The quantization of semiclassical strings in AdS spacetimes yields predictions for the strong-coupling behaviour of the scaling dimensions of the corresponding operators in the planar limit of the dual gauge theory. Finding non-planar corrections requires computing string loops (corresponding to torus and higher genus surfaces), which is a challenging task. It turns out that in the case of the U  k (N) × U −k (N) ABJM theory there is an alternative approach: one may semiclassically quantize M2 branes in AdS4 × S 7 /ℤ k  which are wrapped around the 11d circle of radius 1/k = λ/N . Such M2 branes are the M-theory generalization of the strings in AdS × CP3. In this work, we show that by expanding in large M2 brane tension T 2 ~   <math> <msqrt> <mi>kN</mi> </msqrt> </math>  $$ \sqrt{kN} $$  for fixed k, followed by an expansion in large k, we can predict the large λ asymptotics of the non-planar corrections to the dimensions of the dual ABJM operators. As a specific example, we consider the M2 brane configuration that generalizes the long folded string with large spin in AdS4, and compute the 1-loop correction to its energy. This calculation allows us to determine non-planar corrections to the universal scaling function or cusp anomalous dimension. We extend our analysis to the semiclassical M2 branes that generalize the “short” and “long” circular strings with two equal angular momenta in CP3. The “short” M2 brane corresponds to a dual operator whose dimension at strong coupling scales as ∆ ∼ λ 1/4 + …, and we derive the leading non-planar correction to it.

United States

Simone Giombi

United Kingdom

Stefan Kurlyand

Arkady Tseytlin

Non-planar corrections in ABJM theory from quantum M2 branes

We compute the tree-level connected six-point function of identical scalar fluctuations of the AdS2 string worldsheet dual to the half-BPS Wilson line in planar   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 Super Yang-Mills. The calculation can be carried out analytically in the conformal gauge approach, where the boundary reparametrization mode of the string plays a crucial role. We also study the analytic continuation of the six-point function to an out-of-time-order configuration, which is related to a 3-to-3 scattering amplitude in flat space. As a check of our results, we also numerically compute the six-point function using the Nambu-Goto action in static gauge, finding agreement with the conformal gauge answer.

CERN

Shota Komatsu

Bendeguz Offertaler

Jieru Shan

Boundary reparametrizations and six-point functions on the AdS2 string

We study a surface defect in the free and critical O(N) vector models, defined by adding a quadratic perturbation localized on a two-dimensional subspace of the d-dimensional CFT. We compute the beta function for the corresponding defect renormalization group (RG) flow, and provide evidence that at long distances the system flows to a nontrivial defect conformal field theory (DCFT). We use epsilon and large N expansions to compute several physical quantities in the DCFT, finding agreement across different expansion methods. We also compute the defect free energy, and check consistency with the so-called b-theorem for RG flows on surface defects.

Bowei Liu

Notes on a surface defect in the O(N) model

We study the holographic correlators corresponding to scattering of fluctuations of an open string worldsheet with AdS2 geometry. In the out-of-time-order configuration, the correlators display a Lyapunov growth that saturates the chaos bound. We show that in a double-scaling limit interpolating between the Lyapunov regime and the late time exponential decay, the out-of-time-order correlator (OTOC) can be obtained exactly, and it has the same functional form found in the analogous calculation in JT gravity. The result can be understood as coming from high energy scattering near the horizon of a AdS2 black hole, and is essentially controlled by the flat space worldsheet S-matrix. While previous works on the AdS2 string employed mainly a static gauge approach, here we focus on conformal gauge and clarify the role of boundary reparametrizations in the calculation of the correlators. We find that the reparametrization mode is governed by a non-local action which is distinct from the Schwarzian action arising in JT gravity, and in particular leads to SL(2, ℝ) invariant boundary correlators. The OTOC in the double-scaling limit, however, has the same functional form as that obtained from the Schwarzian, and it can be computed using the reparametrization action and resumming a subset of diagrams that are expected to dominate in the limit. One application of our results is to the defect CFT defined by the half-BPS Wilson loop in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM. In this context, we show that the exact result for the OTOC in the double-scaling limit is in precise agreement with a recent analytic bootstrap prediction to three-loop order at strong coupling.

Chaos and the reparametrization mode on the AdS2 string

We study line defects in the fermionic CFTs in the Gross-Neveu-Yukawa universality class in dimensions 2 &lt; d &lt; 4. These CFTs may be described as the IR fixed points of the Gross-Neveu-Yukawa (GNY) model in d = 4 − ϵ, or as the UV fixed points of the Gross-Neveu (GN) model, which can be studied using the large N expansion in 2 &lt; d &lt; 4. These models admit natural line defects obtained by integrating over a line either the scalar field in the GNY description, or the fermion bilinear operator in the GN description. We compute the beta function for the defect RG flow using both the epsilon expansion and the large N approach, and find IR stable fixed points for the defect coupling, thus providing evidence for a non-trivial IR DCFT. We also compute some of the DCFT observables at the fixed point, and check that the g-function associated with the circular defect is consistent with the g-theorem for the defect RG flow.

Elizabeth Helfenberger

Himanshu Khanchandani

Line defects in fermionic CFTs

The Wilson loop operator in the <math><mi>U</mi><mo>(</mo><mi>N</mi><msub><mo>)</mo><mi>k</mi></msub><mo>×</mo><mi>U</mi><mo>(</mo><mi>N</mi><msub><mo>)</mo><mrow><mo>−</mo><mi>k</mi></mrow></msub></math> Aharony-Bergman-Jafferis-Maldacena theory at large <math><mi>N</mi></math> and fixed level <math><mi>k</mi></math> has a dual description in terms of a wrapped M2-brane in the <math><mi>M</mi></math>-theory given by the product of four-dimensional anti de Sitter space <math><mrow><mo>(</mo><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo></mrow></mrow></math> and <math><mrow><mrow><msup><mrow><mi>S</mi></mrow><mrow><mn>7</mn></mrow></msup><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mrow></math>. We consider the localization result for the <math><mfrac><mn>1</mn><mn>2</mn></mfrac></math>-Bogomol’nyi-Prasad-Sommerfield circular Wilson loop expectation value <math><mi>W</mi></math> in this regime and compare it to the prediction of the M2-brane theory. The leading large <math><mi>N</mi></math> exponential factor is matched as expected by the classical action of the M2-brane solution with <math><mrow><msub><mrow><mi>AdS</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><msup><mi>S</mi><mn>1</mn></msup></mrow></math> geometry. We show that the subleading <math><mi>k</mi></math>-dependent prefactor in <math><mi>W</mi></math> is also exactly reproduced by the one-loop term in the partition function of the wrapped M2-brane (with all Kaluza-Klein modes included). This appears to be the first case of an exact matching of the overall numerical prefactor in the Wilson loop expectation value against the dual holographic result. It provides an example of a consistent quantum M2-brane computation, suggesting various generalizations.

Arkady A. Tseytlin

Wilson Loops at Large <math><mi>N</mi></math> and the Quantum M2-Brane

We study operators with large charge j in the d-dimensional O(N) model with long range interactions that decrease with the distance as 1/r  d+s , where s is a continuous parameter. We consider the double scaling limit of large N, large j with   <math> <mi>j</mi> <mo>/</mo> <mi>N</mi> <mo>=</mo> <mover> <mi>j</mi> <mo>̂</mo> </mover> </math>  $$ j/N=\hat{j} $$  fixed, and identify the semiclassical saddle point that captures the two-point function of the large charge operators in this limit. The solution is given in terms of certain ladder conformal integrals that have recently appeared in the literature on fishnet models. We find that the scaling dimensions for general s interpolate between   <math> <msub> <mtext>∆</mtext> <mi>j</mi> </msub> <mo>∼</mo> <mfrac> <mfenced> <mrow> <mi>d</mi> <mo>−</mo> <mi>s</mi> </mrow> </mfenced> <mn>2</mn> </mfrac> <mi>j</mi> </math>  $$ {\Delta }_j\sim \frac{\left(d-s\right)}{2}j $$  at small   <math> <mover> <mi>j</mi> <mo>̂</mo> </mover> </math>  $$ \hat{j} $$  and   <math> <msub> <mtext>∆</mtext> <mi>j</mi> </msub> <mo>∼</mo> <mfrac> <mfenced> <mrow> <mi>d</mi> <mo>+</mo> <mi>s</mi> </mrow> </mfenced> <mn>2</mn> </mfrac> <mi>j</mi> </math>  $$ {\Delta }_j\sim \frac{\left(d+s\right)}{2}j $$  at large   <math> <mover> <mi>j</mi> <mo>̂</mo> </mover> </math>  $$ \hat{j} $$ , which is a qualitatively different behavior from the one found in the short range version of the O(N) model. We also derive results for the structure constants and 4-point functions with two large charge and one or two finite charge operators. Using a description of the long range models as defects in a higher dimensional local free field theory, we also obtain the scaling dimensions in a complementary way, by mapping the problem to a cylinder in the presence of a chemical potential for the conserved charge.

Long range, large charge, large N

We continue our study of large charge limits of the defect CFT defined by the half-BPS Wilson loop in planar   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 supersymmetric Yang-Mills theory. In this paper, we compute 1/J corrections to the correlation function of two heavy insertions of charge J and two light insertions, in the double scaling limit where the charge J and the ’t Hooft coupling λ are sent to infinity with the ratio J/   <math> <msqrt> <mi>λ</mi> </msqrt> </math>  $$ \sqrt{\lambda } $$  fixed. Holographically, they correspond to quantum fluctuations around a classical string solution with large angular momentum, and can be computed by evaluating Green’s functions on the worldsheet. We derive a representation of the Green’s functions in terms of a sum over residues in the complexified Fourier space, and show that it gives rise to the conformal block expansion in the heavy-light channel. This allows us to extract the scaling dimensions and structure constants for an infinite tower of non-protected dCFT operators. We also find a close connection between our results and the semi-classical integrability of the string sigma model. The series of poles of the Green’s functions in Fourier space corresponds to points on the spectral curve where the so-called quasi-momentum satisfies a quantization condition, and both the scaling dimensions and the structure constants in the heavy-light channel take simple forms when written in terms of the spectral curve. These observations suggest extensions of the results by Gromov, Schafer-Nameki and Vieira on the semiclassical energy of closed strings, and in particular hint at the possibility of determining the structure constants directly from the spectral curve.

Large charges on the Wilson loop in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM. Part II. Quantum fluctuations, OPE, and spectral curve

We study the boundary critical behavior of conformal field theories of interacting fermions in the Gross-Neveu universality class. By a Weyl transformation, the problem can be studied by placing the CFT in an anti de Sitter space background. After reviewing some aspects of free fermion theories in AdS, we use both large N methods and the epsilon expansion near 2 and 4 dimensions to study the conformal boundary conditions in the Gross-Neveu CFT. At large N and general dimension d, we find three distinct boundary conformal phases. Near four dimensions, where the CFT is described by the Wilson-Fisher fixed point of the Gross-Neveu-Yukawa model, two of these phases correspond respectively to the choice of Neumann or Dirichlet boundary condition on the scalar field, while the third one corresponds to the case where the bulk scalar field acquires a classical expectation value. One may flow between these boundary critical points by suitable relevant boundary deformations. We compute the AdS free energy on each of them, and verify that its value is consistent with the boundary version of the F-theorem. We also compute some of the BCFT observables in these theories, including bulk two-point functions of scalar and fermions, and four-point functions of boundary fermions.

Fermions in AdS and Gross-Neveu BCFT

We study the large charge sector of the defect CFT defined by the half-BPS Wilson loop in planar N = 4 supersymmetric Yang-Mills theory. Specifically, we consider correlation functions of two large charge insertions and several light insertions in the double-scaling limit where the ’t Hooft coupling λ and the large charge J are sent to infinity, with the ratio J/   <math> <msqrt> <mi>λ</mi> </msqrt> </math>  $$ \sqrt{\lambda } $$  held fixed. They are holographically dual to the expectation values of light vertex operators on a classical string solution with large angular momentum, which we evaluate in the leading large J limit. We also compute the two-point function of large charge insertions by evaluating the on-shell string action, supplemented by the boundary terms that generalize the one introduced by Drukker, Gross and Ooguri for the Wilson loop without insertions. For a special class of correlation functions, we reproduce the string results from field theory by using supersymmetric localization. The results are given by correlation functions in an “emergent” matrix model whose matrix size is proportional to J and whose spectral curve coincides with that of the classical string. Similar matrix models appeared in the study of extremal correlators in rank-1   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal field theories, but our results hold also for non-extremal cases.