Topics in quantum gravity

We study some aspects of conformal field theory, wormhole physics and two-dimensional random surfaces. Inspite of being rather different, these topics serve as examples of the issues that are involved, both at high and low energy scales, in formulating a quantum theory of gravity. In conformal field theory we show that fusion and braiding properties can be used to determine the operator product coefficients of the non-diagonal Wess-Zumino-Witten models. In wormhole physics we show how Coleman's proposed probability distribution would result in wormholes determining the value of ^��QCD. We attempt such a calculation and find the most probable value of ^��QCD to be ��. This hints at a potential conflict with nature. In random surfaces we explore the behaviour of conformal field theories coupled to gravity and calculate some partition functions and correlation functions. Our results throw some light on the transition that is believed to occur when the central charge of the matter theory gets larger than one.

Entanglement negativity in the harmonic chain out of equilibrium

We study the entanglement in a chain of harmonic oscillators driven out of equilibrium by preparing the two sides of the system at different temperatures, and subsequently joining them together. The steady state is constructed explicitly and the logarithmic negativity is calculated between two adjacent segments of the chain. We find that, for low temperatures, the steady-state entanglement is a sum of contributions pertaining to left-and right-moving excitations emitted from the two reservoirs. In turn, the steady-state entanglement is a simple average of the Gibbs-state values and thus its scaling can be obtained from conformal field theory. A similar averaging behaviour is observed during the entire time evolution. As a particular case, we also discuss a local quench where both sides of the chain are initialized in their respective ground states.

Holographic dual of linear dilaton black hole in Einstein-Maxwell-dilaton-axion gravity

Motivated by the recently proposed Kerr/CFT correspondence, we investigate the holographic dual of the extremal and non-extremal rotating linear dilaton black hole in Einstein-Maxwell-Dilaton-Axion Gravity. For the case of extremal black hole, by imposing the appropriate boundary condition at spatial infinity of the near horizon extremal geometry, the Virasoro algebra of conserved charges associated with the asymptotic symmetry group is obtained. It is shown that the microscopic entropy of the dual conformal field given by Cardy formula exactly agrees with Bekenstein-Hawking entropy of extremal black hole. Then, by rewriting the wave equation of massless scalar field with sufficient low energy as the SLL(2, R) x SLR(2, R) Casimir operator, we find the hidden conformal symmetry of the non-extremal linear dilaton black hole, which implies that the non-extremal rotating linear dilaton black hole is holographically dual to a two dimensional conformal field theory with the non-zero left and right temperatures. Furthermore, it is shown that the entropy of non-extremal black hole can be reproduced by using Cardy formula.

Entropy of Kaluza-Klein black hole from Kerr/CFT correspondence

We extend the recently proposed Kerr/CFT correspondence to examine the dual conformal field theory of four-dimensional Kaluza-Klein black hole in Einstein-Maxwell-Dilaton theory. For the extremal Kaluza-Klein black hole, the central charge and temperature of the dual conformal field are calculated following the approach of Guica, Hartman, Song and Strominger. Meanwhile, we show that the microscopic entropy given by the Cardy formula agrees with Bekenstein-Hawking entropy of extremal Kaluza-Klein black hole. For the non-extremal case, by studying the near-region wave equation of a neutral massless scalar field, we investigate the hidden conformal symmetry of Kaluza-Klein black hole, and find the left and right temperatures of the dual conformal field theory. Furthermore, we find that the entropy of non-extremal Kaluza-Klein black hole is reproduced by Cardy formula. (C) 2010 Elsevier B.V. All rights reserved.

Entanglement entropy dynamics of Heisenberg chains

By means of the time dependent density matrix renormalization group algorithm we study the zero-temperature dynamics of the Von Neumann entropy of a block of spins in a Heisenberg chain after a sudden quench in the anisotropy parameter. In the absence of any disorder the block entropy increases linearly with time and then saturates. We analyse the velocity of propagation of the entanglement as a function of the initial and final anisotropies and compare our results, wherever possible, with those obtained by means of conformal field theory. In the disordered case we find a slower ( logarithmic) evolution which may signal the onset of entanglement localization.

Entanglement Spectrum, Critical Exponents and Order Parameters in Quantum Spin Chains

We investigate the entanglement spectrum near criticality in finite quantum spin chains. Using finite size scaling we show that when approaching a quantum phase transition, the Schmidt gap, i.e., the difference between the two largest eigenvalues of the reduced density matrix ?1, ?2, signals the critical point and scales with universal critical exponents related to the relevant operators of the corresponding perturbed conformal field theory describing the critical point. Such scaling behavior allows us to identify explicitly the Schmidt gap as a local order parameter.

Entanglement properties of spin models in triangular lattices

<p>The different quantum phases appearing in strongly correlated systems as well as their transitions are closely related to the entanglement shared between their constituents. In 1D systems, it is well established that the entanglement spectrum is linked to the symmetries that protect the different quantum phases. This relation extends even further at the phase transitions where a direct link associates the entanglement spectrum to the conformal field theory describing the former. For 2D systems much less is known. The lattice geometry becomes a crucial aspect to consider when studying entanglement and phase transitions. Here, we analyze the entanglement properties of triangular spin lattice models by also considering concepts borrowed from quantum information theory such as geometric entanglement.</p>

��quations diff��rentielles issues des vecteurs singuliers des repr��sentations de l'alg��bre de Virasoro

M��moire num��ris�� par la Division de la gestion de documents et des archives de l'Universit�� de Montr��al

La structure de Jordan des matrices de transfert des mod��les de boucles et la relation avec les hamiltoniens XXZ

Les mod��les sur r��seau comme ceux de la percolation, d���Ising et de Potts servent �� d��crire les transitions de phase en deux dimensions. La recherche de leur solution analytique passe par le calcul de la fonction de partition et la diagonalisation de matrices de transfert. Au point critique, ces mod��les statistiques bidimensionnels sont invariants sous les transformations conformes et la construction de th��ories des champs conformes rationnelles, limites continues des mod��les statistiques, permet un calcul de la fonction de partition au point critique. Plusieurs chercheurs pensent cependant que le paradigme des th��ories des champs conformes rationnelles peut ��tre ��largi pour inclure les mod��les statistiques avec des matrices de transfert non diagonalisables. Ces mod��les seraient alors d��crits, dans la limite d�����chelle, par des th��ories des champs logarithmiques et les repr��sentations de l���alg��bre de Virasoro intervenant dans la description des observables physiques seraient ind��composables. La matrice de transfert de boucles D_N(��, u), un ��l��ment de l���alg��bre de Temperley- Lieb, se manifeste dans les th��ories physiques �� l���aide des repr��sentations de connectivit��s �� (link modules). L���espace vectoriel sur lequel agit cette repr��sentation se d��compose en secteurs ��tiquet��s par un param��tre physique, le nombre d de d��fauts. L���action de cette repr��sentation ne peut que diminuer ce nombre ou le laisser constant. La th��se est consacr��e �� l���identification de la structure de Jordan de D_N(��, u) dans ces repr��sentations. Le param��tre �� = 2 cos �� = ���(q + 1/q) fixe la th��orie : �� = 1 pour la percolation et ���2 pour le mod��le d���Ising, par exemple. Sur la g��om��trie du ruban, nous montrons que D_N(��, u) poss��de les m��mes blocs de Jordan que F_N, son plus haut coefficient de Fourier. Nous ��tudions la non diagonalisabilit�� de F_N �� l���aide des divergences de certaines composantes de ses vecteurs propres, qui apparaissent aux valeurs critiques de ��. Nous prouvons dans ��(D_N(��, u)) l���existence de cellules de Jordan intersectorielles, de rang 2 et couplant des secteurs d, d��� lorsque certaines contraintes sur ��, d, d��� et N sont satisfaites. Pour le mod��le de polym��res denses critique (�� = 0) sur le ruban, les valeurs propres de ��(D_N(��, u)) ��taient connues, mais les d��g��n��rescences conjectur��es. En construisant un isomorphisme entre les modules de connectivit��s et un sous-espace des modules de spins du mod��le XXZ en q = i, nous prouvons cette conjecture. Nous montrons aussi que la restriction de l���hamiltonien de boucles �� un secteur donn�� est diagonalisable et trouvons la forme de Jordan exacte de l���hamiltonien XX, non triviale pour N pair seulement. Enfin nous ��tudions la structure de Jordan de la matrice de transfert T_N(��, ��) pour des conditions aux fronti��res p��riodiques. La matrice T_N(��, ��) a des blocs de Jordan intrasectoriels et intersectoriels lorsque �� = ��a/b, et a, b ��� Z��. L���approche par F_N admet une g��n��ralisation qui permet de diagnostiquer des cellules intersectorielles dont le rang exc��de 2 dans certains cas et peut cro��tre ind��finiment avec N. Pour les blocs de Jordan intrasectoriels, nous montrons que les repr��sentations de connectivit��s sur le cylindre et celles du mod��le XXZ sont isomorphes sauf pour certaines valeurs pr��cises de q et du param��tre de torsion v. En utilisant le comportement de la transformation i_N^d dans un voisinage des valeurs critiques (q_c, v_c), nous construisons explicitement des vecteurs g��n��ralis��s de Jordan de rang 2 et discutons l���existence de blocs de Jordan intrasectoriels de plus haut rang.

Exposants g��om��triques des mod��les de boucles dilu��s et idempotents des TL-modules de la cha��ne de spins XXZ

Cette th��se porte sur les ph��nom��nes critiques survenant dans les mod��les bidimensionnels sur r��seau. Les r��sultats sont l'objet de deux articles : le premier porte sur la mesure d'exposants critiques d��crivant des objets g��om��triques du r��seau et, le second, sur la construction d'idempotents projetant sur des modules ind��composables de l'alg��bre de Temperley-Lieb pour la cha��ne de spins XXZ. Le premier article pr��sente des exp��riences num��riques Monte Carlo effectu��es pour une famille de mod��les de boucles en phase dilu��e. Baptis��s "dilute loop models (DLM)", ceux-ci sont inspir��s du mod��le O(n) introduit par Nienhuis (1990). La famille est ��tiquet��e par les entiers relativement premiers p et p' ainsi que par un param��tre d'anisotropie. Dans la limite thermodynamique, il est pressenti que le mod��le DLM(p,p') soit d��crit par une th��orie logarithmique des champs conformes de charge centrale c(\kappa)=13-6(\kappa+1/\kappa), o�� \kappa=p/p' est li�� �� la fugacit�� du gaz de boucles \beta=-2\cos\pi/\kappa, pour toute valeur du param��tre d'anisotropie. Les mesures portent sur les exposants critiques repr��sentant la loi d'��chelle des objets g��om��triques suivants : l'interface, le p��rim��tre externe et les liens rouges. L'algorithme Metropolis-Hastings employ��, pour lequel nous avons introduit de nombreuses am��liorations sp��cifiques aux mod��les dilu��s, est d��taill��. Un traitement statistique rigoureux des donn��es permet des extrapolations co��ncidant avec les pr��dictions th��oriques �� trois ou quatre chiffres significatifs, malgr�� des courbes d'extrapolation aux pentes abruptes. Le deuxi��me article porte sur la d��composition de l'espace de Hilbert \otimes^nC^2 sur lequel la cha��ne XXZ de n spins 1/2 agit. La version ��tudi��e ici (Pasquier et Saleur (1990)) est d��crite par un hamiltonien H_{XXZ}(q) d��pendant d'un param��tre q\in C^\times et s'exprimant comme une somme d'��l��ments de l'alg��bre de Temperley-Lieb TL_n(q). Comme pour les mod��les dilu��s, le spectre de la limite continue de H_{XXZ}(q) semble reli�� aux th��ories des champs conformes, le param��tre q d��terminant la charge centrale. Les idempotents primitifs de End_{TL_n}\otimes^nC^2 sont obtenus, pour tout q, en termes d'��l��ments de l'alg��bre quantique U_qsl_2 (ou d'une extension) par la dualit�� de Schur-Weyl quantique. Ces idempotents permettent de construire explicitement les TL_n-modules ind��composables de \otimes^nC^2. Ceux-ci sont tous irr��ductibles, sauf si q est une racine de l'unit��. Cette exception est trait��e s��par��ment du cas o�� q est g��n��rique. Les probl��mes r��solus par ces articles n��cessitent une grande vari��t�� de r��sultats et d'outils. Pour cette raison, la th��se comporte plusieurs chapitres pr��paratoires. Sa structure est la suivante. Le premier chapitre introduit certains concepts communs aux deux articles, notamment une description des ph��nom��nes critiques et de la th��orie des champs conformes. Le deuxi��me chapitre aborde bri��vement la question des champs logarithmiques, l'��volution de Schramm-Loewner ainsi que l'algorithme de Metropolis-Hastings. Ces sujets sont n��cessaires �� la lecture de l'article "Geometric Exponents of Dilute Loop Models" au chapitre 3. Le quatri��me chapitre pr��sente les outils alg��briques utilis��s dans le deuxi��me article, "The idempotents of the TL_n-module \otimes^nC^2 in terms of elements of U_qsl_2", constituant le chapitre 5. La th��se conclut par un r��sum�� des r��sultats importants et la proposition d'avenues de recherche qui en d��coulent.

Bistability through triadic closure

We propose and analyse a class of evolving network models suitable for describing a dynamic topological structure. Applications include telecommunication, on-line social behaviour and information processing in neuroscience. We model the evolving network as a discrete time Markov chain, and study a very general framework where, conditioned on the current state, edges appear or disappear independently at the next timestep. We show how to exploit symmetries in the microscopic, localized rules in order to obtain conjugate classes of random graphs that simplify analysis and calibration of a model. Further, we develop a mean ���eld theory for describing network evolution. For a simple but realistic scenario incorporating the triadic closure e���ect that has been empirically observed by social scientists (friends of friends tend to become friends), the mean ���eld theory predicts bistable dynamics, and computational results con���rm this prediction. We also discuss the calibration issue for a set of real cell phone data, and ���nd support for a strati���ed model, where individuals are assigned to one of two distinct groups having di���erent within-group and across-group dynamics.

Quantum current algebra for the AdS(5) x S(5) superstring

The sigma model describing the dynamics of the superstring in the AdS(5) x S(5) background can be constructed using the coset PSU(2, 2 vertical bar 4)/SO(4, 1) x SO(5). A basic set of operators in this two dimensional conformal field theory is composed by the left invariant currents. Since these currents are not (anti) holomorphic, their OPE`s is not determined by symmetry principles and its computation should be performed perturbatively. Using the pure spinor sigma model for this background, we compute the one-loop correction to these OPE`s. We also compute the OPE`s of the left invariant currents with the energy momentum tensor at tree level and one loop.

Phase diagram and spectral properties of a new exactly integrable spin-1 quantum chain

The spectral properties and phase diagram of the exactly integrable spin-1 quantum chain introduced by Alcaraz and Bariev are presented. The model has a U(1) symmetry and its integrability is associated with an unknown R-matrix whose dependence on the spectral parameters is not of a different form. The associated Bethe ansatz equations that fix the eigenspectra are distinct from those associated with other known integrable spin models. The model has a free parameter t(p). We show that at the special point t(p) = 1, the model acquires an extra U(1) symmetry and reduces to the deformed SU(3) Perk-Schultz model at a special value of its anisotropy q = exp(i2 pi/3) and in the presence of an external magnetic field. Our analysis is carried out either by solving the associated Bethe ansatz equations or by direct diagonalization of the quantum Hamiltonian for small lattice sizes. The phase diagram is calculated by exploring the consequences of conformal invariance on the finite-size corrections of the Hamiltonian eigenspectrum. The model exhibits a critical phase ruled by the c = 1 conformal field theory separated from a massive phase by first-order phase transitions.

N=2 sigma models for Ramond-Ramond backgrounds

Using the U(4) hybrid formalism, manifestly N = (2,2) worldsheet supersymmetric sigma models are constructed for the type-IIB superstring in Ramond-Ramond backgrounds. The Kahler potential in these N = 2 sigma models depends on four chiral and antichiral bosonic superfields and two chiral and antichiral fermionic superfields. When the Kahler potential is quadratic, the model is a free conformal field theory which describes a flat ten-dimensional target space with Ramond-Ramond flux and non-constant dilaton. For more general Kahler potentials, the model describes curved target spaces with Ramond-Ramond flux that are not plane-wave backgrounds. Ricci-flatness of the Kahler metric implies the on-shell conditions for the background up to the usual four-loop conformal anomaly.

Multitrace operators and the generalized AdS/CFT prescription

We show that multitrace interactions can be consistently incorporated into an extended AdS conformal field theory (CFT) prescription involving the inclusion of generalized boundary conditions and a modified Legendre transform prescription. We find new and consistent results by considering a self-contained formulation which relates the quantization of the bulk theory to the AdS/CFT correspondence and the perturbation at the boundary by double-trace interactions. We show that there exist particular double-trace perturbations for which irregular modes are allowed to propagate as well as the regular ones. We perform a detailed analysis of many different possible situations, for both minimally and nonminimally coupled cases. In all situations, we make use of a new constraint which is found by requiring consistency. In the particular nonminimally coupled case, the natural extension of the Gibbons-Hawking surface term is generated.