$ \mathcal{N} $ = 4 super-Yang-Mills in LHC superspace part I: classical and quantum theory

We present a formulation of the maximally supersymmetric $ \mathcal{N} $ = 4 gauge theory in Lorentz harmonic chiral (LHC) superspace. It is closely related to the twistor formulation of the theory but employs the simpler notion of Lorentz harmonic variables. They parametrize a two-sphere and allow us to handle efficiently infinite towers of higher-spin auxiliary fields defined on ordinary space-time. In this approach the chiral half of $ \mathcal{N} $ =4 supersymmetry is manifest. The other half is realized non-linearly and the algebra closes on shell. We give a straightforward derivation of the Feynman rules in coordinate space. We show that the LHC formulation of the $ \mathcal{N} $ = 4 super-Yang-Mills theory is remarkably similar to the harmonic superspace formulation of the $ \mathcal{N} $ = 2 gauge and hypermultiplet matter theories. In the twin paper arXiv:1601.06804 we apply the LHC formalism to the study of the non-chiral multipoint correlation functions of the $ \mathcal{N} $ = 4 stress-tensor supermultiplet.