From amplitudes at strong coupling to theories of Class S

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 structures

arise in the construction of theories of class S. Class S theories are super-conformal field

theories (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 can encode data about the flavor symmetry of the Class S theory. The resulting Class S theory is

often 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 algebraically integrable. In the case of theories of Class S, the specific integrable system that appears is a Hitchin system. Hitchin systems are understood as Higgs bundles defined over the same

Riemann surface, which admit solutions to the Hitchen equations. In this way, the low energy

physics 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 a natural hyper-Kahler structure. From the perspective of the N=4 super Yang-Mills at strong

coupling, 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 solves

Thermodynamic 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 methods and gain a different perspective.