Vafa-Witten invariants of projective surfaces

1994年，物理学家瓦法和维滕提出了四维空间的新“不变量”。这些不变量“计数”了四维空间上某个方程（N=4超对称杨-米尔斯方程）的解，并且应该告诉我们一些关于空间的信息。杨-米尔斯场的每一个整数电荷都有一个。受到弦理论中电磁对偶性的推广的启发，瓦法和维滕预测，在一个固定的空间上，人们可以把所有这些不变量放在一个生成级数（泰勒级数或傅里叶级数，系数为瓦法-维滕不变量）中，并得到一个非常特殊的函数，称为“模形式”。特别是不变量应该具有隐藏的对称性，这意味着只有有限数量的不变量决定了所有其他不变量。直到现在，数学家们还无法理解在没有得到无穷大的情况下如何进行这种“计数”。这个项目给出了一个定义，任何空间是“投射”，并为任何收费（包括那些麻烦的“半稳定”或“可约”的解决方案出现）。然后，我们将计算许多具有负曲率的空间的不变量。我们还将产生包含更多信息的“精炼”Vafa-Witten不变量。这些应该是物理学家寻求的不变量，旨在描述“拓扑扭曲的最大超对称5d超杨-米尔斯理论”。

In 1994 the physicists Vafa and Witten introduced new "invariants" of four dimensional spaces. These invariants "count" solutions of a certain equation (the N=4 supersymmetric Yang-Mills equations) over the four dimensional space, and should tell us something about the space. There is one for every integer charge of the Yang-Mills field.Motivated by a generalisation of electromagnetic duality in string theory, Vafa and Witten predicted that on a fixed space, one could put all these invariants together in a generating series (a Taylor series or Fourier series, with coefficients the Vafa-Witten invariants) and get a very special function called a "modular form". In particular the invariants should have hidden symmetries that mean that only a finite number of them determine all the rest.Until now mathematicians have been unable to make sense of how this "counting" should be done without getting infinity. This project gives a definition for any space which is "projective", and for any charge (including ones for which troublesome "semistable" or "reducible" solutions appear). We will then compute the invariants for many such spaces with negative curvature. We will also produce "refined" Vafa-Witten invariants containing more information. These should be the invariants sought by physicists aiming to describe "topologically twisted maximally supersymmetric 5d super Yang-Mills theory".