Gauge Equivalence Research Articles

Inverse problems for Schrödinger equations with Yang–Mills potentials in domains with obstacles and the Aharonov–Bohm effect

We study the inverse boundary value problems for the Schrödinger equations with Yang-Mills potentials in a bounded domain Ω0 ⊂ Rn containing finite number of smooth obstacles Ωj , 1 ⩽ j ⩽ r. We prove that the Dirichlet-to-Neumann operator on ∂Ω0 determines the gauge equivalence class of the Yang-Mills potentials. We also prove that the metric tensor can be recovered up to a diffeomorphism that is identity on ∂Ω0.

Witten triples and the Seiberg–Witten equations on a complex surface

AbstractWe study the solutions of the Seiberg–Witten equations on complex surfaces. We show that for a large class of parameters, the gauge equivalence classes of irreducible solutions of the twisted Seiberg–Witten equations correspond to stable Witten triples. We prove that on Kähler surfaces this correspondence is the set‐theoretical support of an isomorphism of real‐analytic spaces. This makes it possible to take multiplicities into account and generalizes and unifies results previously obtained by Witten. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the Connectedness of Moduli Spaces of Flat Connections over Compact Surfaces

AbstractWe study the connectedness of the moduli space of gauge equivalence classes of flat G-connections on a compact orientable surface or a compact nonorientable surface for a class of compact connected Lie groups. This class includes all the compact, connected, simply connected Lie groups, and some non-semisimple classical groups.

Cohomology of the moduli space of SO( n)-instantons with instanton number 1

Let M(1, SO( n)) denote the moduli space of based gauge equivalence classes of SO( n)-instantons over S 4 with instanton number 1. In this paper, using a description of M(1, SO( n)) in terms of a homogeneous space, we determine the mod 2 cohomology ring with squaring operations and the integral cohomology group of M(1, SO( n)).

Moduli space of potentials in the Aharonov-Bohm effect

We present a simplified derivation of the fact that the set of gauge equivalence classes or moduli space of flat connections (potentials) in the abelian Aharonov-Bohm effect, is isomorphic to the circle. The length of this circle is the absolute value of the electric charge.

The matrix realization of affine Jacobi varieties and the extended Lotka–Volterra lattice

We study completely integrable Hamiltonian systems whose monodromy matrices are related to the representatives for the set of gauge equivalence classes $\boldsymbol{\mathcal{M}}_F$ of polynomial matrices. Let $X$ be the algebraic curve given by the common characteristic equation for $\boldsymbol{\mathcal{M}}_F$. We construct the isomorphism from the set of representatives to an affine part of the Jacobi variety of $X$. This variety corresponds to the invariant manifold of the system, where the Hamiltonian flow is linearized. As the application, we discuss the algebraic completely integrability of the extended Lotka-Volterra lattice with a periodic boundary condition.

The real locus of an involution map on the moduli space of flat connections on a Riemann surface

It is known that every nonorientable surface Σ has an orientable double cover Σ~. The covering map induces an involution on the moduli space ℳ~ of gauge equivalence classes of flat G-connections on Σ~. We identify the relation between the moduli space ℳ and the fixed point set of the moduli space ℳ~. In particular, ℳ is isomorphic to the fixed point set of ℳ~ if and only if the order of the center of G is odd. One important application is that we give a way to construct a minimal Lagrangian submanifold of the moduli space ℳ~.

Gauge equivalence in QCD: The Weyl and Coulomb gauges

The Weyl-gauge ($A_0^a=0)$ QCD Hamiltonian is unitarily transformed to a representation in which it is expressed entirely in terms of gauge-invariant quark and gluon fields. In a subspace of gauge-invariant states we have constructed that implement the non-Abelian Gauss's law, this unitarily transformed Weyl-gauge Hamiltonian can be further transformed and, under appropriate circumstances, can be identified with the QCD Hamiltonian in the Coulomb gauge. We demonstrate an isomorphism that materially facilitates the application of this Hamiltonian to a variety of physical processes, including the evaluation of $S$-matrix elements. This isomorphism relates the gauge-invariant representation of the Hamiltonian and the required set of gauge-invariant states to a Hamiltonian of the same functional form but dependent on ordinary unconstrained Weyl-gauge fields operating within a space of ``standard'' perturbative states. The fact that the gauge-invariant chromoelectric field is not hermitian has important implications for the functional form of the Hamiltonian finally obtained. When this nonhermiticity is taken into account, the ``extra'' vertices in Christ and Lee's Coulomb-gauge Hamiltonian are natural outgrowths of the formalism. When this nonhermiticity is neglected, the Hamiltonian used in the earlier work of Gribov and others results.

Time-independent stochastic quantization, Dyson-Schwinger equations, and infrared critical exponents in QCD

We derive the equations of time-independent stochastic quantization, without reference to an unphysical fifth time, from the principle of gauge equivalence. It asserts that probability distributions P that give the same expectation values for gauge-invariant observables $〈W〉=\ensuremath{\int}\mathrm{dAWP}$ are physically indistinguishable. This method escapes the Gribov critique. We derive an exact system of equations that closely resembles the Dyson-Schwinger equations of Faddeev-Popov theory. The system is truncated and solved nonperturbatively, by means of a power law ansatz, for the critical exponents that characterize the asymptotic form at $k\ensuremath{\approx}0$ of the gluon propagator in Landau gauge. For the transverse and longitudinal parts, we find, respectively, ${D}^{T}\ensuremath{\sim}{(k}^{2}{)}^{\ensuremath{-}1\ensuremath{-}{\ensuremath{\alpha}}_{T}}\ensuremath{\approx}{(k}^{2}{)}^{0.043},$ suppressed and in fact vanishing, though weakly, and ${D}^{L}\ensuremath{\sim}{a(k}^{2}{)}^{\ensuremath{-}1\ensuremath{-}{\ensuremath{\alpha}}_{L}}\ensuremath{\approx}{a(k}^{2}{)}^{\ensuremath{-}1.521},$ enhanced, with ${\ensuremath{\alpha}}_{T}=\ensuremath{-}2{\ensuremath{\alpha}}_{L}.$ Although the longitudinal part vanishes with the gauge parameter a in the Landau-gauge limit $\stackrel{\ensuremath{\rightarrow}}{a}0$ there are vertices of order ${a}^{\ensuremath{-}1}$ so, counterintuitively, the longitudinal part of the gluon propagator does contribute in internal lines in the Landau gauge, replacing the ghost that occurs in Faddeev-Popov theory. We compare our results with the corresponding results in Faddeev-Popov theory.

Gauge equivalence of Dirac structures and symplectic groupoids

Nous etudions les transformations de jauge des structures de Dirac et la relation entre les equivalences de jauge et de Morita pour les varietes de Poisson. Nous decrivons comment la structure symplectique d'un groupoide symplectique est modifiee lors d'une transformation de jauge de la structure de Poisson de la section identite de ce groupoide et nous prouvons que des structures de Poisson integrables equivalentes sous une transformation de jauge sont equivalentes au sens de Morita. Comme exemple, nous etudions certains ensembles generiques de structures de Poisson sur les surfaces de Riemann : nous exhibons des invariants complets d'equivalence de jauge pour de telles structures qui, sur la sphere S 2 , donnent un invariant complet d'equivalence de Morita.

Exceptional discrete mapping class group orbits in moduli spaces

Let M be a four-holed sphere and Γ the mapping class group of M fixing ∂M . The group Γ acts on the space MB(SU(2)) of SU(2)-gauge equivalence classes of flat SU(2)-connections on M with fixed holonomy on ∂M . We give examples of flat SU(2)connections whose holonomy groups are dense in SU(2), but whose Γ-orbits are discrete in MB(SU(2)). This phenomenon does not occur for surfaces with genus greater than zero.

Connected components of the space of surface group representations

Let G be a connected, compact, semisimple Lie group. It is known that for a compact closed orientable surface $\Sigma$ of genus $l >1$, the order of the group $H^2(\Sigma,\pi_1(G))$ is equal to the number of connected components of the space $Hom(\pi_1(\Sigma),G)/G$ which can also be identified with the moduli space of gauge equivalence classes of flat G-bundles over $\Sigma$. We show that the same statement for a closed compact nonorientable surface which is homeomorphic to the connected sum of k copies of the real projective plane, where $k\neq 1,2,4$, can be easily derived from a result in A. Alekseev, A.Malkin and E. Meinrenken's recent work on Lie group valued moment maps.

The extended Lotka–Volterra lattice and affine Jacobi varieties of spectral curves

Based on the work by Smirnov and Zeitlin, we study a simple realization of the matrix construction of the affine Jacobi varieties. We find that the realization is given by a classical integrable model, the extended Lotka–Volterra lattice. We investigate the integrable structure of the representative for the gauge equivalence class of matrices, which is isomorphic to the affine Jacobi variety, and make use of it to discuss the solvability of the model.

On N-wave type systems and their gauge equivalent

The class of nonlinear evolution equations - gauge equivalent to the N-wave equations related to the simple Lie algebra g are derived and analyzed. They are written in terms of the functions S(x,t) satisfying r= rank g nonlinear constraints. The corresponding Lax pairs and the time evolution of the scattering data are found. The Zakharov-Shabat dressing method is appropriately modified to construct their soliton solutions.

Lagrangian sections and holomorphic U(1)-connections

We construct a correspondence between the complex gauge equivalence classes of holomorphic U(1)-connections on a smooth semi-flat special Lagrangian torus fibration and the Hamiltonian deformation classes of Lagrangian sections E in the mirror manifold together with the gauge equivalence classes of flat U(1)-connections on Σ.

Topological dynamics on moduli spaces II

Let $M$ be an orientable genus $g>0$ surface with boundary $\partial M$. Let $\Gamma$ be the mapping class group of $M$ fixing $\partial M$. The group $\Gamma$ acts on ${\mathcal M}_{\mathcal C} = \operatorname {Hom}_{\mathcal C}(\pi _1(M),\operatorname {SU})/\operatorname {SU},$ the space of $\operatorname {SU}$-gauge equivalence classes of flat $\operatorname {SU}$-connections on $M$ with fixed holonomy on $\partial M$. We study the topological dynamics of the $\Gamma$-action and give conditions for the individual $\Gamma$-orbits to be dense in ${\mathcal M}_{\mathcal C}$.

Aharonov-Bohm Effect, Flat Connections, and Green's Theorem

The validity of Green's theorem, and hence of Stokes' theorem, when the involved vector field is differentiable but not continuously differentiable, is crucial for a theoretical explanation of the Aharonov–Bohm (A-B) effect; we review this theorem. We describe the principal bundle in which the A-B effect occurs, and give the geometrical description of the relevant connection. We study the set of gauge equivalence classes of flat connections on a product bundle with abelian structural group, and show that this set has a canonical group structure, which is isomorphic to a quotient of cohomology groups. We apply this result to the A-B bundle and calculate the holonomy groups of all flat connections.

KNOT AND LINK INVARIANTS AND MODULI SPACE OF PARABOLIC BUNDLES

In this paper, we show that the representation variety of the fundamental group of a 2n-punctured S2 with different conjugacy classes in SU(2) along punctures is a symplectic stratified variety from the group cohomology point of view. The representation variety can be identified with the moduli space of s-equivalence classes of stable parabolic bundles over the 2n-punctured S2 with corresponding weights along punctures, and also can be identified with the moduli space of gauge equivalence classes of SU(2)-flat connections with prescribed holonomies along punctures. We obtain an invariant of links (knots) from intersection theory on such a moduli space (a generalization of the signature of the link). We also study a SL2(C)-character variety of a knot K in S3 with fixed holonomy μ + μ-1 along the meridian of π1(S3\ K) (μ ∈ C*). The fixed-trace condition rules out the possibility of reducible representations with non-abelian image and the ideal point of irreducible representations via a generic perturbation. For knots without closed incompressible surfaces in S3 \ K, we show that there is a well-defined SL2(C)-knot invariant which is also related to them A-polynomial for special values μ ∈ U(1)\ {± 1}.

On the prolongation structure and integrability of HNLS equation

Higher order nonlinear Schrödinger equation occurring in the field of nonlinear optics is analysed from the view point of prolongation theory and Painléve analysis. The gauge equivalence of the equation to the complex modified K dV equation is never used. But from our direct analysis we can sort out the various relations as existing between the parameters of the equation for which it is integrable and one can have a Lax pair.

Group cohomology and gauge equivalence of some twisted quantum doubles

Dijkgraaf, Pasquier and Roche introduced twisted quantum doubles of a finite group in the context of conformal field theory. We study equivalences that arise among the braided monoidal categories associated to these quantum doubles, especially in the commutative case. This involves a close study of the cohomology of various complexes introduced by Eilenberg-MacLane. We show among other things that an equivalence of braided monoidal categories is the same as gauge equivalence of the corresponding quantum doubles, and that an invariant for an equivalence class is given by a metabolic quadratic space.