The Nicolai Map and its Application in Supersymmetric Field Theories

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.Comment: PhD thesis, 174 pages, Humboldt University of Berlin, contains arXiv:2005.12324, arXiv:2006.02457, arXiv:2104.06017 and arXiv:2206.0291

Perturbative linearization of super-Yang-Mills theories in general gauges

Supersymmetric Yang-Mills theories can be characterized by a non-local and non-linear transformation of the bosonic fields (Nicolai map) mapping the interacting functional measure to that of a free theory, such that the Jacobi determinant of the transformation equals the product of the fermionic determinants obtained by integrating out the gauginos and ghosts at least on the gauge hypersurface. While this transformation has been known so far only for the Landau gauge and to third order in the Yang-Mills coupling, we here extend the construction to a large class of (possibly non-linear and non-local) gauges, and exhibit the conditions for all statements to remain valid off the gauge hypersurface. Finally, we present explicit results to second order in the axial gauge and to fourth order in the Landau gauge

Perturbative linearization of supersymmetric Yang-Mills theory

Supersymmetric gauge theories are characterized by the existence of a transformation of the bosonic fields (Nicolai map) such that the Jacobi determinant of the transformation equals the product of the Matthews-Salam-Seiler and Faddeev-Popov determinants. This transformation had been worked out to second order in the coupling constant. In this paper, we extend this result (and the framework itself) to third order in the coupling constant. A diagrammatic approach in terms of tree diagrams, aiming to extend this map to arbitrary orders, is outlined. This formalism bypasses entirely the use of anti-commuting variables, as well as issues concerning the (non-)existence of off-shell formulations for these theories. It thus offers a fresh perspective on supersymmetric gauge theories and, in particular, the ubiquitous N = 4 theory. © 2020, The Author(s)

Coupling of branes and twisted self-duality in the Maxwell-Chern-Simons theory

Abstract We study three approaches to electric-magnetic duality in the 4-dimensional Maxwell theory coupled to a dyonic point charge and in the 5-dimensional Maxwell-Chern-Simons (MCS) theory coupled to an electric point charge and a magnetic string charge. The three approaches have been developed by Dirac, Bunster and Henneaux, and Pasti, Sorokin and Tonin (PST). In Dirac’s formulation, the electric magnetic duality is realized only on the level of the equations of motion. The other two formulations introduce a dual (magnetic) gauge potential to induce manifest twisted self-duality in the action. In particular, we study the relations connecting the three approaches. The main results of this paper are the Bunster-Henneaux and PST formulations of the MCS theory with sources. We compare our result to the PST formulation of 11-dimensional supergravity coupled to the M2- and M5-brane by Bandos, Berkovits, and Sorokin

Two Loop Ghost free Quantisation of Wilson Loops in N=4\mathcal{N}=4 supersymmetric Yang-Mills

We report a perturbative calculation of the expectation value of the infinite straight line Maldacena-Wilson loop in $\mathcal{N}=4$ supersymmetric Yang-Mills theory to order $g^6$. Thus, we extend the previous perturbative result by one order. The vacuum expectation value is reformulated in terms of a non-linear and non-local transformation, the Nicolai map, mapping the full functional measure of the interacting theory to that of a free bosonic theory. The results are twofold. The perturbative cancellations of the different contributions to the Maldacena-Wilson loop are by no means trivial and seem to resemble those of the circular Maldacena-Wilson loop at order $g^4$. Furthermore, we argue that our approach to computing quantum correlation functions is competitive with more standard diagrammatic techniques.Comment: 11 pages; v2: two paragraphs added in the introduction, 2 references added, matches published versio