From amplitudes at strong coupling to theories of Class S

对N=4个超杨-米尔斯理论的散射振幅的研究将振幅表示为某些参数空间上的函数，这些参数空间具有簇簇的结构。通过ADS/CFT的对应关系，全息术赋予它们一定的几何结构，通常是(伪)超卡勒结构，它编码了强耦合时的振幅。在S类理论的构造中出现了类似的结构。S类理论是四维超共形场理论，它是通过从6dN=(2，0)超共形场论中得到一个亏格为g的可能截断黎曼曲面而得到的。6dN=(2，0)理论是由M-理论构造的非拉格朗日理论。黎曼表面的穿孔可以编码关于S类理论味道对称性的数据。由此得到的S类理论通常是非拉格朗日的，然而，由于N=2的超对称性，低能物理可以用Seiberg-Witten方法来描述，在该方法中系统变得代数可积。在S类理论中，出现的具体可积系统是Hitchin系统。Hitchin系统被理解为定义在同一Riemann曲面上的Higgs丛，它允许解Hitchen方程。通过这种方式，可以将低能物理编码到紧凑化中使用的黎曼曲面上。这种数学结构的一个重要性质是可能Higgs丛的模空间具有自然的超Kahler结构。从强耦合N=4个超杨-米尔斯的角度出发，通过计算特定极小曲面的面积，即ADS边界上特定零轮廓的边界，得到振幅。极小化问题产生了一个Sinh-Gordon可积系统，它是Hitchen系统的一种特殊实现，面积可以用求解热力学Bethe Anzats(TBA)方程的Y系统的函数来表示。当我们试图在超Kahlermodi空间上构造坐标时，这些TBA方程更普遍地出现在希格斯丛结构中。求解Y系统的函数，以及更一般地，在模空间上的坐标具有簇变化结构。本项目的目的是试图在这两个构式之间建立某种类型的词典，并进一步发展它们的研究。此外，由于模空间的超卡勒性质，有希望尝试和应用扭曲方法并获得不同的观点。

The study of scattering amplitudes for N=4 super Yang-Mills theories expresses amplitudes asfunctions on certain parameter spaces that have the structure of cluster varieties. Holographyvia the AdS/CFT correspondence endows these with certain geometric structures, often(pseudo-) hyper-Kahler, that encodes the amplitude at strong coupling. Analogous structuresarise in the construction of theories of class S. Class S theories are super-conformal fieldtheories (SCFTs) in four dimensions, which are obtained via compactification using a possiblypunctured Riemann surface with genus g from a 6d N=(2,0) SCFT. The 6d N=(2,0) theory is anon-Lagrangian theory constructed via M-theory. The punctures of the Riemann surface canencode data about the flavor symmetry of the Class S theory. The resulting Class S theory isoften non-Lagrangian, however, due to the N=2 supersymmetry, the low energy physics can bedescribed using the Seiberg-Witten approach in which the system becomes algebraicallyintegrable. In the case of theories of Class S, the specific integrable system that appears is aHitchin system. Hitchin systems are understood as Higgs bundles defined over the sameRiemann surface, which admit solutions to the Hitchen equations. In this way, the low energyphysics can be encoded onto the Riemann surface used in the compactification. A key propertyof the mathematical constructions is that the moduli space of possibly Higgs bundles has anatural hyper-Kahler structure. From the perspective of the N=4 super Yang-Mills at strongcoupling, the amplitude is obtained by calculating the area of a particular minimal surface, whichis the boundary of a particular null contour on the boundary of AdS. The minimization problemresults in a sinh-Gordon integrable system which is a special realization of a Hitchen system.The area can be expressed in terms of functions coming from a Y-system that solvesThermodynamic Bethe Anzats (TBA) equations. These TBA equations appear more generally inthe Higgs bundle construction when one attempts to construct coordinates on the hyper-Kahlermoduli space. The functions solving the Y-system and, more generally, the coordinates on themoduli space have a cluster variety structure. The project aims to try to establish some kind ofdictionary between the two constructions and to develop their study further. In addition, due tothe hyper-Kahler nature of the moduli space, there is a hope to try and apply twistor methodsand gain a different perspective.