From amplitudes at strong coupling to theories of Class S

N=4超Yang-Mills理论的散射振幅研究将振幅表达为具有簇变结构的某些参数空间上的函数。通过AdS/CFT对应的全息摄影赋予这些具有一定的几何结构，通常（伪）超卡勒，在强耦合下编码振幅。类似的结构出现在S类理论的构造中。S类理论是四维的超共形场论（SCFTs），它是通过从6d N=(2,0) SCFT中使用具有g属的可能穿孔Riemann曲面的紧化得到的。6d N=(2,0)理论是由m理论构造的非拉格朗日理论。黎曼曲面的穿孔可以编码有关S类理论风味对称性的数据。由此产生的S类理论通常是非拉格朗日的，然而，由于N=2超对称，低能物理可以用Seiberg-Witten方法来描述，在这种方法中，系统变得代数可积。在S类理论中，出现的具体可积系统是阿希钦系统。希钦系统被理解为定义在同一个曼曲面上的希格斯束，它承认希钦方程的解。通过这种方法，低能物理可以被编码到紧化所用的黎曼表面上。数学结构的一个关键性质是可能希格斯束的模空间具有自然的超凯勒结构。从强耦合的N=4超级Yang-Mills的角度出发，通过计算特定最小曲面的面积来获得振幅，该曲面是特定零轮廓在AdS边界上的边界。最小化问题得到sinh-Gordon可积系统，这是Hitchen系统的一种特殊实现。该面积可以用求解热力学Bethe Anzats （TBA）方程的y系函数来表示。当人们试图在超kahlermoduli空间上构造坐标时，这些TBA方程更普遍地出现在希格斯束构造中。求解y系的函数，更一般地说，是模空间上的坐标，具有簇变结构。该项目旨在尝试在这两种结构之间建立某种词典，并进一步发展它们的研究。此外，由于模空间的超kahler性质，有希望尝试和应用扭转方法并获得不同的视角。

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.