Effective Field Theory Description of the Higher Dimensional Quantum Hall Liquid

We derive an effective topological field theory model of the four dimensional quantum Hall liquid state recently constructed by Zhang and Hu. Using a generalization of the flux attachment transformation, the effective field theory can be formulated as a U(1) Chern–Simons theory over the total configuration space CP₃, or as a SU(2) Chern–Simons theory over S⁴. The new quantum Hall liquid supports various types of topological excitations, including the 0-brane (particles), the 2-brane (membranes), and the 4-brane. There is a topological phase interaction among the membranes which generalizes the concept of fractional statistics.     

Reducibility and Gribov Problem in Topological Quantum Field Theory

In spite of its simplicity and beauty, the Mathai–Quillen formulation of cohomological topological quantum field theory with gauge symmetry suffers two basic problems: i) the existence of reducible field configurations on which the action of the gauge group is not free and ii) the Gribov ambiguity associated with gauge fixing, i. e. the lack of global definition on the space of gauge orbits of gauge fixed functional integrals. In this paper, we show that such problems are in fact related and we propose a general completely geometrical recipe for their treatment. The space of field configurations is augmented in such a way to render the action of the gauge group free and localization is suitably modified. In this way, the standard Mathai–Quillen formalism can be rigorously applied. The resulting topological action contains the ordinary action as a subsector and can be shown to yield a local quantum field theory, which is argued to be renormalizable as well. The salient feature of our method is that the Gribov problem is inherent in localization, and thus can be dealt within a completely equivariant setting, whereas gauge fixing is free of Gribov ambiguities. For the stratum of irreducible gauge orbits, the case of main interest in applications, the Gribov problem is solvable. Conversely, for the strata of reducible gauge orbits, the Gribov problem cannot be solved in general and the obstruction may be described in the language of sheaf theory. The formalism is applied to the Donaldson–Witten model. Received: 22 July 1996 / Accepted: 21 October 1996     

Topological investigation of the fractionally quantized Hall conductivity

Using the fiber bundle concept developed in geometry and topology, the fractionally quantized Hall conductivity is discussed in the relevant many-particle configuration space. Electronmagnetic field and electron-electron interactions under FQHE conditions are treated as functional connections over the torus, the torus being the underlying two-dimensional manifold. Relations to the (2 + 1)-dimensional Chern-Simons theory are indicated. The conductivity being a topological invariant is given as e²/h times a linking number which is the quotient of the winding numbers of the self-consistent field and the magnetic field, respectively. Odd denominators are explained by the two spin structures which have been considered for the FQHE correlated electron system.     

The R.I. Pimenov unified gravitation and electromagnetism field theory as semi-Riemannian geometry

More than forty years ago R.I. Pimenov introduced a new geometry—semi-Riemannian one—as a set of geometrical objects consistent with a fibering pr: M _n → M _m . He suggested the heuristic principle according to which the physically different quantities (meter, second, Coulomb, etc.) are geometrically modelled as space coordinates that are not superposed by automorphisms. As there is only one type of coordinates in Riemannian geometry and only three types of coordinates in pseudo-Riemannian one, a multiple-fibered semi-Riemannian geometry is the most appropriate one for the treatment of more than three different physical quantities as unified geometrical field theory. Semi-Euclidean geometry ³ R ₅⁴ with 1-dimensional fiber x ⁵ and 4-dimensional Minkowski space-time as a base is naturally interpreted as classical electrodynamics. Semi-Riemannian geometry ³ V ₅⁴ with the general relativity pseudo-Riemannian space-time ³ V ₄, and 1-dimensional fiber x ⁵, responsible for the electromagnetism, provides the unified field theory of gravitation and electromagnetism. Unlike Kaluza-Klein theories, where the fifth coordinate appears in nondegenerate Riemannian or pseudo-Riemannian geometry, the theory based on semi-Riemannian geometry is free from defects of the former. In particular, scalar field does not arise. The text was submitted by the author in English.     

Topological Invariants in Braid Theory

Many invariants of knots and links have their counterparts in braid theory. Often, these invariants are most easily calculated using braids. A braid is a set of n strings stretching between two parallel planes. This review demonstrates how integrals over the braid path can yield topological invariants. The simplest such invariant is the winding number – the net number of times two strings in a braid wrap about each other. But other, higher-order invariants exist. The mathematical literature on these invariants usually employs techniques from algebraic topology that may be unfamiliar to physicists and mathematicians in other disciplines. The primary goal of this paper is to introduce higher-order invariants using only elementary differential geometry.Some of the higher-order quantities can be found directly by searching for closed one-forms. However, the Kontsevich integral provides a more general route. This integral gives a formal sum of all finite order topological invariants. We describe the Kontsevich integral, and prove that it is invariant to deformations of the braid.Some of the higher-order invariants can be used to generate Hamiltonian dynamics of n particles in the plane. The invariants are expressed as complex numbers; but only the real part gives interesting topological information. Rather than ignoring the imaginary part, we can use it as a Hamiltonian. For n = 2, this will be the Hamiltonian for point vortex motion in the plane. The Hamiltonian for n = 3 generates more complicated motions.     

Spinor Derivation of Quasilocal Mean Curvature Mass in General Relativity

A spinor derivation is presented for quasilocal mean-curvature mass of spacelike 2-surfaces in General Relativity. The derivation is based on the Sen-Witten spinor identity and involves the introduction of novel nonlinear boundary conditions related to the Dirac current of the spinor at the 2-surface and the tangential flux of a boundary Dirac operator, as well the use of a spin basis adapted to the mean curvature frame of the 2-surface normal space. This setting may provide an alternative approach to a positivity proof for mean-curvature mass based on showing that Witten’s equation admits a spinor solution satisfying the proposed nonlinear boundary conditions.     

Geometroelectrodynamics

We present an elementary particle model that can be thought of as a unification of certain topological ideas abstracted from the string model and the standard Yang-Mills theory. The basic dynamical entity of the model is a spacelike 3-surfaceX ³ in some metric spaceH and is interpreted as a particle. The dynamics of the model is based on two ideas. First the model is formally a Yang-Mills theory on the surfaceX ⁴ representing the orbit(s) of the particle(s) inH. Secondly the Yang-Mills structure onX ⁴ is constructed using only the natural geometric structures of the space H by a process which we call induction. It is found that some rather general requirements highly fix the choice of the space H. In fact the minimal model, for which the space H is the product of Minkowski space and the 2-sphere, is obtained by requiring that the symmetry group of the theory is the product of the Poincaré group and the color groupSO(3). The unique feature of the minimal model is that it affords a purely topological mechanism for quark confinement.     

From Weak to Strong Coupling in ABJM Theory

The partition function of N=6{\mathcal{N}=6} supersymmetric Chern–Simons-matter theory (known as ABJM theory) on \mathbbS³{\mathbb{S}^3} , as well as certain Wilson loop observables, are captured by a zero dimensional super-matrix model. This super–matrix model is closely related to a matrix model describing topological Chern–Simons theory on a lens space. We explore further these recent observations and extract more exact results in ABJM theory from the matrix model. In particular we calculate the planar free energy, which matches at strong coupling the classical IIA supergravity action on AdS₄×\mathbbC\mathbbP³{{\rm AdS}_4\times\mathbb{C}\mathbb{P}^3} and gives the correct N ^3/2 scaling for the number of degrees of freedom of the M2 brane theory. Furthermore we find contributions coming from world-sheet instanton corrections in \mathbbC\mathbbP³{\mathbb{C}\mathbb{P}^3} . We also calculate non-planar corrections, both to the free energy and to the Wilson loop expectation values. This matrix model appears also in the study of topological strings on a toric Calabi–Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi–Yau. In particular it suggests that, in addition to the usual perturbative and strong coupling (AdS) expansions, a third natural expansion locus is the line where one of the two ’t Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi–Yau, and leads to an expansion around topological Chern–Simons theory. We present some explicit results for the partition function and Wilson loop observables around this locus.     

Generalized hypergeometric functions on the torus and the adjoint representation ofU q (sl 2)

We study the homology groups with coefficient in local systems arising in the free field representation of minimal models of conformal field theory on an elliptic curve with punctures. We define an action of the quantum enveloping algebraU _q (sl ₂) on a space of relative cycles, extending results obtained previously for the sphere. Absolute cycles are identified with singular vectors. In the case of one puncture, we find that the resulting topological representation is essentially the adjoint representation.     

Exactly soluble diffeomorphism invariant theories

A class of diffeomorphism invariant theories is described for which the Hilbert space of quantum states can be explicitly constructed. These theories can be formulated in any dimension and include Witten's solution to 2+1 dimensional gravity as a special case. Higher dimensional generalizations exist which start with an action similar to the Einstein action inn dimensions. Many of these theories do not involve a spacetime metric and provide examples of topological quantum field theories. One is a version of Yang-Mills theory in which the only quantum states onS ³×R are the vacua. Finally it is shown that the three dimensional Chern-Simons theory (which Witten has shown is intimately connected with knot theory) arises naturally from a four dimensional topological gauge theory.On leave from the Department of Physics, University of California, Santa Barbara, CA, USA     

Limiting case of modified electroweak model for contracted gauge group

The modification of the Electroweak Model with 3-dimensional spherical geometry in the matter fields space is suggested. The Lagrangian of this model is given by the sum of the free (without any potential term) matter fields Lagrangian and the standard gauge fields Lagrangian. The vector boson masses are generated by transformation of this Lagrangian from Cartesian coordinates to coordinates on the sphere S ³. The limiting case of the bosonic part of the modified model, which corresponds to the contracted gauge group SU(2; j) × U(1) is discussed. Within framework of the limit model Z boson and electromagnetic fields can be regarded as external ones with respect to W-boson fields in the sence that W-boson fields do not effect on these external fields. The masses of all particles of the Electroweak Model remain the same, but field interactions in contracted model are more simple as compared with the standard Electroweak Model.     

On Equilibrium Configurations in Continuum Mechanics of Structural Defects

We consider the determination of the theory by a second order tensor field g_ik and affinity Γ^f_ik. By variational principle for Einstein-Hilbert Lagrangian solid state equilibrium positions of the ideal and real crystal will be described. On account of external Galilei-invariance this theory affords an invariant three dimensional geometry at most being able to produce a stable static equilibrium of defects. The motion of defects is related to the theory of invariants of the internal group of field equations produced by this theory in strong analogy to Maxwell's electrodynamics. The elastic ether concept for the theory of light affords the idea of a gauge field approximation of continuum mechanics fitting linearized Einstein-Hilbert Lagrangian approach. The stress and strain space duality has to be understood on this background.     

Abelian model of gravitating magnetic monopole

A self-consistentU(1)-gauge model in gravitational field is investigated. The exact solutions of two types of corresponding field equations are obtained. These solutions can be interpreted as magnetic monopoles. The first solution is regular forr 0 and provides an everywhere regular geometry, the second one has a physical singularity. In order to guarantee the stability of the monopoles we introduce the notion of a gravitational topological charge using de Rham's cohomology theory. This topological charge describes the sizes and the inner structure of the monopole.     

Stable singularities in string theory

We study a topological obstruction of a very stringy nature concerned with deforming the target space of anN=2 non-linear -model. This target space has a singularity which may be smoothed away according to the conventional rules of geometry, but when one studies the associated conformal field theory one sees that such a deformation is not possible without a discontinuous change in some of the correlation functions. This obstruction appears to come from torsion in the homology of the target space (which is seen by deforming the theory by an irrelevant operator). We discuss the link between this phenomenon and orbifolds with discrete torsion as studied by Vafa and Witten.Supported in part by NSF grant DMS-9400873.