Local Mirror Symmetry and Five-dimensional Field Theory

The Universe becomes surprisingly simple at very large and at very small scales. Quantum field theory (QFT) is the simplest framework used in modern physics to describe both the very large and the very small. It is a very successful scientific theory, whose predictions have been confirmed experimentally to an astonishing degree, from the realms of particle physics to astrophysics. Nonetheless, one century after its inception, QFT remains a somewhat ad-hoc structure: Firstly, the physicists' understanding of QFT remains somewhat superficial, and breaks down in the so-called strong-coupling regime. Secondly, QFT is very poorly understood mathematically. This uncomfortable position of QFT is less a problem than a motivation to refine our tools, both in physics and in mathematics.

My research focusses on supersymmetric QFT and on its mathematical applications. Supersymmetry is an elegant idea from theoretical physics which posits an equivalence between particles of forces (like the photon) and particles of matter (like the electron). It is an incredibly useful tool to study QFT at a more fundamental level, because it allows one to relate many QFT phenomena, such as vacuum degeneracies and particle excitations, to geometric concepts such as algebraic varieties (like the zeros of a polynomial) and enumerative geometry (the counting of various geometric objects), which are of interest to pure mathematicians.

This research project consists of two interconnected strands. Firstly, we will study a conjectural map between certain singular geometries, called canonical singularities, and five-dimensional superconformal field theories (5d SCFT), a type of QFT that lives in five space-time dimensions instead of the four we see around us. These theories appear naturally as limits of string theory and of its 11-dimensional completion, M-theory. The main aim is to connect 5d SCFT to local mirror symmetry, which is a string theory relation that has become a very rich area of study in pure mathematics. Mirror symmetry is the statement that two very different geometric objects can be `the same' as far as quantum physics is concerned. This leads to very beautiful and surprising mathematical relations between `shapes' (algebraic geometry) and `volumes' (symplectic geometry), which mathematicians have been working on for many years. This research programme will give a new perspective on some of these mirror symmetry relations, thus furthering the dialogue between string theory and geometry.

The second part of the project concerns the computation of quantum observables in 5d SCFT with cutting-edge tools called supersymmetric localisation techniques. These observables give `quantum invariants', which are objects that the QFT assign to a smooth space-time manifold (such as a sphere) which is locally independent of the metric, generalising Donaldson polynomials. We will uncover and explore new relations between these quantum invariants, on the one hand, and the enumerative geometry of the canonical singularities associated to the 5d SCFTs, on the other hand. This will lead to another bridge between QFT and pure mathematics.

Thus, both strands of this research project aim to create the groundwork for future dialogues between physics and mathematics. Maintaining this dialogue is crucial for the health of both fields of investigation: physical methods uncover new mathematical relations that would not have been guessed otherwise, opening the way to new mathematical theories, and mathematical approaches in turn lead to a deeper understanding of physics. In due time, this may well lead to a deeper understanding of our own Universe.