Niclai maps in super gauge theory and gravity

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In 1980 Hermann Nicolai discovered that globally supersymmetric theories can be characterized without fermionic degrees of freedom: The nonlocal purely bosonic theory obtained after integrating out all fermions (and potential ghosts) at any coupling g admits a nonlinear field transformation to the free theory (at g=0), the so-called Nicolai map. Alternatively, this map can be defined by equating the interacting correlation function of any field functional with the free correlation function of the inversely transformed field functional. In the early 1980s a perturbative construction scheme for the Nicolai map was developed in one, two and four spacetime dimensions, for Wess-Zumino models as well as for super Yang-Mills theory, by exponentiating a "coupling-flow operator" which induces an infinitesimal shift in the coupling. It led to an alternative formulation of supersymmetric perturbation theory, complementary to the standard super-space or component approach. The subject was all but forgotten until 2020, when Hermann Nicolai and collaborators revisited the map for super Yang-Mills theory, now also in higher dimensions, found that it exists only in the critical dimensions three, four, six and ten, and pushed its perturbative construction to (currently) fourth order. Since then, there has been a flurry of new results: a universal formula for the map, its non-uniqueness and ambiguities, criteria for its linearity, its gauge dependence, the large-order perturbation behavior and non-perturbative existence. In addition, we have seen applications to N=4 super Yang-Mills correlators, matrix models and the maximal supermembrane. These findings open a new perspective for a deeper structural analysis of the Nicolai map on one side and for practical applications to correlation functions in supersymmetry, supergauge theory and perhaps supergravity. This project has four parts. First, we plan to expand the Nicolai-map treatment of supersymmetric scalar theories. Specific topics here are the renormalization of the map, an extension to sigma models, the investigation of supersymmetry breaking, and nonrelativistic versions of the Nicolai map. The second part concerns super Yang-Mills theory. Here, we shall generalize the map to harmonic super-space, to supersymmetry-preserving gauge fixings, clarify the failure of linear maps in the light-cone gauge, and optimize the map for N=4 super Yang-Mills. Third, we will try to formulate a Nicolai map for supergravity, and fourth, we intend to revisit the super matrix model and the super-membrane, in order to overcome the technical obstacles with its quantum correlators via the Nicolai map in the small-tension limit. It is expected that successful completion of this project will establish the Nicolai map as a viable tool and a competitive calculational technique in supersymmetric quantum field theories and even offer a new way to quantize previously intractable cases like supergravity or the super-membrane.