Trigonometric Coordinates

In this section we turn to trigonometric coordinates in which the metric of AdS

space has the following form:

ds2= - dτ 2 +Cos[τ ]2[dλ2+Sinh[λ]2(dα2+Sin[α]2dβ2)]

Test of Killing Vectors

This routine is devised to test whether a set of vector fields are Killing vectors for a given metric.

The inputs to be given before running the routine are: ggmunu = metric as n x n matrix;

xmu = set of coordinates as an n-vector;

dxmu = set of coordinate differentials as an n-vector; derdxmu = set of coordinate derivatives as an n-vector; killus = set of Killing vectors to be tested

dim = dimension of manifold kilnu = number of Killing vectors;

the routine is than activated by typing testkillus

Appendix E: Examples of the Use of the Package

NOVAMANIFOLDA

In this appendix we describe some applications of the package Novamanifolda. The MATHEMATICA notebook file with these examples can be downloaded as supplementary material from the Springer distribution site.

MANIFOLDPROVA

In this Notebook we display some examples of the use of the package NOVAMANIFOLDA. Obviously you have to evaluate first the NoteBook Novamanifolda.

The 4-Dimensional Coset CP2

We initialize the programme

start

{Null}

We calculate the structure constant of the SU(3) Lie algebra

cp2stru

{Null}

The result of this calculation is a tensor named fff and stored in the computer memory (if you wanted another group, you had to calculate the structure constants of its Lie algebra and store them in a tri-tensor named also fff. It is important that in ordering the generators the first dim G/H should correspond to the coset generators, while the late dim H should correspond to the stability subgroup H generators)

Next we initial the RUNCOSET programme

initial

=======================================

Welcome to RUNCOSET, a new package built by Petrus on Leonardus technology

It computes various geometric quantities of G/H cosets Please insert the dimensions of the group G and

of the coset G/H

———————————————–

Now you need to provide the structure constants of the group and the rescalings

The structure constants must be given as a tensor cc[[A,B,C]]; The rescalings must be given as r[1]=?, r[2]=?...

———————————————–

{Null}

We supply the calculated SU(3) structure constants cc = fff;

we calculate the spin connection one-form for this coset.

doconnection

12non-zero

13non-zero

14non-zero

21non-zero

23non-zero

24non-zero

31non-zero

32non-zero

34non-zero

41non-zero

42non-zero

43non-zero

I have finished the calculation

The tensor connten[[a,b]] giving the formal expression of the spin connection B[a,b] as a 1-form

is ready for storing on hard disk

Store it in your preferred directory with the name you choose

—————————-

{Null}

We display the result of this calculation. In the formula below om[i], (i=1,. . . ,8) denote the Maurer Cartan one-forms of the SU(3) group ordered and normalized according to the conventions used for the structure constants.

MatrixForm[connten]

Let us now insert the rescaling factors r[i]. These are as many as there are irreducible representations of H in the complementary subspace in the decomposition G=H K. For CP2 the 4 coset generators span just one irreducible representation of the su(2)×u(1) Lie algebra, hence there is only one scaling factor.

r[1] = λ; r[2] = λ; r[3] = λ; r[4] = λ

Next we compute the torsion of the coset

doconncomp

————————————-

Now I calculate the torsion part of the spin connection I have finished the calculation

The tensor contor[a,b]] giving the torsion part B[c,a,b] of the spin connection B[a,b]

is ready for storing on hard disk

Store it in your preferred directory with the name you choose

—————————-

{Null}

The CP2 coset is symmetric and torsionless and this is indeed verified by the computer

contor

{{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}}

Then we calculate the Riemann tensor

doriemann2

————————————-

Now I calculate the Riemann tensor Rie(a,b,c,d) 121 2 non-zero

122 1 non-zero

123 4 non-zero

124 3 non-zero

131 3 non-zero

132 4 non-zero

133 1 non-zero

134 2 non-zero

141 4 non-zero

142 3 non-zero

143 2 non-zero

144 1 non-zero

211 2 non-zero

212 1 non-zero

213 4 non-zero

214 3 non-zero

231 4 non-zero

232 3 non-zero

233 2 non-zero

234 1 non-zero

241 3 non-zero

242 4 non-zero

243 1 non-zero

244 2 non-zero

311 3 non-zero

312 4 non-zero

313 1 non-zero

314 2 non-zero

321 4 non-zero

322 3 non-zero

323 2 non-zero

324 1 non-zero

341 2 non-zero

342 1 non-zero

343 4 non-zero

344 3 non-zero

411 4 non-zero

412 3 non-zero

413 2 non-zero

414 1 non-zero

421 3 non-zero

422 4 non-zero

423 1 non-zero

424 2 non-zero

431 2 non-zero

432 1 non-zero

433 4 non-zero

434 3 non-zero

I have finished the calculation

The tensor Rie(a,b,c,d) is ready for storing on hard disk Store it in your preferred directory with the name you choose

—————————–

————————————

Now I evaluate the curvature 2-form of your space I find the following answer

R[12] = 2( 18 λ2e1**e2 + 18 λ2e3**e4) R[13] = 2( 12 λ2e1**e3 + 14 λ2e2**e4) R[14] = 2( 18 λ2e1**e4 + 18 λ2e2**e3) R[23] = 2( 18 λ2e1**e4 + 18 λ2e2**e3) R[24] = 2( 14 λ2e1**e3 + 12 λ2e2**e4) R[34] = 2( 18 λ2e1**e2 + 18 λ2e3**e4)

The result is encoded in a tensor RR[i,j]

Its components are encoded in a tensor Rie[i,j,a,b] {Null}

and we calculate the explicit form of the curvature two-form

docurvaform

————————————

I evaluate the curvature 2-form of your coset I find the following answer

R[12] = 18 λ2V1**V2 − 18 λ2V2**V1 + 18 λ2V3**V4 − 18 λ2V4**V3 R[13] = 12 λ2V1**V3 + 14 λ2V2**V4 − 12 λ2V3**V1 − 14 λ2V4**V2 R[14] = 18 λ2V1**V4 + 18 λ2V2**V3 − 18 λ2V3**V2 − 18 λ2V4**V1 R[23] = 18 λ2V1**V4 + 18 λ2V2**V3 − 18 λ2V3**V2 − 18 λ2V4**V1

R[24] = 14 λ2V1**V3 + 12 λ2V2**V4 − 14 λ2V3**V1 − 12 λ2V4**V2

R[34] = 18 λ2V1**V2 − 18 λ2V2**V1 + 18 λ2V3**V4 − 18 λ2V4**V3

Now choose a value for the rescaling parameters writing rullina = {....}

Then type redisplay {Null}

Finally we calculate the Ricci tensor