430 10 Conclusion of Volume 2

where ηA are the 6 internal Killing spinors and τ denote the 1-index and 2-index so(6) gamma-matrices. By construction the barred τ s are antisymmetric 6 × 6 matrices, hence so(6) generators in the fundamental representation just as the Kähler form K. Counting these matrices we find that they are 6 + 15 + 1, namely 22, which is too much as a set of independent generators of so(6). This means that there must be linear dependences. By calculating traces of these matrices we find that the 6 matrices τ α are linear independent and orthogonal to the 15, τ αβ , and to the unique K while among these latter 16 matrices only 9 are linear independent.

This observation is important for the following reason. When we write the following formulae:

we are actually decomposing the so(6) connection A AB along an over-complete basis of 15 + 6 = 21 generators of so(6), which is obviously a well defined operation.

It is interesting to establish the inverse formula, namely to express the original connection A AB in terms of the over complete set of objects ΔBα and ΔBαβ . The inverse formula can be established by means of direct calculation in the explicit τ -matrix basis we have chosen and we find what follows:

AB

Appendix D: MATHEMATICA Package NOVAMANIFOLDA

In this section we describe the MATHEMATICA Package NOVAMANIFOLDA that can be downloaded as supplementary material form the Springer distribution site.

This notebook contains various packages for the calculation of the spin connection and the curvatures of various manifolds, both homogeneous (= cosets) and also non-homogeneous. It is divided in various sections.

Coset Manifolds (Euclidian Signature)

Instructions for the Use

This notebook has the following purpose, that of calculating the Riemann tensor and the connection of the several coset manifolds. In particular:

(1) The manifold:

(2) The spheres:

SO(m+1) =

SO(m) Sm

(3) The manifold:

SU(3) = N 010

U (1)

The calculation is done using the RUNCOSET package constructed by Prof. Leonardo Castellani. The input are the structure constants of the corresponding group that are calculated by suitable routines inserted in this package.

First read the two sections of PROGRAMME and then start by the command

start

If you want to calculate the structure constants for CP2, spheres or N010 you just type:

cp2stru, spheres or n010stru

and then initialize the RUNCOSET programme by the command initial

then supply the file cc=fff

and you can calculate with the commands of RUNCOSET that are described in the section below

Description of the Main Commands of RUNCOSET

The available commands one can use at this point are the following ones 1.

doriemann2

This command generates as an output a tensor Rie[[a,b,c,d]] = (Rab)cd where (Rab)cd is the Riemann tensor in the conventions of the old Kaluza-Klein literature, namely Universal mass relations Ann. of Phys. 162, (1985) 372 by D’Auria and Frè.

2. doconnection

This command generates as an output a tensor connten[[a,b]] = Bab where Bab is the spin connection 1-form in the conventions of the old Kaluza-Klein literature, namely Universal mass relations Ann. of Phys. 162, (1985) 372 by D’Auria and Frè.

3. doconcomp

This command generates as an output a tensor contor[[c,a,b]] = (Bab)c where (Bab)c is the torsion part of the spin connection 1-form in the conventions of the old KaluzaKlein literature, namely Universal mass relations Ann. of Phys. 162, (1985) 372 by D’Auria and Frè.