Over the last generation the four-dimensional classical self-dual Yang-Mills equations and their dimensional reductions have had profound consequences for algebraic geometry, differential geometry, low-dimensional topology, integrable systems, and nonlinear PDE. Now a family of six-dimensional quantum field theories is emerging as an object of acute mathematical interest. This Focused Research Group brings together researchers working in diverse parts of mathematics and physics to study these theories. They originally arose as limits of string theories and are usually called 'superconformal (2,0)-theories' to call attention to their symmetries. The simpler appellation 'Theory X' emphasizes how little is known. The projects undertaken here have two overall goals. First, we will make inroads on the structure of Theory X by applying the detailed and profound mathematical understanding of topological and conformal quantum field theories obtained over the past 25 years. Second, we will use expected properties of Theory X and its compactifications to lower dimensions to deduce new conjectures and new organizing principles in geometric representation theory. The rapidly developing web of interactions between the six-dimensional quantum Theory X and a host of central topics in twenty-first century geometry, topology, and geometric representation theory indicates that we are seeing the beginnings of a new revolution, one in which Theory X plays the dominant physical role. Progress towards unraveling its structure and its consequences will have broad ramifications.
Physics has long fueled developments in mathematics, and the past 30 years have been a particularly fruitful period. The depth of mathematics which enters fundamental physical theories has steadily intensified, and at the same time the structure and predictions of these theories have had increasingly profound impacts on mathematics. This project is one of many efforts to mine this intellectually fertile mix of ideas. Our pursuit of Theory X will inevitably illuminate a much broader circle of ideas and contribute to the mathematical understanding of contemporary physics. The work of past generations at the mathematics-physics interface fuels the modern world: our computers, GPS systems, transportation, sophisticated medical tools, and much more owe their existence to basic research in this area which stretches back well over a century. While we cannot predict how current basic research will impact the future, we can say with certainty that the effect will be far-reaching.
