International Journal of Modern Physics A

The scope of this review is to give a pedagogical introduction to some new calculations and methods developed by the author in the context of quantum groups and their applications. The review is self-contained and serves as a "first aid kit" before one ventures into the beautiful but bewildering landscape of Woronowicz's theory. First, we present an up-to-date account of the methods and definitions used in quantum gauge theories. Then, we highlight our new results. The present paper is by no means an exhaustive overview of this swiftly developing subject.

The physical states in a worldvolume model of a noncritical 3-brane are systematically constructed using techniques of four-dimensional conformal field theories on R × S³ developed recently. Invariant combinations of creation modes under a special conformal transformation provide building blocks of physical states. Any state can be created by acting with such building blocks on a conformally invariant vacuum in an invariant way under the other conformal charges: the Hamiltonian and rotation generators on S³. We explicitly construct building blocks for scalar, vector and gravitational fields, and classify them as finite types.

Quasidegenerate binary systems of neutral mesons of the kaon type are investigated in Quantum Field Theory (QFT). General constraints cast by analyticity and discrete symmetries P, C, CP, TCP on the propagator (and on its spectral function) are deduced. Its poles are the physical masses; this unambiguously defines the propagating eigenstates. It is diagonalized and its spectrum thoroughly investigated. The role of "spurious" states, of zero norm at the poles, is emphasized, in particular for unitarity and for the realization of TCP symmetry. The K_L-K_S mass splitting triggers a tiny difference between their CP violating parameters ∊_L and ∊_S, without any violation of TCP. A constant mass matrix like used in Quantum Mechanics (QM) can only be introduced in a linear approximation to the inverse propagator, which respects its analyticity and positivity properties; it is however unable to faithfully describe all features of neutral mesons as we determine them in QFT, nor to provide any sensible parametrization of eventual effects of TCP violation. The suitable way to diagonalize the propagator makes use of a bi-orthogonal basis; it is inequivalent to a bi-unitary transformation (unless the propagator is normal, which cannot occur here). Problems linked with the existence of different "in" and "out" eigenstates are smoothed out. We study phenomenological consequences of the differences between the QFT and QM treatments; the nonvanishing of the semileptonic asymmetry δ_S - δ_L, does not signal, unlike usually claimed, TCP violation, while A_TCP keeps vanishing when TCP is realized. We provide expressions invariant by the rephasing of K⁰ and .

We present an analysis of the Yangian symmetries of various bosonic sectors of the dilatation operator of SYM. The analysis is presented from the point of view of Hamiltonian matrix models. In the various SU(n) sectors, we give a modified presentation of the Yangian generators, which are conserved on states of any size. A careful analysis of the Yangian invariance of the full SO(6) sector of the scalars is also presented in this paper. We also study the Yangian invariance beyond first order perturbation theory. Following this, we derive the continuum limits of the various matrix models and reproduce the sigma model actions for fast moving strings reported in some papers. We motivate the constructions of continuum sigma models (corresponding to both the SU(n) and SO(n) sectors) as variational approximations to the matrix model Hamiltonians. These sigma models retain the semiclassical counterparts of the original Yangian symmetries of the dilatation operator. The semiclassical Yangian symmetries of the sigma models are worked out in detail. The zero curvature representation of the equations of motion and the construction of the transfer matrix for the SO(n) sigma model obtained as the continuum limit of the one loop bosonic dilatation operator is carried out, and the similar constructions for the SU(n) case are also discussed.

The property and gravitational field of global monopole of tachyon are investigated in a four-dimensional static space–time. We give an exact solution of the tachyon field in the flat space–time background. Using the linearized approximation of gravity, we get the approximate solution of the metric. We also solve analytically the coupled Einstein and tachyon field equations which is beyond the linearized approximation to determine the gravitational properties of the monopole solution. We find that the metric of tachyon monopole represents an asymptotically AdS space–time with a small effective mass at the origin. We show that this relatively tiny mass is actually negative, as it is in the case of ordinary scalar field.

We investigate classical dynamics of the bosonic string in the background metric, antisymmetric and dilaton fields. We use canonical methods to find Hamiltonian in terms of energy–momentum tensor components. The later are secondary constraints of the theory. Due to the presence of the dilaton field the Virasoro generators have nonlinear realization. We find that, in the curve space–time, opposite chirality currents do not commute. As a consequence of the two-dimensional general covariance, the energy–momentum tensor components satisfy two Virasoro algebras, even in the curve space–time. We emphasize that background antisymmetric and dilaton fields are the origin of space–time torsion and space–time nonmetricity, respectively.