Finding a systematic expansion of the spectrum of free superstrings on AdS5 × S5, or equivalently strongly coupled   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM in the planar limit, remains an outstanding challenge. No first principle string theory methods are readily available, instead the sole tool at our disposal is the integrability-based Quantum Spectral Curve (QSC). For example, through the QSC the first five orders in the strong coupling expansion of the conformal dimension of an infinite family of short operators have been obtained. However, when using the QSC at strong coupling one must often rely on numerics, and the existing methods for solving the QSC rapidly lose precision as we approach the strong coupling regime.

United Kingdom

Simon Ekhammar

Nikolay Gromov

Paul Ryan

New approach to strongly coupled   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM via integrability

In this paper we show how the Functional Separation of Variables (FSoV) method can be applied to the problem of computing overlaps with integrable boundary states in integrable systems. We demonstrate our general method on the example of a particular boundary state, a singlet of the symmetry group, in an   <math> <mi>su</mi> <mfenced> <mn>3</mn> </mfenced> </math>  $$ \mathfrak{su}(3) $$  rational spin chain in an alternating fundamental-anti-fundamental representation. The FSoV formalism allows us to compute in determinant form not only the overlaps of the boundary state with the eigenstates of the transfer matrix, but in fact with any factorisable state. This includes off-shell Bethe states, whose overlaps with the boundary state have been out of reach with other methods. Furthermore, we also found determinant representations for insertions of so-called Principal Operators (forming a complete algebra of all observables) between the boundary and the factorisable state as well as certain types of multiple insertions of Principal Operators. Concise formulas for the matrix elements of the boundary state in the SoV basis and   <math> <mi>su</mi> <mfenced> <mi>N</mi> </mfenced> </math>  $$ \mathfrak{su}(N) $$  generalisations are presented. Finally, we managed to construct a complete basis of integrable boundary states by repeated action of conserved charges on the singlet state. As a result, we are also able to compute the overlaps of all of these states with integral of motion eigenstates.

Boundary overlaps from Functional Separation of Variables

Despite the rich and fruitful history of the integrability approach to string theory on the AdS3 × S3 × T4 background, it has not been possible to extract many concrete predictions from integrability, except in a strict asymptotic regime of large quantum numbers, due to the severity of wrapping effects. The situation changed radically with two independent and identical proposals for the Quantum Spectral Curve (QSC) for this system in a background of pure Ramond-Ramond flux. In other integrable superstring backgrounds there is compelling evidence that this formulation captures all wrapping effects exactly and describes the full planar spectrum. This great success motivates us to study the new proposed QSC and develop methods to extract from it concrete predictions for spectral data. The AdS3 × S3 × T4 case presents a significant novel feature and challenge compared to its higher-dimensional analogues — massless modes. It has been conjectured that these manifest themselves in a new property of this QSC: the non-quadratic nature of the branch-cut singularities of the QSC Q-functions. This feature implies new technical challenges in solving the QSC equations as compared to the well-studied case of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM. In this paper we resolve these difficulties and obtain the first ever predictions for unprotected string excitations in the planar limit with finite quantum numbers and RR flux. We explain how to extract a systematic expansion around the analogue of the weak ’t Hooft coupling limit in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM and also obtain high-precision numerical results. These concrete data and others obtainable from the QSC could help to identify the so-far mysterious dual CFT.

Italy

Andrea Cavaglià

Sweden

Russia

Exploring the Quantum Spectral Curve for AdS3/CFT2

Integrable   <math> <mi>sl</mi> </math>  $$ \mathfrak{sl} $$ (N) spin chains, which we consider in this paper, are not only the prototypical example of quantum integrable systems but also systems with a wide range of applications. For these models we use the Functional Separation of Variables (FSoV) technique with a new tool called Character Projection to compute all matrix elements of a complete set of operators, which we call principal operators, in the basis diagonalising the tower of conserved charges as determinants in Q-functions. Building up on these results we then derive similar determinant forms for the form-factors of combinations of multiple principal operators between arbitrary factorizable states, which include, in particular, off-shell Bethe vectors and Bethe vectors with arbitrary twists. We prove that the set of principal operators generates the complete spin chain Yangian. Furthermore, we derive the representation of these operators in the SoV bases allowing one to compute correlation functions with an arbitrary number of principal operators. Finally, we show that the available combinations of multiple insertions includes Sklyanin’s SoV B operator. As a result, we are able to derive the B operator for   <math> <mi>sl</mi> </math>  $$ \mathfrak{sl} $$ (N) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.

Nicolò Primi

Form-factors and complete basis of observables via separation of variables for higher rank spin chains

In this paper we study in detail the deformations introduced in [1] of the integrable structures of the AdS2,3 integrable models. We do this by embedding the corresponding scattering matrices into the most general solutions of the Yang-Baxter equation. We show that there are several non-trivial embeddings and corresponding deformations. We work out crossing symmetry for these models and study their symmetry algebras and representations. In particular, we identify a new elliptic deformation of the AdS3 × S3 × M4 string sigma model.

Ireland

Marius Leeuw

Anton Pribytok

Ana Retore

Integrable deformations of AdS/CFT

In this paper we take further steps towards developing the separation of variables program for integrable spin chains with   <math> <mi>gl</mi> <mfenced> <mi>N</mi> </mfenced> </math>  $$ \mathfrak{gl}(N) $$  symmetry. By finding, for the first time, the matrix elements of the SoV measure explicitly we were able to compute correlation functions and wave function overlaps in a simple determinant form. In particular, we show how an overlap between on-shell and off-shell algebraic Bethe states can be written as a determinant. Another result, particularly useful for AdS/CFT applications, is an overlap between two Bethe states with different twists, which also takes a determinant form in our approach. Our results also extend our previous works in collaboration with A. Cavaglia and D. Volin to general values of the spin, including the SoV construction in the higher-rank non-compact case for the first time.

France

Fedor Levkovich-Maslyuk

Determinant form of correlators in high rank integrable spin chains via separation of variables

Separation of variables (SoV) is an extremely efficient and elegant technique for analysing physical systems but its application to integrable spin chains was limited until recently to the simplest <math><mrow><mi>su</mi></mrow><mo>(</mo><mn>2</mn><mo>)</mo></math> cases. In this paper we continue developing the SoV program for higher-rank spin chains and demonstrate how to derive the measure for the <math><mrow><mi>su</mi></mrow><mo>(</mo><mn>3</mn><mo>)</mo></math> case. Our results are a natural consequence of factorisability of the wave function and functional orthogonality relations following from the interplay between Baxter equations for Q-functions and their dual.

Dmytro Volin

Dual separated variables and scalar products

We classify all regular solutions of the Yang-Baxter equation of eight-vertex type. Regular solutions correspond to spin chains with nearest-neighbor interactions. We find a total of four independent solutions. Two are related to the usual six- and eight-vertex models that have <math><mi>R</mi></math> matrices of difference form. We find two new solutions of the Yang-Baxter equation, which are manifestly of nondifference form. These new solutions contain the S-matrices of the <math><msub><mi>AdS</mi><mn>2</mn></msub></math> and <math><msub><mi>AdS</mi><mn>3</mn></msub></math> integrable models as a special case. This can be used as a starting point to study and classify integrable deformations of these holographic integrable systems.