MSc defense By Mikkel Lauritzen

Fermionic Duality for Integrable Super Spin Chains

This thesis examines a fermionic duality present in the super spin chain interpretation of the $\mathfrak{su}(2|1)$ sector of $\mathcal{N}=4$ super Yang-Mills theory. By a direct Feynmann diagram calculation, the two-point correlation functions are determined in the planar limit at one-loop order for the $\mathfrak{so}(6)$ sector and the gauge-invariant operators with definite conformal dimension are shown to correspond to solutions of one-dimensional spin chains for the $\mathfrak{su}(2)$ sector.

The spin chains are then solved by the Coordinate Bethe Ansatz for the $\mathfrak{su}(2)$ sector and by the Nested Bethe Ansatz for the $\mathfrak{su}(2|1)$ sector and are shown to include a fermionic duality transformation. The Algebraic Bethe Ansatz is reviewed and used to prove a closed-form formula that relates the norm of Bethe states to the Gaudin matrix. A defect version of the field theory is investigated where one-point correlation functions can be expressed in terms of the superdeterminant of the Gaudin matrix. The dual spin chain solutions are then shown to correspond to the different choices of grading of the Lie superalgebra of the theory.

The transformation rules of the superdeterminant appearing in the subsector $\mathfrak{su}(2|1)$ are then examined under the fermionic duality transformation of a non-momentum-carrying node of the Dynkin diagram.

Finally, an attempt to prove that the superdeterminant transforms covariantly under this duality is made with promising results.