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。从基本粒子物理的标准模型到物质的奇异相，它在描述各种各样的现象方面取得了惊人的成功。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精度进行预测。一个惊人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的一致性超过10个有效数字。然而，在自然界中有许多强相互作用的系统，比如强核力，这样的计算是无效的。此外，还有一些有趣而重要的量子力学现象只发生在强相互作用的系统中。因此，尽管量子场论在实验上取得了显著的成功，但仍有许多基本问题有待理解。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解决的问题时，研究既可以解决又能表现出感兴趣的现象的简单例子是有用的。超对称在量子场论中扮演着这个角色，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叉融合非常富有成果。在全面理解量子场论的道路上，发现这种联系的根本原因无疑是重要的一步。我的研究是在超对称量子场论和一个被称为枚举几何的数学领域的界面。枚举几何的基本思想是计算几何问题的解的数量，例如平面上两点相交的直线有多少。这只是一个广阔而迷人的数学研究领域的开始，在这个领域中，“计数”的概念呈现出越来越复杂的形式，并融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，来自超对称量子场论的见解有可能在数学中产生新的猜想和计算技术，否则这些猜想和计算技术可能会被发现。另一方面，数学通过将基本的物理原理提炼成精确的数学表述，使我们能够完善我们对物理的理解。

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.