Geometric structures and twisted supersymmetry

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Since the era of Newton, there has been a close relationship between developments in mathematics and theoretical physics. One of the greatest challenges in the modern age is to formulate a unified theory combining the two pillars of our current understanding of the universe: the Standard Model of particle physics and Einstein's General Relativity. Despite their extraordinary successes, these two theories are fundamentally incompatible and it seems that, once again, a resolution of this physical problem will require great advances in mathematics.Superstring theory is a promising candidate for a unified theory, but it exists only in ten-dimensional spacetime. One way to explain the four-dimensional world that we observe via string theory is to view the extra six dimensions as a tiny compact geometric space (like the surface of a sphere, but with more dimensions) whose presence can only be directly detected by particles of such high energy that they are not created in current experiments. In this construction, the shape of this compact space determines the four-dimensional laws of physics we can see around us in experiments. Further, understanding these spaces is not only important for physical models in string theory - they are also the subject of deep questions in pure mathematics research.This project aims to develop new understanding of their geometric properties, via a new physical approach to the mathematics. In the simplest scenarios, the spaces which one requires turn out to be objects, called special holonomy manifolds, that have been studied by mathematicians for decades. Many researchers have focused on spaces in this class called Calabi-Yau manifolds, as they are physically promising and relatively easy to construct. However, more general solutions exist, with additional physical fields called fluxes, and these have physically desirable features. Recently, a new mathematical notion of geometry, "generalised geometry", has been developed which includes these fluxes naturally, and this provides an elegant description of the more general solutions. This is just one of many examples of physics leading to new mathematics and the mixing of these two disciplines has often led to astounding progress on both sides.An important observation, which linked string theory and geometry even more closely, was that on Calabi-Yau manifolds there exist simplified string theories, called topological string theories, whose physics are more tractable and directly encode interesting mathematical features of the geometries. In particular, they encode quantities which are invariant under smooth deformations of the geometry. These invariants are regarded as key mathematical properties of the spaces and are used, for example, to determine if constructed examples of such spaces are really different or not, which can be very hard to ascertain otherwise. Conversely, the invariants encode information about the physics of topological strings, which in turn can provide exact answers to calculations in the full string theory.However, all of this has only been understood in the cases of Calabi-Yau manifolds. There are signs that such theories exist in more general situations, and that generalised geometry is the natural way to approach their constructions. One scenario where this is particularly relevant is actually another type of special holonomy manifold: seven-dimensional spaces called G2 manifolds. These have become a particular focus of the geometry community, and even though they still have zero flux, generalised geometry has been seen to give an elegant "physical" approach to these spaces.This project aims to construct analogues of topological string theories in this wider context and to use these to discover new invariants, which will become key objects in both the mathematical and physical understanding of these spaces.