The renormalization group and spontaneous compactification of a higher-dimensional scalar field theory in curved spacetime

The renormalization group (RG) is used to study the asymptotically free $\phi_6^3$-theory in curved spacetime. Several forms of the RG equations for the effective potential are formulated. By solving these equations we obtain the one-loop effective potential as well as its explicit forms in the case of strong gravitational fields and strong scalar fields. Using zeta function techniques, the one-loop and corresponding RG improved vacuum energies are found for the Kaluza-Klein backgrounds $R^4\times S^1\times S^1$ and $R^4\times S^2$. They are given in terms of exponentially convergent series, appropriate for numerical calculations. A study of these vacuum energies as a function of compactification lengths and other couplings shows that spontaneous compactification can be qualitatively different when the RG improved energy is used.Comment: LaTeX, 15 pages, 4 figure

Spectral analysis and zeta determinant on the deformed spheres

We consider a class of singular Riemannian manifolds, the deformed spheres $S^N_k$, defined as the classical spheres with a one parameter family $g[k]$ of singular Riemannian structures, that reduces for $k=1$ to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian $\Delta_{S^N_k}$, we study the associated zeta functions $\zeta(s,\Delta_{S^N_k})$. We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in $\zeta(s,\Delta_{S^N_k})$. An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular $\zeta(0,\Delta_{S^N_k})$ and $\zeta'(0,\Delta_{S^N_k})$. We give explicit formulas for the zeta regularized determinant in the low dimensional cases, $N=2,3$, thus generalizing a result of Dowker \cite{Dow1}, and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter $k$.Comment: 1 figur

A Conformally Invariant Holographic Two-Point Function on the Berger Sphere

We apply our previous work on Green's functions for the four-dimensional quaternionic Taub-NUT manifold to obtain a scalar two-point function on the homogeneously squashed three-sphere (otherwise known as the Berger sphere), which lies at its conformal infinity. Using basic notions from conformal geometry and the theory of boundary value problems, in particular the Dirichlet-to-Robin operator, we establish that our two-point correlation function is conformally invariant and corresponds to a boundary operator of conformal dimension one. It is plausible that the methods we use could have more general applications in an AdS/CFT context.Comment: 1+49 pages, no figures. v2: Several typos correcte

PYROELECTRIC PROPERTIES OF CHIRAL SMECTIC PHASES

On a mesuré la réponse pyroélectrique des cristaux liquides ferroélectriques (L-p-décyloxy-, L-p-nonyloxybenzilidène p'-amino cinnamate de méthyl-2 butyle et mélange) à des impulsions thermiques de durées de 2 x 10-4 et 3 x 10-8 s. A partir des données obtenues on a calculé les variations du coefficient pyroélectrique γ(T) en fonction de la température. La dispersion de γ et de la polarisation spontanée Ps en fonction de la fréquence a été obtenue et est en accord avec les mesures de dispersion de la susceptibilité diélectrique pour des fréquences dans la gamme 104-2 x 105 Hz. On a mesuré le temps de relaxation de la polarisation spontanée (τ) et on a trouvé une divergence de τ près de la température de transition smectique C-smectique A. On a aussi montré que des impuretés dipolaires ont une influence visible sur les valeurs de γmax et de Pmaxs pour les cristaux étudiés.The temperature dependences of the pyroelectric coefficient γ(T) of ferroelectric liquid crystalline L-p-decyloxy- and L-p-nonyloxybenzylidene-p'-amino-2-methylbutylcinnamate and their mixture were calculated from the measurements of their responses to heat pulses with durations of 2 x 10-4 and 3 x 10-8 s. The frequency dispersion of γ and the spontaneous polarization Ps was discovered which is consistent with the corresponding dispersion of dielectric permittivity over a frequency range 104-2 x 105 Hz. The direct measurements of the relaxation time for spontaneous polarization (τ) were performed and sharp increase in τ near the SmC-SmA phase transition was observed. The effect of dipolar admixtures on γmax and Pmaxs was also discussed