Hyperini KSE Clause Samples
Hyperini KSE. The hyperini KSE is given by iγµϵAV aA = 0 . (2.47) To understand the hyperini KSE, one has to identify the ϵA components of the Killing spinor in the context of spinorial geometry. In our notation ϵ1 = 1 and ϵ2 = e1234 and since ϵ1 = −ϵ2 and ϵ2 = Γ34ϵ1, one has ϵ1 = −e1234 and ϵ2 = e34. Substituting these into the KSE, one finds the conditions + ¯2 ¯1 V aA = 0 , − V a1 + V a2 = 0 , V a1 + V a2 = 0 . (2.48) Expressing the coefficients of the KSEs in terms of the fundamental fields as in (2.7), it is clear from the first condition in (2.48) that D+φI = 0 . (2.49)
Hyperini KSE. Applying the results of chapter 2, in particular (2.48), and using the fact that the scalars of the hypermultiplet do not depend on the coordinates (u, v) means the hyperini KSE implies a1 a2 a1 a2 −V1 + V2 = 0 , V2 + V1 = 0 , (4.54) for ϵ = 1 + e1234.
Hyperini KSE. Substituting the Killing spinors in (2.137) into the hyperini KSE we find, in addition to the conditions in (2.95), that V a1 = 0 , V a2 = 0 . (2.141) α¯ α This means the only non-vanishing component is V aA . (2.142) Imposing the conditions of the hyperini KSE on the physical fields using (2.7), we find that the only non-vanishing derivative on the scalars is D−φI . (2.143) This means that the scalars depend only on one light-cone direction.
Hyperini KSE. To find the conditions arising from the hyperini KSE we have to evaluate it on the four spinors given in (2.151), which is equivalent to simultaneously imposing (2.117) and (2.95). This gives V aA = 0 , a = −, +, 1, ¯1 , (2.154) and V a1 = V a2 = 0 . (2.155) In other words, the only non-vanishing components are V a1 and V a2.
Hyperini KSE. Using ϵ2 = i(1 − e1234) in the hyperini KSE leads to the following restrictions, + ¯2 ¯1 V aA = 0 , V a1 + V a2 = 0 , V a1 − V a2 = 0 . (2.94) Combining these conditions with those in (2.48) obtained for the first Killing spinor gives V aA = 0 , V a1 = 0 , V a2 = 0 . (2.95) + α α¯
Hyperini KSE. Evaluating the hyperini KSE on ϵ2 = e15 + e2345, we find the conditions ¯1 ¯2 V aA = 0 , − V a1 + V a2 = 0 , V a1 + V a2 = 0 . (2.117) Combining these conditions with those coming from the first Killing spinor given in (2.48), we get V aA = 0 , a = −, +, ˜1 , (2.118) where again we have converted back to real coordinates to derive these conditions. The remaining conditions can be derived by substituting (2.118) in either (2.48) or (2.117). Expressing the KSE in terms of the physical fields as in (2.7), one finds that (2.118) implies DaφI = 0 , a = −, +, ˜1 . (2.119) Therefore, the hypermultiplet scalars do not depend on three spacetime directions.