DFG Temporary Position: Integrability in Four Dimensions

Integrable models furnish a fertile arena for studying the theoretical framework that underlies our physical reality. They have a natural home in two dimensions and feature in elegant limits of higher dimensional systems. Notably, integrable models have played a major role throughout the history of physics since Kepler. The planar AdS/CFT correspondence provides the venue for a four-dimensional quantum field theory that is believed to be completely integrable. While this case represents the most popular occurrence of integrability in four dimensional QFT, other examples exist, such as the high energy scattering in quantum chromodynamics. It is the aim of the present project to develop a solid understanding of the principles that underly integrable field theories in four spacetime dimensions and to enlarge the associated integrability toolbox in order to facilitate explicit calculations. To be more explicit, the prototypical AdS/CFT duality belongs to the class of integrable models whose symmetry is the so-called Yangian. This nonlocal quantum group underlies rational solutions to the famous quantum Yang-Baxter equation. It was identified on several observables within the maximally supersymmetric Yang-Mills theory in four dimensions or its dual string theory on the background AdS5 x S5. While the Yangian can naturally be defined in two dimensions, it is a challenge to understand how it is realized in a 4d quantum field theory. Moreover, in two dimensions quantum group symmetries are known to impose powerful constraints, e.g. on the scattering matrix. For the above Yang-Mills theory in four dimensions, however, similar constraints are far from being understood. As recently discovered, the AdS/CFT system has a further nonlocal symmetry which acts as a generator of the so-called spectral parameter. Symmetries with this property are termed master symmetries and appear in many integrable models in 2d. Understanding their role for the AdS/CFT correspondence, however, is just at its beginning. Generically, the spectral parameter plays a crucial role for integrable models. It allows to package conservation laws in compact form and to make important statements using the theorems of complex analysis. While studying the integrability of AdS/CFT results in powerful tools and novel insights, four-dimensional integrability is found in various situations not directly related to this duality. Examples are the Regge limit of scattering processes in QCD or certain field theories with less supersymmetry. The latter cases show similar structures as the AdS/CFT system but a general framework for their study is not known. The following points are addressed by this proposal: To establish the role of the novel master symmetry of AdS/CFT, to identify the Yangian as an integrability tool on various observables and to develop a unified picture of integrable structures in four-dimensional field theories.

Integrable models furnish a fertile arena for studying the theoretical framework that underlies our physical reality. They have a natural home in two dimensions and feature in elegant limits of higher dimensional systems. Notably, integrable models have played a major role throughout the history of physics since Kepler. The planar AdS/CFT correspondence provides the venue for a four-dimensional quantum field theory that is believed to be completely integrable. While this case represents the most popular occurrence of integrability in four dimensional QFT, other examples exist, such as the high energy scattering in quantum chromodynamics. It is the aim of the present project to develop a solid understanding of the principles that underly integrable field theories in four spacetime dimensions and to enlarge the associated integrability toolbox in order to facilitate explicit calculations. To be more explicit, the prototypical AdS/CFT duality belongs to the class of integrable models whose symmetry is the so-called Yangian. This nonlocal quantum group underlies rational solutions to the famous quantum Yang-Baxter equation. It was identified on several observables within the maximally supersymmetric Yang-Mills theory in four dimensions or its dual string theory on the background AdS5 x S5. While the Yangian can naturally be defined in two dimensions, it is a challenge to understand how it is realized in a 4d quantum field theory. Moreover, in two dimensions quantum group symmetries are known to impose powerful constraints, e.g. on the scattering matrix. For the above Yang-Mills theory in four dimensions, however, similar constraints are far from being understood. As recently discovered, the AdS/CFT system has a further nonlocal symmetry which acts as a generator of the so-called spectral parameter. Symmetries with this property are termed master symmetries and appear in many integrable models in 2d. Understanding their role for the AdS/CFT correspondence, however, is just at its beginning. Generically, the spectral parameter plays a crucial role for integrable models. It allows to package conservation laws in compact form and to make important statements using the theorems of complex analysis. While studying the integrability of AdS/CFT results in powerful tools and novel insights, four-dimensional integrability is found in various situations not directly related to this duality. Examples are the Regge limit of scattering processes in QCD or certain field theories with less supersymmetry. The latter cases show similar structures as the AdS/CFT system but a general framework for their study is not known. The following points are addressed by this proposal: To establish the role of the novel master symmetry of AdS/CFT, to identify the Yangian as an integrability tool on various observables and to develop a unified picture of integrable structures in four-dimensional field theories.

Financer

Duration of project

Start date: 10/2017

End date: 09/2019

Research Areas

Research Areas