Week 08.02.2009 – 14.02.2009

The talk is based on a 1990 Phys. Rev. milestone paper Scientists attempt to understand physical phenomena by constructing models. A model serves as a link between scientists and nature, and one basic goal is to develop models whose solutions accurately reflect the nature of the physical process. But a dynamical model uses simplifying assumptions and approximations. The question of whether a model accurately reflects nature is one constantly faced by scientists. I will argue that there exist levels of mathematical difficulty, brought from the theory of dynamical systems, which can limit our ability to represent chaotic processes in nature using deterministic models. Specifically, in such cases, no model of such a system produces solutions of reasonable length that are realized by nature. Furthermore, for these processes, the numerical solutions of the models do not approximate any actual model solutions.

N=4 super Yang-Mills theory has a remarkable feature of being integrable. This discovery has made it possible to achieve strong evidence for the AdS/CFT duality. But what about its siblings: marginal deformations of N=4 SYM that preserve conformal symmetry. In this case much less is known about the conjectured supergravity dual. In particular, the Leigh-Strassler deformations are of this kind, which are only integrable for some special values of the deformation parameters, or in some restricted subsectors. In order to gain more insight into the theory, one would like to understand the symmetries of the theory better. We know that the symmetries of integrable systems can be described by Hopf algebras. Here we would like to show that a Hopf algebra structure actually emerges for the generic Leigh-Strassler deformation.

This talk briefly explains why finding the vacuum solutions of gauged maximal supergravity models is a mathematically challenging problem, and which techniques exist that are powerful enough to actually solve it. As these techniques require joint effort between physicists and computer scientists, the problem is explained in a way that should be accessible to both groups. In models of extended supergravity with more than one gravitino, the symmetry group transforming the gravitini into one another can be promoted to a local symmetry. In order to maintain supersymmetry, such a deformation mandates the introduction of a potential for the scalar fields whose stationary points correspond to vacua with spontaneously broken symmetry. The most famous such model is the SO(8)-gauged N=8 supergravity in four dimensions by de Wit and Nicolai, which also can be obtained by dimensional reduction of M-theory on the 7-sphere. The mathematical problem in the determination of the vacua lies in the complexity of the potential, which is a function on a high-dimensional Riemannian symmetric space based on an E-series exceptional Lie group. While previous group theoretic arguments allowed a partial investigation of the problem only, there are powerful semi-numeric computational techniques that indeed seem to be strong enough to solve the problem in full. These methods are explained using as an example the most challenging supergravity potential, that of N=16 in D=3 with SO(8)xSO(8) gauge group, and some first data on new vacua of N=8 supergravity in D=4 are presented.