Abelian Theory Research Articles

Reducible gauge algebra of BRST-invariant constraints

We show that it is possible to formulate the most general first-class gauge algebra of the operator formalism by only using BRST-invariant constraints. In particular, we extend a previous construction for irreducible gauge algebras to the reducible case. The gauge algebra induces two nilpotent, Grassmann-odd, mutually anti-commuting BRST operators that bear structural similarities with BRST/anti-BRST theories but with shifted ghost number assignments. In both cases we show how the extended BRST algebra can be encoded into an operator master equation. A unitarizing Hamiltonian that respects the two BRST symmetries is constructed with the help of a gauge-fixing boson. Abelian reducible theories are shown explicitly in full detail, while non-Abelian theories are worked out for the lowest reducibility stages and ghost momentum ranks.

GEOMETRIC CONSTRUCTION OF MODULAR FUNCTORS FROM CONFORMAL FIELD THEORY

We give a geometric construct of a modular functor for any simple Lie-algebra and any level by twisting the constructions in [16, 19] by a certain fractional power of the abelian theory first considered in [13] and further studied in [2].

On the expansion of a quantum field theory around a topological sector

The idea of treating quantum general relativistic theories in a perturbative expansion around a topological theory has recently received attention, in the quantum gravity literature. We investigate the viability of this idea by applying it to conventional Yang–Mills theory on flat spacetime. This theory admits indeed a formulation as a modified topological theory, like general relativity. We find that the expansion around the topological theory coincides with the usual expansion around the free abelian theory, though the equivalence is non-trivial. In this context, the technique appears therefore to be viable, but not to bring particularly new insights. On the other hand, we point out that the relation of this expansion with the actual quantum BF theory is far from being transparent. Some implications for gravity are discussed.

Fuzzy sphere dynamics and non-abelian DBI in curved backgrounds

We consider the non-Abelian action for the dynamics of $N Dp'$-branes in the background of $M Dp$-branes, which parameterises a fuzzy sphere using the SU(2) algebra. We find that the curved background leads to collapsing solutions for the fuzzy sphere except when we have $D0$ branes in the $D6$ background, which is a realisation of the gravitational Myers effect. Furthermore we find the equations of motion in the Abelian and non-Abelian theories are identical in the large $N$ limit. By picking a specific ansatz we find that we can incorporate angular momentum into the action, although this imposes restriction upon the dimensionality of the background solutions. We also consider the case of non-Abelian non-BPS branes, and examine the resultant dynamics using world-volume symmetry transformations. We find that the fuzzy sphere always collapses but the solutions are sensitive to the combination of the two conserved charges and we can find expanding solutions with turning points. We go on to consider the coincident $NS$5-brane background, and again construct the non-Abelian theory for both BPS and non-BPS branes. In the latter case we must use symmetry arguments to find additional conserved charges on the world-volumes to solve the equations of motion. We find that in the Non-BPS case there is a turning solution for specific regions of the tachyon and radion fields. Finally we investigate the more general dynamics of fuzzy $\mathbb{S}^{2k}$ in the $Dp$-brane background, and find collapsing solutions in all cases.

On pureness in Abelian groups

Torsion-free Abelian groups G and H are called quasi-equal (G ≈ H) if λG ⊂ H ⊂ G for a certain natural number ≈. It is known (see [3]) that the quasi-equality of torsion-free Abelian groups can be represented as the equality in an appropriate factor category. Thus while dealing with certain group properties it is usual to prove that the property under consideration is preserved under the transition to a quasi-equal group. This trick is especially frequently used when the author investigates module properties of Abelian groups; here a group is considered as a left module over its endomorphism ring. On the other hand, a topical problem in the Abelian group theory is the problem of investigation of pureness in the category of Abelian groups (see [4]). We consider the pureness introduced by P. Cohn [2] for Abelian groups as modules over their endomorphism rings. Particularity of the investigation of the properties of pureness for the Abelian group G as the moduleE(G)G lies in the fact that this is a more general situation than the investigation of pureness for a unitary module over an arbitrary ring R with the identity element. Indeed, if RM is an arbitrary unitary left module and M+ is its Abelian group, then each element from R can be identified with an appropriate endomorphism from the ring E(M+) under the canonical ring homomorphism R → E(M+). Then it holds that if E(M+)N is a pure submodule in E(M+)M+, then RN is a pure submodule in RM. In the present paper the interrelations between pureness, servantness, and quasi-decompositions for Abelian torsion-free groups of finite rank will be investigated.

Stochastic quantization of topological field theory: Generalized Langevin equation with memory kernel

We use the method of stochastic quantization in a topological field theory defined in an Euclidean space, assuming a Langevin equation with a memory kernel. We show that our procedure for the Abelian Chern-Simons theory converges regardless of the nature of the Chern-Simons coefficient.

NON-ABELIAN BORN–INFELD THEORY WITHOUT THE SQUARE-ROOT

A non-Abelian Born–Infeld theory is presented. Its construction does not require the square-root structure that characterizes the Dirac–Born–Infeld (DBI) action. The procedure is based on an Abelian theory proposed by Schrödinger that, as he showed, is equivalent to Born–Infeld theory. Various non-Abelian generalizations are possible. We focus our attention on one of them. For it, it is shown that instantons solutions exist. Our formalism could be of interest in connection with string theory and possible extensions of well-known physical results in the usual Born–Infeld Abelian case.

Chern-Simons theory onS1-bundles: abelianisation and q-deformed Yang-Mills theory

We study Chern-Simons theory on 3-manifolds $M$ that are circle-bundles over 2-dimensional surfaces $\Sigma$ and show that the method of Abelianisation, previously employed for trivial bundles $\Sigma \times S^1$, can be adapted to this case. This reduces the non-Abelian theory on $M$ to a 2-dimensional Abelian theory on $\Sigma$ which we identify with q-deformed Yang-Mills theory, as anticipated by Vafa et al. We compare and contrast our results with those obtained by Beasley and Witten using the method of non-Abelian localisation, and determine the surgery and framing presecription implicit in this path integral evaluation. We also comment on the extension of these methods to BF theory and other generalisations.

The Nature of Electronic Charge

Advances in gauge theories and unified theories have not thrown light on the meaning of electron. The problem of the origin of electronic charge is made precise, new insights gained from Weyl space are summarized, and the origin of charge in terms of fractional spin is suggested. A new perspective on the abelian Chern-Simons theory is presented to explain charge.

Ghost free dual vector theories in 2+1 dimensions

We explore here the issue of duality versus spectrum equivalence in abelian vector theories in 2+1 dimensions. Specifically we examine a generalized self-dual (GSD) model where a Maxwell term is added to the self-dual model. A gauge embedding procedure applied to the GSD model leads to a Maxwell-Chern-Simons (MCS) theory with higher derivatives. We show that the latter contains a ghost mode contrary to the original GSD model. On the other hand, the same embedding procedure can be applied to $N_f$ fermions minimally coupled to the self-dual model. The dual theory corresponds to $N_f$ fermions with an extra Thirring term coupled to the gauge field via a Pauli-like term. By integrating over the fermions at $N_f\to\infty$ in both matter coupled theories we obtain effective quadratic theories for the corresponding vector fields. On one hand, we have a nonlocal type of the GSD model. On the other hand, we have a nonlocal form of the MCS theory. It turns out that both theories have the same spectrum and are ghost free. By figuring out why we do not have ghosts in this case we are able to suggest a new master action which takes us from the local GSD to a nonlocal MCS model with the same spectrum of the original GSD model and ghost free. Furthermore, there is a dual map between both theories at classical level which survives quantum correlation functions up to contact terms. The remarks made here may be relevant for other applications of the master action approach.

MHV techniques for QED processes

Significant progress has been made in the past year in developing new `MHV' techniques for calculating multiparticle scattering amplitudes in Yang-Mills gauge theories. Most of the work so far has focussed on applications to Quantum Chromodynamics, both at tree and one-loop level. We show how such techniques can also be applied to abelian theories such as QED, by studying the simplest tree-level multiparticle process, e^+e^- to n \gamma. We compare explicit results for up to n=5 photons using both the Cachazo, Svrcek and Witten `MHV rules' and the related Britto-Cachazo-Feng `recursion relation' approaches with those using traditional spinor techniques.

Dimension-Two Vacuum Condensates in Gauge-Invariant Theories

We investigate the gauge dependence of dimension-two condensates in the Abelian and non-Abelian Yang-Mills theory.

Abelian Toda field theories on the noncommutative plane

Generalizations of GL(n) abelian Toda and $\widetilde{GL}(n)$ abelian affine Toda field theories to the noncommutative plane are constructed. Our proposal relies on the noncommutative extension of a zero-curvature condition satisfied by algebra-valued gauge potentials dependent on the fields. This condition can be expressed as noncommutative Leznov-Saveliev equations which make possible to define the noncommutative generalizations as systems of second order differential equations, with an infinite chain of conserved currents. The actions corresponding to these field theories are also provided. The special cases of GL(2) Liouville and $\widetilde{GL}(2)$ sinh/sine-Gordon are explicitly studied. It is also shown that from the noncommutative (anti-)self-dual Yang-Mills equations in four dimensions it is possible to obtain by dimensional reduction the equations of motion of the two-dimensional models constructed. This fact supports the validity of the noncommutative version of the Ward conjecture. The relation of our proposal to previous versions of some specific Toda field theories reported in the literature is presented as well.

Remarks on a five-dimensional Kaluza-Klein theory of the massive Dirac monopole

The Gross-Perry-Sorkin spacetime, formed by the Euclidean Taub-NUT space with the time trivially added, is the appropriate background of the Dirac magnetic monopole without an explicit mass term. One remarks that there exists a very simple five-dimensional metric of spacetimes carrying massive magnetic monopoles that is an exact solution of the vacuum Einstein equations. Moreover, the same isometry properties as the original Euclidean Taub-NUT space are preserved. This leads to an Abelian Kaluza-Klein theory whose metric appears as a combinations between the Gross-Perry-Sorkin and Schwarzschild ones. The asymptotic motion of the scalar charged test particles is discussed, now by accounting for the mixing between the gravitational and magnetic effects.

On the short-distance potential in confining theories

The structure of leading nonperturbative corrections to the static Coulomb potential at small distances in Abelian and non-Abelian theories is analyzed. Related problems of validity of the Dirac quantization condition for running charges in Abelian theory and significance of the quantity 〈A2〉 are briefly discussed.

Groups of Arithmetical Functions

Here the sum is taken over all positive integer divisors d of n. This somewhat surprising choice of a product is quite fruitful, allowing one to obtain interesting numbertheoretic formulas from simple computations in the ring. In particular, the useful functions mentioned above can all be expressed in terms of two simple elements of this ring. In this MAGAZINE, Berberian [2] discussed (among other things) the group of units of this ring. He showed that τ , σ , and φ can be expressed in terms of two very simple functions and proved that those two functions are linearly independent. In this article we extend his pair to an uncountably infinite set. In the process, we present answers to other questions posed in his article, including a description of the structure of the group of units. In the interest of accessibility, most of the discussion is confined to real-valued arithmetical functions. Except for a bit of abelian group theory, the algebraic ideas come from introductory linear algebra and abstract algebra. For many readers the only novel concept will be Bell series, a powerful tool developed by E. T. Bell in the early twentieth century.

Towards quantum dielectric branes: curvature corrections in Abelian beta function and non-Abelian Born–Infeld action

We initiate a programme to compute curvature corrections to the non-Abelian Born–Infeld action. This is based on the calculation of derivative corrections to the Abelian Born–Infeld action, describing a maximal brane, to all orders in F = B + 2 π α ′ F . An exact calculation in F allows us to apply the Seiberg–Witten map, reducing the maximal Abelian brane point of view to a minimal non-Abelian brane point of view (replacing 1 / F with [ X , X ] at large F), resulting in matrix model equations of motion in the considered background. We first study derivative corrections to the Abelian Born–Infeld action and compute the two loop beta function for an open bosonic string in a WZW (parallelizable) background. This beta function is the first step in the process of computing open string equations of motion, which can be later obtained by either computing the Weyl anomaly coefficients or the partition function in the given background. The beta function for the gauge field is exact in F and computed to orders O ( H , H 2 , H 3 ) (where H = d B and the curvature is R ∼ H 2 ) and O ( ∇ F , ∇ 2 F , ∇ 3 F ) . In order to carry out this calculation we develop a new regularization method for two loop graphs. We then relate perturbative results for Abelian and non-Abelian Born–Infeld actions, by showing how Abelian derivative corrections yield non-Abelian higher order commutators and vice versa, at large F. We begin the construction of a matrix model describing α ′ corrections to Myers' dielectric effect. This construction is carried out by first setting up a perturbative classification of the relevant non-Abelian tensor structures, which can be considerably narrowed down by the physical constraint of translation invariance in the action and the possibility for generic field redefinitions. The final matrix action is not uniquely determined and depends upon two free parameters. These parameters could be computed via further calculations in the Abelian theory.

On the image of noncommutative local reciprocity map

First steps in the direction of an arithmetic noncommutative local class field theory were described in [2] as an attempt to find an arithmetic generalization of the classical abelian class field theory; see [3] for an exposition of its main features. In particular, [2] clarified and simplified the metabelian local class field theory of H. Koch and E. de Shalit [7], [8]. In the noncommutative local class field theory [2] a direct arithmetic description of Galois extensions of a fixed local field F is given by means of noncommutative reciprocity maps between the Galois group Gal(L/F ) of a totally ramified arithmetically profinite Galois extension L/F and a certain subquotient of formal power series in one variable over the algebraic closure of the residue field of F (which, more precisely, is the completion of the maximal unramified extension of the field of norms of L/F ). One of the reciprocity maps (see below for definitions) is

Extensions of racks and quandles

A rack is a set equipped with a bijective, self-right-distrib- utive binary operation, and a quandle is a rack which satisfies an idempotency condition. In this paper, we introduce a new definition of modules over a rack or quandle, and show that this definition includes the one studied by Etingof and Grana (9) and the more general one given by Andruskiewitsch and Grana (1). We further show that this definition coincides with the appropriate specialisa- tion of the definition developed by Beck (3), and hence that these objects form a suitable category of coefficient objectsin which to develop homology and cohomology theories for racks and quandles. We then develop an Abelian extension theory for racks and quandles which contains the variants developed by Carter, El- hamdadi, Kamada and Saito (6, 7) as special cases.

Renormalization of theσ‐ωmodel within the framework of U(1) gauge symmetry

It is shown that the Sigma-Omega model which is widely used in the study of nuclear relativistic many-body problem can exactly be treated as an Abelian massive gauge field theory. The quantization of this theory can perfectly be performed by means of the general methods described in the quantum gauge field theory. Especially, the local U(1) gauge symmetry of the theory leads to a series of Ward-Takahashi identities satisfied by Green's functions and proper vertices. These identities form an uniquely correct basis for the renormalization of the theory. The renormalization is carried out in the mass-dependent momentum space subtraction scheme and by the renormalization group approach. With the aid of the renormalization boundary conditions, the solutions to the renormalization group equations are given in definite expressions without any ambiguity and renormalized S-matrix elememts are exactly formulated in forms as given in a series of tree diagrams provided that the physical parameters are replaced by the running ones. As an illustration of the renormalization procedure, the one-loop renormalization is concretely carried out and the results are given in rigorous forms which are suitable in the whole energy region. The effect of the one-loop renormalization is examined by the two-nucleon elastic scattering.