Gaugini KSE. Recall that the gaugini KSE is
Gaugini KSE. The gaugini KSE commutes with ρ1, iff µ2 = µ3 = 0 . (2.92) These restrictions are in addition to the conditions given in (2.43). When combined these become
5 a' = 5 a' = 0 , 5 a' α + iµ1 = 0 , 5 a' = 0 . (2.93) +i +− α 12 Once again the 5 a' components are not restricted.
Gaugini KSE. Evaluating the gaugini KSE on ϵ2 = e15 + e2345, we find 1¯1 2¯2 −25 a' + µ2 + iµ3 = 0 , − 5 a' + 5 a' + i(µa' )1 = 0 , 5 a' = 0 . (2.111) Combining the above conditions with those coming from ϵ1 given in (2.43), we get
5 a' = 5 a' = 5 a' = 0 , 5 a' = 0 , 5 a' + i(µa' )1 = 0 , +− +i −i 2¯2 1¯2
5 a' − 5 a' (µa' )2 = 0 , 5 a' + 5 a' + (µa' )2 = 0 , (2.112) To write this in a more compact notation we change to real coordinates to obtain
5 a' = 0 , 5 a' = 0 , 5 a' = −ϵijkµa'k , a = −, +, ˜1 , (2.113) where i = 4, ˜2, 5 and ϵ2˜45 = −1. Each of the indices a and i label 3 real directions, note that we have also used and ˜2 to distinguish the real directions from the complex directions 1 and 2 which naturally appear in the various conditions coming from the KSEs. In addition, the v' = 1, 2, 3 index of µ has been replaced with k = 4, 2, 5 after an appropriate adjustment of the ranges and identification of the components of µ. In particular, µ1 = µ4, µ2 = µ2˜, and µ3 = µ5. We can express these as
5 a' = 1 ϵ 2 µa'kei ∧ ej . (2.114)
Gaugini KSE. The gaugini KSE commutes with ρr' , iff µ1 = µ2 = µ3 = 0 . (2.139) These are of course in addition to the conditions in (2.43). The same conditions can derived by explicitly substituting in the four Killing spinors into the gaugini KSE. This in turn means we have
5 a' = 5 a' e− ∧ ei + (5 asd)a' , (2.140) where we have just imposed the conditions in (2.139) on the expression in (2.84).
Gaugini KSE. The gaugini KSE commutes with ρ1 iff µ2 = µ3 = 0. Combining this with the conditions coming from the N = 2 case in (2.113), we find 5 a¯' + iµa' = 0 , (2.153) where after suppressing the gauge index we set µ = µ1.
