of Mundet i Riera for circle actions to the case of arbitrary compact Lie groups. Our argument relies on an a priori estimate for vor- tices that allows us to apply techniques used by McDu and Salamon in their proof of Gromov compactness for pseudoholomorphic curves. As an intermediate result we prove a removable singularity theorem for symplectic vortices.

Gauge equivalence and the inverse spectral problem for the magnetic Schrödinger operator on the torus

We study the inverse spectral problem for the Schrodinger operator H on the two-dimensional torus with even magnetic field B(x) and even electric potential V(x). Guillemin [11] proved that the spectrum of H determines B(x) and V(x). A simple proof of Guillemin’s results was given by the authors in [3]. In the present paper, we consider gauge equivalent classes of magnetic potentials and give conditions which imply that the gauge equivalence class and the spectrum of H determine the magnetic field and the electric potential. We also show that, generically, the spectrum and the magnetic field determine the “extended” gauge equivalence class of the magnetic potential. The proof is a modification of that in [3] with some corrections and clarifications.

Altered Maxwell equations in the length gauge

The length gauge uses a scalar potential to describe a laser field, thus treating it as a longitudinal field rather than as a transverse field. This distinction is manifested by the fact that the Maxwell equations that relate to the length gauge are not the same as those for transverse fields. In particular, a source term is necessary in the length-gauge Maxwell equations, whereas the Coulomb-gauge description of plane waves possesses the basic property of transverse fields that they propagate with no source terms at all. This difference is shown to be importantly consequential in some previously unremarked circumstances; and it explains why the Göppert-Mayer gauge transformation does not provide the security that might be expected of full gauge equivalence.

Inverse Boundary Problems for Systems in Two Dimensions

We prove identification of coefficients up to gauge equivalence by Cauchy data at the boundary for elliptic systems on oriented compact surfaces with boundary or domains of \({\mathbb{C}}\) . In the geometric setting, we fix a Riemann surface with boundary and consider both a Dirac-type operator plus potential acting on sections of a Clifford module and a connection Laplacian plus potential (i.e. Schrodinger Laplacian with external Yang–Mills field) acting on sections of a Hermitian bundle. In either case we show that the Cauchy data determine both the connection and the potential up to a natural gauge transformation: conjugation by an endomorphism of the bundle which is the identity at the boundary. For domains of \({\mathbb{C}}\) , we recover zeroth order terms up to gauge from Cauchy data at the boundary in first order elliptic systems.

Cartan equivalence problem for third order differential operators

This article is dedicated to solving the equivalence problem for a pair of third-order differential operators on the line under general fiber-preserving transformation using the Cartan method of equivalence. We will treat 2 versions of equivalence problems: first, the direct equivalence problem, and second, an equivalence problem to determine conditions on 2 differential operators such that there exists a fiber-preserving transformation mapping one to the other according to gauge equivalence.