Projectdetails

The rapid developments in modern day theoretical physics generate an abundance of new ideas in mathematics. The theme of this project is to combine exciting developments in physics and mathematics. Specifically, the physics includes topics in quantum field theory and string theory such as Seiberg-Witten theory, the AdS/CFT correspondence, quantum gravity, Dijkgraaf-Vafa theory and the Nekrasov instanton partition function. We plan to focus on the interplay of physics with themes in integrable systems such as matrix models, WDVV equations, Poisson structures and symmetric spaces.

A simple but fundamental observation in the development of physics and mathematics today is that breakthroughs in quantum field theory and string theory are usually characterized by the presence of integrable systems. Out of an abundance of examples we mention three. Two-dimensional topological field theory was solved by Witten in the early '90s and the genus zero part of its partition function was shown to be a tau function of the dispersionless Korteweg-de Vries hiererarchy. The second example is formed by Seiberg and Witten's complete solution of four-dimensional N=2 supersymmetric Yang-Mills theory. Here, the effective low-energy partition functions were shown, by Marshakov among others, to be tau functions of the Whitham hierarchy related to Hitchin type integrable systems. The third example is the AdS/CFT correspondence formulated by Maldacena in 1997, stating that a string theory on anti-deSitter space is equivalent to a conformal field theory on its boundary. It has recently become clear, by works of Arutyunov and Marshakov among others, that on both sides of this correspondence integrable systems, like spin chains and sigma models, play an important role. These integrable systems may be used to check the correspondence.

We propose extending and prolonging the interactions between quantum physics and integrable systems in a number of ways. We identify three subprograms summarized here and presented further in question 12.

The first is centered around the AdS/CFT correspondence and the occurrence of matrix models in string theory and gauge theory. As mentioned before, integrability can be of great use in checking the important AdS/CFT correspondence and can be a guideline to new developments of AdS/CFT.

Matrix models are useful for the description and solution of numerous problems in physics. Besides playing a major role in Witten's two-dimensional topological field theory, they also occur in more recent developments such as the AdS/CFT correspondence, Nekrasov's instanton partition function and Dijkgraaf-Vafa theory.

The second subprogram concerns nonlinear integrable PDE?s. We study the Witten-Dijkgraaf-Verlinde-Verlinde equations which govern various partition functions in physics (Witten's topological field theory, Seiberg-Witten theory, Dijkgraaf-Vafa theory). Furthermore we want to generalize the inverse scattering method to make it suitable for quantum integrable systems as well as particular versions of the KP hierarchy relevant for physics. Finally we intend to study and generalize Poisson structures for general integrable PDEs.

The third subprogram puts emphasis on representation theoretical aspects and is concerned with symmetric spaces, the geometric Langlands program and current algebras. Relations to physics include Berezin quantization and symmetries of quantum field theories.