Pontrjagin forms, Chern Simons classes, Codazzi transformations, and affine hypersurfaces

We show that the primary and secondary characteristic classes vanish in the context of affine differential geometry. This gives rise to obstructions to realizing a conformal class of metrics on a manifold either as the first or as the second fundamental form of an affine immersion.     

At the Dawn of a Unifying Theory

I present a brief summary of the recent advancements on the unification of abelian quantum Hall states that covers the effects of disorder as well as the Coulomb interactions. I also present some exact results on the CP^N–1 model at large N. In contrast to the previous expectations, this theory shows all the basic features of the quantum Hall effect, including the massless bulk excitations at =.     

An Algebraic Proof on the Finiteness of Yang–Mills–Chern–Simons Theory in D=3

A rigorous algebraic proof of the full finiteness in all orders of perturbation theory is given for the Yang–Mills–Chern–Simons theory in a general three-dimensional Riemannian manifold. We show the validity of a trace identity, playing the role of a local form of the Callan–Symanzik equation, in all loop orders, which yields the vanishing of the -functions associated to the topological mass and gauge coupling constant as well as the anomalous dimensions of the fields.     

Loop Observables for BF Theories in Any Dimension and the Cohomology of Knots

A generalization of Wilson loop observables for BF theories in any dimension is introduced within the Batalin–Vilkovisky framework. The expectation values of these observables are cohomology classes of the space of imbeddings of a circle. One of the resulting theories discussed in the Letter has only trivalent interactions and, irrespective of the actual dimension, looks like a three-dimensional Chern–Simons theory.     

Variational formulation of ideal fluid flows according to gauge principle

On the basis of the gauge principle of field theory, a new variational formulation is presented for flows of an ideal fluid. The fluid is defined thermodynamically by mass density and entropy density, and its flow fields are characterized by symmetries of translation and rotation. The rotational transformations are regarded as gauge transformations as well as the translational ones. In addition to the Lagrangians representing the translation symmetry, a structure of rotation symmetry is equipped with a Lagrangian Λ_A including the vorticity and a vector potential bilinearly. Euler's equation of motion is derived from variations according to the action principle. In addition, the equations of continuity and entropy are derived from the variations. Equations of conserved currents are deduced as the Noether theorem in the space of Lagrangian coordinate a. Without Λ_A, the action principle results in the Clebsch solution with vanishing helicity. The Lagrangian Λ_A yields non-vanishing vorticity and provides a source term of non-vanishing helicity. The vorticity equation is derived as an equation of the gauge field, and the Λ_A characterizes topology of the field. The present formulation is comprehensive and provides a consistent basis for a unique transformation between the Lagrangian a space and the Eulerian x space. In contrast, with translation symmetry alone, there is an arbitrariness in the transformation between these spaces.     

Topological mass generation four-dimensional gauge theory

The Lagrangian of non-Abelian tensor gauge fields describes the interaction of the Yang–Mills and massless tensor bosons of increasing helicities. We have found a metric-independent gauge invariant density which is a four-dimensional analog of the Chern–Simons density. The Lagrangian augmented by this Chern–Simons-like invariant describes massive Yang–Mills boson, providing a gauge-invariant mass gap for a four-dimensional gauge field theory. We present invariant densities which can provide masses to the high-rank tensor bosons.