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A bstract We propose a holographic description of $$ \mathcal{N} $$ N = 4 super Yang-Mills on the four-dimensional real projective space $$ \mathbbm{RP} $$ RP 4 . We first construct the dual background in the framework of five-dimensional $$ \mathcal{N} $$ N = 8 gauged supergravity, and then uplift it to a new one-half BPS solution of type IIB supergravity. A salient feature of our solution is the presence of a bulk naked singularity whose local behavior resembles that of an O1 − plane in flat space. 

A bstract We revisit from a modern bootstrap perspective the longstanding problem of solving QCD in the large N limit. We derive universal bounds on the effective field theory of massless pions by imposing the full set of positivity constraints that follow from 2 → 2 scattering. Some features of our exclusion plots have intriguing connections with hadronic phenomenology. The exclusion boundary exhibits a sharp kink, raising the tantalizing scenario that large N QCD may sit at this kink. We critically examine this possibility, developing in the process a partial analytic understanding of the geometry of the bounds. 

A bstract There is a well-known map from 4d $$ \mathcal{N} $$ N = 2 superconformal field theories (SCFTs) to 2d vertex operator algebras (VOAs). The 4d Schur index corresponds to the VOA vacuum character, and must be a solution with integral coefficients of a modular differential equation. This suggests a classification program for 4d $$ \mathcal{N} $$ N = 2 SCFTs that starts with modular differential equations and proceeds by imposing all known constraints that follow from the 4d → 2d map. This program becomes fully algorithmic once one specifies the order of the modular differential equation and the rank (complex dimension of the Coulomb branch) of the $$ \mathcal{N} $$ N = 2 theory. As a proof of concept, we apply the algorithm to the study of rank-two $$ \mathcal{N} $$ N = 2 SCFTs whose Schur indices satisfy a fourth-order untwisted modular differential equation. Scanning over a large number of putative cases, only 15 satisfy all of the constraints imposed by our algorithm, six of which correspond to known 4d SCFTs. More sophisticated constraints can be used to argue against the existence of the remaining nine cases. Altogether, this indicates that our knowledge of such rank-two SCFTs is surprisingly complete. 

A bstract We revisit the leading irrelevant deformation of $$ \mathcal{N} $$ N = 4 Super Yang-Mills theory that preserves sixteen supercharges. We consider the deformed theory on S 3 × ℝ . We are able to write a closed form expression of the classical action thanks to a formalism that realizes eight supercharges off shell. We then investigate integrability of the spectral problem, by studying the spin-chain Hamiltonian in planar perturbation theory. While there are some structural indications that a suitably defined deformation might preserve integrability, we are unable to settle this question by our two-loop calculation; indeed up to this order we recover the integrable Hamiltonian of undeformed $$ \mathcal{N} $$ N = 4 SYM due to accidental symmetry enhancement. We also comment on the holographic interpretation of the theory. 

A bstract We formulate a series of conjectures relating the geometry of conformal manifolds to the spectrum of local operators in conformal field theories in d > 2 spacetime dimensions. We focus on conformal manifolds with limiting points at infinite distance with respect to the Zamolodchikov metric. Our central conjecture is that all theories at infinite distance possess an emergent higher-spin symmetry, generated by an infinite tower of currents whose anomalous dimensions vanish exponentially in the distance. Stated geometrically, the diameter of a non-compact conformal manifold must diverge logarithmically in the higher-spin gap. In the holographic context our conjectures are related to the Distance Conjecture in the swampland program. Interpreted gravitationally, they imply that approaching infinite distance in moduli space at fixed AdS radius, a tower of higher-spin fields becomes massless at an exponential rate that is bounded from below in Planck units. We discuss further implications for conformal manifolds of superconformal field theories in three and four dimensions. 

A bstract It is a long-standing conjecture that any CFT with a large central charge and a large gap ∆ gap in the spectrum of higher-spin single-trace operators must be dual to a local effective field theory in AdS. We prove a sharp form of this conjecture by deriving numerical bounds on bulk Wilson coefficients in terms of ∆ gap using the conformal bootstrap. Our bounds exhibit the scaling in ∆ gap expected from dimensional analysis in the bulk. Our main tools are dispersive sum rules that provide a dictionary between CFT dispersion relations and S-matrix dispersion relations in appropriate limits. This dictionary allows us to apply recently-developed flat-space methods to construct positive CFT functionals. We show how AdS 4 naturally resolves the infrared divergences present in 4D flat-space bounds. Our results imply the validity of twice-subtracted dispersion relations for any S-matrix arising from the flat-space limit of AdS/CFT. 

A bstract We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension. In a unitary CFT, the crossing equation (for, say, the s - and t -channel expansions) can be thought of as a vector equation in an infinite-dimensional space of complex analytic functions in two variables, which satisfy a boundedness condition at infinity. We identify a useful basis for this space of functions, consisting of the set of s- and t-channel conformal blocks of double-twist operators in mean field theory. We describe two independent algorithms to construct the dual basis of linear functionals, and work out explicitly many examples. Our basis of functionals appears to be closely related to the CFT dispersion relation recently derived by Carmi and Caron-Huot. 

A bstract We reconsider the problem of bounding higher derivative couplings in consistent weakly coupled gravitational theories, starting from general assumptions about analyticity and Regge growth of the S-matrix. Higher derivative couplings are expected to be of order one in the units of the UV cutoff. Our approach justifies this expectation and allows to prove precise bounds on the order one coefficients. Our main tool are dispersive sum rules for the S-matrix. We overcome the difficulties presented by the graviton pole by measuring couplings at small impact parameter, rather than in the forward limit. We illustrate the method in theories containing a massless scalar coupled to gravity, and in theories with maximal supersymmetry. 

A bstract We give a unified treatment of dispersive sum rules for four-point correlators in conformal field theory. We call a sum rule “dispersive” if it has double zeros at all double-twist operators above a fixed twist gap. Dispersive sum rules have their conceptual origin in Lorentzian kinematics and absorptive physics (the notion of double discontinuity). They have been discussed using three seemingly different methods: analytic functionals dual to double-twist operators, dispersion relations in position space, and dispersion relations in Mellin space. We show that these three approaches can be mapped into one another and lead to completely equivalent sum rules. A central idea of our discussion is a fully nonperturbative expansion of the correlator as a sum over Polyakov-Regge blocks . Unlike the usual OPE sum, the Polyakov-Regge expansion utilizes the data of two separate channels, while having (term by term) good Regge behavior in the third channel. We construct sum rules which are non-negative above the double-twist gap; they have the physical interpretation of a subtracted version of “superconvergence” sum rules. We expect dispersive sum rules to be a very useful tool to study expansions around mean-field theory, and to constrain the low-energy description of holographic CFTs with a large gap. We give examples of the first kind of applications, notably we exhibit a candidate extremal functional for the spin-two gap problem. 

We analyze the N = 2 superconformal field theories that arise when a pair of D3-branes probe an F-theory singularity from the perspective of the associated vertex operator algebra. We identify these vertex operator algebras for all cases; we find that they have a completely uniform description, parameterized by the dual Coxeter number of the corresponding global symmetry group. We further present free field realizations for these algebras in the style of recent work by three of the authors. These realizations transparently reflect the algebraic structure of the Higgs branches of these theories. We find fourth-order linear modular differential equations for the vacuum characters/Schur indices of these theories, which are again uniform across the full family of theories and parameterized by the dual Coxeter number.We comment briefly on expectations for the still higher-rank cases.