scholarly journals VERTEX OPERATORS FOR THE BF SYSTEM AND ITS SPIN–STATISTICS THEOREMS

1994 ◽  
Vol 09 (10) ◽  
pp. 1569-1629 ◽  
Author(s):  
A. P. BALACHANDRAN ◽  
P. TEOTONIO-SOBRINHO
Keyword(s):  
Simons Theory ◽  
Vertex Operators ◽  
Edge States ◽  
Flux Tubes ◽  
Vortex Loops ◽  
Spin Excitations ◽  
Large Length ◽  
A Charge ◽  
Aharonov Bohm ◽  
Chern Simons

Let B and [Formula: see text] be two-forms, Fµν being the field strength of an Abelian connection A. The topological BF system is given by the integral of B ∧ F. With "kinetic energy" terms added for B and A, it generates a mass for A, thereby suggesting an alternative to the Higgs mechanism, and also gives the London equations. The BF action, being the large length and time scale limit of this augmented action, is thus of physical interest. In earlier work, it has been studied on spatial manifolds Σ with boundaries ∂Σ, and the existence of edge states localized at ∂Σ has been established. They are analogous to the conformal family of edge states to be found in a Chern–Simons theory in a disc. Here we introduce charges and vortices (thin flux tubes) as sources in the BF system and show that they acquire an infinite number of spin excitations due to renormalization, just as a charge coupled to a Chern–Simons potential acquires a conformal family of spin excitations. For a vortex, these spins are transverse and attached to each of its points, so that it resembles a ribbon. Vertex operators for the creation of these sources are constructed and interpreted in terms of a Wilson integral involving A and a similar integral involving B. The standard spin–statistics theorem is proved for these sources. A new spin–statistics theorem, showing the equality of the "interchange" of two identical vortex loops and 2π rotation of the transverse spins of a constituent vortex, is established. Aharonov–Bohm interactions of charges and vortices are studied. The existence of topologically nontrivial vortex spins is pointed out and their vertex operators are also discussed.

scholarly journals THE EDGE STATES OF THE BF SYSTEM AND THE LONDON EQUATIONS

1993 ◽  
Vol 08 (04) ◽  
pp. 723-752 ◽  
Author(s):  
A.P. BALACHANDRAN ◽  
P. TEOTONIO-SOBRINHO
Keyword(s):  
Scalar Field ◽  
Scaling Limit ◽  
Quantum Hall ◽  
Simons Theory ◽  
Massless Scalar ◽  
Electromagnetic Current ◽  
Massless Scalar Field ◽  
Edge States ◽  
Chern Simons ◽  
London Equations

It is known that the 3D Chern–Simons interaction describes the scaling limit of a quantum Hall system and predicts edge currents in a sample with boundary, the currents generating a chiral U(1) Kac-Moody algebra. It is no doubt also recognized that, in a somewhat similar way, the 4D BF interaction (with B a two-form, dB the dual *j of the electromagnetic current, and F the electromagnetic field form) describes the scaling limit of a superconductor. We show in this paper that there are edge excitations in this model as well for manifolds with boundaries. They are the modes of a scalar field with invariance under the group of diffeomorphisms (diffeos) of the bounding spatial two-manifold. Not all diffeos of this group seem implementable by operators in quantum theory, the implementable group being a subgroup of volume-preserving diffeos. The BF system in this manner can lead to the w1+∞ algebra and its variants. Lagrangians for fields on the bounding manifold which account for the edge observables on quantization are also presented. They are the analogs of the (1+1)-dimensional massless scalar field Lagrangian describing the edge modes of an Abelian Chern-Simons theory with a disk as the spatial manifold. We argue that the addition of “Maxwell” terms constructed from F∧*F and dB∧*dB does not affect the edge states, and that the augmented Lagrangian has an infinite number of conserved charges—the aforementioned scalar field modes—localized at the edges. This Lagrangian is known to describe London equations and a massive vector field. A (3+1)-dimensional generalization of the Hall effect involving vortices coupled to B is also proposed.

scholarly journals Towards Noncommutative Linking Numbers via the Seiberg-Witten Map

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
H. García-Compeán ◽  
O. Obregón ◽  
R. Santos-Silva
Keyword(s):  
Three Dimensional ◽  
Simons Theory ◽  
Dimensional Manifold ◽  
First Order ◽  
Linking Number ◽  
Linking Numbers ◽  
Bohm Effect ◽  
Aharonov Bohm ◽  
Chern Simons ◽  
Noncommutative Parameter

Some geometric and topological implications of noncommutative Wilson loops are explored via the Seiberg-Witten map. In the abelian Chern-Simons theory on a three-dimensional manifold, it is shown that the effect of noncommutativity is the appearance of6nnew knots at thenth order of the Seiberg-Witten expansion. These knots are trivial homology cycles which are Poincaré dual to the higher-order Seiberg-Witten potentials. Moreover the linking number of a standard 1-cycle with the Poincaré dual of the gauge field is shown to be written as an expansion of the linking number of this 1-cycle with the Poincaré dual of the Seiberg-Witten gauge fields. In the process we explicitly compute the noncommutative “Jones-Witten” invariants up to first order in the noncommutative parameter. Finally in order to exhibit a physical example, we apply these ideas explicitly to the Aharonov-Bohm effect. It is explicitly displayed at first order in the noncommutative parameter; we also show the relation to the noncommutative Landau levels.

ANYONS AND GAUSSIAN CONFORMAL FIELD THEORIES

1991 ◽  
Vol 06 (04) ◽  
pp. 289-294 ◽  
Author(s):  
DILEEP P. JATKAR ◽  
SUMATHI RAO
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Mass Parameter ◽  
Simons Theory ◽  
Conformal Field Theories ◽  
Conformal Blocks ◽  
Conformal Field ◽  
A Charge ◽  
Chern Simons ◽  
Chern Simons Theory

We identify the spin of the anyons with the holomorphic dimension of the primary fields of a Gaussian conformal field theory. The angular momentum addition rules for anyons go over to the fusion rules for the primary fields and the r↔1/2r duality of the Gaussian CFT is reproduced by a charge-flux duality of the anyons. For a U(1) Chern-Simons theory with topological mass parameter k=2n, N-anyon states on the torus have 2n components, which correspond to the 2n conformal blocks of an N-point function in the Gaussian conformal field theory.

scholarly journals THE CHERN-SIMONS SOURCE AS A CONFORMAL FAMILY AND ITS VERTEX OPERATORS

1992 ◽  
Vol 07 (23) ◽  
pp. 5855-5876 ◽  
Author(s):  
A.P. BALACHANDRAN ◽  
G. BIMONTE ◽  
K.S. GUPTA ◽  
A. STERN
Keyword(s):  
Wilson Line ◽  
Relativistic Quantum ◽  
Spatial Location ◽  
Vertex Operators ◽  
Edge States ◽  
Functional Integrals ◽  
Internal States ◽  
Canonical Approach ◽  
Chern Simons ◽  
Gauge Conditions

In a previous work, a straighforward canonical approach to the source-free quantum Chern—Simons dynamics was developed. It makes use of neither gauge conditions nor functional integrals and needs only ideas known from QCD and quantum gravity. It gives Witten’s conformal edge states in a simple way when the spatial slice is a disc. Here we extend the formalism by including sources as well. The quantum states of a source with a fixed spatial location are shown to be those of a conformal family, a result also discovered first by Witten. The internal states of a source are not thus associated with just a single ray of a Hilbert space. Vertex operators for both abelian and nonabelian sources are constructed. The regularized abelian Wilson line is proved to be a vertex operator. We also argue in favor of a similar nonabelian result. The spin-statistics theorem is established for Chern-Simons dynamics even though the sources are not described by relativistic quantum fields. The proof employs geometrical methods which we find are strikingly transparent and pleasing. It is based on the research of European physicists about “fields localized on cones.”

scholarly journals Edge States from Defects on the Noncommutative Plane

2003 ◽  
Vol 18 (33n35) ◽  
pp. 2509-2516 ◽  
Author(s):  
A. Pinzul ◽  
A. Stern
Keyword(s):  
Hilbert Space ◽  
Point Defects ◽  
Nonlinear Deformation ◽  
Simons Theory ◽  
Edge States ◽  
Boundary States ◽  
Chern Simons ◽  
Spatial Support ◽  
Noncommutative Plane ◽  
Chern Simons Theory

We illustrate how boundary states are recovered when going from a noncommutative manifold to a commutative one with a boundary. Our example is the noncommutative plane with a defect, whose commutative limit was found to be a punctured plane - so here the boundary is one point. Defects were introduced by removing states from the standard harmonic oscillator Hilbert space. For Chern-Simons theory, the defect acts as a source, which was found to be associated with a nonlinear deformation of the w∞ algebra. The undeformed w∞ algebra is recovered in the commutative limit, and here we show that its spatial support is in a tiny region near the puncture.

LAUGHLIN'S WAVE FUNCTION AND ANGULAR MOMENTUM

2011 ◽  
Vol 25 (10) ◽  
pp. 1301-1357 ◽  
Author(s):  
KESHAV N. SHRIVASTAVA
Keyword(s):  
Wave Function ◽  
Angular Momentum ◽  
First Principles ◽  
Condensed Matter ◽  
Quantum Hall ◽  
Edge States ◽  
Coulomb Interactions ◽  
Bohm Effect ◽  
A Charge ◽  
Aharonov Bohm

In 1983, Laughlin reported a wave function which while using the first-principles kinetic energy and Coulomb interactions fractionalizes the charge of the electron so that a charge such as 1/3 occurs. Since then this wave function has been applied to many problems in condensed matter physics. An effort is made to review the literature dealing with Aharonov–Bohm effect, ground state, confinement, phase transitions, Wigner and Luttinger solids, edge states, Anderson's theory, statistics and anyons, etc. The importance of the angular momentum is pointed out and it is shown that Landau levels play an important role in understanding the fractions at which the plateaus occur in the quantum Hall effect.

scholarly journals CONFORMAL EDGE CURRENTS IN CHERN-SIMONS THEORIES

1992 ◽  
Vol 07 (19) ◽  
pp. 4655-4670 ◽  
Author(s):  
A.P. BALACHANDRAN ◽  
G. BIMONTE ◽  
K.S. GUPTA ◽  
A. STERN
Keyword(s):  
Quantum Gravity ◽  
Gauge Theories ◽  
Field Theories ◽  
Vertex Operators ◽  
Edge States ◽  
Moody Algebra ◽  
Functional Integrals ◽  
Topological Field ◽  
Chern Simons ◽  
Spin And Statistics

We develop elementary canonical methods for the quantization of Abelian and non-Abelian Chern-Simons actions, using well-known ideas in gauge theories and quantum gravity. Our approach does not involve choice of gauge or clever manipulations of functional integrals. When the spatial slice is a disc, it yields Witten’s edge states carrying a representation of the Kac-Moody algebra. The canonical expressions for the generators of diffeomorphisms on the boundary of the disc are also found, and it is established that they are the Chern-Simons version of the Sugawara construction. This paper is a prelude to our future publications on edge states, sources, vertex operators, and their spin and statistics in 3D and 4D topological field theories.

scholarly journals EDGE STATES IN 4D AND THEIR 3D GROUPS AND FIELDS

1993 ◽  
Vol 08 (14) ◽  
pp. 1305-1313 ◽  
Author(s):  
A. P. BALACHANDRAN ◽  
G. BIMONTE ◽  
P. TEOTONIO-SOBRINHO
Keyword(s):  
Field Theory ◽  
Group Representation ◽  
Coadjoint Orbit ◽  
Simons Theory ◽  
Chiral Field ◽  
Massless Scalar ◽  
Edge States ◽  
Four Dimensions ◽  
Chern Simons ◽  
Chern Simons Theory

It is known that the Lagrangian for the edge states of a Chern-Simons theory describes a coadjoint orbit of a Kac-Moody (KM) group with its associated Kirillov symplectic form and group representation. It can also be obtained from a chiral sector of a non-chiral field theory. We study the edge states of the Abelian BF system in four dimensions (4D) and show the following results in almost exact analogy: 1) The Lagrangian for these states is associated with a certain 2D generalization of the KM group. It describes a coadjoint orbit of this group as a Kirillov symplectic manifold and also the corresponding group representation. 2) It can be obtained from with a "self-dual" or "anti-self-dual" sector of a Lagrangian describing a massless scalar and a Maxwell field (the phrase "self-dual" here being used essentially in its sense in monopole theory). There are similar results for the non-Abelian BF system as well. These shared features of edge states in 3D and 4D suggest that the edge Lagrangians for BF systems are certain natural generalizations of field theory Lagrangians related to KM groups.

scholarly journals From the fuzzy disc to edge currents in Chern-Simons Theory

2003 ◽  
Vol 18 (33n35) ◽  
pp. 2381-2387 ◽  
Author(s):  
F. LIZZI ◽  
P. VITALE ◽  
A. ZAMPINI
Keyword(s):  
Finite Algebra ◽  
Simons Theory ◽  
Edge States ◽  
Chern Simons ◽  
Chern Simons Theory

We present a brief review of the fuzzy disc, the finite algebra approximating functions on a disc, which we have introduced earlier. We also present a comparison with recent papers of Balachandran, Gupta and Kürkçüoǧlu, and of Pinzul and Stern, aimed at the discussion of edge states of a Chern-Simons thoery.

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