$$ \frac{1}{2} $$ -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

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 $\textit{U}$ ($\textit{N}$) × $\textit{U}$ ($\textit{N}$) ABJM theory there is an alternative approach: one may semiclassically quantize M2 branes in AdS × $\textit{S}$ $\textit{/}$ℤ which are wrapped around the 11d circle of radius 1$\textit{/k}$ = $\textit{λ/N}$ . Such M2 branes are the M-theory generalization of the strings in AdS × CP . In this work, we show that by expanding in large M2 brane tension $\textit{T}$ ~ $$ \sqrt{kN} $$ for fixed $\textit{k}$, followed by an expansion in large $\textit{k}$, we can predict the large $\textit{λ}$ 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 AdS , 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 CP . The “short” M2 brane corresponds to a dual operator whose dimension at strong coupling scales as ∆ ∼ $\textit{λ}$ + …, and we derive the leading non-planar correction to it.

United States

Simone Giombi

Non-planar corrections in ABJM theory from quantum M2 branes

We compute the one-loop 2 → 2 scattering amplitude of massless scalars on the world volume of an infinite $\textit{D}$ = 11 supermembrane quantized in the static gauge. The resulting expression is manifestly finite and turns out to be much simpler than in the bosonic membrane case in arXiv:2308.12189 being proportional to the tree-level scattering amplitude. We also consider the case of $\textit{ℝ}$ × $\textit{S}$ membrane with one dimension compactified on a circle of radius R and demonstrate how the supermembrane scattering amplitude reduces to the one on an infinite $\textit{D}$ = 10 Green-Schwarz superstring in the limit of R$\textit{→}$ 0.

Fiona Seibold

Scattering on the supermembrane

We consider a   <math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math>  $$ \frac{1}{2} $$ -BPS solution for a D3 brane probe in AdS5 × S 5 that has world-volume geometry of AdS3 × S 1. It intersects the boundary over a surface that represents a dimension 2 defect in the boundary   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM theory. The effective action of the probe brane is proportional to the logarithmically divergent volume of AdS3 and may thus be interpreted as computing conformal anomaly of supersymmetric S 2 defect. The classical action scales as N. We compute the 1-loop correction to it due to quantum fluctuations of the D3 brane world-volume fields and compare the result to an earlier suggested expression for the defect anomaly. We also perform a similar analysis of a   <math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math>  $$ \frac{1}{2} $$ -BPS M5 brane probe solution in AdS7 × S 4 with the world-volume geometry of AdS5 × S 1 that represents a dimension 4 defect in the boundary (2,0) 6d theory. Here the classical M5 brane action computes the leading order N 2 term in a-anomaly of the supersymmetric S 4 defect. We perform a detailed computation of the 1-loop correction to the M5 brane effective action and thus provide a prediction for the subleading constant in the S 4 defect a-anomaly coefficient.

Hongliang Jiang

On co-dimension 2 defect anomalies in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM and (2,0) theory via brane probes in AdS/CFT

We consider type IIB string theory on an AdS 5 × S 5/ℤ2 orbifold background, which should be dual to 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal SU(N) × SU(N) gauge theory with two bi-fundamental hypermultiplets. The correlator of two chiral BPS operators from the twisted sector of this quiver CFT exhibits non-trivial dependence on the ’t Hooft coupling λ already in the planar limit. This dependence was recently determined using localisation and the expansion at large λ contains a subleading contribution proportional to ζ(3)λ  −3/2. We address the question of how to reproduce this correction on the string theory side by starting with the ζ(3)α′3 term in the type IIB string effective action. We find a regular solution of type IIB supergravity which represents a resolution of the AdS 5 × S 5/ℤ2 orbifold and demonstrate that the relevant light twisted sector states may be identified as additional supergravity 2-form modes “wrapping” a finite 2-cycle in the resolution space. Reproducing the structure of the gauge theory result becomes more transparent in the large R-charge or BMN-like limit in which the resolved background takes a pp-wave form with the transverse space being a product of ℝ 4 and the Eguchi-Hanson space.

Torben Skrzypek

On AdS/CFT duality in the twisted sector of string theory on AdS 5 × S 5 /ℤ2 orbifold background

We construct a family of type IIB string backgrounds that are deformations of AdS3 × S3 × T4 with a “squashed” AdS3 × S3 metric supported by a combination of NSNS and RR fluxes. They have global SU(1, 1) × SU(2) symmetry, regular curvature, constant dilaton and preserve 8 supercharges. Upon compactification to 4 dimensions they reduce to   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 supersymmetric AdS2 × S2 solutions with electric and magnetic Maxwell fluxes. These type IIB supergravity solutions can be found from the undeformed AdS3 × S3 × T4 background by a combination of T-dualities and S-duality. In contrast to T-duality, S-duality transformations of a type IIB supergravity background do not generally preserve the classical integrability of the corresponding Green-Schwarz superstring sigma model. Nevertheless, we show that integrability is preserved in the present case. Indeed, we find that these backgrounds can be obtained, up to T-dualities, from an integrable inhomogeneous Yang-Baxter deformation (with unimodular Drinfel’d-Jimbo R-matrix) of the original AdS3 × S3 supercoset model.

Ben Hoare

Integrable supersymmetric deformations of AdS3 × S3 × T4

Motivated by a possibility of solving non-supersymmetric type 0 string theory in AdS 5 × S 5 background using integrability, we revisit the construction of type 0 string spectrum in some solvable examples of backgrounds with RR fluxes that are common to type IIB and type 0B theories. The presence of RR fluxes requires the use of a Green-Schwarz description for type 0 string theory. Like in flat space, the spectrum of type 0 theory can be derived from the type II theory spectrum by a (−1) F  orbifolding, i.e. combining the untwisted sector where GS fermions are periodic with the twisted sector where GS fermions are antiperiodic (and projecting out all spacetime fermionic states). This construction of the type 0 spectrum may also be implemented using Melvin background that allows to continuously interpolate between the type II and type 0 theories. As an illustration, we discuss the type 0B spectrum in the pp-wave background which is the Penrose limit of AdS 5 × S 5 with RR 5-form flux and also in the pp-wave background which is the Penrose limit of AdS 3 × S 3 × T4 supported by mixed RR and NSNS 3-form fluxes. We show that increasing the strength of the RR flux increases the value of the effective normal ordering constant (which determines the mass of the type 0 tachyon in flat space) and thus effectively decreases the momentum-space domain of instability of the ground state. We also comment on the semiclassical sector of states of type 0B theory in AdS 5 × S 5.

On type 0 string theory in solvable RR backgrounds

In this paper we consider the Klebanov-Tseytlin background and its non-Abelian <math><mi>T</mi></math>-dual geometry along a suitably chosen <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></math> subgroup of isometries. We analyze the Penrose limits along various null geodesics of both geometries. We observe that the Klebanov-Tseytlin geometry does not admit any pp-wave solutions. However, the <math><mi>T</mi></math>-dual background gives rise to a pp-wave solution upon taking the Penrose limit along some appropriate null geodesic. We comment on the possible gauge theory dual for our pp-wave background.

India

Sourav Roychowdhury

Prasanta K. Tripathy

Penrose limits in non-Abelian <math><mi>T</mi></math> dual of Klebanov-Tseytlin background

We consider a class of 2d σ-models on products of group spaces that provide new examples of a close connection between integrability and stability under the RG flow. We first study the integrable G × G model derived from the affine Gaudin construction (for which the 1-loop β-functions were found in  arXiv:2010.07879  ) and show that its condition of integrability is preserved also by the 2-loop RG flow. We then investigate the RG flow in the gauged G × G/H model, in particular the integrable T 1,1 model found in  arXiv:2010.05573  . We also construct a new class of integrable G × G/H models in the case when the subgroup H is abelian. In the simplest case of G = SU 2 , H = U 1 this leads to an integrable σ-model on the T 1,q  space (with a particular B-field). This model is also shown to be stable under the 2-loop RG flow, and we relate this property to its invariance under T-duality in an isometric U 1 direction. This T 1,q  model may be interpreted as an integrable deformation of the GMM model (of two coupled WZW theories with generic levels) away from the conformal point.