Supersymmetric Gauge Theory and Enumerative Geometry

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Quantum field theory is the fundamental framework used by physicists to describe the world around us. It is spectacularly successful in describing a diverse range of phenomena, from the standard model of elementary particle physics to exotic phases of matter. In theories of weakly interacting particles, the outcomes of experiments such as the scattering and decay of particles can be predicted with astounding accuracy. A stunning example is the quantum theory of the electromagnetic field, where the theoretical prediction for the magnetic dipole moment of the electron agrees with experiment to more than 10 significant figures. However, there are many strongly interacting systems in nature, such as the strong nuclear force, where such calculations are not valid. Moreover, there are fascinating and important quantum mechanical phenomenon that only occur in strongly interacting systems. Therefore, despite the remarkable experimental success of quantum field theory, many fundamental questions remain to be understood. As in many other areas of science and mathematics, when faced with a seemingly insurmountable problem, it is useful to study simple examples that are both solvable and exhibit the phenomenon of interest. Supersymmetry plays this role in quantum field theory and has provided tremendous insight into strongly interacting systems. Furthermore, supersymmetric quantum field theories have deep connections to pure mathematics and the cross-fertilisation of ideas between these disciplines has been extremely fruitful. Discovering the underlying reason for this connection is surely an important step on the road to a full understanding of quantum field theory. My research lies at the interface of supersymmetric quantum field theory and an area of mathematics known as enumerative geometry. The basic idea of enumerative geometry is to count the number of solutions to geometric problems, such as how many lines intersect two points in a plane. This is just the beginning of a vast and fascinating area of mathematical research where the notion of `counting' takes on ever more sophisticated forms and incorporates the symmetries of geometric problems. It turns out that enumerative geometry appears in profound and surprising ways in supersymmetric quantum field theories. The interaction between these disciplines is beneficial in both directions. On one hand, insight from supersymmetric quantum field theory has the potential to generate new conjectures and computational techniques in mathematics that might otherwise lay undiscovered. On the other, mathematics allows us to refine our physical understanding by distilling the underlying physical principles into precise mathematical statements.