The Bethe/Gauge Correspondence

3.5. The nonabelian case 57Here, the (component determining the) nonabelian gauge field strength is given by the expressionF 01 = ∂ 0 A 1 − ∂ 1 A 0 + [A 0 , A 1 ] = ∗(dA + 1 2[A, A]), and the dots contain further commutators ofthe component fields.Lagrangians. As usual, (3.32) is changed by taking the trace, and also involves gauge covariantderivatives and commutators:L gauge = − 1 ∫2 e 2 d 4 θ tr ¯ΣΣ= 1 e 2 tr (− ∇ µ¯σ ∇ µ σ + i ¯λ − (∇ 0 + ∇ 1 ) λ − + i ¯λ + (∇ 0 − ∇ 1 ) λ + + 1 2 D2 + 1 2 F 2 01− 1 2 [σ, ¯σ]2 − √ 2λ + [σ, ¯λ − ] + √ 2[¯σ, λ − ]¯λ +).The ϑ-term is changed in the same way:(3.66)L ϑ = ϑ2π tr F 01 . (3.67)For abelian gauge theory with G = U(1), the gauge field strength F 01 sources an electric field.In Section 1.3.2 we noticed that in two dimensions, such a field is constant, and the Colemaneffect explains why the parameter ϑ ∈ S 1 is periodic. In general, if G is nonabelian, the vacuumangle ϑ is still periodic with period 2π. Indeed, the gauge field strength can be written asF = dA + 1 2[A, A], and the first Chern class is given by4c 1 = 1 ∫2π tr F = 1 ∫tr dA = 1 ∫tr F 01 d 2 x .2π2πImposing appropriate boundary conditions we have that c i ∈ Z. Since ∫ L ϑ d 2 x = ϑ c 1 , the pathintegral e iS is invariant under shifts ϑ ↦→ ϑ + 2π, and therefore so is the physics.Recall that for G = U(1) the Fayet-Iliopoulos term is given by L r = −r D. For generalgauge group G, such a term can be turned on for each U(1)-factor in the centre of G, addingthe corresponding FI-terms. For G = U(N) we therefore haveL r = −r tr D .We can again combine the FI- and ϑ-terms in a tree level twisted superpotential∫L ˜W= iτ d 2 ϑ tr Σ∣ + h.c.ϑ± =0Chiral covariant superfields. We can use the gauge supercovariant derivatives (3.65) todefine chiral covariant superfields Φ ′ by¯D ± Φ ′ = e V ¯D± (e −V Φ ′ ) = 0 . (3.68)From this we see that Φ ′ can be written as Φ ′ = e V Φ with Φ an ordinary chiral superfield (3.11).(Notice that, due to the difference between the two expressions in (3.65), we cannot do somethingsimilar for a twisted chiral covariant superfield Σ.)The Lagrangian for Φ ′ is given by the usual expression L kin = ∫ d 4 θ ¯Φ ′ Φ ′ , which is equal to∫d 4 θ ¯Φ e 2V Φ. Hence we see that switching from the ordinary chiral superfield Φ to the chiralcovariant Φ ′ amounts to applying the minimal substitution prescription. This means we canstick with ordinary chiral superfields in the remainder.4 In the mathematical literature, the normalization factor is i/2π. The difference comes from the fact that inthe physics literature, elements of the Lie algebra are usually rescaled by a factor of i in order to make themself-adjoint. Since the gauge field (connection) A µ and its field strength (curvature) F 01 are g-valued, we do notget the i in the normalization of c 1 .