$$ \frac{1}{2} $$ -BPS surface operator viewed as a conformal defect in rank $\textit{N}$ 6d (2,0) theory is expected to have a holographic description in terms of a probe M2 brane wrapped on AdS in the AdS × $\textit{S}$ M-theory background. The M2 brane has the effective tension T = $$ \frac{2}{\pi } $$ $\textit{N}$ so that the large tension expansion corresponds to the 1/$\textit{N}$ expansion. The value of the defect conformal anomaly coefficient in SU($\textit{N}$) (2,0) theory was previously argued to be b= 12$\textit{N}$ – 9 – 3$\textit{N}$ . Semiclassically quantizing M2 brane it was found in arXiv:2004.04562 that the first two terms in b are indeed reproduced by the classical and 1-loop corrections to the M2 free energy. Here we address the question if the 2-loop term in the M2 brane free energy reproduces the $\textit{N}$ term in b. Remarkably, despite the general non-renormalizability of the standard BST M2 brane action we find that the 2-loop correction to the free energy of the AdS M2 brane in AdS × $\textit{S}$ is UV finite (modulo power divergences that can be removed by an analytic regularization). Moreover, the 2-loop correction vanishes in the dimensional and $\textit{ζ}$-function regularizations. This result appears to be in disagreement with the non-vanishing of the coefficient of the $\textit{N}$ term in the expected expression for the anomaly coefficient b. We discuss possible resolutions of this puzzle, including the one that the M2 brane probe computation may be capturing the surface defect anomaly in the U($\textit{N}$) rather than the SU($\textit{N}$) boundary 6d CFT.

Italy

Matteo Beccaria

United Kingdom

Stefan Kurlyand

Russia

Arkady Tseytlin

2-loop free energy of M2 brane in AdS7 × $\textit{S}$ 4 and surface defect anomaly in (2,0) theory

Planar parts of conformal dimensions of primary operators in $\textit{U}$ ($\textit{N}$) × $\textit{U}$ ($\textit{N}$) ABJM theory are controlled by integrability. Strong coupling asymptotics of planar dimensions of operators with large spins can be found from the energy of semiclassical strings in AdS × CP but computing non-planar corrections requires understanding higher genus string corrections. As was pointed out in arXiv:2408.10070, there is an alternative way to find the non-planar corrections by quantizing M2 branes in AdS × $\textit{S}$ /ℤ which are wrapped around the 11d circle of radius 1/$\textit{k}$ = $\textit{λ/N}$ and generalize spinning strings in AdS × CP . Computing the 1-loop correction to the energy of M2 brane that corresponds to the long folded string with large spin $\textit{S}$ in AdS allowed to obtain a prediction for the large $\textit{λ}$ limit of non-planar corrections to the cusp anomalous dimension. Similar predictions were found for non-planar dimensions of operators dual to M2 branes that generalize the “short” and “long” circular strings with two equal spins $\textit{J}$ = $\textit{J}$ in CP . Here we consider two more non-trivial examples of 1-loop M2 brane computations that correspond to: (i) long folded string with large spin $\textit{S}$ in AdS and orbital momentum $\textit{J}$ in CP whose energy determines the generalized cusp anomalous dimension, and (ii) circular string with spin $\textit{S}$ in AdS and spin $\textit{J}$ in CP . We find the leading terms of the expansion of the corresponding 1-loop M2 brane energies in 1/$\textit{k}$. We also discuss similar semiclassical 1-loop M2 brane computation in flat 11d background and comment on possible relation to higher genus corrections to energies in 10d string theory.

On non-planar ABJM anomalous dimensions from M2 branes in AdS

We show that the Bekenstein-Hawking entropy of large supersymmetric black holes in AdS5 × S 5 emerges from remarkable cancellations in the giant graviton expansions recently proposed by Imamura, and Gaiotto and Lee, independently. A similar cancellation mechanism is shown to happen in the exact expansion in terms of free fermions recently put-forward by Murthy. These two representations can be understood as sums over independent systems of giant D3-branes and free fermions, respectively. At large charges, the free energy of each independent system localizes to its asymptotic expansion near the leading singularity. The sum over the independent systems maps their localized free energy to the localized free energy of the superconformal index of U(N)  $$\mathcal{N}$$  = 4 SYM. This result constitutes a non-perturbative test of the giant graviton expansion valid at any value of N. Moreover, in the holographic scaling limit N → ∞ at fixed ratio  $$\frac{{\text{Entropy}}}{{N}^{2}}$$ , it recovers the 1/16 BPS black hole entropy by a saddle-point approximation of the giant graviton expansion.

Alejandro Cabo-Bizet

Large black hole entropy from the giant brane expansion

Boundary correlators of elementary fields in some 2d conformal field theories defined on AdS2 have a particularly simple structure. For example, the correlators of the Liouville scalar happen to be the same as the correlators of the chiral component of the stress tensor on a plane restricted to the real line. Here we show that an analogous relation is true also in the WZW model: boundary correlators of the WZW scalars have the same structure as the correlators of chiral Kac-Moody currents. This is checked at the level of the tree and one-loop Witten diagrams in AdS2. We also compute some tree-level correlators in a generic -model defined on AdS2 and show that they simplify only in the WZW case where an extra Kac-Moody symmetry appears. In particular, the terms in 4- point correlators having logarithmic dependence on 1d cross-ratio cancel only at the WZW point. One motivation behind this work is to learn how to compute AdS2 loop corrections in 2d models with derivative interactions related to the study of correlators of operators on Wilson loops in string theory in AdS.

Switzerland

Hongliang Jiang

Boundary correlators in WZW model on AdS

Conformal theories with a global symmetry may be studied in the double scaling regime where the interaction strength is reduced while the global charge increases. Here, we study generic 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SU(N ) gauge theories with conformal matter content at large R-charge Q R → ∞  with fixed ’t Hooft-like coupling   <math> <mi>κ</mi> <mo>=</mo> <msub> <mi>Q</mi> <mi>R</mi> </msub> <msubsup> <mi>g</mi> <mi>YM</mi> <mn>2</mn> </msubsup> <mo>.</mo> </math>  $$ \kappa ={Q}_{\mathrm{R}}{g}_{\mathrm{YM}}^2. $$  Our analysis concerns two distinct classes of natural scaling functions. The first is built in terms of chiral/anti-chiral two-point functions. The second involves one-point functions of chiral operators in presence of   <math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math>  $$ \frac{1}{2} $$  -BPS Wilson-Maldacena loops. In the rank-1 SU (2) case, the two-point sector has been recently shown to be captured by an auxiliary chiral random matrix model. We extend the analysis to SU(N) theories and provide an algorithm that computes arbitrarily long perturbative expansions for all considered models, parametric in the rank. The leading and next-to-leading contributions are cross-checked by a three- loops computation in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 superspace. This perturbative analysis identifies maximally non-planar Feynman diagrams as the relevant ones in the double scaling limit. In the Wilson-Maldacena sector, we obtain closed expressions for the scaling functions, valid for any rank and κ. As an application, we analyze quantitatively the large ’t Hooft coupling limit κ ≫ 1 where we identify all perturbative and non-perturbative contributions. The latter are associated with heavy electric BPS states and the precise correspondence with their mass spectrum is clarified.

Francesco Galvagno

Azeem Hasan

<math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 conformal gauge theories at large R-charge: the SU(N) case

It was recently observed that boundary correlators of the elementary scalar field of the Liouville theory on AdS2 background are the same (up to a non-trivial proportionality coefficient) as the correlators of the chiral stress tensor of the Liouville CFT on the complex plane restricted to the real line. The same relation generalizes to the conformal abelian Toda theory: boundary correlators of Toda scalars on AdS2 are directly related to the correlation functions of the chiral   <math> <mi>W</mi> </math>  $$ \mathcal{W} $$ -symmetry generators in the Toda CFT and thus are essentially controlled by the underlying infinite-dimensional symmetry. These may be viewed as examples of AdS2/CFT1 duality where the CFT1 is the chiral half of a 2d CFT; we shall to this as   <math> <msub> <mi>AdS</mi> <mn>2</mn> </msub> <mo>/</mo> <msubsup> <mi>CFT</mi> <mn>2</mn> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msubsup> </math>  $$ {\mathrm{AdS}}_2/{\mathrm{CFT}}_2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} $$ . In this paper we demonstrate that this duality applies also to the non-abelian Toda theory containing a Liouville scalar coupled to a 2d σ-model originating from the SL(2, ℝ)/U(1) gauged WZW model. Here the Liouville scalar is again dual to the chiral stress tensor T while the other two scalars are dual to the parafermionic operators V ± of the non-abelian Toda CFT. We explicitly check the duality at the next-to-leading order in the large central charge expansion by matching the chiral CFT correlators of (T, V +, V −) (computed using a free field representation) with the boundary correlators of the three Toda scalars given by the tree-level and one-loop Witten diagrams in AdS2.

Non-abelian Toda theory on AdS2 and   <math> <msub> <mi>AdS</mi> <mn>2</mn> </msub> <mo>/</mo> <msubsup> <mi>CFT</mi> <mn>2</mn> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msubsup> </math>  $$ {\mathrm{AdS}}_2/{\mathrm{CFT}}_2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} $$  duality

We consider the Liouville theory in fixed Euclidean AdS2 background. Expanded near the minimum of the potential the elementary field has mass squared 2 and (assuming the standard Dirichlet b.c.) corresponds to a dimension 2 operator at the boundary. We provide strong evidence for the conjecture that the boundary correlators of the Liouville field are the same as the correlators of the holomorphic stress tensor (or the Virasoro generator with the same central charge) on a half-plane or a disc restricted to the boundary. This relation was first observed at the leading semiclassical order (tree-level Witten diagrams in AdS2) in [19] and here we demonstrate its validity also at the one-loop level. We also discuss arguments that may lead to its general proof.

On boundary correlators in Liouville theory on AdS2

In a series of recent papers, a special kind of AdS2/CFT1 duality was observed: the boundary correlators of elementary fields that appear in the Lagrangian of a 2d conformal theory in rigid AdS2 background are the same as the correlators of the corresponding primary operators in the chiral half of that 2d CFT in flat space restricted to the real line. The examples considered were: (i) the Liouville theory where the operator dual to the Liouville scalar in AdS2 is the stress tensor; (ii) the abelian Toda theory where the operators dual to the Toda scalars are the   <math> <mi>W</mi> </math>  $$ \mathcal{W} $$  -algebra generators; (iii) the non-abelian Toda theory where the Liouville field is dual to the stress tensor while the extra gauged WZW theory scalars are dual to non-abelian parafermionic operators. By direct Witten diagram com- putations in AdS2 one can check that the structure of the boundary correlators is indeed consistent with the Virasoro (or higher) symmetry. Here we consider a supersymmetric generalization: the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 superconformal Liouville theory in AdS2. We start with the super Liouville theory coupled to 2d supergravity and show that a consistent restriction to rigid AdS2 background requires a non-zero value of the supergravity auxiliary field and thus a modification of the Liouville potential from its familiar flat-space form. We show that the Liouville scalar and its fermionic partner are dual to the chiral half of the stress tensor and the supercurrent of the super Liouville theory on the plane. We perform tests supporting the duality by explicitly computing AdS2 Witten diagrams with bosonic and fermionic loops.

Supersymmetric Liouville theory in AdS2 and AdS/CFT

We consider the conformal A n  Toda theory in AdS2. Due to the bulk full Virasoro symmetry, this system provides an instance of a non-gravitational AdS2/CFT1 correspondence where the 1d boundary theory enjoys enhanced   <math> <mo>“</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>−</mo> <mtext>Virasoro</mtext> <mo>”</mo> </math>  $$ ``\frac{1}{2}- Virasoro" $$  symmetry. General boundary correlators are expected to be captured by the restriction of chiral correlators in a suitable WAn Virasoro extension. At next-to-leading order in weak coupling expansion they have been conjectured to match the subleading terms in the large central charge expansion of the dual 𝒲A n  correlators. We explicitly test this conjecture on the boundary four point functions of the Toda scalar fields dual to 𝒲A n  generators with next-to-minimal spin 3 and 4. Our analysis is valid in the generic rank case and extends previous results for specific rank-2 Toda theories. On the AdS side, the extension is straightforward and requires the computation of a finite set of tree Witten diagrams. This is due to simple rank dependence and selection rules of cubic and quartic couplings. On the boundary, we exploit crossing symmetry and specific meromorphic properties of the W-algebra correlators at large central charge. We present the required 4-point functions in closed form for any rank and verify the bulk-boundary correspondence in full details.

Giulio Landolfi

Toda theory in AdS2 and 𝒲An-algebra structure of boundary correlators

Correlators of local operators inserted on a straight or circular Wilson loop in a conformal gauge theory have the structure of a one-dimensional “defect” CFT. As was shown in arXiv:1706.00756, in the case of supersymmetric Wilson-Maldacena loop in   <math> <mi>N</mi> <mo>=</mo> <mn>4</mn> </math>  $$ \mathcal{N}=4 $$  SYM one can compute the leading strong-coupling contributions to 4-point correlators of the simplest “protected” operators by starting with the AdS5 × S 5 string action expanded near the AdS2 minimal surface and evaluating the corresponding tree-level AdS2 Witten diagrams. Here we perform the analogous computations in the non-supersymmetric case of the standard Wilson loop with no coupling to the scalars. The corresponding non-supersymmetric “defect” CFT1 should have an unbroken SO(6) global symmetry. The elementary bosonic operators (6 SYM scalars and 3 components of the SYM field strength) are dual respectively to the S 5 embedding coordinates and AdS5 coordinates transverse to the minimal surface ending on the line at the boundary. The SO(6) symmetry is preserved on the string side provided the 5-sphere coordinates satisfy Neumann boundary conditions (as opposed to Dirichlet in the supersymmetric case); this implies that one should integrate over the S 5 expansion point. The massless S 5 fluctuations then have logarithmic propagator, corresponding to the boundary scalar operator having dimension   <math> <mi>Δ</mi> <mo>=</mo> <mfrac> <mn>5</mn> <msqrt> <mi>λ</mi> </msqrt> </mfrac> <mo>+</mo> <mo>…</mo> </math>  $$ \Delta =\frac{5}{\sqrt{\lambda }}+\dots $$  at strong coupling. The resulting functions of 1d cross-ratio appearing in the 4-point functions turn out to have a more complicated structure than in the supersymmetric case, involving polylog (Li3 and Li2) functions. We also discuss consistency with the operator product expansion which allows extracting the leading strong coupling corrections to the anomalous dimensions of the operators appearing in the intermediate channels.