Non-abelian Structures in String Theory

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Einstein's theory of General Relativity describes gravitational forces between large objects: the attraction between the Earth and the Sun that holds the Earth in its orbit around the Sun, or the forces that have been determining the rate of expansion of the Universe since the Big Bang. General Relativity is an elegant and accurate theory and its predictions have been thoroughly tested by observations. Its central idea is that space and time are dynamical entities and gravity is nothing more than space-time being deformed.Quantum Mechanics is the theory of the small: the atom, as well as the atom's constituents. It governs everything from superconductivity to the speed of chemical reactions, from the behaviour of electrons to that of the recently discovered Higgs particle. Quantum theory predictions agree with experimental results to an unprecedented accuracy.General Relativity and Quantum Mechanics are the two paradigms through which we describe our Universe; however, it is not understood how to combine them into a single framework. We lack a unified approach to describing physical phenomena, and this limits our ability to study some of the most fascinating objects in the Universe (such as Black Holes) which are at once quantum mechanical and gravitational in nature.String Theory is currently our best candidate for a framework through which these two theories can be brought together. It has some profound consequences for what we think might be possible in our Universe, such as existence of extra dimensions. The details of how String Theory allows General Relativity and Quantum Mechanics to come together are not well understood: one idea is that space and time are not fundamental concepts, instead, space-time and its dynamics (i.e., gravity) might emerge from some microscopic degrees of freedom in a way similar to that in which movement of water emerges from the collective behaviour of a large number of water molecules. General Relativity, then, should emerge together with space-time from a Quantum Mechanical model of some objects more fundamental than space and time themselves. Such a formulation would be the ultimate theory of Quantum Gravity. Towards this goal, my work explores emergent geometries and their properties through String Theoretic tools.In this Proposal, I outline a research program bringing together several recent advancements in Quantum Gravity research such as holography, entanglement entropy, noncommutative geometry and matrix models. The central theme here is non-abelian objects: mathematical entities for which x times y is not the same as y times x, of which simplest example is square matrices. Such non-abelian entities appear in String Theory as coordinates of fundamental objects called D-branes, providing us with clues towards the true nature of space-time. Unravelling these clues is the main purpose of my work.