Stongly Coupled Field Theories, String Theory and Gravity

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This project is concerned with string theory and quantum field theory (QFT). There are two broad aims. Part I is to use tools inspired from string theory to describe otherwise intractable regimes of QFT. Part II is to use new geometric tools within string theory to describe aspects of gravity and our observable universe. QFT describes interactions at the sub-atomic level of our universe exceptionally well, underpinning all experimentally verified particles and interactions. The equations of QFT, except in special circumstances are unwieldy and not amenable to a direct analysis. The standard approach is to approximate the equations in a manner known as perturbation theory. This requires the interactions between particles be weak, not a universal situation. Often, the interactions are strong, a situation known as strong coupling, and the perturbation theory approximation breaks down. This problem is a major limitation for understanding many aspects of particle physics. Moreover, QFT does not describe macroscopic interactions, in particular gravity. If we are dealing with very dense objects such as black holes, then we need to find a theory that incorporates Einstein's theory of general relativity and QFT. The leading candidate that does this is string theory. In order to work, it requires stringent mathematical conditions be imposed. For example, in addition to the three dimensions we observe, there must exist six additional dimensions, whose geometry is very small and so not visible to present day experiment. A rough analogy is with a hose: from a distance it looks one-dimensional, but on closer inspection there is an additional circular direction. Describing the physics of the observable universe becomes a problem closely tied to the geometry of certain spaces, and conversely, demanding sensible physics as an output of string theory leads to new geometric techniques. One can then understand quantum corrections as coming from the string theory itself. String theory has led to new ideas in our understanding of QFT and gravity. We start with a strongly coupled QFT. Holography is an equivalence between two theories, let us call them theory A and theory B. Theory A is a d+1-dimensional gravity theory while theory B is QFT in flat (without gravity) d-dimensional space. Holography means that theory A can be utilised to learn about strong coupling aspects of theory B and vice versa. For example, theory B can be integrable, meaning it is completely soluble, and information about the strong coupling regime is determined purely by symmetry. Holography means we can determine (perhaps obscure) properties of theory A, the gravity theory. Conversely, via classical gravity, theory A can be used to describe regimes of strong coupling in theory B. Even if theory A nor theory B are not realistic models - one typically makes simplifying assumptions for the dualities to work -- one might hope the resulting features are universal, teaching us some new lessons on otherwise difficult problems in particle physics. Part I of this proposal is concerned with developing holography in a new paradigm of examples, as well as using integrability to explore properties of QFT. Next, in the context of gravity, new ideas have arisen in understanding black holes. Symmetries that derive from string theory, e.g. supersymmetry, have led to novel techniques for obtaining new types of black holes, as well as understanding their geometric and physical properties. Many interesting questions arise: what is the role of quantum corrections to these black hole solutions? Are they stable? A different but related question is how can we use string theory to describe quasi-realistic phenomenological models? Doing so requires understanding the geometry of spaces. What types of geometries lead to realistic models of our universe? What is the role of quantum corrections? These are the types of questions that form part II of this proposal.