International Journal of Modern Physics A

Connes' gauge theory on M₄ × Z₂ is reformulated in the Lagrangian level. It is pointed out that the field strength in Connes' gauge theory is not unique. We explicitly construct a field strength different from Connes' and prove that our definition leads to the generation-number independent Higgs potential. It is also shown that the nonuniqueness is related to the assumption that two different extensions of the differential geometry are possible when the extra one-form basis χ is introduced to define the differential geometry on M₄ × Z₂. Our reformulation is applied to the standard model based on Connes' color-flavor algebra. A connection between the unimodularity condition and the electric charge quantization is then discussed in the presence or absence of ν_R.

The dimensional reduction of the effective ten-action of the heterotic superstring theory to the physical four-action S[g_ij] results in the appearance of three moduli B_(a), whose real parts , set equal for simplicity, define the radius and shape of the compact internal six-space , in addition to the dilaton , the ten-interval being . These scalar fields are massless at tree level and can be put into canonical form with coefficients for the positive kinetic-energy terms, when higher-derivative terms are ignored, as found by Witten, so that . Previously, we have shown that σ_A and σ_B acquire a potential from the higher-derivative terms and , which becomes large close to the Planck era. Here, we discuss the renormalization of the kinetic-energy terms due to which, after diagonalization, results in a mixing of ∇σ_B with ∇σ_A, while the remaining coefficient of (∇σ_B)² vanishes at t_c ≈ t_P/12 in a radiation-dominated Universe, corresponding to a temperature T_c ≈ 5 × 10¹⁷GeV, where the four-theory is still classical. At earlier times, the energy is unbounded from below, signalling that the four-theory has become unphysical, and that the string must still be in its uncompactified form with one dilaton ϕ, whose canonical kinetic energy is positive in the Einstein metric. This mechanism depends upon the equation of state of the source for the Friedmann expansion assumed, and is only effective for values of the adiabatic index in the range 1.14 < γ < 2.63, which thus includes radiation (γ = 4/3) and the Zel'dovich equation of state (γ = 2).

Using a Hamiltonian approach, we construct the classical and quantum theory of open WZW strings on a strip. (These are the strings which end on WZW branes.) The development involves non-Abelian generalized Dirichlet images in an essential way. At the classical level, we find a new non-commutative geometry in which the equal-time coordinate brackets are non-zero at the world-sheet boundary, and the result is an intrinsically non-Abelian effect which vanishes in the Abelian limit. Using the classical theory as a guide to the quantum theory, we also find the operator algebra and the analogue of the Knizhnik–Zamolodchikov equations for the conformal field theory of open WZW strings.

The role of conformal anomaly in AdS/CFT and related issues is clarified. The comparison of holographic and QFT conformal anomalies (with the account of brane quantum gravity contribution) indicates the possibility for brane quantum gravity to occur within the AdS/CFT setup. Three-dimensional quantum induced inflationary (or hyperbolic) brane-world is shown to be realized in the frames of AdS3/CFT2 correspondence where the role of 2d brane cosmological constant is played by effective tension due to two-dimensional conformal anomaly. The dynamical equations to describe 4d FRW universe with the account of quantum effects produced by conformal anomaly are obtained. The quantum corrected energy, pressure and entropy are found. The dynamical evolution of entropy bounds in the inflationary universe is estimated and its comparison with quantum corrected entropy is done. It is demonstrated that entropy bounds for quantum corrected entropy are getting the approximate ones and occur for some limited periods of evolution of the inflationary universe.

The Casimir force is calculated analytically for configurations of two parallel plates and a spherical lens (sphere) above a plate with an account of nonzero temperature, finite conductivity of the boundary metal and surface roughness. The permittivity of the metal is described by the plasma model. It is proved that in case of the plasma model the scattering formalism of quantum field theory in Matsubara formulation underlying Lifshitz formula is well defined and no modifications are needed concerning the zero-frequency contribution. The temperature correction to the Casimir force is found completely with respect to temperature and perturbatively (up to the second order in the relative penetration depth of electromagnetic zero-point oscillations into the metal) with respect to finite conductivity. The asymptotics of low and high temperatures are presented and contributions of longitudinal and perpendicular modes are determined separately. Serving as an example, aluminium test bodies are considered, showing good agreement between the obtained analytical results and previously performed numerical computations. The roughness correction is formally included and formulas are given permitting us to calculate the Casimir force under the influence of all relevant factors.

We study the knot invariant based on the quantum dilogarithm function. This invariant can be regarded as a noncompact analog of Kashaev's invariant, or the colored Jones invariant, and is defined by an integral form. The three-dimensional picture of our invariant originates from the pentagon identity of the quantum dilogarithm function, and we show that the hyperbolicity consistency conditions in gluing polyhedra arise naturally in the classical limit as the saddle point equation of our invariant.