Local Mirror Symmetry and Five-dimensional Field Theory

宇宙在非常大和非常小的尺度上变得令人惊讶地简单。量子场论(QFT)是现代物理学中用来描述极大和极小的最简单的框架。这是一个非常成功的科学理论，它的预测在实验上得到了惊人的证实，从粒子物理学领域到天体物理学领域。尽管如此，在QFT诞生一个世纪后，它仍然是一种有点特殊的结构：首先，物理学家对QFT的理解仍然有些肤浅，并在所谓的强耦合制度下崩溃。其次，量化宽松在数学上很难理解。QFT的这种令人不快的地位与其说是一个问题，不如说是激励我们改进我们的工具，无论是在物理上还是在数学上。我的研究重点是超对称QFT及其数学应用。超对称性是理论物理学中的一个优雅概念，它假设作用力粒子(如光子)和物质粒子(如电子)之间存在等价性。它是在更基本的水平上研究QFT的一个非常有用的工具，因为它允许人们将许多QFT现象，如真空简并和粒子激发，与几何概念，如代数簇(如多项式的零点)和计数几何(对各种几何对象的计数)，这些都是纯粹的数学家感兴趣的。首先，我们将研究某些奇异几何之间的猜想映射，称为正则奇点，和五维超共形场理论(5D SCFT)，这是一种存在于五个时空维度而不是我们周围看到的四个维度的QFT。这些理论自然而然地表现为弦理论及其11维完备的M理论的极限。其主要目的是将5DSCFT与局部镜像对称联系起来，这是一种弦理论关系，已成为纯数学中一个非常丰富的研究领域。镜像对称性是指，就量子物理而言，两个截然不同的几何物体可以是“相同的”。这导致了数学家们多年来一直致力于的“形状”(代数几何)和“体积”(辛几何)之间非常美丽和令人惊讶的数学关系。这项研究计划将对其中一些镜像对称关系提供一个新的视角，从而促进弦理论和几何之间的对话。项目的第二部分涉及使用称为超对称局域化技术的尖端工具计算5D SCFT中的量子可见。这些观测给出了量子不变量，量子不变量是QFT分配给局部独立于度规的光滑时空流形(如球体)的对象，推广了Donaldson多项式。我们将一方面揭示和探索这些量子不变量与与5D SCFT相关的正则奇点的计数几何之间的新关系。这将导致QFT和纯数学之间的另一座桥梁。因此，这项研究项目的两个方面都旨在为未来物理和数学之间的对话奠定基础。保持这种对话对这两个研究领域的健康至关重要：物理方法揭示了否则无法猜测的新的数学关系，为新的数学理论开辟了道路，而数学方法反过来又导致了对物理学的更深层次的理解。在适当的时候，这很可能会导致对我们自己宇宙的更深层次的理解。

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.