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In this paper we study expansions of infinite dimensional Hilbert spaces with a unitary representation of a group. When the group is finite, we prove the theory of the corresponding expansion is $\aleph_0$-categorical, $\aleph_0$-stable and is SFB. On the other hand, when the group involved is a product of the form $H\times \mathbb{Z}^n$, where $H$ is a finite group and $n\geq 1$, the theory of the Hilbert space expanded by the representation of this group is, in general, stable not $\aleph_0$-stable, not $\aleph_0$-categorical, but it is $\aleph_0$-categorical up to perturbations and $\aleph_0$-stable up to perturbations., Comment: 19 pages

We analyze the bootstrap approach (a dual optimization method to the variational approach) to one-dimensional spin chains, leveraging semidefinite programming to extract numerical results. We study how correlation functions in the ground state converge to their true values at and away from criticality and at relaxed optimality. We consider the transverse Ising model, the three state Potts model, and other non-integrable spin chains and investigate to what extent semidefinite methods can reliably extract numerical emergent physical data, including conformal central charges, correlation lengths and scaling dimensions. We demonstrate procedures to extract these data and show preliminary results in the various models considered. We compare to exact analytical results and to exact diagonalization when the system volume is small enough. When we attempt to go to the thermodynamic limit, the semidefinite numerical method with translation invariance imposed as a constraint finds the solution with periodic boundary conditions even if these have not been specified. This implies that the determination of all conformal data in correlators has to be handled at finite volume. Our investigation reveals that the approach has practical challenges. In particular, the correlation functions extracted from the optimal solution, which function as slack variables in the optimization, have convergence issues that suggest an underlying exponential complexity in the system size., Comment: 41 pages, plus many figures

The aim of this work is to build a theory of homomorphisms of Hecke and Artin monoids with possible applications to representation theory. Our original motivation was the striking observation that parabolic projections of Hecke monoids respect all parabolic elements. We found other classes of homomorphisms of Hecke monoids with the same property and discovered that many of them lift to homomorphisms of covering Artin monoids with a similar property. It turned out that they belong to a much larger class (in fact, a category) of homomorphisms of Artin monoids, most of which appear to be new., Comment: AMSLaTeX, 112 pages

We describe a novel way to think about bosonic lattice theories in Hamiltonian form where each lattice site has only a half boson degree of freedom. The construction requires a non-trivial Poisson bracket between neighboring sites and leads to gapless theories with non-invertible symmetries. We also describe a bosonic version of Kahler-Dirac fermions, dubbed Kahler-Dirac bosons that can be performed on any triangulation of a manifold. This also leads to a straightforward implementation of supersymmetry on the lattice and one immediately deduces the Dirac equation of the corresponding Kahler-Dirac fermions., Comment: 8 pages, Proceedings of the Corfu Summer Institute 2023, Workshop on Noncommutative and Generalized Geometry in String theory, Gauge theory and Related Physical Models

We generalize the electrical Lie algebras originally introduced by Lam and Pylyavskyy in several ways. To each Kac-Moody Lie algebra $\mathfrak{g}$ we associate two types (vertex type and edge type) of the generalized electrical algebras. The electrical Lie algebras of vertex type are always subalgebras of $\mathfrak{g}$ and are flat deformations of the nilpotent Lie subalgebra of $\mathfrak{g}$. In many cases including $sl_n$, $so_n$, and $sp_{2n}$ we find new (edge) models for our generalized electrical Lie algebras of vertex type. Finding an edge model in general is an interesting an open problem., Comment: AmsLaTeX, 20 pages, few more misprints corrected

The aim of this paper is to build a theory of commutative and noncommutative injective valuations of various algebras. The targets of our valuations are (well-)ordered commutative and noncommutative (partial or entire) semigroups including any sub-semigroups of the free monoid $F_n$ on $n$ generators and various quotients. In the case when the (partial) valuation semigroup is finitely generated, we construct a generalization of the standard monomial bases for the so-valued algebra, which seems to be new in noncommutative case. Quite remarkably, for any pair of well-ordered valuations one has canonical bijections between the valuation semigroups, which serve as analogs of the celebrated Jordan-H\"older correspondences and these bijections are "almost" homomorphisms of the involved (partial and entire) semigroups., Comment: Ams LaTeX 72 pages

We study systems of staggered boson Hamiltonians in a one dimensional lattice and in particular how the translation symmetry by one unit in these systems is in reality a non-invertible symmetry closely related to T-duality. We also study the simplest systems of clock models derived from these staggered boson Hamiltonians. We show that the non-invertible symmetries of these lattice models together with the discrete ${\mathbb Z}_N$ symmetry predict that these are critical points with a $U(1)$ current algebra at $c=1$ and radius $\sqrt{2N}$ whenever $N>4$., Comment: 39 pages. v2: typos fixed, references added

A Gallai coloring of the complete graph is an edge-coloring with no rainbow triangle. This concept first appeared in the study of comparability graphs and anti-Ramsey theory. We introduce a transitive analogue for acyclic directed graphs, and generalize both notions to Coxeter systems, matroids and commutative algebras. It is shown that for any finite matroid (or oriented matroid), the maximal number of colors is equal to the matroid rank. This generalizes a result of Erd\H{o}s-Simonovits-S\'os for complete graphs. The number of Gallai (or transitive) colorings of the matroid that use at most $k$ colors is a polynomial in $k$. Also, for any acyclic oriented matroid, represented over the real numbers, the number of transitive colorings using at most 2 colors is equal to the number of chambers in the dual hyperplane arrangement. We count Gallai and transitive colorings of the root system of type A using the maximal number of colors, and show that, when equipped with a natural descent set map, the resulting quasisymmetric function is symmetric and Schur-positive., Comment: 31 pages, 5 figures

We study a spin chain for a confining string that arises at first order in degenerate perturbation from the strong-coupling expansion of the Kogut-Susskind Hamiltonian on a square lattice in the leading large $N$ expansion. We show some subsectors are integrable and that with a relaxed constraint related to zigzag symmetry, the full spin chain is integrable in arbitrary dimensions., Comment: 5 pages, 5 figures. v2: updated references, minor edits

We describe the application of the quantum mechanical bootstrap to the solution of one-dimensional scattering problems. By fixing a boundary and modulating the Robin parameter of the boundary conditions we are able to extract the reflection coefficient for various potentials and compare to physical expectations. This includes an application of semidefinite programming to solving a half-line Schrodinger problem with arbitrary Robin boundary conditions. Finally, the WKB approximation is used to numerically determine the scattering behavior of the exponential potential of Liouville theory., Comment: 7 pages, 4 figures

We study the matrix quantum mechanics of two free hermitian $N\times N$ matrices subject to a singlet constraint in the microcanonical ensemble. This is the simplest example of a theory that at large $N$ has a confinement/deconfinement transition. In the microcanonical ensemble, it also exhibits partial confinement with a Hagedorn density of states. We argue that the entropy of these configurations, calculated by a counting of states based on the fact that Young diagrams are dominated by Young diagrams that have the VKLS shape. When the shape gets to the maximal depth allowed for a Young diagram of $SU(N)$, namely $N$, we argue that the system stops exhibiting the Hagedorn behavior. The number of boxes (energy) at the transition is $N^2/4$, independent of the charge of the state., Comment: 20 pages, 8 figures. v2: typos corrected, references added

We study null geodesics of the ten-dimensional LLM geometries. In particular, we show that there are a subset of these null geodesics that are confined to the LLM plane. The effective dynamics of these in-plane geodesics is that of a Hamiltonian system with two degrees of freedom (a phase space of dimension 4). We show that these are chaotic. In the two-coloring of the LLM plane, if they start in the empty region, they cannot penetrate the filled region and viceversa. The dynamical problem is therefore very similar to that of a billiards problem with fixed obstacles. We study to what extent LLM geometries with many droplets may be treated as an incipient black hole and draw analogies with the fuzzball proposal. We argue that for in-plane null geodesics deep in the interior of a region with a lot of droplets, in order to exit towards the $AdS$ boundary they will need to undergo a process that resembles diffusion. This mechanism can account for signals getting lost in the putative black hole for a very long time., Comment: 18 pages, 8 figures, uses JHEP. v2: Typos corrected, references added

A model with a half boson degree of freedom per lattice site in one dimension is developed. The boson is protected from developing a gap by translation symmetry: while the left movers are at zero quasi-momentum, the associated right movers are at the midpoint of the quasi-momentum period. The model has different properties depending on if a periodic lattice has an even or an odd number of sites and similar features are found for open boundary conditions. A special case of the non-linear half boson model where even and odd lattice sites contribute differently to the Hamiltonian gives rise to the Toda chain and a more symmetric generalization of the Toda chain is found. Upon periodic identifications of the half bosons degrees of freedom under a shift, the total Hilbert space has a finite dimension and can be encoded in finitely many qubits per unit length. This way one finds interesting critical spin chains, examples of which include the critical Ising model in a transverse magnetic field and the 3-state Potts model at criticality. Extensions to higher dimensions are considered. Models obtained this way automatically produce dynamical systems of gapless fractons., Comment: 27 pages, 4 figures

This expository paper treats the model theory of probability spaces using the framework of continuous $[0,1]$-valued first order logic. The metric structures discussed, which we call probability algebras, are obtained from probability spaces by identifying two measurable sets if they differ by a set of measure zero. The class of probability algebras is axiomatizable in continuous first order logic; we denote its theory by $Pr$. We show that the existentially closed structures in this class are exactly the ones in which the underlying probability space is atomless. This subclass is also axiomatizable; its theory $APA$ is the model companion of $Pr$. We show that $APA$ is separably categorical (hence complete), has quantifier elimination, is $\omega$-stable, and has built-in canonical bases, and we give a natural characterization of its independence relation. For general probability algebras, we prove that the set of atoms (enlarged by adding $0$) is a definable set, uniformly in models of $Pr$. We use this fact as a basis for giving a complete treatment of the model theory of arbitrary probability spaces. The core of this paper is an extensive presentation of the main model theoretic properties of $APA$. We discuss Maharam's structure theorem for probability algebras, and indicate the close connections between the ideas behind it and model theory. We show how probabilistic entropy provides a rank connected to model theoretic forking in probability algebras. In the final section we mention some open problems., Comment: 58 pages; to appear in the volume "Model theory of operator algebras" as part of DeGruyter's Logic and its Application Series

A complete theory $T$ has the Schr\"oder-Bernstein property or simply the SB-property if any pair of elementarily bi-embeddable models are isomorphic. This property has been studied in the discrete first-order setting and can be seen as a first step towards classification theory. This paper deals with the SB-property on continuous theories. Examples of complete continuous theories that have this property include Hilbert spaces and any completion of the theory of probability algebras. We also study a weaker notion, the SB-property up to perturbations. This property holds if any two elementarily bi-embeddable models are isomorphic up to perturbations. We prove that the theory of Hilbert spaces expanded with a bounded self-adjoint operator has the SB-property up to perturbations of the operator and that the theory of atomless probability algebras with a generic automorphism have the SB-property up to perturbations of the automorphism. We also study how the SB-property behaves with respect to randomizations. Finally we prove, in the continuous setting, that if $T$ is a strictly stable theory then $T$ does not have the SB-property.

We revisit the problem of analytically computing the one point functions for scalar fields in planar AdS black holes of arbitrary dimension, which are sourced by the Weyl squared tensor. We analyze the problem in terms of power series expansions around the boundary using the method of Frobenius. We clarify the pole structure of the final answer in terms of operator mixing, as argued previously by Grinberg and Maldacena. We generalize the techniques to also obtain analytic results for slowly modulated spatially varying sources to first non-trivial order in the wave vector for arbitrary dimension. We also study the first order corrections to the one point function of the global AdS black hole at large mass, where we perturb in terms that correspond to the curvature of the horizon., Comment: v2: references added, minor typos fixed

We present a semidefinite program (SDP) algorithm to find eigenvalues of Schr\"{o}dinger operators within the bootstrap approach to quantum mechanics. The bootstrap approach involves two ingredients: a nonlinear set of constraints on the variables (expectation values of operators in an energy eigenstate), plus positivity constraints (unitarity) that need to be satisfied. By fixing the energy we linearize all the constraints and show that the feasability problem can be presented as an optimization problem for the variables that are not fixed by the constraints and one additional slack variable that measures the failure of positivity. To illustrate the method we are able to obtain high-precision, sharp bounds on eigenenergies for arbitrary confining polynomial potentials in 1-D., Comment: 4 pages, 2 figures

We study carefully the problem of the bootstrap on the half line. We show why one needs the full set of constraints derived from the Stieltjes theorem on the moment problem by reexamining previous results on the hydrogen atom. We also study the hydrogen atom at continuous angular momentum. We show that the constraints on the moment problem alone do not fix the boundary conditions in all cases and at least one of the positive matrices needs to be slightly enlarged to remove unphysical branches. We explain how to solve the more general problem of the bootstrap for Robin boundary conditions. The recursion relations that are usually used receive additional anomalous contributions. These corrections are necessary to compute the moments of the measure. We apply these to the linear potential and we show how the bootstrap matches the analytical results, based on the Airy function, for this example., Comment: 31 pages, 11 figures. v2: We added an anomaly for n=0 that was omitted accidentally. Fixed a problem of conventions of factors of 2 between the text on the paper and the code we developed for the Airy function. v3: typos fixed

We show that braided Cherednik algebras introduced by the first two authors are cocycle twists of rational Cherednik algebras of the imprimitive complex reflection groups $G(m,p,n)$, when $m$ is even. This gives a new construction of mystic reflection groups which have Artin-Schelter regular rings of quantum polynomial invariants. As an application of this result, we show that a braided Cherednik algebra has a finite-dimensional representation if and only if its rational counterpart has one., Comment: 18 pages; v2: fixed a typo (epsilon -> 1/epsilon in reflection formula), results unchanged

We study the Feynman propagator of free scalar fields in $AdS_3$ with a conical defect. The propagator is built by solving the bulk equation of motion, summing over the modes of the field, and taking the boundary limit. We then perform several consistency checks. In the dual CFT, the operator responsible for the defect creates a highly excited state. We consider the exchange of the Virasoro identity block in the heavy-light limit to obtain an expression for the propagator sensitive to the mass of the defect. In $AdS_3/\mathbb{Z}_n$, we treat the propagator by the method of images and in the geodesic approximation. More generally, we argue that long-range correlations of the scalar are suppressed as the defect becomes more massive: we find a continuous phase transition in the correlator at the BTZ threshold and examine its critical behavior. Finally, we apply our results to holographic entanglement entropy using an analogy between our scalars and replica twist fields., Comment: 33 pages, 5 figures, JHEP style. v2: Section 4 was rewritten: an error was corrected that arose from using an approximation that did not apply. The crossing relations are now verified to all orders and do not present a puzzle any longer. The conclusions are updated based on the new calculations. Figures and references added. v3: Number of typos reduced. Supersedes the published version

We study expansions of a vector space $V$ over a field $\mathbb F$, possibly with extra structure, with a generic submodule over a subring of $\mathbb F$. We construct a natural expansion by existentially defined functions so that the expansion in the extended language satisfies quantifier elimination. We show that this expansion preserves tame model theoretic properties such as stability, NIP, NTP$_1$, NTP$_2$ and NSOP$_1$. We also study induced independence relations in the expansion., Comment: 41 pages

We introduce coherent states averaged over a gauge group action to study correlators of half BPS states in ${\cal N}=4 $ SYM theory. The overlaps of these averaged coherent states are a generating function of correlators and can be written in terms of the Harish-Chandra-Itzykzon-Zuber (HCIZ) integral. We show that this formula immediately leads to a computation of the normalization of two point functions in terms of characters obtained originally in the work of Corley, Jevicki and Ramgoolam. We also find various generalizations for $A_{n-1}$ quivers that follow directly from other solvable integrals over unitary groups. All of these can be computed using localization methods. When we promote the parameters of the generating function to collective coordinates, there is a dominant saddle that controls the effective action of these coherent states in the regime where they describe single AdS giant gravitons. We also discuss how to add open strings to this formulation. These will produce calculations that rely on correlators of matrix components of unitaries in the ensemble that is determined by the HCIZ integral to determine anomalous dimensions. We also discuss how sphere giants arise from Grassman integrals, how one gets a dominant saddle and how open strings are added in that case. The fact that there is a dominant saddle helps to understand how a $1/N$ expansion arises for open strings. We generalize the coherent state idea to study $1/4$ and $1/8$ BPS states as more general integrals over unitary groups., Comment: 36 pages. v2: added reference

This paper is motivated by the study of probability measure-preserving (pmp) actions of free groups using continuous model theory. Such an action is treated as a metric structure that consists of the measure algebra of the probability measure space expanded by a family of its automorphisms. We prove that the existentially closed pmp actions of a given free group form an elementary class, and therefore the theory of pmp $\mathbb{F}_k$-actions has a model companion. We show this model companion is stable and has quantifier elimination. We also prove that the action of $\mathbb{F}_k$ on its profinite completion with the Haar measure is metrically generic and therefore, as we show, it is existentially closed. We deduce our main result from a more general theorem, which gives a set of sufficient conditions for the existence of a model companion for the theory of $\mathbb{F}_k$-actions on a separably categorical, stable metric structure., Comment: 41 pages

We construct $(p,q)$ string junction solutions suspended between both sphere and AdS giant gravitons in $AdS_5\times S^5$. Our results extend easily to more general half BPS geometries of LLM type. These carry angular momentum in the directions of the worldvolume of the giant gravitons. We argue that these are charged under a central extension of the supersymmetry algebra similar to the one that has appeared in the works of Beisert for the ${\cal N}=4 $ spin chain. We also argue that they are BPS with respect to this central extension. We show that apart from some kinematical details, the junctions end up solving the same minimization problem that appears in the Coulomb branch of ${\cal N}=4 $ SYM. Their mass and shape is independent of the angular momentum $Q$ that the junction carries., Comment: 12 pages

A new quantum link microstructure was proposed for the lattice quantum chromodynamics (QCD) Hamiltonian, replacing the Wilson gauge links with a bilinear of fermionic qubits, later generalized to D-theory. This formalism provides a general framework for building lattice field theory algorithms for quantum computing. We focus mostly on the simplest case of a quantum rotor for a single compact $U(1)$ field. We also make some progress for non-Abelian setups, making it clear that the ideas developed in the $U(1)$ case extend to other groups. These in turn are building blocks for $1 + 0$-dimensional ($1 + 0$-D) matrix models, $1 + 1$-D sigma models and non-Abelian gauge theories in $2+1$ and $3+1$ dimensions. By introducing multiple flavors for the $U(1)$ field, where the flavor symmetry is gauged, we can efficiently approach the infinite-dimensional Hilbert space of the quantum $O(2)$ rotor with increasing flavors. The emphasis of the method is on preserving the symplectic algebra exchanging fermionic qubits by sigma matrices (or hard bosons) and developing a formal strategy capable of generalization to $SU(3)$ field for lattice QCD and other non-Abelian $1 + 1$-D sigma models or $3 +3$-D gauge theories. For $U(1)$, we discuss briefly the qubit algorithms for the study of the discrete $1+1$-D Sine-Gordon equation., Comment: 19 pages, 10 figures

Some issues of recoil effects in AdS/CFT are studied from the point of view of OPE expansions for generalized free fields. We show that the conformal group structure encodes the relative energies and momenta at a collision center. This is done by being careful with the analysis of Clebsch-Gordan coefficients for an $SL(2)$ subalgebra of the conformal group. The collision fraction of kinetic energy carried by the particles is derived from a probability distribution that arises from these coefficients. We specifically identify a precise statement of when recoil of a heavy particle in AdS can be ignored: the maximum probability is for the heavy particle to be in its ground state. We also argue how a notion of reduced mass appears in these collisions, in the limit where the particles are moving slowly with respect to each other. This controls the notion of the impact parameter of the collision., Comment: 19 pages. Version accepted for publication. Notice that the title change is to have the names of the accepted versions and the arXiv version match

We test the bootstrap approach for determining the spectrum of one dimensional Hamiltonians. In this paper we focus on problems that have a two parameter search space in the bootstrap approach: the double well and a periodic potential associated to the Mathieu equation. For the double well, we compare the energies with contributions from perturbative and non-perturbative results, finding good agreement. For the periodic potentials, we notice that the bootstrap approach gives the band structure of the periodic potential, but it has trouble finding the quasi-momentum of the system. To make further progress on the dispersion relation of the bands, new techniques are needed.

We test the bootstrap approach for determining the spectrum of one dimensional Hamiltonians, following the recent approach of Han, Hartnoll, and Kruthoff. We focus on comparing the bootstrap method data to known analytical predictions for the hydrogen atom and the harmonic oscillator. We resolve many energy levels for each, and more levels are resolved as the size of the matrices used to solve the problem increases. Using the bootstrap approach we find the spectrum of the Coulomb and harmonic potentials converge exponentially fast., Comment: 14 pages, 9 figures. v2: minor changes, references added

A generalized set of Clifford cellular automata, which includes all Clifford cellular automata, result from the quantization of a lattice system where on each site of the lattice one has a $2k$-dimensional torus phase space. The dynamics is a linear map in the torus variables and it is also local: the evolution depends only on variables in some region around the original lattice site. Moreover it preserves the symplectic structure. These are classified by $2k\times 2k$ matrices with entries in Laurent polynomials with integer coefficients in a set of additional formal variables. These can lead to fractal behavior in the evolution of the generators of the quantum algebra. Fractal behavior leads to non-trivial Lyapunov exponents of the original linear dynamical system. The proof uses Fourier analysis on the characteristic polynomial of these matrices., Comment: 4 pages, plus supplementary material. v2: references added

We introduce the symplectic group $\mathrm{Sp}_2(A,\sigma)$ over a noncommutative algebra $A$ with an anti-involution $\sigma$. We realize several classical Lie groups as $\mathrm{Sp}_2$ over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups $\mathrm{Sp}_2(A,\sigma)$ act. We introduce the space of isotropic $A$-lines, which generalizes the projective line. We describe the action of $\mathrm{Sp}_2(A,\sigma)$ on isotropic $A$-lines, generalize the Kashiwara-Maslov index of triples and the cross ratio of quadruples of isotropic $A$-lines as invariants of this action. When the algebra $A$ is Hermitian or the complexification of a Hermitian algebra, we introduce the symmetric space $X_{\mathrm{Sp}_2(A,\sigma)}$, and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as $\mathrm{Sp}_2(A,\sigma)$) and their complexifications, we obtain different models of the symmetric space as noncommutative generalizations of models of the hyperbolic plane and of the three-dimensional hyperbolic space. We also provide a partial classification of Hermitian algebras in Appendix A., Comment: 87 pages

We study real time classical matrix mechanics of a simplified $2\times 2$ matrix model inspired by the black hole evaporation problem. This is a step towards making a quantitative model of real time evaporation of a black hole, which is realized as a bound state of D0-branes in string theory. The model we study is the reduction of Yang Mills in $2+1$ dimension to $0+1$ dimensions, which has been corrected with an additional potential that can be interpreted as a zero point energy for fermions. Our goal is to understand the lifetime of such a classical bound state object in the classical regime. To do so, we pay particular attention to when D-particles separate to check that the "off diagonal modes" of the matrices become adiabatic and use that information to improve on existing models of evaporation. It turns out that the naive expectation value of the lifetime with the fermionic correction is infinite. This is a logarithmic divergence that arises from very large excursions in the separation between the branes near the threshold for classical evaporation. The adiabatic behavior lets us get some analytic control of the dynamics in this regime to get this estimate. This divergence is cutoff in the quantum theory due to quantization of the adiabatic parameter, resulting in a long lifetime of the bound state, with a parametric dependence of order $\log(1/\hbar)$., Comment: 17 pages, 4 figures. v2: added references

We consider the dynamics of local entropy and nearest neighbor mutual information of a 1-D lattice of qubits via the repeated application of nearest neighbor CNOT quantum gates. This is a quantum version of a cellular automaton. We analyze the entropy dynamics for different initial product states, both for open boundary conditions, periodic boundary conditions and we also consider the infinite lattice thermodynamic limit. The dynamics gives rise to fractal behavior, where we see the appearance of the Sierpinski triangle both for states in the computational basis and for operator dynamics in the Heisenberg picture. In the thermodynamics limit, we see equilibration with a time dependence controlled by $\exp(-\alpha t^{h-1})$ where $h$ is the fractal dimension of the Sierpinski triangle, and $\alpha$ depends on the details of the initial state. We also see log-periodic reductions in the one qubit entropy where the approach to equilibrium is only power law. For open boundary conditions we see time periodic oscillations near the boundary, associated to subalgebras of operators localized near the boundary that are mapped to themselves by the dynamics., Comment: 24 pages, 12 figures. v2: improved discussion of periodic oscillations, references added

We pose and resolve a holographic puzzle regarding an apparent violation of causality in AdS/CFT. If a point in the bulk of $AdS$ moves at the speed of light, the boundary subregion that encodes it may need to move superluminally to keep up. With $AdS_3$ as our main example, we prove that the finite extent of the encoding regions prevents a paradox. We show that the length of the minimal-size encoding interval gives rise to a tortoise coordinate on $\mathrm{AdS}$ that measures the nonlocality of the encoding. We use this coordinate to explore circular and radial motion in the bulk before passing to the analysis of bulk null geodesics. For these null geodesics, there is always a critical encoding where the possible violation of causality is barely avoided. We show that in any other encoding, the possible violation is subcritical., Comment: 28 pages, 14 figures

We compute sigma model solutions for rigidly rotating open strings suspended between giant gravitons in general LLM geometries. These solutions are confined to the LLM plane. These all have a dispersion relation for $\Delta-J$ that is consistent with saturation of a BPS bound of the centrally extended spin chain. For the special case of circularly symmetric LLM geometries, we can further evaluate the amount of angular momentum $J$ carried by these strings. This quantity diverges for string configurations that try to move between different "coloring regions" in the LLM plane. All of these quantities have a perturbative expansion in the t'Hooft coupling. For the strings suspended between AdS giants, we can compute in field theory the leading result of $J$ carried by the string via an analytic continuation of the $SU(2)$ result, with the help of the Bethe Ansatz for the $SL(2)$ sector. We thus provide additional information on how the radial direction of $AdS$ arises from (open) spin chain calculations., Comment: 30 pages

We study stable circular orbits in spherically symmetric AdS black holes in various dimensions and their limiting innermost stable circular orbits (ISCOs). We provide analytic expressions for their size, angular velocity and angular momentum in a large black hole mass regime. The dual interpretation is in terms of meta-stable states not thermalising in typical thermal scales and whose existence is due to non-perturbative effects on the spatial curvature. Our calculations reproduce the binding energy known in the literature, but also include a binding energy in the radial fluctuations corresponding to near circular trajectories. We also describe how particles are placed on these orbits from integrated operators on the boundary: they tunnel inside in a way that can be computed from both complex geodesics in the black hole background and from the WKB approximation of the wave equation. We explain how these two computations are related., Comment: 15 pages, 1 figure. v2: Added references, improved discussion on relations of this work to the Bootstrap

We study classical solutions to the Nambu-Goto string on $AdS_2 \times S^1$ and $AdS_3 \times S^1$ corresponding to strings stretched between wrapped branes extending in $AdS$. The solutions are obtained by analytic continuation of giant magnon solutions, cut at the position of the branes. These solutions carry one or two $SO(2,4)$ charges and a single $SO(6)$ charge. We compute their energies and show their relation to $\frac{1}{2}$-BPS geometries. Their relevance to the $SL(2)$ sector of $\mathcal{N}=4$ SYM is also discussed., Comment: 20 pages

The quantum link~\cite{Brower:1997ha} Hamiltonian was introduced two decades ago as an alternative to Wilson's Euclidean lattice QCD with gauge fields represented by bi-linear fermion/anti-fermion operators. When generalized this new microscopic representation of lattice field theories is referred as {\tt D-theory}~\cite{Brower:2003vy}. Recast as a Hamiltonian in Minkowski space for real time evolution, D-theory leads naturally to quantum Qubit algorithms. Here to explore digital quantum computing for gauge theories, the simplest example of U(1) compact QED on triangular lattice is defined and gauge invariant kernels for the Suzuki-Trotter expansions are expressed as Qubit circuits capable of being tested on the IBM-Q and other existing Noisy Intermediate Scale Quantum (NISQ) hardware. This is a modest step in exploring the quantum complexity of D-theory to guide future applications to high energy physics and condensed matter quantum field theories., Comment: 7 pages , 5 figures, 37th International Symposium on Lattice Field Theory - Lattice2019, 16-22 June 2019, Wuhan, China