There is large and expanding array of problems in fundamental physics (ranging from sub-nuclear structure of matter through astrophysics and cosmology) which deal with systems whose theoretical description involves continuously many variables with strong interconnection between them. Quantum field theory is a unifying mathematical language for such systems. Appropriately, this language is difficult, and in general its 'grammatical rules' are not even yet established. In this situation, any instances, or 'models', of quantum field theory whose equations can be worked out all the way through are especially valuable. Such models are known as 'integrable quantum field theories'. Mainly, they serve as the testing ground for verification and development of ideas and mathematical methods, but remarkably many integrable 'models' directly apply to theoretical understanding of properties of complex materials. That is why today the integrable quantum field theories constitute one of the most important and actively developed area of mathematical physics. Moreover, since the discovery of 'gauge/string duality' and the role of integrability in this relation, this field of research has found itself at the cutting edge of theoretical high-energy physics. Unfortunately, the integrable models emerging in this context still resist full solution. This research project proposes further development of integrable quantum field theories, with emphasis on the so-called integrable sigma models. This is exactly the general class of models which emerge in the context of the gauge/string duality. Also, similar models are believed to provide theoretical understanding of particularly complicated disordered systems in physics of materials. It is proposed to develop a certain new class of integrable sigma-models, with novel mathematical structure, and find full solutions. This is an intermediate step in the approach to the systems of direct application in the gauge/string dualities. Interestingly, mathematical methods developed for integrable quantum field theories turn out to have broader significance in mathematical physics, and often help to investigate a wider class of non-integrable systems. This area will be explored as well.The activity consists of three projects. (1) Analysis of a new class of integrable sigma models whose target spaces are deformed group manifolds. With the most general known deformations, quantum integrability of such models require significant modifications of standard tools such as the Lax representations, the Yang-Baxter algebras, and Baxter's Q operators and their relations. The project's goal is to develop all such modifications, and use that as the basis for finding full solutions. The key to our approach will be the so-called ODE/IQFT correspondence, the new mathematical tool of integrable quantum field theories ('IQFT side') which allows to 'encode' its algebraic structures into a system of ordinary differential equations ('ODE' side). This project lays in the mainstream of the current research in mathematical physics; its significance is in the relation to the gauge/string dualities, and possibly to physics of disordered systems. The other two parts the research is to explore applications of the methods of integrability in non-integrable systems. (2) The relation between the 't Hooft's equation in 1+1 QCD and the Baxter's T-Q equation will be used to analyze the meson decays and scattering amplitudes in the large-N expansion, with the ultimate goal of gaining universal tools for analysis of decays and amplitudes in generic 1+1 systems with confining interactions, which are very common in condensed matter physics. (3) The third part addresses possibility of emergence of near-integrable subsystems in otherwise non-integrable quantum field theories, the possibility which was not previously explored.
|Effective start/end date||9/1/14 → 8/31/17|
- National Science Foundation (National Science Foundation (NSF))