arXiv subjects

Paul Ryan

Publications and source records attributed to Paul Ryan.

At least 19 recordsLinked to original sources

arXiv2026

Structure Constants from Q-Systems and Separation of Variables

We introduce a novel method to compute structure constants from Q-functions in the scalar sector of planar N=4 super Yang-Mills (SYM) and related theories. The method derives from operatorial as well as functional separation of variables, and the structure constants are expressed as determinants of matrices whose entries are integrals over products of Q-functions. In this framework, each operator is twisted by an external angle, mirroring the cusped Maldacena-Wilson loop. The structure constants of local single-trace operators in N=4 SYM are recovered in the untwisting limit, where we obtain a one-to-one correspondence between our key building blocks and those of the Hexagon formalism. Retaining appropriate twists, our structure constants also perfectly match those of the orbifold points of N=4 SYM. Our results thus far are valid at leading order in the weak-coupling expansion, but their formulation in terms of Q-functions provides a natural starting point for including loop corrections. Many of the methods we develop in this work apply more generally to the computation of correlation functions in integrable models.

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arXiv2025

Integrability and the spectrum of two-dimensional fishnet CFT

We formulate a closed set of equations for the spectrum of two-dimensional bi-scalar fishnet conformal field theory, comprising Baxter equations and quantisation conditions, which we derive operatorially from the underlying sl(2) spin chain. These equations are reminiscent of the Quantum Spectral Curve (QSC) framework found in other holographic CFTs and are expected to provide a complete non-perturbative description of the spectrum at arbitrary coupling. We solve the QSC numerically at finite coupling and uncover a rich analytic structure, including state collisions and complex energy levels. Analytically, we introduce a new method to derive the Asymptotic Bethe Ansatz equations, which control the spectrum up to wrapping order and incorporate spinning states. We further extend our results to the twisted case, which may be particularly useful for future separation of variables analyses of correlation functions in this theory.

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arXiv2025

Orthogonality of Q-Functions up to Wrapping in Planar N=4 Super Yang-Mills Theory

We construct orthogonality relations in the Separation of Variables framework for the sl(2) sector of planar N=4 supersymmetric Yang-Mills theory. Specifically, we find simple universal measures that make Q-functions of operators with different spins vanish at all orders in perturbation theory, prior to wrapping corrections. To analyze this rank-one sector, we relax some of the assumptions thus far considered in the Separation of Variables framework. Our findings may serve as guidelines for extending this formalism to other sectors of the theory as well as other integrable models

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arXiv2024

How to Manage My Data? With Machine--Interpretable GDPR Rights!

The EU GDPR is a landmark regulation that introduced several rights for individuals to obtain information and control how their personal data is being processed, as well as receive a copy of it. However, there are gaps in the effective use of rights due to each organisation developing custom methods for rights declaration and management. Simultaneously, there is a technological gap as there is no single consistent standards-based mechanism that can automate the handling of rights for both organisations and individuals. In this article, we present a specification for exercising and managing rights in a machine-interpretable format based on semantic web standards. Our approach uses the comprehensive Data Privacy Vocabulary to create a streamlined workflow for individuals to understand what rights exist, how and where to exercise them, and for organisations to effectively manage them. This work pushes the state of the art in GDPR rights management and is crucial for data reuse and rights management under technologically intensive developments, such as Data Spaces.

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arXiv2024

New Approach to Strongly Coupled N = 4 SYM via Integrability

Finding a systematic expansion of the spectrum of free superstrings on AdS${}_5\times $S${}^5$, or equivalently strongly coupled 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. In this paper, we introduce a new framework that utilises a novel set of QSC variables with a regular strong coupling expansion. We demonstrate how to use this approach to construct a new numerical algorithm that remains stable even at a 't Hooft coupling as large as $10^6$ (or g ~ 100). Employing this approach, we derive new analytic results for some states in the sl(2) sector and beyond. We present a new analytic prediction for a coefficient in the strong coupling expansion of the conformal dimension for the lowest trajectory at a given twist L. For non-lowest trajectories, we uncover a novel feature of mixing with operators outside the sl(2) sector, which manifests as a new type of analytic dependence on the twist.

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arXiv2024

Data Privacy Vocabulary (DPV) -- Version 2

The Data Privacy Vocabulary (DPV), developed by the W3C Data Privacy Vocabularies and Controls Community Group (DPVCG), enables the creation of machine-readable, interoperable, and standards-based representations for describing the processing of personal data. The group has also published extensions to the DPV to describe specific applications to support legislative requirements such as the EU's GDPR. The DPV fills a crucial niche in the state of the art by providing a vocabulary that can be embedded and used alongside other existing standards such as W3C ODRL, and which can be customised and extended for adapting to specifics of use-cases or domains. This article describes the version 2 iteration of the DPV in terms of its contents, methodology, current adoptions and uses, and future potential. It also describes the relevance and role of DPV in acting as a common vocabulary to support various regulatory (e.g. EU's DGA and AI Act) and community initiatives (e.g. Solid) emerging across the globe.

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arXiv2023

Boundary Overlaps from Functional Separation of Variables

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 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 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.

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arXiv2022

Exploring the Quantum Spectral Curve for AdS${}_3$/CFT${}_2$

Despite the rich and fruitful history of the integrability approach to string theory on the $AdS_3\times S^3\times T^4$ 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. This formulation is expected to capture all wrapping effects exactly and describe the full planar spectrum. Massless modes conjecturally 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 N=4 SYM. In this paper we resolve these difficulties and obtain the first ever predictions for generic unprotected string excitations. We explain how to extract a systematic expansion around the analogue of the weak 't Hooft coupling limit in N=4 SYM and also obtain high-precision numerical results. This concrete data and others obtainable from the QSC could help to identify the so-far mysterious dual CFT.

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arXiv2022

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

Integrable 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 sl(N) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.

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arXiv2022

Integrable systems, separation of variables and the Yang-Baxter equation

This article, based on the author's PhD thesis, reviews recent advancements in the field of quantum integrability, in particular the separation of variables (SoV) program for high-rank integrable spin chains and the boost mechanism for solving the Yang-Baxter equation. We begin with a general overview of quantum integrable systems with special emphasis on their description in terms of quantum algebras. We then provide a detailed account of the Yangian of $\mathfrak{gl}(n)$ in particular the Bethe algebra, fusion, and T- and Q-systems. We then introduce the notion of separation of variables in integrable systems and build on Sklyanin's work in rank 1 models and extend to higher rank. By exploiting a novel link between SoV and quantum algebra representation theory we construct the separated variables for $\mathfrak{gl}(n)$ spin chains for arbitrary compact representations of the symmetry algebra and develop various new tools along the way. Next, we build on the previous part and develop a new technique for the computation of scalar products in the SoV framework which we call Functional SoV or FSoV. Unlike the work in the previous, operatorial, part this approach is based on the Baxter TQ equations. After developing this technique we supplement it with a new operator construction providing a unified view of functional and operatorial SoV. Then, we generalise the results of the previous part from compact spin chains to non-compact spin chains. The final part of this work is based on the development of tools for solving the Yang- Baxter equation. We develop a bottom-up approach for this based on the so-called Boost automorphism and uses the spin chain Hamiltonian as a starting point. Our approach allows us to classify numerous families of solutions in particular a complete classification of 4 x 4 solutions which preserve fermion number which have applications in the AdS/CFT correspondence.

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arXiv2021

Integrable deformations of AdS/CFT

In this paper we study in detail the deformations introduced in [1] of the integrable structures of the AdS$_{2,3}$ 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 $\rm AdS_3 \times S^3 \times M^4$ string sigma model.

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arXiv2021

A Common Semantic Model of the GDPR Register of Processing Activities

The creation and maintenance of a Register of Processing Activities (ROPA) is an essential process for the demonstration of GDPR compliance. We analyse ROPA templates from six EU Data Protection Regulators and show that template scope and granularity vary widely between jurisdictions. We then propose a flexible, consolidated data model for consistent processing of ROPAs (CSM-ROPA). We analyse the extent that the Data Privacy Vocabulary (DPV) can be used to express CSM-ROPA. We find that it does not directly address modelling ROPAs, and so needs additional concept definitions. We provide a mapping of our CSM-ROPA to an extension of the Data Privacy Vocabulary.

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arXiv2020

Determinant Form of Correlators in High Rank Integrable Spin Chains via Separation of Variables

In this paper we take further steps towards developing the separation of variables program for integrable spin chains with 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.

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arXiv2020

Yang-Baxter and the Boost: splitting the difference

In this paper we continue our classification of regular solutions of the Yang-Baxter equation using the method based on the spin chain boost operator developed in \cite{deLeeuw:2019zsi}. We provide details on how to find all non-difference form solutions and apply our method to spin chains with local Hilbert space of dimensions two, three and four. We classify all $16\times 16$ solutions which exhibit $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$ symmetry, which include the one-dimensional Hubbard model and the $S$-matrix of the ${\rm AdS}_5 \times {\rm S}^5$ superstring sigma model. In all cases we find interesting novel solutions of the Yang-Baxter equation.

math-ph↗

arXiv2020

Towards a Semantic Model of the GDPR Register of Processing Activities

A core requirement for GDPR compliance is the maintenance of a register of processing activities (ROPA). Our analysis of six ROPA templates from EU data protection regulators shows the scope and granularity of a ROPA is subject to widely varying guidance in different jurisdictions. We present a consolidated data model based on common concepts and relationships across analysed templates. We then analyse the extent of using the Data Privacy Vocabulary - a vocabulary specification for GDPR. We show that the DPV currently does not provide sufficient concepts to represent the ROPA data model and propose an extension to fill this gap. This will enable creation of a pan-EU information management framework for interoperability between organisations and regulators for GDPR compliance.

cs.CY↗

arXiv2020

Design Challenges for GDPR RegTech

The Accountability Principle of the GDPR requires that an organisation can demonstrate compliance with the regulations. A survey of GDPR compliance software solutions shows significant gaps in their ability to demonstrate compliance. In contrast, RegTech has recently brought great success to financial compliance, resulting in reduced risk, cost saving and enhanced financial regulatory compliance. It is shown that many GDPR solutions lack interoperability features such as standard APIs, meta-data or reports and they are not supported by published methodologies or evidence to support their validity or even utility. A proof of concept prototype was explored using a regulator based self-assessment checklist to establish if RegTech best practice could improve the demonstration of GDPR compliance. The application of a RegTech approach provides opportunities for demonstrable and validated GDPR compliance, notwithstanding the risk reductions and cost savings that RegTech can deliver. This paper demonstrates a RegTech approach to GDPR compliance can facilitate an organisation meeting its accountability obligations.

cs.CY↗

arXiv2020

Classifying nearest-neighbour interactions and deformations of AdS

We classify all regular solutions of the Yang-Baxter equation of eight-vertex type. Regular solutions correspond to spin chains with nearest-neighbour interactions. We find a total of four independent solutions. Two are related to the usual six- and eight-vertex models that have R-matrices of difference form. We find two completely new solutions of the Yang-Baxter equation, which are manifestly of non-difference form. These new solutions contain the S-matrices of the AdS2 and AdS3 integrable models as a special case. Consequently, we can classify all possible integrable deformations of eight-vertex type of these holographic integrable systems.

hep-th↗

arXiv2020

Separation of variables for rational gl(n) spin chains in any compact representation, via fusion, embedding morphism and Backlund flow

We propose a way to separate variables in a rational integrable $\mathfrak{gl}(n)$ spin chain with an arbitrary finite-dimensional irreducible representation at each site and with generic twisted periodic boundary conditions. Firstly, we construct a basis that diagonalises a higher-rank version of the Sklyanin B-operator; the construction is based on recursive usage of an embedding of a $\mathfrak{gl}(k)$ spin chain into a $\mathfrak{gl}(k+1)$ spin chain which is induced from a Yangian homomorphism and controlled by dual diagonals of Gelfand-Tsetlin patterns. Then, we show that the same basis can be equivalently constructed by action of Backlund-transformed fused transfer matricies, whence the Bethe wave functions factorise into a product of ascending Slater determinants in Baxter Q-functions. Finally, we construct raising and lowering operators -- the conjugate momenta -- as normal-ordered Wronskian expressions in Baxter Q-operators evaluated at zeros of B -- the separated variables. It is an immediate consequence of the proposed construction that the Bethe algebra comprises the maximal possible number of mutually commuting charges -- a necessary property for Bethe equations to be complete.

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