The Bethe/Gauge Correspondence

viPrefaceA note on notationMany aspects of supersymmetry are more nicely formulated in a two-component (Van der Waerden)notation for spinors (involving θ α and ¯θ ˙α ). However, we won’t need this notation when wetalk about supersymmetry in two dimensions. To avoid explaining the two-component notation,we use another, more intuitive notation for spinors in the general discussion of supersymmetryin Section 1.2.2: a four-component notation θ a for (Majorana) spinors which is adapted fromFigueroa-O’Farrill [5]. The aim of Section 1.2.2 is to introduce the main features of supersymmetryand the general structure of the supersymmetry algebra, and the four-component notationallows us to do this without getting into details.On the other hand, in order to work out how N = (2, 2) arises as a dimensional reductionfrom four dimensions, it is useful to be acquainted with the two-component notation as well.For more about this notation see e.g. Appendix A of Wess and Bagger [6], and Appendices A.5and A.6 of [5], or [7] for a very gentle introduction. In terms of the conventions of [6], the twoandfour-component way of writing spinors are related by( )θ a θα= ¯θ ˙α , θ a = (θ α , ¯θ ˙α ) and θ a ψ a = −(θ α ψ α + ¯θ ˙α ¯ψ ˙α ) .The notation in the papers [1, 2] varies a bit from place to place. Since it mostly follows theconventions and notation of Witten [8], who in turn follows Wess and Bagger, we will also usemost of the notation and conventions of the latter. For example, we will take the metric to havesignature (−1, 1, 1, 1), useσ µ := (− 1, ⃗σ) and ¯σ µ := (− 1, −⃗σ) , (0.1)and employ the Weyl basis for the gamma matrices:( )(γ µ ) a 0 σµb =¯σ µ . (0.2)0This basis is related to the canonical basis, in which γ 0 is given by diag(− 1, 1), by a similaritytransformation using the orthogonal matrix( )1 1 − 1√ .2 1 1With our choice of signature, the Clifford algebra thus comes with a minus sign:{γ µ , γ ν } = −2 η µν 1 . (0.3)BackgroundThis thesis is mostly the result of literature research, and none of the results are new. What Ihave done is to try and write a pedagogical and self-contained introduction to the Bethe/gaugecorrespondence, aimed at fellow Master’s students who do not have a background in integrabilityor supersymmetry. This means that• all the necessary prerequisites are covered, and I give references to further backgroundinformation;• when a new topic or quantity is introduced, I try to motivate its use or relevance;• important calculations are worked out in more detail, again providing references wherenecessary.The following references were especially useful. The general exposition of quantum integrabilityin Section 1.1 has been inspired by [9, 10], and some parts of Section 1.2.2 aboutsupersymmetry by [5, 6].