All Publications

We study the Rényi entropies of N disjoint intervals in the conformal field theories describing the free compactified boson and the Ising model. They are computed as the 2N-point function of twist fields, by employing the partition function of the model on a particular class of Riemann surfaces. The results are written in terms of Riemann theta functions. The prediction for the free boson in the decompactification regime is checked against exact results for the harmonic chain. For the Ising model, matrix product state computations agree with the conformal field theory result once the finite size corrections have been taken into account. © 2014 IOP Publishing Ltd and SISSA Medialab srl.

We study causal wedges associated with a given sub-region in the boundary of asymptotically AdS spacetimes. Part of our motivation is to better understand the recently proposed holographic observable, causal holographic information, χ, which is given by the area of a bulk co-dimension two surface lying on the boundary of the causal wedge. It has been suggested that χ captures the basic amount of information contained in the reduced density matrix about the bulk geometry. To explore its properties further we examine its behaviour in time-dependent situations. As a simple model we focus on null dust collapse in an asymptotically AdS spacetime, modeled by the Vaidya-AdS geometry. We argue that while χ is generically quasi-telelogical in time-dependent backgrounds, for suitable choice of sub-regions in conformal field theories, the temporal evolution of χ is entirely causal. We comment on the implications of this observation and more generally on features of causal constructions and contrast our results with the behaviour of holographic entanglement entropy. Along the way we also derive the rate of early time growth and late time saturation (to the thermal value) of both χ and entanglement entropy in these backgrounds. © 2013 SISSA, Trieste, Italy.

We study the scaling of the traces of the integer powers of the partially transposed reduced density matrix Tr(ρT2A)n and of the entanglement negativity for two spin blocks as a function of their length and separation in the critical Ising chain. For two adjacent blocks, we show that tensor network calculations agree with universal conformal field theory (CFT) predictions. In the case of two disjoint blocks the CFT predictions are recovered only after taking into account the finite size corrections induced by the finite length of the blocks. © 2013 IOP Publishing Ltd and SISSA Medialab srl.

We report on a systematic approach for the calculation of the negativity in the ground state of a one-dimensional quantum field theory. The partial transpose ρAT2 of the reduced density matrix of a subsystem A = A1 ∪ A2 is explicitly constructed as an imaginary-time path integral and from this the replicated traces Tr(ρAT2)n are obtained. The logarithmic negativity ε = log ∥ρAT2x∥ is then the continuation to n → 1 of the traces of the even powers. For pure states, this procedure reproduces the known results.We then apply this method to conformally invariant field theories (CFTs) in several different physical situations for infinite and finite systems and without or with boundaries. In particular, in the case of two adjacent intervals of lengths ℓ1; ℓ2 in an infinite system, we derive the result ε ∼ (c/4)ln(ℓ1ℓ2=(ℓ1+ℓ2) ), where c is the central charge. For the more complicated case of two disjoint intervals, we show that the negativity depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We explicitly calculate the scale invariant functions for the replicated traces in the case of the CFT for the free compactified boson, but we have not so far been able to obtain the n→1 continuation for the negativity even in the limit of large compactification radius. We have checked all our findings against exact numerical results for the harmonic chain which is described by a non-compactified free boson. © 2013 IOP Publishing Ltd and SISSA Medialab srl.

We examine general features of causal wedges in asymptotically AdS spacetimes and show that in a wide variety of cases they have non-trivial topology. We also prove some general results regarding minimal area surfaces on the causal wedge boundary and thereby derive constraints on the causal holographic information. We go on to demonstrate that certain properties of the causal wedge impact significantly on features of extremal surfaces which are relevant for computation of holographic entanglement entropy. © SISSA 2013.

We consider the entanglement entropy for holographic field theories in finite volume. We show that the Araki-Lieb inequality is saturated for large enough subregions, implying that the thermal entropy can be recovered from the knowledge of the region and its complement. We observe that this actually is forced upon us in holographic settings due to non-trivial features of the causal wedges associated with a given boundary region. In the process, we present an infinite set of extremal surfaces in Schwarzschild-AdS geometry anchored on a given entangling surface. We also offer some speculations regarding the homology constraint required for computing holographic entanglement entropy. © 2013 SISSA, Trieste, Italy.

We develop a systematic method to extract the negativity in the ground state of a 1+1 dimensional relativistic quantum field theory, using a path integral formalism to construct the partial transpose ρAT 2 of the reduced density matrix of a subsystem A=A 1A 2, and introducing a replica approach to obtain its trace norm which gives the logarithmic negativity E=lnρAT 2. This is shown to reproduce standard results for a pure state. We then apply this method to conformal field theories, deriving the result E∼(c/4)ln[ 12/( 1+ 2)] for the case of two adjacent intervals of lengths 1, 2 in an infinite system, where c is the central charge. For two disjoint intervals it depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We check our findings against exact numerical results in the harmonic chain. © 2012 American Physical Society.

We compute the time evolution of the mutual information in out of equilibrium quantum systems whose gravity duals are Vaidya spacetimes in three and four dimensions, which describe the formation of a black hole through the collapse of null dust. We find the holographic mutual information to be non monotonic in time and always monogamous in the ranges explored. We also find that there is a region in the configuration space where it vanishes at all times. We show that the null energy condition is a necessary condition for both the strong subadditivity of the holographic entanglement entropy and the monogamy of the holographic mutual information. © SISSA 2012.

We expand the results of arXiv:1105.5165, where a holographic description of a conformal field theory defined on a manifold with boundaries (so called BCFT) was proposed, based on AdS/CFT. We construct gravity duals of conformal field theories on strips, balls and also time-dependent boundaries. We show a holographic g-theorem in any dimension. As a special example, we can define a 'boundary central charge' in three dimensional conformal field theories and our holographic g-theorem argues that it decreases under RG ows. We also computed holographic one-point functions and confirmed that their scaling property agrees with field theory calculations. Finally, we give an example of string theory embedding of this holography by inserting orientifold 8-planes in AdS 4×CP3. © 2011 SISSA.

The existence of a linearized SUSY invariant for N = 8 supergravity whose gravitational components are usually called R4 was established long ago by on-shell superspace arguments. Superspace and string theory methods have also established analogous higher dimensional D2kR4 invariants. However, very little is known about the SUSY completions of these operators which involve other fields of the theory. In this paper we find the detailed component expansion of the linearized R4 invariant starting from the corresponding superamplitude which generates all component matrix elements of the operator. It is then quite straightforward to extend results to the entire set of D2kR4 operators. © SISSA 2011.

We study the finite term of the holographic entanglement entropy for thecharged black hole in AdSd2 and other examples of black holes when the spatial regionin the boundary theory is given by one or two parallel strips. For one large strip it scaleslike the width of the strip. The divergent term of its expansion as the turning point ofthe minimal surface approaches the horizon is determined by the near horizon geometry.Examples involving a Lifshitz scaling are also considered. For two equal strips in theboundary we study the transition of the mutual information given by the holographicprescription. In the case of the charged black hole, when the width of the strips becomeslarge this transition provides a characteristic finite distance depending on the temperature.© SISSA 2011.© SISSA 2011.

We continue the study of the entanglement entropy of two disjoint intervals in conformal field theories that we started in Calabrese et al 2009 J. Stat. Mech. P11001. We compute Tr ρnA for any integer n for the Ising universality class and the final result is expressed as a sum of Riemann-Siegel theta functions. These predictions are checked against existing numerical data. We provide a systematic method that gives the full asymptotic expansion of the scaling function for small four-point ratio (i.e. short intervals). These formulas are compared with the direct expansion of the full results for a free compactified boson and Ising model. We finally provide the analytic continuation of the first term in this expansion in a completely analytic form. © 2011 IOP Publishing Ltd and SISSA.

We study regular spacelike warped black holes in the three dimensional general massive gravity model, which contains both the gravitational Chern-Simons term and the linear combination of curvature squared terms characterizing the new massive gravity besides the Einstein-Hilbert term. The parameters of the metric are found by solving a quartic equation, constrained by an inequality that imposes the absence of closed timelike curves. Explicit expressions for the central charges are suggested by exploiting the fact that these black holes are discrete quotients of spacelike warped AdS 3 and a known formula for the entropy. Previous results obtained separately in topological massive gravity and in new massive gravity are recovered as special cases. © 2010 SISSA, Trieste, Italy.

We investigate the Brown-York stress tensor for curvature-squared theories. This requires a generalized Gibbons-Hawking term in order to establish a well-posed variational principle, which is achieved in a universal way by reducing the number of derivatives through the introduction of an auxiliary tensor field. We examine the boundary stress tensor thus defined for the special case of 'massive gravity' in three dimensions, which augments the Einstein-Hilbert term by a particular curvature-squared term. It is shown that one obtains finite results for physical parameters on AdS upon adding a 'boundary cosmological constant' as a counterterm, which vanishes at the so-called chiral point. We derive known and new results, like the value of the central charges or the mass of black hole solutions, thereby confirming our prescription for the computation of the stress tensor. Finally, we inspect recently constructed Lifshitz vacua and a new black hole solution that is asymptotically Lifshitz, and we propose a novel and covariant counterterm for this case. © SISSA 2010.

We study the entanglement of two disjoint intervals in the conformal field theory of the Luttinger liquid (free compactified boson). Tr ρAn for any integer n is calculated as the four-point function of twist fields of a particular type and the final result is expressed in a compact form in terms of the Riemann-Siegel theta functions. In the decompactification limit we provide the analytic continuation valid for all model parameters and from this we extract the entanglement entropy. These predictions are checked against existing numerical data. © 2009 IOP Publishing Ltd.

Based on recent developments, in this letter we find 2+1 dimensional gauge theories with scale invariance and = 8 supersymmetry. The gauge theories are defined by a Lagrangian and are based on an infinite set of 3-algebras, constructed as an extension of ordinary Lie algebras. Recent no-go theorems on the existence of 3-algebras are circumvented by relaxing the assumption that the invariant metric is positive definite. The gauge group is non compact, and its maximally compact subgroup can be chosen to be any ordinary Lie group, under which the matter fields are adjoints or singlets. Interestingly, the theories are parity invariant and do not admit any tunable coupling constant. © 2009 SISSA.

We find explicit solutions for giant magnons and spiky strings on the squashed three dimensional sphere. For a special value of the squashing parameter the solutions describe strings moving in a sector of the conifold, while for another value of the squashing parameter we recover the known results on the round three dimensional sphere. A new feature is that the energy and the momenta enter in the dispersion relation of the conifold in a transcendental way. © 2009 SISSA.

In an arbitrary unitary 4D CFT we consider a scalar operator φ, and the operator φ2 defined as the lowest dimension scalar which appears in the OPE φ × φ with a nonzero coefficient. Using general considerations of OPE, conformal block decomposition, and crossing symmetry, we derive a theory-independent inequality [φ2] f([φ]) for the dimensions of these two operators. The function f(d) entering this bound is computed numerically. For d1 we have f(d) = 2+O((d-1)1/2), which shows that the free theory limit is approached continuously. We perform some checks of our bound. We find that the bound is satisfied by all weakly coupled 4D conformal fixed points that we are able to construct. The Wilson-Fischer fixed points violate the bound by a constant O(1) factor, which must be due to the subtleties of extrapolating to 4- dimensions. We use our method to derive an analogous bound in 2D, and check that the Minimal Models satisfy the bound, with the Ising model nearly-saturating it. Derivation of an analogous bound in 3D is currently not feasible because the explicit conformal blocks are not known in odd dimensions. We also discuss the main phenomenological motivation for studying this set of questions: constructing models of dynamical ElectroWeak Symmetry Breaking without flavor problems. © 2008 SISSA.

We study the near-flat space limit for strings on AdS 5 × M 5, where the internal manifold M 5 is equipped with a generic metric with U(1) 3 isometry. In the bosonic sector, the limiting sigma model is similar to the one found for AdS 5 × S 5, as the global symmetries are reduced in the most general case. When M 5 is a Sasaki-Einstein space like T 1,1, Y p,q and L p,q,r, whose dual CFT's have = 1 supersymmetry, the near-flat space limit gives the same bosonic sector of the sigma model found for AdS 5 × S 5. This indicates the generic presence of integrable subsectors in AdS/CFT.

We develop a general technique for solving the Riemann-Hilbert problem in presence of a number of "heavy charges" and a small one thus providing the exact Green functions of Liouville theory for various non trivial backgrounds. The non invariant, regularization suggested by Zamolodchikov and Zamolodchikov gives the correct quantum dimensions; this is shown to one loop in the sphere topology and for boundary Liouville theory and to all loop on the pseudosphere. The method is also applied to give perturbative checks of the one point functions derived in the bootstrap approach by Fateev Zamolodchikov and Zamolodchikov in boundary Liouville theory and by Zamolodchikov and Zamolodchikov on the pseudosphere, obtaining complete agreement. © 2008 World Scientific Publishing Co. Pte. Ltd.