On Torsion and Nieh-Yan Form

Using the well-known Chern-Weil formula and its generalization, we systematically construct the Chern-Simons forms and their generalization induced by torsion as well as the Nieh-Yan (N-Y) forms. We also give an argument on the vanishing of integration of N-Y form on any compact manifold without boundary. A systematic construction of N-Y forms in D = 4n dimensions is also given.     

Chern-Simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space ofT 2

We describe a cut-and-paste method for computing Chern-Simons invariant of flatG-connections on 3-manifolds decomposed along tori, especially forG=SU(2) andSL(2,C). We use this method to make computations ofSU(2) Chern-Simons invariants of graph manifolds which generalize Fintushel and Stern's computations for Seifert-fibered spaces. We also use this technique to give a simple derivation of a formula of Yoshida relating the flatSL(2,C) Chern-Simons invariant of the holonomy representation to the volume and the metric Chern-Simons invariant for cusped hyperbolic 3-manifolds.     

Chern-Simons solitons,Toda theories and the chiral model

The two-dimensional self-dual Chern-Simons equations are equivalent to the conditions for static, zero-energy solutions of the (2+1)-dimensional gauged nonlinear Schrödinger equation with Chern-Simons matter-gauge dynamics. In this paper we classify all finite chargeSU(N) solutions by first transforming the self-dual Chern-Simons equations into the two-dimensional chiral model (or harmonic map) equations, and then using the Uhlenbeck-Wood classification of harmonic maps into the unitary groups. This construction also leads to a new relationship between theSU(N) Toda andSU(N) chiral model solutions.This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract #DE-AC02-76ER03069, and NSF grant #87-08447     

0-extended supergravity and Chern-Simons theories

We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in three dimensions by gauging its intrinsic global SO(N) symmetry. We call these ₀ (Aleph-null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ₀ supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ₀ globally and locally supersymmetric theories in three dimensions. As an interesting application, we present ₀ supergravity Liouville theory in two dimensions after appropriate dimensional reduction from three dimensions.     

ℵ0-extended supergravity and Chern-Simons theories

We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in three dimensions by gauging its intrinsic global SO(N) symmetry. We call these ℵ₀ (Aleph-null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ℵ₀ supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ℵ₀ globally and locally supersymmetric theories in three dimensions. As an interesting application, we present ℵ₀ supergravity Liouville theory in two dimensions after appropriate dimensional reduction from three dimensions.     

Torsion and gauge-invariant massive electrodynamics

We study electrodynamics in Einstein-Cartan space-time, that is, in space-time with torsion, and show an analogy with the Chern-Simons gauge-invariant massive electrodynamics. In our case, however, there is no arbitrary parameter, the torsionQ playing the role of the Chern-Simons parameter. This leads to bounds on the photon mass, charge, and torsion coupling.     

Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories

We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H⁴(BG, ℤ) for a compact semi-simple Lie group G. The Chern-Simons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply these notions to refine the Dijkgraaf-Witten correspondence between three dimensional Chern-Simons functionals and Wess-Zumino-Witten models associated to the group G. We do this by introducing a lifting to the level of bundle gerbes of the natural map from H⁴(BG, ℤ) to H³(G, ℤ). The notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for non-simply connected Lie groups. The implications for Wess-Zumino-Witten models are also discussed.The authors acknowledge the support of the Australian Research Council. ALC thanks MPI für Mathematik in Bonn and ESI in Vienna and BLW thanks CMA of Australian National University for their hospitality during part of the writing of this paper.     

Torsion,Scalar Field,Mass and FRW Cosmology

In the Einstein–Cartan space U₄, an axial vector torsion together with a scalar field connected to a local scale factor have been considered. By combining two particular terms from the SO(4, 1) Pontryagin density and then modifying it in a SO(3, 1) invariant way, we get a Lagrangian density with Lagrange multipliers. Then under FRW-cosmological background, where the scalar field is connected to the source of gravitation, the Euler–Lagrange equations ultimately give the constancy of the gravitational constant together with only three kinds of energy densities representing mass, radiation and cosmological constant. The gravitational constant has been found to be linked with the geometrical Nieh-Yan density.     

The existence of generalised self-dual Chern-Simons vortices

We establish the existence and uniqueness of radially symmetric self-dual topological vortices in thep=2 members of the hierarchies of (generalised) Chern-Simons Higgs and Abelian Higgs models. We also obtain all possible symmetric nontopological vortices in the Chern-Simons model characterised by an additional parameter governing the decay rates of the fields.Supported in part by CEC under grant HCM-ERBCHRXCT930362.Supported in part by NSF under grant DMS-9400243.     

Moyal Deformations of Gravity via SU(∞) Gauge Theories, Branes and Topological Chern-Simons Matrix Models

Moyal noncommutative star-product deformations of higher-dimensional gravitational Einstein-Hilbert actions via lower-dimensional SU(), W gauge theories are constructed explicitly based on the holographic reduction principle. New reparametrization invariant p-brane actions and their Moyal star product deformations follows. It is conjectured that topological Chern-Simons brane actions associated with higher-dimensional knots have a one-to-one correspondence with topological Chern-Simons Matrix models in the large N limit. The corresponding large N limit of Topological BF Matrix models leads to Kalb-Ramond couplings of antisymmetric-tensor fields to p-branes. The former Chern-Simons branes display higher-spin W symmetries which are very relevant in the study of W Gravity, the Quantum Hall effect and its higher-dimensional generalizations. We conclude by arguing why this interplay between condensed matter models, higher-dimensional extensions of the Quantum Hall effect, Chern-Simons Matrix models and Gravity needs to be investigated further within the framework of W Gauge theories.     

Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes

We investigate the generic 3D topological field theory within the AKSZ-BV framework. We use the Batalin-Vilkovisky (BV) formalism to construct explicitly cocycles of the Lie algebra of formal Hamiltonian vector fields and we argue that the perturbative partition function gives rise to secondary characteristic classes. We investigate a toy model which is an odd analogue of Chern-Simons theory, and we give some explicit computation of two point functions and show that its perturbation theory is identical to the Chern-Simons theory. We give a concrete example of the homomorphism taking Lie algebra cocycles to Q-characteristic classes, and we reinterpret the Rozansky-Witten model in this light.     

Anomalous generation of Chern-Simons terms in D=3, N=2 supersymmetric gauge theories

Parity-invariant three-dimensional gauge theories with N=2 extended supersymmetry are studied by the heat kernel method. The parity-anomalous part of the one-loop effective action is exactly found. It is expressed in terms of the N=2 supersymmetric Chern-Simons term and is identified as a N=2 superspace Atiyah-Patodi-Singer eta-invariant.     

Dual formulation of classicalW-algebras

By extending the concept of Maurer-Cartan equations, a dual formulation of (classical) nonlinear extensions of the Virasoro algebra is introduced. This dual formulation is closely related to three-dimensional actions which are analogous to a Chern-Simons action. An explicit construction in terms of superfields of theN = 2 superW ₄-algebra is presented.     

Abelian Chern-Simons term in superfluid 3He-A

We show in this paper that an Abelian Chern-Simons term is induced in (2 + 1)- and (3 + 1)-dimensional rotating superfluid ³He-A and that it plays an important role in its dynamics. Because U(1) symmetry is spontaneously broken in ³He-A, a Goldstone mode appears and contributes to the induced Chern-Simons term. We found that the coefficient of the Chern-Simons term, which is equivalent to Hall conductance, depends on an infra-red cut-off of the Goldstone mode, and that the orbital angular momentum of ³He-A in a cylinder geometry is derived from the Chern-Simons term.     

The Branch Processes of Chern-Simons (CS) <Emphasis Type="Italic">p</Emphasis>-Branes

In this paper, by making use of Duan’s topological current theory, the branch process of Chern-Simons (CS) p-branes is discussed in detail. Chern-Simons (CS) p-branes are found generating or annihilating at the limit points and encountering, splitting, or merging at the bifurcation points and higher degenerated points systematically of the vector order parameter field . Furthermore, it is also shown that CS p-branes are found splitting or merging at the degenerate point of field function but the total topological charges of the CS p-branes are still unchanged.     

Gauge fixing of Chern-Simons <Emphasis Type="Italic">N</Emphasis>-extended supergravity

We treat N-extended supergravity in 2 + 1 space-time dimensions as a Yang-Mills gauge field with Chern-Simons action associated to the N-extended Poincaré supergroup. We fix the gauge of this theory within the Batalin-Vilkovisky scheme.Received: 26 January 2004, Published online: 25 June 2004W. Spalenza: Supported in part by the Conselho Nacional de Desenvolvimento Científico e Tecnológico CNPq, Brazil     

Torsion Structure in Riemann-Cartan Manifold and Dislocation

The U(1) gauge structure of torsion and dislocation in three dimensional Riemann-Cartan manifold have been studied. The local topological structure of dislocation have been presented by so-called topological method in which the quantum number is by Hopf indices and Brouwer degree. Furthermore, the relationship between the dislocation lines and Wilson lines of the U(1) gauge theory is discussed by using the Chern-Simons theory.     

The Ka -Moody anomaly. An obstruction to a gauged Wess-Zumino-Witten model

We present a gauged Wess-Zumino-Witten model, in which the topological term is a difference of Chern-Simons forms. Part of the Ka -Moody currents become first-class constraints, thus generating local Ka -Moody symmetries. This model provides a lagrangian relization of the Goddard-Kent-Olive coset construction. At the quantum level, however, the Ka -Moody anomaly vanishes only for a negative value of the central charge k.     

Random-phase approximation for the grand-canonical potential of composite fermions in the half-filled lowest Landau level

We reconsider the theory of the half-filled lowest Landau level using the Chern-Simons formulation and study the grand-canonical potential in the random-phase approximation (RPA). Calculating the unperturbed response functions for current- and charge-density exactly, without any expansion with respect to frequency or wave vector, we find that the integral for the ground-state energy converges rapidly (algebraically) at large wave vectors k, but exhibits a logarithmic divergence at small k. This divergence originates in the k^-2 singularity of the Chern-Simons interaction and it is already present in lowest-order perturbation theory. A similar divergence appears in the chemical potential. Beyond the RPA, we identify diagrams for the grand-canonical potential (ladder-type, maximally crossed, or a combination of both) which diverge with powers of the logarithm. We expand our result for the RPA ground-state energy in the strength of the Coulomb interaction. The linear term is finite and its value compares well with numerical simulations of interacting electrons in the lowest Landau level. Received: 19 February 1998 / Revised: 25 March 1998 / Accepted: 17 April 1998