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We construct infinitely many new $\frac{1}{16}$-BPS cohomologies of the 4d maximal super-Yang-Mills theory and interpret them as a black hole wrapped by dual giant graviton hairs. Since the black hole inside a dual giant feels the RR 5-form flux reduced by one unit, its microstate should essentially be an $SU(N-1)$ cohomology. However, due to the fortuitous nature of the black hole microstates, promoting an $SU(N-1)$ black hole state to $SU(N)$ generally fails to yield a cohomology. We show at $N=3$ that suitable fusion products with the dual giants yield cohomologies. The core black hole size is probed by the minimal size of the dual giant which can wrap it. We also discuss two types of large black hole hairs: large conformal descendants of gravitons and large dual giants. We prove that any $SU(N)$ black hole cohomology admits infinitely many hairs of the first type., Comment: 42 pages

Laguerre-Gaussian (LG) modes are solutions of the paraxial Helmholtz equation in cylindrical coordinates and are associated with light fields carrying orbital angular momentum (OAM). It is customary to modulate such beams using phase-only vortex profiles, for example, when increasing (laddering up) or decreasing (laddering down) the OAM content of some given LG mode. However, the resulting beams have been shown to be hypergeometric-Gaussian modes, due to the changing radial amplitudes on propagation. In this work, we show that these beams in fact have the angular spectrum of elegant Laguerre-Gaussian (eLG) modes, and therefore map back to LG-type modes. Accordingly, the fields obtain new OAM and radial quantum numbers that depend on the initial OAM and additional OAM gained during modulation., Comment: 4 pages, 2 figures

The topology of a pure state of two entangled photons is leveraged to provide a discretization of quantum information. Since discrete signals are inherently more resilient to the effects of perturbations, this discrete class of entanglement observables may offer an advantage against noise. Establishing this is the primary objective of this paper. We develop a noise model that exploits the specific form of such topological wave functions - an entangled state of two photons with one in an orbital angular momentum state and the other in a polarization state. We show that noise affecting both photons can be recast as a position-dependent perturbation affecting only the photon in the polarization state. This approach allows us to utilize both the language and concepts used in studying noisy qubits, as well as recent advances in quantum polarimetry. By adding noise to a finite-dimensional Hilbert space of polarization states, we can describe the noise using quantum operations expressed through appropriate Krauss operators, whose structure is determined by quantum polarimetry. For non-depolarizing noise, we provide an argument based on homotopic maps that demonstrates the topology's resilience to noise. For depolarizing noise, numerical studies using the quantum channel description show that the discrete entanglement signal remains completely resilient. Finally, we identify sources of local noise that can destabilize the topology. This foundational work establishes a framework for understanding how topology enhances the resilience of quantum information, directly impacting the distribution of information through entanglement in noisy environments, such as quantum computers and quantum networks., Comment: 24 pages, 15 figures

Recent advances in the study of microstates for 1/16-BPS black holes have inspired renewed interest in the analysis of heavy operators. For these operators, traditional techniques that work effectively in the planar limit are no longer applicable. Methods that are sensitive to finite N effects are required. In particular, trace relations that connect different multi-trace operators must be carefully considered. A powerful approach to tackling this challenge, which utilizes the representation theory of the symmetric group, is provided by restricted Schur polynomials. In this review, we develop these methods with the goal of providing the background needed for their application to 1/16-BPS black holes., Comment: 50 pages, 45 figures

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 $X^{\pm}$ 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., Comment: 25+1 pages

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$_{d+1}\times$S$^{k-1}\times$S$^{(d-2)(k-2)}\times$S$^{d-3}$ spacetime. The modes of a harmonic expansion on the S$^{k-1}\times$S$^{(d-2)(k-2)}\times$S$^{d-3}$ portion of the spacetime leads to the spinning bulk fields of the dual gravity theory.

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 AdS$_4\times$S$^{k-2}\times$S$^{k-1}$ spacetime. The modes of a harmonic expansion on the S$^{k-2}\times$S$^{k-1}$ portion of the spacetime leads to the spinning bulk fields of the dual gravity theory.

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., Comment: 1+31 pages

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 ${1\over N}$. 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., Comment: 1+16 pages

An open challenge in the context of quantum information processing and communication is improving the robustness of quantum information to environmental contributions of noise, a severe hindrance in real-world scenarios. Here, we show that quantum skyrmions and their nonlocal topological observables remain resilient to noise even as typical entanglement witnesses and measures of the state decay. This allows us to introduce the notion of digitization of quantum information based on our new discrete topological quantum observables, foregoing the need for robustness of entanglement. We compliment our experiments with a full theoretical treatment that unlocks the quantum mechanisms behind the topological behaviour, explaining why the topology leads to robustness. Our approach holds exciting promise for intrinsic quantum information resilience through topology, highly applicable to real-world systems such as global quantum networks and noisy quantum computers.

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 $\hbar$ 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., Comment: 24+1 pages

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., Comment: 1+45 pages; v2: minor typos corrected

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 ${1\over N}$ 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., Comment: 71+1 pages; Version 2 is JHEP version

The structure of half-BPS representations of psu$(2,2|4)$ leads to the definition of a super-polynomial ring $\mathcal{R}(8|8)$ 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 $\mathcal{R}(8|8)$. 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., Comment: 60 pages, no figures; v2: Mathematica notebook added to submission

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., Comment: 29+1 pages, 9 Figures

In the early 1960s, inspired by developing notions of topological structure, Tony Skyrme suggested that sub-atomic particles be described as natural excitations of a single quantum field. Although never adopted for its intended purpose, the notion of a skyrmion as a topologically stable field configuration has proven highly versatile, finding application in condensed matter physics, acoustics and more recently optics, but all realised as localised fields and particles. Here we report the first non-local quantum entangled state with a non-trivial topology that is skyrmionic in nature, even though each individual photon has no salient topological structure. We demonstrate how the topology of the quantum wavefunction makes such quantum states robust to entanglement decay, remaining intact until the entanglement itself vanishes. Our work points to a nascent connection between entanglement classes and topology, holding exciting promise for the creation and preservation of quantum information by topologically structured quantum states that persist even when entanglement is fragile.

Two and three-point functions of primary fields in four dimensional CFT have a simple space-time dependences factored out from the combinatoric structure which enumerates the fields and gives their couplings. This has led to the formulation of two dimensional topological field theories with $so(4,2)$ equivariance which are conjectured to be equivalent to higher dimensional conformal field theories. We review this CFT4/TFT2 construction in the simplest possible setting of a free scalar field, which gives an algebraic construction of the correlators in terms of an infinite dimensional $so(4,2)$ equivariant algebra with finite dimensional subspaces at fixed scaling dimension. Crossing symmetry of the CFT4 is related to associativity of the algebra. This construction is then extended to describe perturbative CFT4, by making use of deformed co-products. Motivated by the Wilson-Fisher CFT we outline the construction of U(so($d$,2)) equivariant TFT2 for non-integer $d$, in terms of diagram algebras and their representations., Comment: 19 pages, written version of invited talk given by SR at the XIV International Workshop LIE THEORY AND ITS APPLICATIONS IN PHYSICS, Sofia

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., Comment: 30+1 pages; v2: refs added

In this work we revisit the problem of solving multi-matrix systems through numerical large $N$ methods. The framework is a collective, loop space representation which provides a constrained optimization problem, addressed through master-field minimization. This scheme applies both to multi-matrix integrals ($c=0$ systems) and multi-matrix quantum mechanics ($c=1$). The complete fluctuation spectrum is also computable in the above scheme, and is of immediate physical relevance in the later case. The complexity (and the growth of degrees of freedom) at large $N$ have stymied earlier attempts and in the present work we present significant improvements in this regard. The (constrained) minimization and spectrum calculations are easily achieved with close to $10^4$ variables, giving solution to Migdal-Makeenko, and collective field equations. Considering the large number of dynamical (loop) variables and the extreme nonlinearity of the problem, high precision is obtained when confronted with solvable cases. Through numerical results presented, we prove that our scheme solves, by numerical loop space methods, the general two matrix model problem., Comment: 38 pages. v2: minor typos corrected and refs added. v3: Published version

A remarkable result at the intersection of number theory and group theory states that the order of a finite group $G$ (denoted $|G|$) is divisible by the dimension $d_R$ of any irreducible complex representation of $G$. We show that the integer ratios ${ |G|^2 / d_R^2 } $ are combinatorially constructible using finite algorithms which take as input the amplitudes of combinatoric topological strings ($G$-CTST) of finite groups based on 2D Dijkgraaf-Witten topological field theories ($G$-TQFT2). The ratios are also shown to be eigenvalues of handle creation operators in $G$-TQFT2/$G$-CTST. These strings have recently been discussed as toy models of wormholes and baby universes by Marolf and Maxfield, and Gardiner and Megas. Boundary amplitudes of the $G$-TQFT2/$G$-CTST provide algorithms for combinatoric constructions of normalized characters. Stringy S-duality for closed $G$-CTST gives a dual expansion generated by disconnected entangled surfaces. There are universal relations between $G$-TQFT2 amplitudes due to the finiteness of the number $K $ of conjugacy classes. These relations can be labelled by Young diagrams and are captured by null states in an inner product constructed by coupling the $G$-TQFT2 to a universal TQFT2 based on symmetric group algebras. We discuss the scenario of a 3D holographic dual for this coupled theory and the implications of the scenario for the factorization puzzle of 2D/3D holography raised by wormholes in 3D., Comment: 50 pages, 4 Figures; v2 - refs added, typos corrected

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., Comment: 31 pages, 6 figures

Promising resolutions of the generalization puzzle observe that the actual number of parameters in a deep network is much smaller than naive estimates suggest. The renormalization group is a compelling example of a problem which has very few parameters, despite the fact that naive estimates suggest otherwise. Our central hypothesis is that the mechanisms behind the renormalization group are also at work in deep learning, and that this leads to a resolution of the generalization puzzle. We show detailed quantitative evidence that proves the hypothesis for an RBM, by showing that the trained RBM is discarding high momentum modes. Specializing attention mainly to autoencoders, we give an algorithm to determine the network's parameters directly from the learning data set. The resulting autoencoder almost performs as well as one trained by deep learning, and it provides an excellent initial condition for training, reducing training times by a factor between 4 and 100 for the experiments we considered. Further, we are able to suggest a simple criterion to decide if a given problem can or can not be solved using a deep network.

Acting on operators with a bare dimension $\Delta\sim N^2$ the dilatation operator of $U(N)$ ${\cal N}=4$ 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 $p\sim N$ 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 $t\sim{p\over\lambda}$ with $\lambda$ the 't Hooft coupling., Comment: v2: Reference added

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 ${\cal N}=4$ super Yang-Mills theory, have a bare dimension $\sim N$ and are a linear combination of restricted Schur polynomials with $p\sim O(1)$ 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 problem 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., Comment: 34 pages

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 $ \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 $ {\bf 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 ${\bf U}_{\star ,2}$, 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., Comment: 67 pages, many figures; V2-minor typos corrected

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 $su(2|2)\ltimes\mathbb{R}$ 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., Comment: 35+1 pages

In this article we explore the holographic duals of tensor models using collective field theory. We develop a description of the gauge invariant variables of the tensor model. This is then used to develop a collective field theory description of the dynamics. We consider matrix like subsectors that develop an extra holographic dimension. In contrast to this, we argue that melonic large $N$ limits do not develop an extra dimension. The finite $N$ physics of the model is also developed and non-perturbative effects in the $1/N$ expansion are exhibited., Comment: 27+1 pages, 7 figures; small changes, refs added and typos corrected

Efficient and powerful approaches to the computation of correlation functions involving determinant, sub-determinant and permanent operators, as well as traces, have recently been developed in the setting of ${\cal N}=4$ super Yang-Mills theory. In this article we show that they can be extended to ABJM and ABJ theory. After making use of a novel identity which follows from character orthogonality, an integral representation of certain projection operators used to define Schur polynomials is given. This integral representation provides an effective description of the correlation functions of interest. The resulting effective descriptions have ${1\over N}$ as the loop counting parameter, strongly suggesting their relevance for holography., Comment: 1+22 pages, 3 Figures

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., Comment: 50 pages, 4 figures. v2: refs added

Although there has been a rapid development of practical applications, theoretical explanations of deep learning are in their infancy. Deep learning performs a sophisticated coarse graining. Since coarse graining is a key ingredient of the renormalization group (RG), RG may provide a useful theoretical framework directly relevant to deep learning. In this study we pursue this possibility. A statistical mechanics model for a magnet, the Ising model, is used to train an unsupervised restricted Boltzmann machine (RBM). The patterns generated by the trained RBM are compared to the configurations generated through an RG treatment of the Ising model. Although we are motivated by the connection between deep learning and RG flow, in this study we focus mainly on comparing a single layer of a deep network to a single step in the RG flow. We argue that correlation functions between hidden and visible neurons are capable of diagnosing RG-like coarse graining. Numerical experiments show the presence of RG-like patterns in correlators computed using the trained RBMs. The observables we consider are also able to exhibit important differences between RG and deep learning., Comment: This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2020.3000901, IEEE Access

We prove that in bosonic quantum mechanics the two-point spectral form factor can be obtained as an average of the two-point out-of-time ordered correlation function, with the average taken over the Heisenberg group. In quantum field theory, there is an analogous result with the average taken over the tensor product of many copies of the Heisenberg group, one copy for each field mode. The resulting formula is expressed as a path integral over two fields, providing a promising approach to the computation of the spectral form factor. We develop the formula that we have obtained using a coherent state description from the JC model and also in the context of the large-$N$ limit of CFT and Yang-Mills theory from the large-$N$ matrix quantum mechanics., Comment: 16 pages, minor changes, reference added

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., Comment: 20 pages

We consider the computation of out-of-time-ordered correlators (OTOCs) in the fishnet theories, with a mass term added. These fields theories are not unitary. We compute the growth exponent, in the planar limit, at any value of the coupling and show that the model exhibits chaos. At strong coupling the growth exponent violates the Maldacena-Shenker-Stanford bound. We also consider the mass deformed versions of the six dimensional honeycomb theories, which can also be solved in the planar limit. The honeycomb theory shows a very similar behavior to that exhibited by the fishnet theory., Comment: 1+14 pages, 6 Figures

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 ${}_2F_1(a,b;c;1)$, we are able to rephrase the $1/N$ expansion of the correlator as a semi-classical expansion for a Schr\"odinger 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., Comment: 33pages,no figures

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., Comment: 23 pages plus appendices, 3 figures; v2: various improvements, references and comments added, minor typos corrected

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 ${\cal N}=4$ super Yang-Mills. There are three observations supporting this conclusion: (i) we argue for an isomorphism between the planar Hilbert space of the original ${\cal N}=4$ 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 ${\cal N}=4$ super Yang-Mills. Despite the fact that the planar limit of the emergent gauge theory is planar ${\cal N}=4$ super Yang-Mills, we explain why the emergent gauge theory is not ${\cal N}=4$ super Yang-Mills theory., Comment: 30 pages plus Appendices

We define lowest weight polynomials (LWPs), motivated by $so(d,2)$ representation theory, as elements of the polynomial ring over $ d \times 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 \times (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., Comment: 50 pages + Appendices; V2 : minor typos corrected

We consider operators in ${\cal N}=4$ 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., Comment: v2 matches published version

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 for 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., Comment: v2: matches published version

Operators dual to strings attached to giant graviton branes in AdS$_5\times$S$^5$ can be described rather explicitly in the dual ${\cal N} = 4$ super Yang-Mills theory. They have a bare dimension of order $N$ so that for these operators the large $N$ limit and the planar limit are distinct: summing only the planar diagrams will not capture the large $N$ dynamics. Focusing on the one-loop $SU(3)$ sector of the theory, we consider operators that are a small deformation of a ${1\over 2}-$BPS multi-giant graviton state. The diagonalization of the dilatation operator at one loop has been carried out, but explicit formulas for the operators of a good scaling dimension are only known when certain terms which were argued to be small, are neglected. In this article we include the terms which were neglected. The diagonalization is achieved by a novel mapping which replaces the problem of diagonalizing the dilatation operator with a system of bosons hopping on a lattice. The giant gravitons define the sites of this lattice and the open strings stretching between distinct giant gravitons define the hopping terms of the Hamiltonian. Using the lattice boson model, we argue that the lowest energy giant graviton states are obtained by distributing the momenta carried by the $X$ and $Y$ fields evenly between the giants with the condition that any particular giant carries only $X$ or $Y$ momenta, but not both., Comment: 26+1 pages

We identify the operators in ${\cal N}=4$ super Yang-Mills theory that correspond to ${1\over 8}$-BPS giant gravitons in AdS$_5\times$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., Comment: 16 pages; v2: matches published version

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., Comment: 32+1 pages; v2: refs added, minor typos corrected. v3: matches JHEP version

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., Comment: 52+1 pages

We develop general counting formulae for primary fields in free four dimensional (4D) scalar conformal field theory (CFT). Using a duality map between primary operators in scalar field theory and multi-variable polynomial functions subject to differential constraints, we identify a sector of holomorphic primary fields corresponding to polynomial functions on a class of permutation orbifolds. These orbifolds have palindromic Hilbert series, which indicates they are Calabi-Yau. We construct the top-dimensional holomorphic form expected from the Calabi-Yau property. This sector includes and extends previous constructions of infinite families of primary fields. We sketch the generalization of these results to free 4D vector and matrix CFTs., Comment: 6 pages

By performing explicit computations of correlation functions, we find evidence that there is a sector of the two matrix model defined by the $SU(2)$ sector of ${\cal N}=4$ super Yang-Mills theory, that can be reduced to eigenvalue dynamics. There is an interesting generalization of the usual Van der Monde determinant that plays a role. The observables we study are the BPS operators of the $SU(2)$ sector and include traces of products of both matrices, which are genuine multi matrix observables. These operators are associated to supergravity solutions of string theory., Comment: 28+1 pages

We study the worldsheet S-matrix of a string attached to a D-brane in AdS$_5\times$S$^5$. The D-brane is either a giant graviton or a dual giant graviton. In the gauge theory, the operators we consider belong to the $su(2|3)$ sector of the theory. Magnon excitations of open strings can exhibit both elastic (when magnons in the bulk of the string scatter) and inelastic (when magnons at the endpoint of an open string participate) scattering. Both of these $S$-matrices are determined (up to an overall phase) by the $su(2|2)^2$ global symmetry of the theory. In this note we study the $S$-matrix for inelastic scattering. We show that it exhibits poles corresponding to boundstates of bulk and boundary magnons. A crossing equation is derived for the overall phase. It reproduces the crossing equation for maximal giant gravitons, in the appropriate limit. Finally, scattering in the $su(2)$ sector is computed to two loops. This two loop result, which determines the overall phase to two loops, will be useful when a unique solution to the crossing equation is to be selected., Comment: 10+1 pages; v2: small typos corrected

Large $N$ but non-planar limits of ${\cal N}=4$ super Yang-Mills theory can be described using restricted Schur polynomials. Previous investigations demonstrate that the action of the one loop dilatation operator on restricted Schur operators, with classical dimension of order $N$ and belonging to the $su(2)$ sector, is largely determined by the $su(2)$ ${\cal R}$ symmetry algebra as well as structural features of perturbative field theory. Studies presented so far have used the form of ${\cal R}$ symmetry generators when acting on small perturbations of half-BPS operators. In this article, as a first step towards going beyond small perturbations of the half-BPS operators, we explain how the exact action of symmetry generators on restricted Schur polynomials can be determined., Comment: 22+1 pages

We consider excitations of LLM geometries described by coloring the LLM plane with concentric black rings. Certain closed string excitations are localized at the edges of these rings. The string theory predictions for the energies of magnon excitations of these strings depends on the radii of the edges of the rings. In this article we construct the operators dual to these closed string excitations and show how to reproduce the string theory predictions for magnon energies by computing one loop anomalous dimensions. These operators are linear combinations of restricted Schur polynomials. The distinction between what is the background and what is the excitation is accomplished in the choice of the subgroup and the representations used to construct the operator., Comment: 42 pages, 4 figures

We consider the anomalous dimensions of restricted Schur polynomials constructed using n~O(N) complex adjoint scalars Z and m complex adjoint scalars Y. We fix m<

In a recent paper we showed that the correlators of free scalar field theory in four dimensions can be constructed from a two dimensional topological field theory based on so(4,2) equivariant maps (intertwiners). The free field result, along with results of Frenkel and Libine on equivariance properties of Feynman integrals, are developed further in this paper. We show that the coefficient of the log term in the 1-loop 4-point conformal integral is a projector in the tensor product of so(4,2) representations. We also show that the 1-loop 4-point integral can be written as a sum of four terms, each associated with the quantum equation of motion for one of the four external legs. The quantum equation of motion is shown to be related to equivariant maps involving indecomposable representations of so(4,2), a phenomenon which illuminates multiplet recombination. The harmonic expansion method for Feynman integrals is a powerful tool for arriving at these results. The generalization to other interactions and higher loops is discussed., Comment: 42 pages, 3 figures