The Nicolai Map and its Application in Supersymmetric Field Theories

Abstract

In this thesis, we study the Nicolai maps of the 2-dimensional Wess-Zumino model, $\mathcal{N}=1$ super Yang-Mills and $\mathcal{N}=4$ super Yang-Mills. We compute the Nicolai map of the 2-dimensional Wess-Zumino model up to the fifth order in the coupling. In $\mathcal{N}=1$ super Yang-Mills, we introduce the notion of on- and off-shell Nicolai maps. The on-shell Nicolai map of $\mathcal{N}=1$ super Yang-Mills exists in d=3, 4, 6 and 10 dimensions but is constrained to the Landau gauge. We compute this map up to the fourth order. The off-shell Nicolai map exists only in d=4 dimensions but for general gauges. We compute it in the axial gauge up to the second order. We show that the $\mathcal{N}=4$ super Yang-Mills Nicolai map can be obtained from the Nicolai map of 10-dimensional $\mathcal{N}=1$ super Yang-Mills by dimensional reduction. Inverse Nicolai maps allow for a fermion (and ghost) free quantization of supersymmetric (gauge) theories. We apply this property to compute the vacuum expectation value of the infinite straight line Maldacena-Wilson loop in $\mathcal{N}=4$ super Yang-Mills to the sixth order. In the second part of this thesis, we derive the explicit field content of the 1/2-BPS stress tensor multiplet in $\mathcal{N}=4$ super Yang-Mills, which contains the R-symmetry current and the energy-momentum tensor. The original version of this thesis, as submitted in May 2023 to the Humboldt University of Berlin, is available under the DOI https://doi.org/10.18452/26406.