Periodic Reporting for period 1 - Bootstrability (Solving holographic CFTs through the synergy of integrability and conformal bootstrap.)
Reporting period: 2023-10-02 to 2025-10-01
Quantum Field Theory (QFT) is the universal framework describing the fundamental interactions of nature. It provides the theoretical foundation for both the Standard Model of particle physics and for the study of critical phenomena in condensed-matter systems.
Despite numerous achievements, our theoretical understanding of this model is mostly limited to the perturbative regime, where the interaction among fundamental particles is sufficiently weak. Many of the most intriguing aspects of nature lie instead in the strongly coupled regime, which remains largely inaccessible to conventional analytical tools.
A particularly fruitful setting to explore these challenges is offered by Conformal Field Theories (CFTs). CFTs describe physical systems at criticality and represent the fixed points of the renormalisation-group flow. Moreover, some CFTs possess a holographic dual description in terms of quantum gravity on an Anti-de Sitter (AdS) background, an idea formalised in the AdS/CFT correspondence. This duality has revolutionised our understanding of strongly coupled systems by relating them to weakly coupled gravitational theories, but it still leaves many intermediate-coupling phenomena unexplored.
Hope that the deadlock of non-perturbative physics could be finally resolved appeared almost two decades ago with the discovery of integrability in the flagship holographic models together with the revival of the conformal bootstrap program by a change of perspective compared to the unsuccessful attempts in the early days of CFTs.These methodologies are to some extent complementary in their regime of validity. Integrability allows one to compute the spectrum of scaling dimensions of operators exactly, even at finite coupling, through the Quantum Spectral Curve (QSC) formalism. The conformal bootstrap exploits the consistency of the theory and conformal symmetry to derive rigorous constraints on CFT data. Together, they offer powerful but still separate windows into non-perturbative physics.
The Bootstrability project was designed to bring these two paradigms together for the first time. Its main goal is to build an analytical and numerical framework capable of computing not only spectra but also structure constants, the quantities that encode interactions among operators and determine all correlation functions in a CFT.
This ambitious program has as its long-term final goal the solution of the prototypical holographic integrable superconformal field theories: N = 4 SYM and ABJM.
The project’s specific objectives are to merge QSC spectral data with conformal-bootstrap equations to extract the structure constants developing both analytical and numerical implementations covering the full range of coupling regimes.
In its first two years, the project has achieved significant milestones, obtaining analytical and numerical non-perturbative results for the structure constants and correlation functions of defect operators in N=4 SYM. Moreover, with the long-term goal of incorporating integrability data beyond the spectrum of local operators, a completely new method was developed to compute non-perturbative Regge trajectories that link different local operators across the spectrum. These advances mark a step forward in extending the Bootstrability program beyond existing integrability techniques. The results have been disseminated through peer-reviewed publications, invited conference presentations, and the release of open-access computational tools.
Beyond its scientific results, Bootstrability demonstrates the value of cross-disciplinary research at the interface of mathematical physics, high-energy theory, and computational science. Ultimately, the project advances toward a complete non-perturbative description of quantum field theories — a step of profound importance for our understanding of nature and for future progress in quantum gravity, particle physics, and mathematical modelling.
Despite numerous achievements, our theoretical understanding of this model is mostly limited to the perturbative regime, where the interaction among fundamental particles is sufficiently weak. Many of the most intriguing aspects of nature lie instead in the strongly coupled regime, which remains largely inaccessible to conventional analytical tools.
A particularly fruitful setting to explore these challenges is offered by Conformal Field Theories (CFTs). CFTs describe physical systems at criticality and represent the fixed points of the renormalisation-group flow. Moreover, some CFTs possess a holographic dual description in terms of quantum gravity on an Anti-de Sitter (AdS) background, an idea formalised in the AdS/CFT correspondence. This duality has revolutionised our understanding of strongly coupled systems by relating them to weakly coupled gravitational theories, but it still leaves many intermediate-coupling phenomena unexplored.
Hope that the deadlock of non-perturbative physics could be finally resolved appeared almost two decades ago with the discovery of integrability in the flagship holographic models together with the revival of the conformal bootstrap program by a change of perspective compared to the unsuccessful attempts in the early days of CFTs.These methodologies are to some extent complementary in their regime of validity. Integrability allows one to compute the spectrum of scaling dimensions of operators exactly, even at finite coupling, through the Quantum Spectral Curve (QSC) formalism. The conformal bootstrap exploits the consistency of the theory and conformal symmetry to derive rigorous constraints on CFT data. Together, they offer powerful but still separate windows into non-perturbative physics.
The Bootstrability project was designed to bring these two paradigms together for the first time. Its main goal is to build an analytical and numerical framework capable of computing not only spectra but also structure constants, the quantities that encode interactions among operators and determine all correlation functions in a CFT.
This ambitious program has as its long-term final goal the solution of the prototypical holographic integrable superconformal field theories: N = 4 SYM and ABJM.
The project’s specific objectives are to merge QSC spectral data with conformal-bootstrap equations to extract the structure constants developing both analytical and numerical implementations covering the full range of coupling regimes.
In its first two years, the project has achieved significant milestones, obtaining analytical and numerical non-perturbative results for the structure constants and correlation functions of defect operators in N=4 SYM. Moreover, with the long-term goal of incorporating integrability data beyond the spectrum of local operators, a completely new method was developed to compute non-perturbative Regge trajectories that link different local operators across the spectrum. These advances mark a step forward in extending the Bootstrability program beyond existing integrability techniques. The results have been disseminated through peer-reviewed publications, invited conference presentations, and the release of open-access computational tools.
Beyond its scientific results, Bootstrability demonstrates the value of cross-disciplinary research at the interface of mathematical physics, high-energy theory, and computational science. Ultimately, the project advances toward a complete non-perturbative description of quantum field theories — a step of profound importance for our understanding of nature and for future progress in quantum gravity, particle physics, and mathematical modelling.
During the first two years, the project has focused on developing and applying the Bootstrability framework to concrete examples of conformal field theories.
The main scientific activities centred on two complementary directions.
The first addressed the conformal defect theory living on the supersymmetric Wilson line in N=4 Super Yang–Mills (SYM). This system provides an ideal testing ground where conformal symmetry and integrability coexist in a simplified one-dimensional setting. We combined integrability data gathered with the QSC technique with conformal bootstrap constraints to compute non-perturbative correlation functions and structure constants in the defect theory. These results were presented in two publications (arXiv:2312.11604 and arXiv:2412.07624). The first paper shows how parity symmetry can be incorporated into the Bootstrability analysis. We also studied the problem of bounding directly a 4-point function at generic cross ratio instead of single structure constants, showing how to adapt for this purpose the numerical bootstrap algorithms based on semidefinite programming. The second extended this framework to mixed-correlator sectors, obtaining precise bounds for operator product expansion (OPE) coefficients as a function of the coupling. Together, these works confirmed the feasibility and power of the hybrid method, opening the way to its future analytical generalisation.
The second line of research extended the scope of integrability beyond the spectrum of local single-trace operators, towards the description of non-local observables such as Regge trajectories in N=4 SYM. In this direction, we developed an entirely new approach based on QSC together with the Asymptotic Bethe Anzatz that we named Asymptotic Baxter–Bethe Ansatz (ABBA). This new method is capable of computing non-perturbative Regge trajectories that interpolate between local operators across the conformal spectrum. The corresponding results were published in arXiv:2406.18639 and arXiv:2507.15983 which together introduced the first consistent framework to describe the trajectories both analytically at weak coupling and numerically at any coupling regime. This development represents a decisive step towards incorporating Regge data as a new class of integrability input within the Bootstrability programme.
The main scientific activities centred on two complementary directions.
The first addressed the conformal defect theory living on the supersymmetric Wilson line in N=4 Super Yang–Mills (SYM). This system provides an ideal testing ground where conformal symmetry and integrability coexist in a simplified one-dimensional setting. We combined integrability data gathered with the QSC technique with conformal bootstrap constraints to compute non-perturbative correlation functions and structure constants in the defect theory. These results were presented in two publications (arXiv:2312.11604 and arXiv:2412.07624). The first paper shows how parity symmetry can be incorporated into the Bootstrability analysis. We also studied the problem of bounding directly a 4-point function at generic cross ratio instead of single structure constants, showing how to adapt for this purpose the numerical bootstrap algorithms based on semidefinite programming. The second extended this framework to mixed-correlator sectors, obtaining precise bounds for operator product expansion (OPE) coefficients as a function of the coupling. Together, these works confirmed the feasibility and power of the hybrid method, opening the way to its future analytical generalisation.
The second line of research extended the scope of integrability beyond the spectrum of local single-trace operators, towards the description of non-local observables such as Regge trajectories in N=4 SYM. In this direction, we developed an entirely new approach based on QSC together with the Asymptotic Bethe Anzatz that we named Asymptotic Baxter–Bethe Ansatz (ABBA). This new method is capable of computing non-perturbative Regge trajectories that interpolate between local operators across the conformal spectrum. The corresponding results were published in arXiv:2406.18639 and arXiv:2507.15983 which together introduced the first consistent framework to describe the trajectories both analytically at weak coupling and numerically at any coupling regime. This development represents a decisive step towards incorporating Regge data as a new class of integrability input within the Bootstrability programme.
The Bootstrability project has established a new framework that unites integrability and the conformal bootstrap, two of the most powerful non-perturbative methods in modern theoretical physics. This synthesis, achieved for the first time, enables the computation of observables at finite coupling, advancing far beyond previous analytical or numerical approaches.
Technically, the project produced the first non-perturbative determination of structure constants and correlation functions in the defect theory of the supersymmetric Wilson line in N=4 SYM, integrating QSC data with conformal bootstrap constraints. In parallel, it introduced the Asymptotic Baxter–Bethe Ansatz (ABBA), a novel method to compute Regge trajectories that extend integrability beyond local operators and capture the high-energy regime of gauge theories.
These results provide a solid foundation for a complete description of strongly coupled quantum field theories. Further research will focus on generalising the framework to non-planar effects and less supersymmetric models.
Technically, the project produced the first non-perturbative determination of structure constants and correlation functions in the defect theory of the supersymmetric Wilson line in N=4 SYM, integrating QSC data with conformal bootstrap constraints. In parallel, it introduced the Asymptotic Baxter–Bethe Ansatz (ABBA), a novel method to compute Regge trajectories that extend integrability beyond local operators and capture the high-energy regime of gauge theories.
These results provide a solid foundation for a complete description of strongly coupled quantum field theories. Further research will focus on generalising the framework to non-planar effects and less supersymmetric models.