String theory approach to problems of strongly coupled gauge theories

Most of modern quantum field theory is based on the remarkableframework of Yang and Mills, who used structures that also occur ingeometry to describe the dynamics of elementary particles. However,despite the fact that the predictions of this theory have beenrigorously tested experimentally, its mathematical foundation is stillnot fully understood. Namely, an important open problem exists on howto analytically compute the spectrum of this theory at strong coupling. Though there are computer simulations which reproduce the observed spectrum,a theoretical understanding of it is still missing.The main obstacle in computing the spectrum of Yang-Mills theory, isthat one needs to understand a system at strong coupling; nonatural small parameter exists which would allow for a standardperturbative approach. Some ten years ago, an ingenious conjecturewas proposed by Maldacena on how to address this long-standingproblem. The idea is very simple, though conceptually challenging:instead of analysing the theory in four space-time dimensions of ourworld, he proposed to consider string theory in a higher-dimensional,curved space. He conjectured that this higher-dimensional stringtheory is equivalent to the lower-dimensional theory of Yang and Millsand proposed a specific map between the two. The key point of thismap is that it inverts the coupling between the two theories: hencestrongly coupled (and hard to address) phenomena in one theory aremapped to weakly coupled (and easy to compute) phenomena in the othertheory.Though the conjecture has so far been checked in specific limits, its proof isstill lacking. Recent discoveries of integrable structures inmaximally supersymmetric Yang-Mills theory (N=4 SYM)have introduced new ideas on how one could prove theconjecture. The main goal of my research is to work towards a proof ofthe string/gauge theory correspondence using these insights.The methods which will be used require a combination of severalinterdisciplinary techniques coming from integrability, quantum fieldtheory, string theory and group theory. As such, the project is verychallenging and a construction of the proof is likely to lead to newdevelopments in mathematics and physics.Although the initial conjecture was formulated for a very specificcase of Yang-Mills theory (the N=4 SYM theory) there are strongindications that the ideas should hold much more generally. Pushingthe limits of the conjecture and understanding where it breaks isanother goal of my research. Since the conjecture opened up a conceptuallynew way of looking at field theory and gravitational(i.e. geometrical) phenomena, understanding to what kind of systems itcan be applied will teach us about fundamental properties ofYang-Mills theories.