Conformal Field Theories in Fractional Dimensions

We study the conformal bootstrap in fractional space-time dimensions, obtaining rigorous bounds on operator dimensions. Our results show strong evidence that there is a family of unitary CFTs connecting the 2D Ising model, the 3D Ising model, and the free scalar theory in 4D. We give numerical predictions for the leading operator dimensions and central charge in this family at different values of D and compare these to calculations of phi^4 theory in the epsilon-expansion.Comment: 11 pages, 4 figures - references updated - one affiliation modifie

The effective bootstrap

We study the numerical bounds obtained using a conformal-bootstrap method - advocated in ref. [1] but never implemented so far - where different points in the plane of conformal cross ratios z and z¯ are sampled. In contrast to the most used method based on derivatives evaluated at the symmetric point z=z¯=1/2, we can consistently "integrate out" higher-dimensional operators and get a reduced simpler, and faster to solve, set of bootstrap equations. We test this "effective" bootstrap by studying the 3D Ising and O(n) vector models and bounds on generic 4D CFTs, for which extensive results are already available in the literature. We also determine the scaling dimensions of certain scalar operators in the O(n) vector models, with n=2,3,4, which have not yet been computed using bootstrap techniques. ArXI

Solving the 3D Ising Model with the Conformal Bootstrap

We study the constraints of crossing symmetry and unitarity in general 3D Conformal Field Theories. In doing so we derive new results for conformal blocks appearing in four-point functions of scalars and present an efficient method for their computation in arbitrary space-time dimension. Comparing the resulting bounds on operator dimensions and OPE coefficients in 3D to known results, we find that the 3D Ising model lies at a corner point on the boundary of the allowed parameter space. We also derive general upper bounds on the dimensions of higher spin operators, relevant in the context of theories with weakly broken higher spin symmetries.Comment: 32 pages, 11 figures; v2: refs added, small changes in Section 5.3, Fig. 7 replaced; v3: ref added, fits redone in Section 5.

The information paradox: A pedagogical introduction

The black hole information paradox is a very poorly understood problem. It is often believed that Hawking's argument is not precisely formulated, and a more careful accounting of naturally occurring quantum corrections will allow the radiation process to become unitary. We show that such is not the case, by proving that small corrections to the leading order Hawking computation cannot remove the entanglement between the radiation and the hole. We formulate Hawking's argument as a `theorem': assuming `traditional' physics at the horizon and usual assumptions of locality we will be forced into mixed states or remnants. We also argue that one cannot explain away the problem by invoking AdS/CFT duality. We conclude with recent results on the quantum physics of black holes which show the the interior of black holes have a `fuzzball' structure. This nontrivial structure of microstates resolves the information paradox, and gives a qualitative picture of how classical intuition can break down in black hole physics.Comment: 38 pages, 7 figures, Latex (Expanded form of lectures given at CERN for the RTN Winter School, Feb 09), typo correcte

G2 Hitchin functionals at one loop

We consider the quantization of the effective target space description of topological M-theory in terms of the Hitchin functional whose critical points describe seven-manifolds with G2 structure. The one-loop partition function for this theory is calculated and an extended version of it, that is related to generalized G2 geometry, is compared with the topological G2 string. We relate the reduction of the effective action for the extended G2 theory to the Hitchin functional description of the topological string in six dimensions. The dependence of the partition functions on the choice of background G2 metric is also determined.Comment: 58 pages, LaTeX; v2: Acknowledgments adde

Holographic Renormalization of Foliation Preserving Gravity and Trace Anomaly

From the holographic renormalizationg group viewpoint, while the scale transformation plays a primary role in the duality by providing the extra dimension, the special conformal transformation seems to only play a secondary role. We, however, claim that the space-time diffeomorphism is crucially related to the latter. For its demonstration, we study the holographic renormalization group flow of a foliation preserving diffeomophic theory of gravity (a.k.a. space-time flipped Horava gravity). We find that the dual field theory, if any, is only scale invariant but not conformal invariant. In particular, we show that the holographic trace anomaly in four-dimension predicts the Ricci scalar squared term that would be incompatible with the Wess-Zumino consistency condition if it were conformal. This illustrates how the foliation preserving diffeomophic theory of gravity could be inconsistent with a theorem of the dual unitary quantum field theory.Comment: 18 pages, v2: reference added, v3: comments on more recent literature added in response to referee's reques

Conformal symmetry of the critical 3D Ising model inside a sphere

We perform Monte-Carlo simulations of the three-dimensional Ising model at the critical temperature and zero magnetic field. We simulate the system in a ball with free boundary conditions on the two dimensional spherical boundary. Our results for one and two point functions in this geometry are consistent with the predictions from the conjectured conformal symmetry of the critical Ising model.We are grateful to Slava Rychkov for useful discussions and for suggesting this work. The research leading to these results has received funding from the [European Union] Seventh Framework Programme [FP7-People-2010-IRSES] and [FP7/2007-2013] under grant agreements No 269217, 317089 and No 247252, and from the grant CERN/FP/123599/2011. Centro de Física do Porto is partially funded by the Foundation for Science and Technology of Portugal (FCT). J.V.P.L. acknowledges funding from projecto Operacional Regional do Norte, within Quadro de Referência Estratégico Nacional (QREN) and through Fundo Europeu de Desenvolvimento Regional (FEDER), Ref. NORTE-07-0124-FEDER- 00003