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Abstract Collective field theory offers a constructive framework for exploring the AdS/CFT duality. In this article, we focus on constructing rotations within the light-front quantized collective field theory for the full set of spatial coordinates in the dual bulk AdS spacetime. Two intricate aspects require attention: how rotations involving the emergent holographic coordinate are implemented, and how rotations that involve the spatial coordinates participating in the construction of the light-cone coordinates X ± are realized. Our construction is in agreement with Metsaev’s construction directly in the gravity theory. Additionally, we derive the eigenfunctions of the AdS mass operator, which dictate the GKPW rule for the emergent higher-dimensional theory.

Abstract We consider the collective field theory description of the singlet sector of a free and massless matrix field in d dimensions. The k-local collective fields are functions of (d − 1)k + 1 coordinates. We provide a map between the collective fields and fields in the dual gravitational theory defined on AdS d+1 spacetime. The coordinates of the collective field have a natural interpretation: the k-local collective field is a field defined on an AdS d+1×S k−1×S(d−2)(k−2)×S d−3 spacetime. The modes of a harmonic expansion on the S k−1×S(d−2)(k−2)×S d−3 portion of the spacetime leads to the spinning bulk fields of the dual gravity theory.

Abstract In this paper we consider the collective field theory description of the singlet sector of a free matrix field in 2+1 dimensions. This necessarily involves the study of k-local collective fields, which are functions of 2k + 1 coordinates. We argue that these coordinates have a natural interpretation: the k-local collective field is a field defined on an AdS4×S k−2×S k−1 spacetime. The modes of a harmonic expansion on the S k−2×S k−1 portion of the spacetime leads to the spinning bulk fields of the dual gravity theory.

Abstract In this paper we consider the collective field theory description of a single free massless scalar matrix theory in 2+1 dimensions. The collective fields are given by k-local operators obtained by tracing a product of k-matrices. For k = 2 and k = 3 we argue that the collective field packages the fields associated to a single and two Regge trajectories respectively. We also determine the coordinate transformation between the coordinates of the collective field theory and the bulk AdS space time. This is used to verify that the bulk equations of motion holds in the collective field theory description.

Abstract Bilocal holography provides a constructive approach to the vector model/higher spin gravity duality. It has two ingredients: a change of field variables and a change of space time coordinates. The change of field variables ensures that the loop expansion parameter becomes 1 N $$ \frac{1}{N} $$ . The change of coordinates solves the Clebsch-Gordan problem of moving from the tensor product basis (in which the collective bilocal field is written) to the direct sum basis (appropriate for the description of the gravity fields). We argue that the change of space time coordinates can be deduced by requiring that operators constructed in the bilocal collective field theory are dual to local operators in the AdS bulk.

Abstract We consider the relevance of a collective field theory description for the AdS/CFT correspondence. Collective field theory performs a systematic reorganization of the degrees of freedom of a (non-gravitational) field theory, replacing the original loop expansion parameter ℏ with 1/N. Collective fields are over complete signalling a redundancy inherent in the theory. We propose that this over completeness is the mechanism by which one arrives at a holographic description, to be identified with the gravity dual. We find evidence for this by studying the redundancy of the collective field theory, showing that degrees of freedom in the bulk can be expressed as a linear combination of degrees of freedom contained in an arbitrarily small neighbourhood of the boundary.

Abstract We study the holographic duality between the free O(N) vector model and higher spin gravity. Conserved spinning primary currents of the conformal field theory (CFT) are dual to spinning gauge fields in the gravity. Reducing to independent components of the conserved CFT currents one finds two components at each spin. After gauge fixing the gravity and then reducing to independent components, one finds two components of the gauge field at each spin. Collective field theory provides a systematic way to map between these two sets of degrees of freedom, providing a complete and explicit identification between the dynamical degrees of freedom of the CFT and the dual gravity. The resulting map exhibits many features expected of holographic duality: it provides a valid bulk reconstruction, it reproduces insights expected from the holography of information and it provides a microscopic derivation of entanglement wedge reconstruction.

Abstract The structure of half-BPS representations of psu(2, 2|4) leads to the definition of a super-polynomial ring R $$ \mathcal{R} $$ (8|8) which admits a realisation of psu(2, 2|4) in terms of differential operators on the super-ring. The character of the half-BPS fundamental field representation encodes the resolution of the representation in terms of an exact sequence of modules of R $$ \mathcal{R} $$ (8|8). The half-BPS representation is realized by quotienting the super-ring by a quadratic ideal, equivalently by setting to zero certain quadratic polynomials in the generators of the super-ring. This description of the half-BPS fundamental field irreducible representation of psu(2, 2|4) in terms of a super-polynomial ring is an example of a more general construction of lowest-weight representations of Lie (super-) algebras using polynomial rings generated by a commuting subspace of the standard raising operators, corresponding to positive roots of the Lie (super-) algebra. We illustrate the construction using simple examples of representations of su(3) and su(4). These results lead to the definition of a notion of quantum mechanical emergence for oscillator realisations of symmetries, which is based on ideals in the ring of polynomials in the creation operators.

Abstract We construct generating functions for operators dual to systems of giant gravitons with open strings attached. These operators have a bare dimension of order N so that the usual methods used to solve the planar limit are not applicable. The generating functions are given as integrals over auxiliary variables, which implement symmetrization and antisymmetrization of the indices of the fields from which the operator is composed. Operators of a good scaling dimension (eigenstates of the dilatation operator) are known as Gauss graph operators. Our generating functions give a simple construction of the Gauss graph operators which were previously obtained using a Fourier transform on a double coset. The new description provides a natural starting point for a systematic 1 N $$ \frac{1}{N} $$ expansion for these operators as well as the action of the dilatation operator on them, in terms of a saddle point evaluation of their integral representation.

Abstract The principle of the holography of information states that in a theory of quantum gravity a copy of all the information available on a Cauchy slice is also available near the boundary of the Cauchy slice. This redundancy in the theory is already present at low energy. In the context of the AdS/CFT correspondence, this principle can be translated into a statement about the dual conformal field theory. We carry out this translation and demonstrate that the principle of the holography of information holds in bilocal holography.

Abstract We define circuits given by unitary representations of Lorentzian conformal field theory in 3 and 4 dimensions. Our circuits start from a spinning primary state, allowing us to generalize formulas for the circuit complexity obtained from circuits starting from scalar primary states. These results are nicely reproduced in terms of the geometry of coadjoint orbits of the conformal group. In contrast to the complexity geometry obtained from scalar primary states, the geometry is more complicated and the existence of conjugate points, signaling the saturation of complexity, remains open.

Abstract Bilocal holography is a constructive approach to the higher spin theory holographically dual to O(N ) vector models. In contrast to other approaches to bulk reconstruction, bilocal holography does not take input from the dual gravitational theory. The resulting map is a complete bulk/boundary mapping in that it maps the complete set of O(N ) invariant degrees of freedom in the CFT, to the complete set of higher spin degrees of freedom. After restricting to a suitable code subspace we demonstrate that bilocal holography naturally reproduces the quantum error correcting properties of holography and it gives a robust bulk (entanglement wedge) reconstruction. A gauge invariant entangled pair of CFT degrees of freedom are naturally smeared over a semicircle in the bulk spacetime, which is highly suggestive of bit threads. Finally, we argue that finite N relations in the CFT, when interpreted in the dual AdS spacetime, can provide relations between degrees of freedom located near the boundary and degrees of freedom deep in the bulk.

Abstract Acting on operators with a bare dimension ∆ ∼ N 2 the dilatation operator of U(N) N $$ \mathcal{N} $$ = 4 super Yang-Mills theory defines a 2-local Hamiltonian acting on a graph. Degrees of freedom are associated with the vertices of the graph while edges correspond to terms in the Hamiltonian. The graph has p ∼ N vertices. Using this Hamiltonian, we study scrambling and equilibration in the large N Yang-Mills theory. We characterize the typical graph and thus the typical Hamiltonian. For the typical graph, the dynamics leads to scrambling in a time consistent with the fast scrambling conjecture. Further, the system exhibits a notion of equilibration with a relaxation time, at weak coupling, given by t ∼ ρ λ $$ \frac{\rho }{\lambda } $$ with λ the ’t Hooft coupling.

Abstract We study the spectrum of anomalous dimensions of operators dual to giant graviton branes. The operators considered belong to the su(2|3) sector of N $$ \mathcal{N} $$ = 4 super Yang-Mills theory, have a bare dimension ∼ N and are a linear combination of restricted Schur polynomials with p ∼ O(1) long rows or columns. In the same way that the operator mixing problem in the planar limit can be mapped to an integrable spin chain, we find that our problems maps to particles hopping on a lattice. The detailed form of the model is in precise agreement with the expected world volume dynamics of p giant graviton branes, which is a U(p) Yang-Mills theory. The lattice model we find has a number of noteworthy features. It is a lattice model with all-to-all sites interactions and quenched disorder.

Abstract The state space of excited giant graviton brane systems is given by the Gauss graph operators. After restricting to the su(2|3) sector of the theory, we consider this state space. Our main result is the decomposition of this state space into irreducible representations of the su(2|2) ⋉ ℝ global symmetry. Excitations of the giant graviton branes are charged under a central extension of the global symmetry. The central extension generates gauge transformations so that the action of the central extension vanishes on physical states. Indeed, we explicitly demonstrate that the central charge is set to zero by the Gauss Law of the brane world volume gauge theory.

Abstract The correlators of free four dimensional conformal field theories (CFT4) have been shown to be given by amplitudes in two-dimensional so(4, 2) equivariant topological field theories (TFT2), by using a vertex operator formalism for the correlators. We show that this can be extended to perturbative interacting conformal field theories, using two representation theoretic constructions. A co-product deformation for the conformal algebra accommodates the equivariant construction of composite operators in the presence of non-additive anomalous dimensions. Explicit expressions for the co-product deformation are given within a sector of N $$ \mathcal{N} $$ = 4 SYM and for the Wilson-Fischer fixed point near four dimensions. The extension of conformal equivariance beyond integer dimensions (relevant for the Wilson-Fischer fixed point) leads to the definition of an associative diagram algebra U ★, abstracted from Uso(d) in the limit of large integer d, which admits extension of Uso(d) representation theory to general real (or complex) d. The algebra is related, via oscillator realisations, to so(d) equivariant maps and Brauer category diagrams. Tensor representations are constructed where the diagram algebra acts on tensor products of a fundamental diagram representation. A similar diagrammatic algebra U ★, 2, related to a general d extension for Uso(d, 2) is defined, and some of its lowest weight representations relevant to the Wilson-Fischer fixed point are described.

Abstract A new approach to the computation of correlation functions involving two determinant operators as well as one non-protected single trace operator has recently been developed by Jiang, Komatsu and Vescovi. This correlation function provides the holographic description of the absorption of a closed string by a giant graviton. The analysis has a natural interpretation in the framework of group representation theory, which admits a generalization to general Schur polynomials and restricted Schur polynomials. This generalizes the holographic description to any giant or dual giant gravitons which carry more than one angular momentum on the sphere. For a restricted Schur polynomial labeled by a column with N boxes (dual to a maximal giant graviton) we find evidence in favor of integrability. Since this restricted Schur polynomial is a 1 2 $$ \frac{1}{2} $$ -BPS operator, this integrability is a corollary of the results of Jiang, Komatsu and Vescovi.

Abstract We present in detail the basic ingredients contained in bi-local holography, representing a constructive scheme for reconstructing AdS bulk theories in Vectorial/AdS duality. Explicit Mapping to bulk AdS and higher spin fields is seen to be given by a double Fourier transform. All order interactions are explicitly specified through the collective action. This generates bulk Feynman (Witten) diagrams (at tree and loop level). We give details of the four-point case evaluation. It is noted that the bi-local construction goes beyond the assumptions in various discussions of non-locality.

Abstract We consider the gauged free fermionic matrix model, for a single fermionic matrix. In the large N limit this system describes a c = 1/2 chiral fermion in 1 + 1 dimensions. The Gauss’ law constraint implies that to obtain a physical state, indices of the fermionic matrices must be fully contracted, to form a singlet. There are two ways in which this can be achieved: one can consider a trace basis formed from products of traces of fermionic matrices or one can consider a Schur function basis, labeled by Young diagrams. The Schur polynomials for the fermions involve a twisted character, as a consequence of Fermi statistics. The main result of this paper is a proof that the trace and Schur bases coincide up to a simple normalization coefficient that we have computed.

Abstract The large N expansion of giant graviton correlators is considered. Giant gravitons are described using operators with a bare dimension of order N . In this case the usual 1/N expansion is not applicable and there are contributions to the correlator that are non-perturbative in character. By writing the (square of the) correlators in terms of the hypergeometric function 2 F 1(a, b; c; 1), we are able to rephrase the 1/N expansion of the correlator as a semi-classical expansion for a Schrödinger equation. In this way we are able to argue that the 1/N expansion of the correlator is Borel summable and that it exhibits a parametric Stokes phenomenon as the angular momentum of the giant graviton is varied.

Abstract We define lowest weight polynomials (LWPs), motivated by so(d, 2) representation theory, as elements of the polynomial ring over d × n variables obeying a system of first and second order partial differential equations. LWPs invariant under S n correspond to primary fields in free scalar field theory in d dimensions, constructed from n fields. The LWPs are in one-to-one correspondence with a quotient of the polynomial ring in d × (n − 1) variables by an ideal generated by n quadratic polynomials. The implications of this description for the counting and construction of primary fields are described: an interesting binomial identity underlies one of the construction algorithms. The product on the ring of LWPs can be described as a commutative star product. The quadratic algebra of lowest weight polynomials has a dual quadratic algebra which is non-commutative. We discuss the possible physical implications of this dual algebra.

Abstract We study excitations of LLM geometries. These geometries arise from the backreaction of a condensate of giant gravitons. Excitations of the condensed branes are open strings, which give rise to an emergent Yang-Mills theory at low energy. We study the dynamics of the planar limit of these emergent gauge theories, accumulating evidence that they are planar N=4 $$ \mathcal{N}=4 $$ super Yang-Mills. There are three observations supporting this conclusion: (i) we argue for an isomorphism between the planar Hilbert space of the original N=4 $$ \mathcal{N}=4 $$ super Yang-Mills and the planar Hilbert space of the emergent gauge theory, (ii) we argue that the OPE coefficients of the planar limit of the emergent gauge theory vanish and (iii) we argue that the planar spectrum of anomalous dimensions of the emergent gauge theory is that of planar N=4 $$ \mathcal{N}=4 $$ super Yang-Mills. Despite the fact that the planar limit of the emergent gauge theory is planar N=4 $$ \mathcal{N}=4 $$ super Yang-Mills, we explain why the emergent gauge theory is not N=4 $$ \mathcal{N}=4 $$ super Yang-Mills theory.

Abstract We consider operators in N=4 $$ \mathcal{N}=4 $$ super Yang-Mills theory dual to closed string states propagating on a class of LLM geometries. The LLM geometries we consider are specified by a boundary condition that is a set of black rings on the LLM plane. When projected to the LLM plane, the closed strings are polygons with all corners lying on the outer edge of a single ring. The large N limit of correlators of these operators receives contributions from non-planar diagrams even for the leading large N dynamics. Our interest in these fluctuations is because a previous weak coupling analysis argues that the net effect of summing the huge set of non-planar diagrams, is a simple rescaling of the ’t Hooft coupling. We carry out some nontrivial checks of this proposal. Using the su(2|2)2 symmetry we determine the two magnon S-matrix and demonstrate that it agrees, up to two loops, with a weak coupling computation performed in the CFT. We also compute the first finite size corrections to both the magnon and the dyonic magnon by constructing solutions to the Nambu-Goto action that carry finite angular momentum. These finite size computations constitute a strong coupling confirmation of the proposal.

Abstract We carry out a systematic study of primary operators in the conformal field theory of a free Weyl fermion. Using SO(4, 2) characters we develop counting formulas for primaries constructed using a fixed number of fermion fields. By specializing to particular classes of primaries, we derive very explicit formulas giving the generating functions for the number of primaries in these classes. We present a duality map between primary operators in the fermion field theory and polynomial functions. This allows us to construct the primaries that were counted. Next we show that these classes of primary fields correspond to polynomial functions on certain permutation orbifolds. These orbifolds have palindromic Hilbert series.

Abstract We identify the operators in N = 4 super Yang-Mills theory that correspond to 18 $$ \frac{1}{8} $$ -BPS giant gravitons in AdS5 × S 5. Our evidence for the identification comes from (1) counting these operators and showing agreement with independent counts of the number of giant graviton states, and (2) by demonstrating a correspondence between correlation functions of the super Yang-Mills operators and overlaps of the giant graviton wave functions.

Abstract Motivated by the close connection of tensor models to the SYK model, we use representation theory to construct the complete set of gauge invariant observables for bosonic and fermionic tensor models. Correlation functions of the gauge invariant operators in the free theory are computed exactly. The gauge invariant operators close a ring. The structure constants of the ring are described explicitly. Finally, we construct a collective field theory description of the bosonic tensor model.

Abstract Counting formulae for general primary fields in free four dimensional conformal field theories of scalars, vectors and matrices are derived. These are specialised to count primaries which obey extremality conditions defined in terms of the dimensions and left or right spins (i.e. in terms of relations between the charges under the Cartan subgroup of SO(4, 2)). The construction of primary fields for scalar field theory is mapped to a problem of determining multi-variable polynomials subject to a system of symmetry and differential constraints. For the extremal primaries, we give a construction in terms of holomorphic polynomial functions on permutation orbifolds, which are shown to be Calabi-Yau spaces.