4.2 Non-nilpotent four-point invariants
4.2.2 Other non-nilpotent invariants
As we have seen, the construction of the invariants starts from a homogeneously transforming interval, which isx¯ij.
There are many possibilities for combinations of the intervalsx¯ij for 1≤i, j≤4, which are invariant. Still they are built from basic variables, which result from the elimination of the transformation terms.
The variablesX1±andX1(i)±are already invariant under the transformations of all points butz1. Non-nilpotent functions of four points with this property have to be built with these variables and invariants. It is useful for the further calculations to introduce a normalization,
withi= 1,2. These are already vectors, which transform as Xˆ1±µ 0 Hence any contraction of one of these vectors with another is a scalar non-nilpotent invariant. However, we have seen in equation (4.1.13), that the difference between a plus and a minus sign in the indices is only a nilpotent term. Thus the insertion of this or the analogous equation into any of the invariants will give a relation between two non-nilpotent invariants, in which their difference is a nilpotent invariant.
4.2. NON-NILPOTENT FOUR-POINT INVARIANTS 43 To write down these relations explicitly we first need to introduce the spinors,
Θ˜1(i−1) = i ˜x−1¯41θ¯41−˜x¯−1i1 θ¯i1
, (4.2.18)
˜¯
Θ1(i−1) = i θ1i˜x−1¯1i −θ14˜x−1¯14
, (4.2.19)
analogous to eqns. (4.1.9) and (4.1.10). Furthermore it is also here convenient to introduce a normalization also for all these spinoral functions of three points:
Θˆ1 = Θ1
X1(1)+2 X1(1)−2 18
, (4.2.20)
ˆ¯
Θ1 = Θ¯1
X1(1)+2 X1(1)−2 18
, (4.2.21)
Θˆ1(i) = Θ1(i)
X1(1)+2 X1(1)−2
18 , (4.2.22)
ˆ¯
Θ1(i) =
Θ¯1(i)
X1(1)+2 X1(1)−2 18
. (4.2.23)
Now an example to the mentioned relations can be given compactly: With the equation of ˆX1(2)+µ and ˆX1(2)−µ analogous to eq. (4.1.13) we get the following differ-ence of two non-nilpotent invariants:
Xˆ1(1)+·Xˆ1(2)−−Xˆ1(1)+·Xˆ1(2)+= 2i ˆX1(1)+µ Θˆ1σµΘˆ¯1. (4.2.24) As there are two invariants on the left hand side of the equation, the right hand side must be a nilpotent invariant. Analogously a couple of similar relations between other non-nilpotent invariants can be written down.
Three contractions of the vector invariants, however, are actually nilpotent apart from a constant leading term. These are ˆX1(1)±·Xˆ1(1)±, which are√
J1,p
J1−1 and a function ofJ1 and another nilpotent invariant.
Moreover ˆX1+µ , ˆΘ1 and ˆΘ¯1 can be expressed in terms of ˆX1(i)±µ , ˆΘ1(i) and ˆΘ¯1(i): X1+µ = X1(1)+µ −X1(2)−µ + 2iΘ1(1)σµΘ¯1(2), (4.2.25)
Θ1 = Θ1(1)−Θ1(2), (4.2.26)
Θ¯1 = Θ¯1(1)−Θ¯1(2). (4.2.27)
Eq. (4.1.13) then also determines ˆX1−µ . Hence the two invariants, Xˆ1(2)−2
, Xˆ1(2)−·Xˆ1(1)+ , (4.2.28) are a set of two independent non-nilpotent scalar invariants. All other contractions of the vector invariants in eq. (4.2.13) or functions of these – in fact this means all
44 CHAPTER 4. INVARIANTS
Figure 4.1: A part of the tree starting fromx¯122/x¯132for the construction of invariants as ratios of superconformal intervals. The continuous boxes are the invariants, which end a branch, because they have an index2in the interval last added to the denominator.
The dashed boxes contain an invariant, which is given by the ratio of two three point functions,Xˆ1± andXˆ1(i)± and thus end the branch.
scalar non-nilpotent invariants – can be expressed in terms of them and nilpotent invariants. The cross ratios, of course, are no exception because one only has to put hats on all X’s in eqns. (4.2.6), (4.2.7) and (4.2.8). As the X’s appear only in quotients of two of them there, the normalization cancels down. These equations of invariants can be solved for the cross ratios, so that we see, that all non-nilpotent invariants can also be expressed by two cross ratios and nilpotent invariants.
This whole construction building invariant functions can be started in a way, in which analogously to eq. (4.2.13) either X2(i)±, X3(i)± or X4(i)± are defined and so one is left with non-trivial transformations only for the coordinates z2, z3 and z4, respectively. But this would result in the same calculations with indices permuted. For the cross ratios we have already noted, that a permutation of the indices again gives one of the cross ratios given by (4.2.2). We have seen, that all the other non-nilpotent invariants in this section can be expressed in terms of two superconformal cross ratios and nilpotent invariants. Thus a permutation of the indices in these invariants does not lead to invariants, which are independent of those already written down here and the mentioned different ways lead to the same result.
As mentioned above it is already obvious from ordinary conformal invariants, that there can be only two non-nilpotent invariants, which are independent. Here we have seen, how the asymmetry of the superconformal invariants only gives rise to a lot of non-nilpotent invariants related to each other by nilpotent invariants.
4.2. NON-NILPOTENT FOUR-POINT INVARIANTS 45 More invariant ratios. An especially interesting subset of non-nilpotent invari-ants are those, which are ratios of squares of superconformal intervals,x¯ij2, or square roots of these ratios. While squares of the variables, ˆX1± and ˆX1(i)±, their inverses and then, of course, cross ratios belong to it, mixed contractions of the former vari-ables are not of this form.
With 56 invariant ratios of squared superconformal intervals, which themselves can be formed with ˆX1±2 and ˆX1(i)±2 , all other invariant ratios can be formed.
The calculations in Maple, which lead to this result can be found in D.5. In summary one can start with an arbitrary squared superconformal interval – herex¯122
– and devide it in a first step by x¯132 because this cancels the transformation term associated to the index ¯1. All other starting possibilities are given by permutations of the indices throughout the calculations.
One continues to multiply alternately to the numerator and the denominator the possible intervals, which just eliminate the transformation terms of the last added factor, which has not yet been eliminated, but do not cancel down with other intervals in the ratio. The list of possible terms forms a tree and this tree grows despite of the terminating condition for single branches: A new term in the denominator, which just has 2 as second index, ends a branch.
Further terminating conditions are added to get to the goal. The variables Xˆ1±2 and ˆX1(i)±2 can only appear in pairs with their inverses. So the hats cancels.
Whenever any variableX1±2 orX1(i)±2 appears together with any of the inverses in the ratio of a branch, it is terminated. Although part of this ratio is not expressed by these variables, nothing is missed, as this part appears in all possible other combinations in other branches (possibly belonging to a tree with permuted indices).
This condition does not lead to invariants, but prevents endless loops. A part of the tree starting from x¯122/x¯132 is depicted in figure 4.1.
The simplest of the ends of branches are the appearances of cross ratios. At the end there are seven branches, which end with the first terminating condition and so with finished invariant ratios in this tree.
x¯122x¯432
x¯132x¯422 =
Xˆ1(2)−2
Xˆ1(1)−2 , (4.2.29)
x¯122x¯232x¯312
x¯132x¯212x¯322 = Xˆ1+2
Xˆ1−2 , (4.2.30)
x¯122x¯232x¯412
x¯132x¯212x¯422 =
Xˆ1+2
Xˆ1(1)−2 , (4.2.31)
x¯122x¯232x¯342
x¯132x¯242x¯322 =
Xˆ1+2 Xˆ1(2)+2
Xˆ1(1)+2 Xˆ1−2 , (4.2.32) x¯122x¯432x¯312
x¯132x¯412x¯322 =
Xˆ1(2)−2
Xˆ1−2 , (4.2.33)
46 CHAPTER 4. INVARIANTS The second one isJ1 and so essentially a nilpotent invariant.
As all have the form of ratios of the variables ˆX1±2 and ˆX1(i)±2 , this shows together with the permuted trees from the other starting possibilities, that all invariant ratios of squared superconformal intervals can be expressed in this form.
At last in this section we want to give the following relations of these three-point variables to cross ratio,J1 and J1(i): These equations are very useful to find bugs in procedures calculating with these entities.