We continue our study of the defect CFT on a Maldacena-Wilson line in $$ \mathcal{N} $$ = 4 Super-Yang-Mills theory using Bootstrability — the conformal bootstrap supplemented with exact integrability data. In this paper, we extend this program to charged sectors of the theory, considering a mixed-correlator setup first introduced by Liendo, Meneghelli, and Mitev. The exact spectrum in all channels is given by integrability at any coupling. Additionally, we use exact expressions for some structure constants fixed by localisation and leverage the exact discrete symmetries of the theory. We analyse the remaining data with the numerical bootstrap, developing an algorithm optimised to scan over a large multidimensional space of OPE coefficients and carve the allowed region with the “cutting surface” procedure. We compute upper and lower bounds for 12 OPE coefficients for several values of the coupling. Our results are sharp for the lowest states in each sector but become quite wide for the excited states due to their near degeneracy. This highlights the need for studying the system with non-protected external states and for further input from integrability in the form of integrated correlators of non-protected operators.

Italy

Andrea Cavaglià

United Kingdom

Nikolay Gromov

France

Julius Julius

United States

Michelangelo Preti

Nika Sokolova

Probing line defect CFT with mixed-correlator bootstrability

We consider the boundary dual of AdS × $\textit{S}$ × $\textit{K}$3 for NS5-flux $$ {Q}_5^{NS} $$ = 1, which is described by a sigma model with target space given by the $\textit{d}$-fold symmetric product of $\textit{K}$3. Building on results in algebraic geometry, we address the problem of deforming it away from the orbifold point from the viewpoint of topological strings. We propose how the ’t Hooft expansion can be geometrized in terms of Gromov-Witten invariants and, in favorable settings, how it can be summed up to all orders in closed form. We consider an explicit example in detail for which we discuss the genus expansion around the orbifold point, as well as the divergence in the strong coupling regime. We find that within the domain of convergence, scale separation does not occur. However, in order for the mathematical framework to be applicable in the first place, we need to consider “reduced” Gromov-Witten invariants that fit, as we argue, naturally to topologically twisted $\textit{N}$ = 4 strings. There are some caveats and thus to what extent this toy model captures the physics of strings on AdS × $\textit{S}$ × $\textit{K}$3 remains to be seen.

Switzerland

Wolfgang Lerche

Gromov-Witten/Hilbert versus AdS

The combination of integrability and crossing symmetry has proven to give tight non-perturbative bounds on some planar structure constants in $$ \mathcal{N} $$ =4 SYM, particularly in the setup of defect observables built on a Wilson-Maldacena line. Whereas the precision is good for the low lying states, higher in the spectrum it drops due to the degeneracies at weak coupling when considering a single correlator. As this could be a clear obstacle in restoring higher point functions, we studied the problem of bounding directly a 4-point function at generic cross ratio, showing how to adapt for this purpose the numerical bootstrap algorithms based on semidefinite programming. Another tool we are using to further narrow the bounds is a parity symmetry descending from the $$ \mathcal{N} $$ =4 SYM theory, which allowed us to reduce the number of parameters. We also give an interpretation for the parity in terms of the Quantum Spectral Curve at weak coupling. Our numerical bounds give an accurate determination of the 4-point function for physical values of the cross ratio, with at worst 5-6 digits precision at weak coupling and reaching more than 11 digits for ’t Hooft coupling $$ \frac{\sqrt{\lambda }}{4\pi}\sim 4 $$ .

Computing four-point functions with integrability, bootstrap and parity symmetry

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.

Simon Ekhammar

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

Integrability methods give us access to a number of observables in the planar   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM. Among them, the Quantum Spectral Curve (QSC) governs the spectrum of anomalous dimensions. Low lying states were successfully studied in the past using the QSC. However, with the increased demand for a systematic study of a large number of states for various applications, there is a clear need for a fast QSC solver which can easily access a large number of excited states. Here, we fill this gap by developing a new algorithm and applied it to study all 219 states with the bare dimension ∆0 ≤ 6 in a wide range of couplings.

Hungary

Árpád Hegedűs

Fast QSC solver: tool for systematic study of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 Super-Yang-Mills spectrum

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.

Sweden

Russia

Exploring the Quantum Spectral Curve for AdS3/CFT2

We present a systematic method for the derivation of a relation which connects the correlation function of operators on the straight Maldacena-Wilson line with the integrability data for the cusp anomalous dimension. As we show, the derivation requires very careful treatment of the UV divergences. Our method opens a way to derive infinitely many constraints on integrals of multi-point correlation functions, relating them with the integrability data for the generalised cusp anomalous dimension governed by the Quantum Spectral Curve. Such constraints have been shown recently to be very powerful in combination with the numerical conformal bootstrap, leading to very narrow non-perturbative bounds on conformal data beyond the spectrum.

Integrated correlators from integrability: Maldacena-Wilson line in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM

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

We continue to develop Bootstrability — a method merging Integrability and Conformal Bootstrap to extract CFT data in integrable conformal gauge theories such as   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM. In this paper, we consider the 1D defect CFT defined on a   <math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math>  $$ \frac{1}{2} $$ -BPS Wilson line in the theory, whose non-perturbative spectrum is governed by the Quantum Spectral Curve (QSC). In addition, we use that the deformed setup of a cusped Wilson line is also controlled by the QSC. In terms of the defect CFT, this translates into two nontrivial relations connecting integrated 4-point correlators to cusp spectral data, such as the Bremsstrahlung and Curvature functions — known analytically from the QSC. Combining these new constraints and the spectrum of the 10 lowest-lying states with the Numerical Conformal Bootstrap, we obtain very sharp rigorous numerical bounds for the structure constant of the first non-protected state, giving this observable with seven digits precision for the ’t Hooft coupling in the intermediate coupling region   <math> <mfrac> <msqrt> <mi>λ</mi> </msqrt> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> <mo>~</mo> <mn>1</mn> </math>  $$ \frac{\sqrt{\lambda }}{4\pi}\sim 1 $$ , with the error decreasing quickly at large ’t Hooft coupling. Furthermore, for the same structure constant we obtain a 4-loop analytic result at weak coupling. We also present results for excited states.