We set up a perturbative framework for the ’t Hooft line in the $$ \mathcal{N} $$ = 4 super-Yang-Mills theory, and apply it to correlators thereof with Wilson loops and local operators. Using this formalism we obtain a number of perturbative and non-perturbative results that directly connect to localization, holography and integrability.

Denmark

Charlotte Kristjansen

Sweden

Konstantin Zarembo

’t Hooft loops in N=4 super-Yang-Mills

The two-dimensional Gross-Neveu model is anticipated to undergo a crystalline phase transition at high baryon charge densities. This conclusion is drawn from the mean-field approximation, which closely resembles models of Peierls instability. We demonstrate that this transition indeed occurs when both the rank of the symmetry group and the dimension of the particle representation contributing to the baryon density are large (the large <math><mi>N</mi></math> limit). We derive this result through the exact solution of the model, developing the large <math><mi>N</mi></math> limit of the Bethe ansatz. Our analytical construction of the large-<math><mi>N</mi></math> solution of the Bethe ansatz equations aligns perfectly with the periodic (finite-gap) solution of the Korteweg–de Vries (KdV) of the mean-field analysis.

Valdemar Melin

Yuta Sekiguchi

United States

Paul Wiegmann

Peierls Transition in Gross-Neveu Model from Bethe Ansatz

We report the explicit solution for the vacuum state of the two-dimensional <math><mrow><mi>SU</mi></mrow><mo>(</mo><mi>N</mi><mo>)</mo></math> principal chiral model at large <math><mrow><mi>N</mi></mrow></math> for an arbitrary set of chemical potentials and any interaction strength, a unique result of such kind for an asymptotically free quantum field theory. The solution matches one-loop perturbative calculation at weak coupling, and in the opposite strong-coupling regime exhibits an emergent spacial dimension from the continuum limit of the <math><mrow><mi>SU</mi></mrow><mo>(</mo><mi>N</mi><mo>)</mo></math> Dynkin diagram.

France

Vladimir Kazakov

Evgeny Sobko

Large-<math><mi>N</mi></math> Principal Chiral Model in Arbitrary External Fields

We consider the defect CFT defined by a ’t Hooft line embedded in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super Yang-Mills theory. By explicitly quantizing around the given background we exactly reproduce a prediction from S-duality for the correlators between the ’t Hooft line and chiral primaries in the bulk and pave the way for higher loop analyses for non-protected operators. Furthermore, we demonstrate at the leading perturbative order that correlators between the ’t Hooft line and non-protected bulk operators can be efficiently computed using integrability. As a byproduct we find new integrable overlaps in   <math> <mi>sl</mi> </math>  $$ \mathfrak{sl} $$ (2) spin chains in different representations.

’t Hooft loops and integrability

One-point functions of local operators are studied, at weak and strong coupling, for the ABJM theory in the presence of a 1/2 BPS domain wall. In the underlying quantum spin chain the domain wall is represented by a boundary state which we show is integrable yielding a compact determinant formula for one-point functions of generic operators.

Dinh-Long Vu

Russia

Integrable domain walls in ABJM theory

The encoding of all possible sets of Bethe equations for a spin chain with SU(N|M) symmetry into a QQ-system calls for an expression of spin chain overlaps entirely in terms of Q-functions. We take a significant step towards deriving such a universal formula in the case of overlaps between Bethe eigenstates and integrable boundary states, of relevance for AdS/dCFT, by determining the transformation properties of the overlaps under fermionic as well as bosonic dualities which allows us to move between any two descriptions of the spin chain encoded in the QQ-system. An important part of our analysis involves introducing a suitable regularization for singular Bethe root configurations.

Dennis Müller

Duality relations for overlaps of integrable boundary states in AdS/dCFT

The D3-D5 probe-brane system is holographically dual to a defect CFT which is known to be integrable. The evidence comes mainly from the study of correlation functions at weak coupling. In the present work we shed light on the emergence of integrability on the string theory side. We do so by constructing the double row transfer matrix which is conserved when the appropriate boundary conditions are imposed. The corresponding reflection matrix turns out to be dynamical and depends both on the spectral parameter and the string embedding coordinates.

Greece

Georgios Linardopoulos

String integrability of defect CFT and dynamical reflection matrices

The   <math> <mi>psu</mi> <mfenced> <mn>2</mn> <mrow> <mfenced> <mn>2</mn> </mfenced> <mn>4</mn> </mrow> </mfenced> </math>  $$ \mathfrak{psu}\left(2,\left.2\right|4\right) $$  integrable super spin chain underlying the AdS/CFT correspondence has integrable boundary states which describe set-ups where k D3-branes get dissolved in a probe D5-brane. Overlaps between Bethe eigenstates and these boundary states encode the one-point functions of conformal operators and are expressed in terms of the superdeterminant of the Gaudin matrix that in turn depends on the Dynkin diagram of the symmetry algebra. The different possible Dynkin diagrams of super Lie algebras are related via fermionic dualities and we determine how overlap formulae transform under these dualities. As an application we show how to consistently move between overlap formulae obtained for k = 1 from different Dynkin diagrams.

Overlaps and fermionic dualities for integrable super spin chains

A D3-D5 intersection gives rise to a defect CFT, wherein the rank of the gauge group jumps by k units across a domain wall. The one-point functions of local operators in this set-up map to overlaps between on-shell Bethe states in the underlying spin chain and a boundary state representing the D5 brane. Focussing on the k = 1 case, we extend the construction to gluonic and fermionic sectors, which was prohibitively difficult for k &gt; 1. As a byproduct, we test an all-loop proposal for the one-point functions in the su(2) sector at the half-wrapping order of perturbation theory.

Integrable boundary states in D3-D5 dCFT: beyond scalars

We initiate a systematic, non-perturbative study of the large-<math><mi>N</mi></math> expansion in the two-dimensional <math><mrow><mi>SU</mi><mo>(</mo><mi>N</mi><mo>)</mo><mo>×</mo><mi>SU</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow></math> principal Chiral model (PCM). Starting with the known infinite-<math><mi>N</mi></math> solution for the ground state at fixed chemical potential, we devise an iterative procedure to solve the Bethe ansatz equations order by order in <math><mn>1</mn><mo>/</mo><mi>N</mi></math>. The first few orders, which we explicitly compute, reveal a systematic enhancement pattern at strong coupling calling for the near-threshold resummation of the large-<math><mi>N</mi></math> expansion. The resulting double-scaling limit bears striking similarities to the <math><mi>c</mi><mo>=</mo><mn>1</mn></math> noncritical string theory and suggests that the double-scaled PCM is dual to a noncritical string with a (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-dimensional target space where an additional dimension emerges dynamically from the <math><mrow><mi>SU</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow></math> Dynkin diagram.