Kallol Sen

Analytic bootstrap at large spin

A bstractWe use analytic conformal bootstrap methods to determine the anomalous dimensions and OPE coefficients for large spin operators in general conformal field theories in four dimensions containing a scalar operator of conformal dimension Δϕ. It is known that such theories will contain an infinite sequence of large spin operators with twists approaching 2Δϕ + 2n for each integer n. By considering the case where such operators are separated by a twist gap from other operators at large spin, we analytically determine the n, Δϕ dependence of the anomalous dimensions. We find that for all n, the anomalous dimensions are negative for Δϕ satisfying the unitarity bound. We further compute the first subleading correction at large spin and show that it becomes universal for large twist. In the limit when n is large, we find exact agreement with the AdS/CFT prediction corresponding to the Eikonal limit of a 2-2 scattering with dominant graviton exchange.

A Mellin space approach to the conformal bootstrap

A bstractWe describe in more detail our approach to the conformal bootstrap which uses the Mellin representation of CFTd four point functions and expands them in terms of crossing symmetric combinations of AdSd+1 Witten exchange functions. We consider arbitrary external scalar operators and set up the conditions for consistency with the operator product expansion. Namely, we demand cancellation of spurious powers (of the cross ratios, in position space) which translate into spurious poles in Mellin space. We discuss two contexts in which we can immediately apply this method by imposing the simplest set of constraint equations. The first is the epsilon expansion. We mostly focus on the Wilson-Fisher fixed point as studied in an epsilon expansion about d = 4. We reproduce Feynman diagram results for operator dimensions to O(ϵ3) rather straightforwardly. This approach also yields new analytic predictions for OPE coefficients to the same order which fit nicely with recent numerical estimates for the Ising model (at ϵ = 1). We will also mention some leading order results for scalar theories near three and six dimensions. The second context is a large spin expansion, in any dimension, where we are able to reproduce and go a bit beyond some of the results recently obtained using the (double) light cone expansion. We also have a preliminary discussion about numerical implementation of the above bootstrap scheme in the absence of a small parameter.

Universal anomalous dimensions at large spin and large twist

A bstractIn this paper we consider anomalous dimensions of double trace operators at large spin (ℓ) and large twist (τ) in CFTs in arbitrary dimensions (d ≥ 3). Using analytic conformal bootstrap methods, we show that the anomalous dimensions are universal in the limit ℓ ≫ τ ≫ 1. In the course of the derivation, we extract an approximate closed form expression for the conformal blocks arising in the four point function of identical scalars in any dimension. We compare our results with two different calculations in holography and find perfect agreement.

Conformal Bootstrap in Mellin Space

We propose a new approach towards analytically solving for the dynamical content of conformal field theories (CFTs) using the bootstrap philosophy. This combines the original bootstrap idea of Polyakov with the modern technology of the Mellin representation of CFT amplitudes. We employ exchange Witten diagrams with built-in crossing symmetry as our basic building blocks rather than the conventional conformal blocks in a particular channel. Demanding consistency with the operator product expansion (OPE) implies an infinite set of constraints on operator dimensions and OPE coefficients. We illustrate the power of this method in the ε expansion of the Wilson-Fisher fixed point by reproducing anomalous dimensions and, strikingly, obtaining OPE coefficients to higher orders in ε than currently available using other analytic techniques (including Feynman diagram calculations). Our results enable us to get a somewhat better agreement between certain observables in the 3D Ising model and the precise numerical values that have been recently obtained.

A strongly coupled anisotropic fluid from dilaton driven holography

A bstractWe consider a system consisting of 5 dimensional gravity with a negative cosmological constant coupled to a massless scalar, the dilaton. We construct a black brane solution which arises when the dilaton satisfies linearly varying boundary conditions in the asymptotically AdS5 region. The geometry of this black brane breaks rotational symmetry while preserving translational invariance and corresponds to an anisotropic phase of the system. Close to extremality, where the anisotropy is big compared to the temperature, some components of the viscosity tensor become parametrically small compared to the entropy density. We study the quasi normal modes in considerable detail and find no instability close to extremality. We also obtain the equations for fluid mechanics for an anisotropic driven system in general, working upto first order in the derivative expansion for the stress tensor, and identify additional transport coefficients which appear in the constitutive relation. For the fluid of interest we find that the parametrically small viscosity can result in a very small force of friction, when the fluid is enclosed between appropriately oriented parallel plates moving with a relative velocity.

Constraining gravity using entanglement in AdS/CFT

A bstractWe investigate constraints imposed by entanglement on gravity in the context of holography. First, by demanding that relative entropy is positive and using the Ryu-Takayanagi entropy functional, we find certain constraints at a nonlinear level for the dual gravity. Second, by considering Gauss-Bonnet gravity, we show that for a class of small perturbations around the vacuum state, the positivity of the two point function of the field theory stress tensor guarantees the positivity of the relative entropy. Further, if we impose that the entangling surface closes off smoothly in the bulk interior, we find restrictions on the coupling constant in Gauss-Bonnet gravity. We also give an example of an anisotropic excited state in an unstable phase with broken conformal invariance which leads to a negative relative entropy.

More on analytic bootstrap for O(N) models

A bstractThis note is an extension of a recent work on the analytical bootstrapping of O(N) models. An additonal feature of the O(N) model is that the OPE contains trace and antisymmetric operators apart from the symmetric-traceless objects appearing in the OPE of the singlet sector. This in addition to the stress tensor (Tμν ) and the ϕiϕi scalar, we also have other minimal twist operators as the spin-1 current Jμ and the symmetric-traceless scalar in the case of O(N). We determine the effect of these additional objects on the anomalous dimensions of the corresponding trace, symmetric-traceless and antisymmetric operators in the large spin sector of the O(N) model, in the limit when the spin is much larger than the twist. As an observation, we also verified that the leading order results for the large spin sector from the ϵ−expansion are an exact match with our n = 0 case. A plausible holographic setup for the special case when N = 2 is also mentioned which mimics the calculation in the CFT.

On critical exponents without Feynman diagrams

In order to achieve a better analytic handle on the modern conformal bootstrap program, we re-examine and extend the pioneering 1974 work of Polyakovs, which was based on consistency between the operator product expansion and unitarity. As in the bootstrap approach, this method does not depend on evaluating Feynman diagrams. We show how this approach can be used to compute the anomalous dimensions of certain operators in the O(n) model at the Wilson-Fisher fixed point in 4 - epsilon dimensions up to O(epsilon(2)).

Counterterms, critical gravity, and holography

We consider counterterms for odd dimensional holographic conformal field theories (CFTs). These counterterms are derived by demanding cutoff independence of the CFT partition function on S-d and S-1 x Sd-1. The same choice of counterterms leads to a cutoff independent Schwarzschild black hole entropy. When treated as independent actions, these counterterm actions resemble critical theories of gravity, i.e., higher curvature gravity theories where the additional massive spin-2 modes become massless. Equivalently, in the context of AdS/CFT, these are theories where at least one of the central charges associated with the trace anomaly vanishes. Connections between these theories and logarithmic CFTs are discussed. For a specific choice of parameters, the theories arising from counterterms are nondynamical and resemble a Dirac-Born-Infeld generalization of gravity. For even dimensional CFTs, analogous counterterms cancel log-independent cutoff dependence.

c-theorems in arbitrary dimensions

The dilaton action in 3 + 1 dimensions plays a crucial role in the proof of the a-theorem. This action arises using Wess-Zumino consistency conditions and crucially relies on the existence of the trace anomaly. Since there are no anomalies in odd dimensions, it is interesting to ask how such an action could arise otherwise. Motivated by this we use the AdS/CFT correspondence to examine both even and odd dimensional conformal field theories. We find that in even dimensions, by promoting the cutoff to a field, one can get an action for this field which coincides with the Wess-Zumino action in flat space. In three dimensions, we observe that by finding an exact Hamilton-Jacobi counterterm, one can find a non-polynomial action which is invariant under global Weyl rescalings. We comment on how this finding is tied up with the F-theorem conjectures.