Supersymmetric Gauge Theory and Enumerative Geometry

量子场论是物理学家用来描述我们周围世界的基本框架。它在描述从基本粒子物理学的标准模型到物质的奇异相的各种现象方面都取得了巨大的成功。在弱相互作用粒子的理论中，粒子的散射和衰变等实验结果可以以惊人的准确性预测。一个令人震惊的例子是电磁场的量子理论，其中对电子磁偶极矩的理论预测与实验符合超过10位有效数字。然而，在自然界中有许多强相互作用系统，例如强核力，在那里这种计算是无效的。此外，还有一些迷人而重要的量子力学现象，只发生在强相互作用系统中。因此，尽管量子场论在实验上取得了显著的成功，但许多基本问题仍有待理解。正如在科学和数学的许多其他领域一样，当面对一个看似无法克服的问题时，研究既可解决又能展示有趣现象的简单例子是有用的。超对称性在量子场论中扮演了这个角色，并为强相互作用系统提供了巨大的洞察力。此外，超对称量子场论与纯数学有着深刻的联系，这些学科之间的思想交流非常富有成效。发现这种联系的根本原因无疑是全面理解量子场论的重要一步。我的研究是在超对称量子场论和数学领域称为枚举几何的接口。枚举几何的基本思想是计算几何问题的解的个数，例如平面上有多少条直线与两点相交。这只是数学研究中一个广阔而迷人的领域的开始，在这个领域中，“计数”的概念呈现出越来越复杂的形式，并结合了几何问题的对称性。事实证明，枚举几何在超对称量子场论中以深刻而令人惊讶的方式出现。这些学科之间的相互作用在两个方向上都是有益的。一方面，超对称量子场论的洞察力有可能在数学中产生新的理论和计算技术，否则可能会被发现。另一方面，数学使我们能够通过将潜在的物理原理提炼成精确的数学陈述来完善我们的物理理解。

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.