The Bethe/Gauge Correspondence

3.3. Supersymmetric abelian gauge theory 413.3.1 Superfields IIBefore we take a look at vector superfields, we discuss another type of superfield which plays animportant role in supersymmetric gauge theory in two dimensions.Twisted chiral superfields. There is a neat trick to find the most general form of twisted(anti)chiral fields. Recall that chiral superfields obey the twisted chiral conditions (3.9):D − Σ = ¯D + Σ = 0 .Define ‘twisted’ odd coordinates [16]ϑ − := θ − , ϑ + := ¯θ + , ¯ϑ− := ¯θ − , ¯ϑ+ := θ + (3.24)and corresponding ‘twisted’ supercovariant derivatives˜D ± :=∂∂ϑ ± − i ¯ϑ ± ∂ ± ,¯˜D ± := −∂∂ ¯ϑ + ± iϑ± ∂ ± .Notice that ˜D − and ¯˜D− coincide with their untwisted counterparts, while the other two areinterchanged (up to a sign):¯˜D + = −D + and ˜D + = − ¯D + . In terms of the new coordinates,then, twisted chiral superfields satisfy antichiral-looking conditions: ˜D ± Σ = 0. This immediatelytells us that Σ contains two complex scalar fields, say σ and E, and a Dirac spinor which wedenote by (˜χ − , ˜χ + ) = (χ − , ¯χ + ) in accordance with (3.24). We read off from the componentexpansion (3.13) of ¯Φ thatΣ(ỹ ± , ¯ϑ ± ) = σ(ỹ ± ) + √ 2 ¯ϑ − ˜χ − (ỹ ± ) + √ 2 ¯ϑ + ˜χ + (ỹ ± ) + 2 ¯ϑ − ¯ϑ+ E(ỹ ± ) ,ỹ ± = x ± + i ϑ ± ¯ϑ± .(Notice that ỹ + = y + , in accordance with ¯D + Φ = ¯D + Σ = 0.) In terms of θ ± , ¯θ ± and χ ± theresult isΣ(ỹ ± , θ + , ¯θ − ) = σ(ỹ ± ) + √ 2 ¯θ − χ − (ỹ ± ) + √ 2 θ + ¯χ + (ỹ ± ) − 2 θ + ¯θ− E(ỹ ± ) ,ỹ ± := x ± ∓ i θ ± ¯θ± .(3.25)From this we can work out the full expansion into component fields:Σ = σ(x µ ) + i θ − ¯θ− (∂ 0 − ∂ 1 ) σ(x µ ) − i θ + ¯θ+ (∂ 0 + ∂ 1 ) σ(x µ ) − θ − θ + ¯θ− ¯θ+ (∂ 2 0 − ∂ 2 1) σ(x µ )+ √ 2 ¯θ − χ − (x µ ) + √ 2i θ + ¯θ− ¯θ+ (∂ 0 + ∂ 1 ) χ − (x µ )+ √ 2 θ + ¯χ + (x µ ) − √ 2i θ − θ + ¯θ− (∂ 0 − ∂ 1 ) ¯χ + (x µ )− 2 θ + ¯θ− E(x µ ) .The sign of the last term is not important, and can be absorbed via a redefinition of E.Schematically, supersymmetry transformations act on the components asχ −σE¯χ +Twisted antichiral superfields. Like for (anti)chiral superfields, twisted antichiral superfieldsare conjugate to twisted chiral fields, so that we can e.g. write the expansion of an arbitrarytwisted antichiral superfield as¯Σ(¯ỹ ± , θ − , ¯θ + ) = ¯σ(¯ỹ ± ) + √ 2 θ − ¯χ − (¯ỹ ± ) + √ 2 ¯θ + χ + (¯ỹ ± ) − 2 θ − ¯θ+ Ē(¯ỹ ± ) ,¯ỹ ± = x ± ± i θ ± ¯θ± .(3.26)