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The nonlocal bosonic theory obtained from integrating out all anticommutingand auxiliary variables in a globally supersymmetric theory is characterized bythe Nicolai map. We present a universal formula for the latter in terms of anordered exponential of the integrated coupling flow operator, which can becanonically constructed. Also for supersymmetric gauge theories, this allows usto perturbatively construct the Nicolai map explicitly in terms of treediagrams. For off-shell supersymmetry this works in any gauge, in the on-shellcase the Landau gauge is required. The dimensional reduction of $D{=}10$ superYang-Mills to maximally supersymmetric SU($N$) matrix mechanics (the BFSSmodel) is known to provide a regularization of the $D{=}11$ supermembrane inlight-cone gauge via its incarnation as a one-dimensional gauge theory ofarea-preserving membrane diffeomorphisms. We show how a well-definedcorresponding Nicolai map perturbatively linearizes the supermembrane in thesmall-tension regime and points the way for a computation of vertex-operatorcorrelators.

The Nicolai map for super Yang-Mills theory and application to the supermembrane