The flat space limit of bulk scalar fields in anti–de Sitter (AdS) is discussed within a Lorentzian canonical quantization setup. We construct their Lorentzian wave functions and use time dependent perturbation theory to compute amplitudes. These can be computed once the wave functions of the incoming and outgoing states, relevant to set up an scattering problem, have been determined. To get the flat limit we need states tailored so that the flat S-matrix arises naturally. This limit is provided by the algebraic action of the Ìnönü-Wigner contraction on the wave functions we construct. We develop the embedding formalism to describe the bulk AdS scalar primary wave functions as holomorphic functions. Flat space massive particle states are built out of the AdS primary together with AdS boosted wave functions. We compute their inner products and show that these become orthogonal in the flat limit, resulting in the correct continuous spectrum for a standard unitary representation of the Lorentz group. In this same limit the original AdS descendants become null states. We also argue how the flat space S-matrix emerges from standard perturbation theory in the interaction picture. To obtain flat space massless particles requires to consider a double scaled limit in which the boost rapidity is scaled to infinity keeping the average particle energy in the flat space limit fixed. We comment on how this limit generates interesting massless state wave functions with nontrivial shape profiles that remember the dimension of the AdS operator. We discuss some of the puzzles attached to these.

United States

David Berenstein

United Kingdom

Joan Simón

Aspects of the bulk flat space limit in AdS/CFT

We show how to construct Hamiltonian lattice theories with one exact supersymmetry on arbitrary triangulations of curved space in any number of dimensions. Both bosons and fermions satisfy discrete Kähler-Dirac equations. The quantization of the fermions proceeds by imposing conventional anticommutation relations while the bosons require a modification of the usual canonical commutator. On regular lattices we construct parity, time reversal and translation-by-one (shift) symmetries. We argue that the latter are generically noninvertible symmetries. We also show how to couple these degrees of freedom to background gauge fields which leads to a theory with enhanced supersymmetry.

Simon Catterall

Supersymmetric lattice theories on curved space

We discuss chaos and ensemble averaging in 1/2 BPS bubbling AdS spaces of Lin, Lunin and Maldacena (LLM) by studying trapped and escaping null geodesics and estimating their decay rates. We find typical chaotic scattering behavior and confirm the Pesin relation between escape rates, Lyapunov exponents and Kolmogorov-Sinai entropy. On the other hand, for geodesics in coarse-grained (grayscale) LLM geometries (which exhibit a naked singularity) chaos is strongly suppressed, which is consistent with orbits and escape rates averaged over microscopic backgrounds. Also the singularities in these grayscale geometries produce an attractive potential and have some similarities to black hole throats trapping geodesics for a long time. Overall, averaging over the ensembles of LLM geometries brings us closer toward the typical behavior of geodesics in black hole backgrounds, but some important differences remain, in particular the existence of a threshold timescale when the averaging fails.

Netherlands

Serbia

Mihailo Čubrović

Vladan Ðukić

Trapping, chaos and averaging in bubbling AdS spaces

I derive a new charge constraint in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mrow></math>-Bogomol’nyi-Prasad-Sommerfield sectors of the Berenstein, Maldacena, and Nastase supersymmetric, gauged <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> matrix quantum mechanics. It was previously shown that the supersymmetric index of the model exhibits an exponential dependence on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>N</mi><mn>2</mn></msup></math> from the vacuum that preserves the full gauge group. I show that the reality of the associated entropy implies that the Cartan of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></math> vanishes for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mfrac><mn>1</mn><mn>16</mn></mfrac></math>-Bogomol’nyi-Prasad-Sommerfield sector with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> gauge symmetry. The Berenstein-Maldacena-Nastase charge constraint could be useful to determine whether there exists a dual supersymmetric black hole solution asymptoting to a plane-wave background.

Yoav Zigdon

Charge constraint in the Berenstein-Maldacena-Nastase model

We study simplified bootstrap problems for probability distributions on the infinite line and the circle. We show that the rapid convergence of the bootstrap method for problems on the infinite line is related to the fact that the smallest eigenvalue of the positive matrices in the exact solution becomes exponentially small for large matrices, while the moments grow factorially. As a result, the positivity condition is very finely tuned. For problems on the circle we show instead that the entries of the positive matrix of Fourier modes of the distribution depend linearly on the initial data of the recursion, with factorially growing coefficients. By positivity, these matrix elements are bounded in absolute value by one, so the initial data must also be fine-tuned. Additionally, we find that we can largely bypass the semi-definite program (SDP) nature of the problem on a circle by recognizing that these Fourier modes must be asymptotically exponentially small. With a simple ansatz, which we call the shoestring bootstrap, we can efficiently identify an interior point of the set of allowed matrices with much higher precision than conventional SDP bounds permit. We apply this method to solving unitary matrix model integrals by numerically constructing the orthogonal polynomials associated with the circle distribution.

Victor Rodriguez

Goldilocks and the bootstrap

We investigate the thermal phase structure of the Berenstein-Maldacena-Nastase matrix model using nonperturbative lattice Monte Carlo calculations. Our main analyses span 3 orders of magnitude in the coupling, involving systems with sizes up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>τ</mi></msub><mo>=</mo><mn>24</mn></math> lattice sites and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow></math> gauge groups with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>8</mn><mo>≤</mo><mi>N</mi><mo>≤</mo><mn>16</mn></math>. In addition, we carry out extended checks of discretization artifacts for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>τ</mi></msub><mo>≤</mo><mn>128</mn></math> and gauge group SU(4). We find results for the deconfinement temperature that interpolate between the perturbative prediction at weak coupling and the large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi></math> dual supergravity calculation at strong coupling. While we confirm that the phase transition is first order for strong coupling, it appears to be continuous for weaker couplings.

Raghav G. Jha

South Africa

Anosh Joseph

David Schaich

Finite-temperature phase diagram of the Berenstein-Maldacena-Nastase matrix model on the lattice

We investigate the large-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>N</mi></mrow></math> limit of the Berenstein-Maldacena-Nastase (BMN) matrix model with classical bosonic membranes which have spherical topologies and spin inside the 11-dimensional maximally supersymmetric plane-wave background. First we classify all possible M2-brane configurations based on the distribution of their components inside the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>×</mo><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></math> symmetric plane-wave spacetime. We then formulate a number of simple but very representative Ansätze of dielectric tops that rotate in this space. We examine the leading-order radial and angular/multipole stability for a wide range of these configurations. By analyzing perturbations at the next-to-leading order, we find that they exhibit the phenomenon of turbulent cascading of instabilities. Thereby, long-wavelength perturbations generate higher-order multipole instabilities through their nonlinear couplings.

Greece

Minos Axenides

Emmanuel Floratos

South Korea

Georgios Linardopoulos

Turbulent aspects of Berenstein-Maldacena-Nastase membrane dynamics

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 noninvertible 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 noninvertible symmetries of these lattice models together with the discrete <math><msub><mi>Z</mi><mi>N</mi></msub></math> symmetry predict that these are critical points with a <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> current algebra at <math><mi>c</mi><mo>=</mo><mn>1</mn></math> and radius <math><msqrt><mrow><mn>2</mn><mi>N</mi></mrow></msqrt></math> whenever <math><mi>N</mi><mo>&gt;</mo><mn>4</mn></math>. We also present an independent computation of this value that arises directly from the staggered boson variables and does not use these additional symmetries. We also present a theoretical estimate of the values of critical coupling constants away from the self-dual symmetry point in these clock models.

P. N. Thomas Lloyd

One dimensional staggered bosons, clock models, and their noninvertible symmetries

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.

Elliot Maderazo

Robinson Mancilla

Italy

Anayeli Ramírez

Chaotic LLM billiards

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 <math><mn>1</mn><mo>+</mo><mn>0</mn></math>-dimensional (<math><mn>1</mn><mo>+</mo><mn>0</mn></math>-D) matrix models, <math><mn>1</mn><mo>+</mo><mn>1</mn></math>-D sigma models and non-Abelian gauge theories in <math><mn>2</mn><mo>+</mo><mn>1</mn></math> and <math><mn>3</mn><mo>+</mo><mn>1</mn></math> 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 a <math><mi>SU</mi><mo>(</mo><mn>3</mn><mo>)</mo></math> field for lattice QCD and other non-Abelian <math><mn>1</mn><mo>+</mo><mn>1</mn></math>-D sigma models or <math><mn>3</mn><mo>+</mo><mn>1</mn></math>-D gauge theories. For U(1), we discuss briefly the qubit algorithms for the study of the discrete <math><mn>1</mn><mo>+</mo><mn>1</mn></math>-D sine-Gordon equation.