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A bstract The worldsheet string theory dual to free 4d N = 4 super Yang-Mills theory was recently proposed in [ 1 ]. It is described by a free field sigma model on the twistor space of AdS 5 × S 5 , and is a direct generalisation of the corresponding model for tensionless string theory on AdS 3 × S 3 . As in the case of AdS 3 , the worldsheet theory contains spectrally flowed representations. We proposed in [ 1 ] that in each such sector only a finite set of generalised zero modes (‘wedge modes’) are physical. Here we show that after imposing the appropriate residual gauge conditions, this worldsheet description reproduces precisely the spectrum of the planar gauge theory. Specifically, the states in the sector with w units of spectral flow match with single trace operators built out of w super Yang-Mills fields (‘letters’). The resulting physical picture is a covariant version of the BMN light-cone string, now with a finite number of twistorial string bit constituents of an essentially topological worldsheet.

A bstract It was recently argued that string theory on AdS 3 × S 3 × 𝕋 4 with one unit ( k = 1) of NS-NS flux is exactly dual to the symmetric orbifold CFT Sym N (𝕋 4 ). In this paper we show how to directly relate the n -point correlators of the two sides to one another. In particular, we argue that the correlators of the world-sheet theory are delta-function- localised in string moduli space to those configurations that allow for a holomorphic covering map of the S 2 -boundary of AdS 3 by the world-sheet. This striking feature can be seen both from a careful Ward identity analysis, as well as from semi-classically exact AdS 3 solutions that are pinned to the boundary. The world-sheet correlators therefore have exactly the same structure as in the Lunin-Mathur construction of symmetric orbifold CFT correlators in terms of a covering surface — which now gets identified with the world-sheet. Together with the results of [ 1 , 2 ] this essentially demonstrates how the k = 1 AdS 3 string theory becomes equivalent to the spacetime orbifold CFT in the genus expansion.

A bstract We elaborate on some general aspects of the crossing symmetric approach of Polyakov to the conformal bootstrap, as recently formulated in Mellin space. This approach uses, as building blocks, Witten diagrams in AdS. We show the necessity for having contact Witten diagrams, in addition to the exchange ones, in two different contexts: a) the large c expansion of the holographic bootstrap b) in the ϵ expansion at subleading orders to the ones studied already. In doing so, we use alternate simplified representations of the Witten diagrams in Mellin space. This enables us to also obtain compact, explicit expressions (in terms of a 7 F 6 hypergeometric function!) for the analogue of the crossing kernel for Witten diagrams i.e., the decomposition into s -channel partial waves of crossed channel exchange diagrams.

A bstract The spectrum of superstrings on AdS 3 × S 3 × M 4 with pure NS-NS flux is analysed for the background where the radius of the AdS space takes the minimal value ( k = 1). Both for M 4 = S 3 × S 1 and M 4 = T 4 we show that there is a special set of physical states, coming from the bottom of the spectrally flowed continuous representations, which agree in precise detail with the single particle spectrum of a free symmetric product orbifold. For the case of AdS 3 × S 3 × T 4 this relies on making sense of the world-sheet theory at k = 1, for which we make a concrete proposal. We also comment on the implications of this striking result.

A bstract Superstring theory on AdS 3 × S 3 × T 4 with the smallest amount of NS-NS flux (“ k = 1”) is shown to be dual to the spacetime CFT given by the large N limit of the free symmetric product orbifold Sym N T 4 . To define the worldsheet theory at k = 1, we employ the hybrid formalism in which the AdS 3 × S 3 part is described by the p s u 1 , 1 | 2 1 WZW model (which is well defined). Unlike the case for k ≥ 2, it turns out that the string spectrum at k = 1 does not exhibit the long string continuum, and perfectly matches with the large N limit of the symmetric product. We also demonstrate that the fusion rules of the symmetric orbifold are reproduced from the worldsheet perspective. Our proposal therefore affords a tractable worldsheet description of a tensionless limit in string theory, for which the dual CFT is also explicitly known.

A bstract We describe in more detail our approach to the conformal bootstrap which uses the Mellin representation of CFT d four point functions and expands them in terms of crossing symmetric combinations of AdS d +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 ( ϵ 3 ) 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.

A bstract We employ the free field realisation of the psu 1 1 2 1 world-sheet theory to constrain the correlators of string theory on AdS 3 × S 3 × 𝕋 4 with unit NS-NS flux. In particular, we directly obtain the unusual delta function localisation of these correlators onto branched covers of the boundary S 2 by the (genus zero) world-sheet — this is the key property which makes the equivalence to the dual symmetric orbifold manifest. In our approach, this feature follows from a remarkable ‘incidence relation’ obeyed by the correlators, which is reminiscent of a twistorial string description. We also illustrate our results with explicit computations in various special cases.

A bstract The non-linear symmetry algebra underlies the duality between the minimal model CFTs and the hs[ μ ] higher spin theory on AdS 3 . It is shown how the structure of this symmetry algebra at the quantum level, i.e. for finite central charge, can be determined completely. The resulting algebra exhibits an exact equivalence (a ‘triality’) between three (generically) distinct values of the parameter μ . This explains, among other things, the agreement of symmetries between the minimal models and the bulk higher spin theory. We also study the consequences of this triality for some of the simplest representations, thereby clarifying the analytic continuation between the ‘light states’ of the minimal models and conical defect solutions in the bulk. These considerations also lead us to propose that one of the two scalar fields in the bulk actually has a non-perturbative origin.

A bstract Correlators in symmetric orbifold CFTs are given by a finite sum of admissible branched covers of the 2d spacetime. We consider a Gross-Mende like limit where all operators have large twist, and show that the corresponding branched covers can be described via a Penner-like matrix model. The limiting branched covers are given in terms of the spectral curve for this matrix model, which remarkably turns out to be directly related to the Strebel quadratic differential on the covering space. Interpreting the covering space as the world-sheet of the dual string theory, the spacetime CFT correlator thus has the form of an integral over the entire world-sheet moduli space weighted with a Nambu-Goto-like action. Quite strikingly, at leading order this action can also be written as the absolute value of the Schwarzian of the covering map. Given the equivalence of the symmetric product CFT to tensionless string theory on AdS 3 , this provides an explicit realisation of the underlying mechanism of gauge-string duality originally proposed in [1] and further refined in [2].

A bstract The symmetric orbifold of 𝕋 4 is exactly dual to string theory on AdS 3 × S 3 × 𝕋 4 with minimal ( k = 1) NS-NS flux. In this paper we study the perturbation of the symmetric orbifold that is dual to switching on R-R flux, and hence to deforming the theory away from the tensionless point. More specifically, we determine systematically the action of a centrally extended supersymmetry algebra on the CFT states, and deduce from it the anomalous conformal dimensions. In the w -twisted sector with large w the structure is similar to what was found for N = 4 SYM: the basic excitations are multi-magnons whose individual dispersion relation is fixed by symmetry, and the comparison with the BMN answer suggests that the result is true to all orders in perturbation theory. Finally we show that the multi-magnon states interact via an integrable S -matrix and possess a natural family of bound states.