Abelian Theory Research Articles

Nonabelian Toda Theories from Parafermionic Reductions of the WZW Model

We investigate a class of conformal nonabelian-Toda models representing noncompact SL(2, R)/U(1) parafermions (PF) interacting with specific abelian Toda theories and having a global U(1) symmetry. A systematic derivation of the conserved currents, their algebras, and the exact solution of these models are presented. An important property of this class of models is the affine SL(2, R)q algebra spanned by charges of the chiral and antichiral nonlocal currents and the U(1) charge. The classical (Poisson brackets) algebras of symmetries VGn of these models appear to be of mixed PF-WGn type. They contain together with the local quadratic terms specific for the Wn-algebras the nonlocal terms similar to the ones of the classical PF-algebra. The renormalization of the spins of the nonlocal currents is the main new feature of the quantum VAn-algebras. The quantum VA2-algebra and its degenerate representations are studied in detail

Landau-Ginzburg theories for non-abelian quantum Hall states

We construct Landau-Ginzburg effective field theories for fractional quantum Hall states - such as the Pfaffian state - which exhibit non-abelian statistics. These theories rely on a Meissner construction which increases the level of a non-abelian Chem-Simons theory while simultaneously projecting out the unwanted degrees of freedom of a concomitant enveloping abelian theory. We describe this construction in the context of a system of bosons at Landau level filling factor ν = l, where the non-abelian symmetry is a dynamically generated SU(2) continuous extension of the discrete particle-hole symmetry of the lowest Landau level. We show how the physics of quasiparticles and their non-abelian statistics arises in this Landau-Ginzburg theory. We describe its relation to edge theories - where a coset construction plays the role of the Meissner projection — and discuss extensions to other states.

NON-ABELIAN DUAL MAPS IN PATH SPACE

We study an extension of the procedure to construct duality transformations among Abelian gauge theories to the non-Abelian case using a path space formulation. We define a pre-dual functional in path space and introduce a particular nonlocal map among Lie algebra valued one-form functionals which reduces to the ordinary Hodge-⋆ duality map of the Abelian theories. Further, we establish a full set of equations on path space representing the ordinary Yang–Mills equations and Bianchi identities of non-Abelian gauge theories of four-dimensional Euclidean space.

On mirror symmetry in three dimensional Abelian gauge theories

We present an identity relating the partition function of N=4 supersymmetric QED to that of its dual under mirror symmetry. The identity is a generalized Fourier transform. Many known properties of abelian theories can be derived from this formula, including the mirror transforms for more general gauge and matter content. We show that N=3 Chern-Simons QED and N=4 QED with BF-type couplings are conformal field theories with exactly marginal couplings. Mirror symmetry acts on these theories as strong-weak coupling duality. After identifying the mirror of the gauge coupling (sometimes called the ``magnetic coupling'') we construct a theory which is exactly mirror -- at all scales -- to N=4 SQED. We also study vortex-creation operators in the large $N_f$ limit.

Existence Proofs

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsFred RichmanFRED RICHMAN has an AB from Princeton University and a PhD from the University of Chicago. After teaching at New Mexico State University for many years, dividing his research time between infinite abelian group theory and constructive mathematics, he worked off and on at TCI Software Research as a developer of Scientific Word and Scientific Workplace. He has coauthored texts on modern algebra, mathematics for liberal arts students, constructive algebra, and varieties of constructive mathematics.

A new approach to axial vector model calculations II

We further develop the new approach, proposed in part I (hep-th/9807072), to computing the heat kernel associated with a Fermion coupled to vector and axial vector fields. We first use the path integral representation obtained for the heat kernel trace in a vector-axialvector background to derive a Bern-Kosower type master formula for the one-loop amplitude with $M$ vectors and $N$ axialvectors, valid in any even spacetime dimension. For the massless case we then generalize this approach to the full off-diagonal heat kernel. In the D=4 case the SO(4) structure of the theory can be broken down to $SU(2) \times SU(2)$ by use of the 't Hooft symbols. Various techniques for explicitly evaluating the spin part of the path integral are developed and compared. We also extend the method to external fermions, and to the inclusion of isospin. On the field theory side, we obtain an extension of the second order formalism for fermion QED to an abelian vector-axialvector theory.

Aharonov-Bohm effect, center monopoles and center vortices in SU(2) lattice gluodynamics

SU(2) gluodynamics is investigated numerically and analytically in the (Indirect) Maximal Center gauge at finite temperature. The center vortices are shown to be condensed in the confinement phase and dilute in the deconfinement phase. A new physical object, center monopole, is constructed. We show that the center monopole condensate is the order parameter of deconfinement phase transition. The linking of the vortex worldsheets and quark trajectories is identified with the Aharonov-Bohm interaction in an effective Abelian Higgs theory. We conclude that the confinement in the Maximal Center gauge can be explained by a new mechanism called “ the real superconductor mechanism” .

Equivalence of the self-dual model and Maxwell-Chern-Simons theory on arbitrary manifolds

Using a group-invariant version of the Faddeev-Popov method we explicitly obtain the partition functions of the self-dual model and Maxwell-Chern-Simons theory. We show that their ratio coincides with the partition function of Abelian Chern-Simons theory to within a phase factor depending on the geometrical properties of the manifold.

Strings of opposite magnetic charges in a gauge field theory

We show that there are soliton–like solutions representing arbitrarily located M vortices and N antivortices with opposite magnetic charges in an Abelian gauge field theory. We establish an existence and uniqueness theorem and prove that the total magnetic flux is proportional to M − N but the total energy is proportional to M + N . In the presence of Einstein's gravity, we establish an existence theorem under a necessary and sufficient condition that ensures the geodesic completeness of the corresponding gravitational metric. The coexisting vortices and antivortices are now cosmic strings and antistrings and the total magnetic flux and matter–gauge energy are given by similar expressions as those for vortices. Besides the energy, the total curvature also provides an exact, quantized, measurement of the number of the strings of the two types.

Hamiltonian formulation of the W1 + ∞ minimal models

The W 1 + ∞ minimal models are conformal field theories which can describe the edge excitations of the hierarchical plateaus in the quantum Hall effect. In this paper, these models are described in very explicit terms by using a bosonic Fock space with constraints, or, equivalently, with a non-trivial Hamiltonian. The Fock space is that of the multi-component abelian conformal theories, which provide another possible description of the hierarchical plateaus; in this space, the minimal models are shown to correspond to the sub-set of states which satisfy the constraints. This reduction of degrees of freedom can also be implemented by adding a relevant interaction to the Hamiltonian, leading to a renormalization-group flow between the two theories. Next, a physical interpretation of the constraints is obtained by representing the quantum incompressible Hall fluids as generalized Fermi seas. Finally, the non-abelian statistics of the quasi-particles in the W 1 + ∞. minimal models is described by computing their correlation functions in the Coulomb gas approach.

Numerical study of hierarchical quantum Hall edge states in the disk geometry

We present a detailed analysis of the exact numerical spectrum of up to ten interacting electrons in the first Landau level on the disk geometry. We study the edge excitations of the hierarchical plateaus and check the predictions of two relevant conformal field theories: the multi-component Abelian theory and the W-infinity minimal theory of the incompressible fluids. We introduce two new criteria for identifying the edge excitations within the low-lying states: the plot of their density profiles and the study of their overlaps with the Jain wave functions in a meaningful basis. We find that the exact bulk and edge excitations are very well reproduced by the Jain states; these, in turn, can be described by the multi-component Abelian conformal theory. Most notably, we observe that the edge excitations form sub-families of the low-lying states with a definite pattern, which is explained by the W-infinity minimal conformal theory. Actually, the two conformal theories are related by a projection mechanism whose effects are observed in the spectrum. Therefore, the edge excitations of the hierarchical Hall states are consistently described by the W-infinity minimal theory, within the finite-size limitations.

Constraints from extended supersymmetry in quantum mechanics

We consider quantum mechanical gauge theories with sixteen supersymmetries. The Hamiltonians or Lagrangians characterizing these theories can contain higher derivative terms. In the operator approach, we show that the free theory is essentially the unique abelian theory with up to four derivatives in the following sense: any small deformation of the free theory, which preserves the supersymmetries, can be gauged away by a unitary conjugation. We also present a method for deriving constraints on terms appearing in an effective Lagrangian. We apply this method to the effective Lagrangian describing the dynamics of two well-separated clusters of D0-branes. As a result, we prove a non-renormalization theorem for the ν 4 interaction.

Issues in topological gauge theory

We discuss topological theories, arising from the general N = 2 twisted gauge theories. We initiate a program of their study in the Gromov-Witten paradigm. We re-examine the low-energy effective abelian theory in the presence of sources and study the mixing between the various p-observables. We present the twisted superfield formalism which makes duality transformations transparent. We propose a scheme which uniquely fixes all the contact terms. We derive a formula for the correlation functions of p-observables on the manifolds of generalized simple type for 0 ⪕ p ⪕ 4 and on some manifolds with b 2 + = 1. We study the theories with matter and explore the properties of universal instantons. We also discuss the compactifications of higher dimensional theories. Some relations to sigma models of type A and B are pointed out and exploited.

Charges in Gauge Theories

In this article we investigate charged particles in gauge theories. After reviewing the physical and theoretical problems, a method to construct charged particles is presented. Explicit solutions are found in the Abelian theory and a physical interpretation is given. These solutions and our interpretation of these variables as the true degrees of freedom for charged particles, are then tested in the perturbative domain and are demonstrated to yield infra-red finite, on-shell Green's functions at all orders of perturbation theory. The extension to collinear divergences is studied and it is shown that this method applies to the case of massless charged particles. The application of these constructions to the charged sectors of the standard model is reviewed and we conclude with a discussion of the successes achieved so far in this programme and a list of open questions.

More on lattice BRST invariance

In the gauge-fixing approach to (chiral) lattice gauge theories, the action in the U(1) case implicitly contains a free ghost term, in accordance with the continuum abelian theory. On the lattice there is no BRST symmetry and, without fermions, the partition function is strictly positive. Recently, Neuberger pointed out in hep-lat/9801029 that a different choice of the ghost term would lead to a BRST-invariant lattice model, which is ill-defined nonperturbatively. We show that such a lattice model is inconsistent already in perturbation theory, and clearly different from the gauge-fixing approach.

String holonomy and extrinsic geometry in four-dimensional topological gauge theory

The most general gauge-invariant marginal deformation of four-dimensional abelian BF-type topological field theory is studied. It is shown that the deformed quantum field theory is topological and that its observables compute, in addition to the usual linking numbers, smooth intersection indices of immersed surfaces which are related to the Euler and Chern characteristic classes of their normal bundles in the underlying space-time manifold. Canonical quantization of the theory coupled to non-dynamical particle and string sources is carried out in the Hamiltonian formalism and explicit solutions of the Schr̈odinger equation are obtained. The wavefunctions carry a one-dimensional unitary representation of the particle-string exchange holonomies and of non-topological string-string intersection holonomies given by adiabatic limits of the world-sheet Euler numbers. They also carry a multi-dimensional projective representation of the deRham complex of the underlying spatial manifold and define a generalization of the presentation of its motion group from Euclidean space to an arbitrary 3-manifold. Some potential physical applications of the topological field theory as a dual model for effective vortex strings are discussed.

Partition function versus boundary conditions and confinement in the Yang-Mills theory

We analyse dependence of the partition function on the boundary condition for the longitudinal component of the electric field strength in gauge field theories. In a physical gauge the Gauss law constraint may be resolved explicitly expressing this component via an integral of the physical transversal variables. In particular, we study quantum electrodynamics with an external charge and SU(2) gluodynamics. We find that only a charge distribution slowly decreasing at spatial infinity can produce a nontrivial dependence in the Abelian theory. However, in gluodynamics for temperatures below some critical value the partition function acquires a delta-function like dependence on the boundary condition, which leads to colour confinement.

CURRENT ALGEBRA IN THE PATH INTEGRAL FRAMEWORK

In this letter we describe an approach to the current algebra based on the path integral formalism. We use this method for Abelian and non-Abelian quantum field theories in (1+1) and (2+1) dimensions and the correct expressions are obtained. Our results show the independence of the regularization of the current algebras.

Charges in gauge theories

In this article we investigate charged particles in gauge theories. After reviewing the physical and theoretical problems, a method to construct charged particles is presented. Explicit solutions are found in the Abelian theory and a physical interpretation is given. These solutions and our interpretation of these variables as the true degrees of freedom for charged particles, are then tested in the perturbative domain and are demonstrated to yield infra-red finite, on-shell Green's functions at all orders of perturbation theory. The extension to collinear divergences is studied and it is shown that this method applies to the case of massless charged particles. The application of these constructions to the charged sectors of the standard model is reviewed and we conclude with a discussion of the successes achieved so far in this programme and a list of open questions.

Chern-Simons number diffusion with hard thermal loops

We construct an extension of the standard Kogut-Susskind lattice model for classical 3+1 dimensional Yang-Mills theory, in which ``classical particle'' degrees of freedom are added. We argue that this will correctly reproduce the ``hard thermal loop'' effects of hard degrees of freedom, while giving a local implementation which is numerically tractable. We prove that the extended system is Hamiltonian and has the same thermodynamics as dimensionally reduced hot Yang-Mills theory put on a lattice. We present a numerical update algorithm and study the abelian theory to verify that the classical gauge theory self-energy is correctly modified. Then we use the extended system to study the diffusion constant for Chern-Simons number. We verify the Arnold-Son-Yaffe picture that the diffusion constant is inversely proportional to hard thermal loop strength. Our numbers correspond to a diffusion constant of Gamma = 29 +- 6 alpha_w^5 T^4 for m_D^2 = 11 g^2 T^2/6.