The Bethe/Gauge Correspondence

36 Chapter 3. Gaugeexpansion we will write down the possible terms in the Lagrangian, including the kinetic terms,interactions, and mass terms. There is no reason to restrict ourselves to just one chiral multiplet,and we will see how we can accommodate for multiple fields in pure chiral theory, and brieflydiscuss supersymmetric sigma models.3.2.1 Superfields ITo get a better feeling for the superfields, we start by working out their expansion in componentfields. The full expansion does not look very nice, but there is also a more compact way ofwriting the expansion, which clearly shows the field content of the supersymmetry multiplets.Chiral superfields. To find the most general chiral superfield Φ obeying (3.7), notice that thecoordinatesy 0 := t − i (θ − ¯θ− + θ + ¯θ+ ) , y 1 := x + i (θ − ¯θ− − θ + ¯θ+ ) ,satisfy ¯D ± y µ = 0. It’s convenient to form the right and left moving combinationsy ± := y0 ± y 12= x ± − i θ ± ¯θ± .In terms of the coordinates y ± , θ ± , ¯θ ± the supercovariant derivatives readD ± =∂∂θ ± − 2i¯θ ± ∂∂y ± ,¯D± = − ∂∂ ¯θ ±which means that the most general Φ only depends on y ± and θ ± :Φ(y ± , θ ± ) = φ(y ± ) + √ 2 θ − ψ − (y ± ) + √ 2 θ + ψ + (y ± ) + 2 θ − θ + F (y ± ) . (3.11)Here φ and F are complex scalar fields, and (ψ − , ψ + ) is a Dirac spinor. We’ll learn more aboutthese fields when we write down the Lagrangian in the next section. (The factors of √ 2 aretraditional; they can of course be absorbed via a redefinition of the ψ ± .)We can further work out the compact expansion (3.11) to get the full result:Φ = φ(x µ ) − i θ − ¯θ− (∂ 0 − ∂ 1 ) φ(x µ ) − i θ + ¯θ+ (∂ 0 + ∂ 1 ) φ(x µ ) + θ − θ + ¯θ− ¯θ+ (∂ 2 0 − ∂ 2 1) φ(x µ )+ √ 2 θ − ψ − (x µ ) − √ 2i θ − θ + ¯θ+ (∂ 0 + ∂ 1 ) ψ − (x µ )+ √ 2 θ + ψ + (x µ ) + √ 2i θ − θ + ¯θ− (∂ 0 − ∂ 1 ) ψ + (x µ )+ 2 θ − θ + F (x µ ) .Let’s explicitly find the action of a supersymmetry transformation on Φ,δ ε Φ = (−ε − Q + + ε + Q − + ¯ε − ¯Q+ − ¯ε + ¯Q− ) Φ ,where ε is a spinorial parameter. As for the supercovariant derivatives, we can express Q ±and ¯Q ± in terms of y ± , θ ± , ¯θ ± :Q ± =∂∂θ ± ,¯Q± = − ∂ − ∂ 2iθ±∂ ¯θ± ∂y ± .From this it’s easy to work out that the component fields transform asδ ε φ = √ 2 (ε + ψ − − ε − ψ + ) ,δ ε ψ ± = ± √ 2i ¯ε ∓ ∂ ± φ + √ 2 ε ± F , (3.12)δ ε F = − √ 2i (¯ε − ∂ + ψ − + ¯ε + ∂ − ψ + ) .We can represent the way the component fields are mixed by supersymmetry as