E_6 and the bipartite entanglement of three qutrits

TL;DR

This paper explores the analogy between black hole entropy in string theory and bipartite entanglement in quantum information, focusing on three qutrits and the E_6 symmetry group.

Contribution

It introduces a new example linking five-dimensional N=8 black holes to bipartite entanglement of three qutrits with E_6 symmetry, expanding the black hole-entanglement analogy.

Findings

Black hole entropy relates to E_6 invariant.

Bipartite entanglement of three qutrits exhibits E_6 symmetry.

Analogies extend to other supergravity black holes.

Abstract

Recent investigations have established an analogy between the entropy of four-dimensional supersymmetric black holes in string theory and entanglement in quantum information theory. Examples include: (1) N=2 STU black holes and the tripartite entanglement of three qubits (2-state systems), where the common symmetry is [SL(2)]^3 and (2) N=8 black holes and the tripartite entanglement of seven qubits where the common symmetry is E_7 which contains [SL(2)]^7. Here we present another example: N=8 black holes (or black strings) in five dimensions and the bipartite entanglement of three qutrits (3-state systems), where the common symmetry is E_6 which contains [SL(3)]^3. Both the black hole (or black string) entropy and the entanglement measure are provided by the Cartan cubic E_6 invariant. Similar analogies exist for ``magic'' N=2 supergravity black holes in both four and five dimensions.

Figures

Equations

$$|\Psi ⟩=a_{ABC}|ABC⟩$$

$${\tau }_3(ABC)=4|\mathrm{Det}{a}_{ABC}|.$$

$$\mathrm{Det}a=-\frac{1}{2}{ϵ}^{A_1{A}_2}{ϵ}^{B_1{B}_2}{ϵ}^{A_3{A}_4}{ϵ}^{B_3{B}_4}{ϵ}^{C_1{C}_4}{ϵ}^{C_2{C}_3}{a}_{A_1{B}_1{C}_1}{a}_{A_2{B}_2{C}_2}{a}_{A_3{B}_3{C}_3}{a}_{A_4{B}_4{C}_4}$$

$$S=\pi \sqrt{|\mathrm{Det}{a}_{ABC}|}.$$

$$|\Psi ⟩\sim |111⟩+|000⟩$$

$$|\Psi ⟩\sim |100⟩+|010⟩+|001⟩$$

$$|\Psi ⟩=-|000⟩+|011⟩+|101⟩+|110⟩$$

$$|\Psi ⟩=a_{AB}|AB⟩$$

$${\tau }_2(AB)=C^2(AB)$$

$$C(AB)=2|\det a_{AB}|$$

$$S=\pi |\det a_{AB}|$$

$$|\Psi ⟩\sim |11⟩+|00⟩$$

$$S=\pi \sqrt{|J_4|}$$

$$J_4$$

$$E_7(C)\supset [SL(2,C){]}^7$$

$$56\to$$

$$(2,2,1,2,1,1,1)$$

$$+(1,2,2,1,2,1,1)$$

$$+(1,1,2,2,1,2,1)$$

$$+(1,1,1,2,2,1,2)$$

$$+(2,1,1,1,2,2,1)$$

$$+(1,2,1,1,1,2,2)$$

$$+(2,1,2,1,1,1,2)$$

$$|\Psi ⟩=$$

$$a_{ABD}|ABD⟩$$

$$+b_{BCE}|BCE⟩$$

$$+c_{CDF}|CDF⟩$$

$$+d_{DEG}|DEG⟩$$

$$+e_{EFA}|EFA⟩$$

$$+f_{FGB}|FGB⟩$$

Full text

1. 1 $D=4$ black holes and qubits

2. 1.1 $N=2$ black holes and the tripartite entanglement of three qubits

3. 1.2 $N=2$ black holes and the bipartite entanglement of two qubits

4. 1.3 $N=8$ black holes and the tripartite entanglement of seven qubits

5. 1.4 Magic supergravities in $D=4$

6. 2 Five-dimensional supergravity

7. 3 $D=5$ black holes and qutrits

8. 3.1 $N=2$ black holes and the bipartite entanglement of two qutrits

9. 3.2 $N=8$ black holes and the bipartite entanglement of three qutrits

10. 3.3 Magic supergravities in $D=5$

11. 4 Conclusions

12. 5 Acknowledgements

1 $D=4$ black holes and qubits

It sometimes happens that two very different areas of theoretical physics share the same mathematics. This may eventually lead to the realisation that they are, in fact, dual descriptions of the same physical phenomena, or it may not. Either way, it frequently leads to new insights in both areas. Recent papers [1, 2, 3, 4, 5, 6] have established an analogy between the entropy of certain four-dimensional supersymmetric black holes in string theory and entanglement measures in quantum information theory. In this paper we extend the analogy from four dimensions to five which also involves going from two-state systems (qubits) to three-state systems (qutrits).

We begin by recalling the four-dimensional examples:

1.1 $N=2$ black holes and the tripartite entanglement of three qubits

The three qubit system (Alice, Bob, Charlie) is described by the state

[TABLE]

where $A=0,1$, so the Hilbert space has dimension $2^{3}=8$. The complex numbers $a_{ABC}$ transforms as a $(2,2,2)$ under $SL(2,C)_{A}\times SL(2,C)_{B}\times SL(2,C)_{C}$. The tripartite entanglement is measured by the 3-tangle [7, 8]

[TABLE]

where ${\rm Det}~{}a_{ABC}$ is Cayley’s hyperdeterminant [9].

[TABLE]

The hyperdeterminant is invariant under $SL(2)_{A}\times SL(2)_{B}\times SL(2)_{C}$ and under a triality that interchanges $A$, $B$ and $C$.

In the context of stringy black holes the 8 $a_{ABC}$ are the 4 electric and 4 magnetic charges of the $N=2$ STU black hole [10] and hence take on real (integer) values. The $STU$ model corresponds to $N=2$ supergravity coupled to three vector multiplets, where the symmetry is $[SL(2,Z)]^{3}$. The Bekenstein-Hawking entropy of the black hole, $S$, was first calculated in [11]. The connection to quantum information theory arises by noting [1] that it can also be expressed in terms of Cayley’s hyperdeterminant

[TABLE]

One can then establish a dictionary between the classification of various entangled states (separable A-B-C; bipartite entangled A-BC, B-CA, C-AB; tripartite entangled W; tripartite entangled GHZ) and the classfication of various “small” and “large” BPS and non-BPS black holes [1, 2, 3, 4, 5, 6]. For example, the GHZ state [12]

[TABLE]

with ${\rm Det}~{}a_{ABC}\geq 0$ corresponds to a large non-BPS 2-charge black hole; the W-state

[TABLE]

with ${\rm Det}~{}a_{ABC}=0$ corresponds to a small-BPS 3-charge black hole; the GHZ-state

[TABLE]

corresponds to a large BPS 4-charge black hole.

1.2 $N=2$ black holes and the bipartite entanglement of two qubits

An even simpler example [2] is provided by the two qubit system (Alice and Bob) described by the state

[TABLE]

where $A=0,1$, and the Hilbert space has dimension $2^{2}=4$. The $a_{AB}$ transforms as a $(2,2)$ under $SL(2,C)_{A}\times SL(2,C)_{B}$. The entanglement is measured by the 2-tangle

[TABLE]

where

[TABLE]

is the concurrence. The determinant is invariant under $SL(2,C)_{A}\times SL(2,C)_{B}$ and under a duality that interchanges $A$ and $B$. Here it is sufficient to look at $N=2$ supergravity coupled to just one vector multiplet and the 4 $a_{AB}$ are the 2 electric and 2 magnetic charges of the axion-dilaton black hole with entropy

[TABLE]

For example, the Bell state

[TABLE]

with ${\rm det}~{}a_{AB}\geq 0$ corresponds to a large non-BPS 2-charge black hole.

1.3 $N=8$ black holes and the tripartite entanglement of seven qubits

We recall that in the case of $D=4,N=8$ supergravity, the the 28 electric and 28 magnetic charges belong to the $56$ of $E_{7(7)}$. The black hole entropy is [15, 18]

[TABLE]

where $J_{4}$ is Cartan’s quartic $E_{7}$ invariant [13, 14]. It may be written

[TABLE]

where $P^{ij}$ and $Q_{jk}$ are $8\times 8$ antisymmetric matrices.

The qubit interpretation [4] relies on the decomposition

[TABLE]

under which

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

suggesting the tripartite entanglement of seven qubits (Alice, Bob, Charlie, Daisy, Emma, Fred and George) described by the state.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $A=0,1$, so the Hilbert space has dimension $7.2^{3}=56$. The $a,b,c,d,e,f,g$ transform as a $56$ of $E_{7}(C)$. The entanglement may be represented by a heptagon where the vertices A,B,C,D,E,F,G represent the seven qubits and the seven triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC represent the tripartite entanglement. See Figure 1.

Alternatively, we can use the Fano plane. See Figure 2. The Fano plane also corresponds to the multiplication table of the octonions333Not the “split” octonions as was incorrectly stated in the published version of [4].

The measure of the tripartite entanglement of the seven qubits is provided by the 3-tangle

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where products like

[TABLE]

[TABLE]

exclude four individuals (here Charlie, Emma, Fred and George), products like

[TABLE]

[TABLE]

exclude two individuals (here Charlie and Emma), and products like

[TABLE]

[TABLE]

exclude one individual (here George)444This corrects the corresponding equation in the published version of [4] which had the wrong index contraction..

Once again large non-BPS, small BPS and large BPS black holes correspond to states with $J_{4}>0$, $J_{4}=0$ and $J_{4}<0$, respectively.

1.4 Magic supergravities in $D=4$

The black holes described by Cayley’s hyperdeterminant are those of $N=2$ supergravity coupled to three vector multiplets, where the symmetry is $[SL(2,Z)]^{3}$. In [4] the following four-dimensional generalizations were considered:

1. $N=2$ supergravity coupled to $l$ vector multiplets where the symmetry is $SL(2,Z)\times SO(l-1,2,Z)$ and the black holes carry charges belonging to the $(2,l+1)$ representation ($l+1$ electric plus $l+1$ magnetic).

2. $N=4$ supergravity coupled to $m$ vector multiplets where the symmetry is $SL(2,Z)\times SO(6,m,Z)$ where the black holes carry charges belonging to the $(2,6+m)$ representation ($m+6$ electric plus $m+6$ magnetic).

3. $N=8$ supergravity where the symmetry is the non-compact exceptional group $E_{7(7)}(Z)$ and the black holes carry charges belonging to the fundamental $56$-dimensional representation (28 electric plus 28 magnetic).

In all three case there exist quartic invariants akin to Cayley’s hyperdeterminant whose square root yields the corresponding black hole entropy. In [4] we succeeded in giving a quantum theoretic interpretation in the $N=8$ case together with its truncations to $N=4$ (with $m=6$) and $N=2$ (with $l=3$, the case we already knew [1]).

However, as suggested by Levay [5], one might also consider the “magic” supergravities [22, 23, 24]. These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) $N=2,D=4$ supergravity coupled to $6,9,15$ and $27$ vector multiplets with symmetries $Sp(6,Z),SU(3,3),SO^{*}(12)$ and $E_{7(-25)}$, respectively. Once again, as has been shown just recently [20], in all cases there are quartic invariants whose square root yields the corresponding black hole entropy.

Here we demonstrate that the black-hole/qubit correspondence does indeed continue to hold for magic supergravities. The crucial observation is that, although the black hole charges $a_{ABC}$ are real (integer) numbers and the entropy (1.13) is invariant under $E_{7}(7)(Z)$, the coefficients $a_{ABC}$ that appear in the qutrit state (1.17) are complex. So the three tangle (1.18) is invariant under $E_{7}(C)$ which contains both $E_{7(7)}(Z)$ and $E_{7(-25)}(Z)$ as subgroups. To find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic $N=2$ supergravity rather than the conventional $N=8$ supergravity. The fact that

[TABLE]

but

[TABLE]

is irrelevant. All that matters is that

[TABLE]

The same argument holds for the magic real, complex and quaternionic $N=2$ supergravities which are, in any case truncations of $N=8$ (in contrast to the octonionic) .

Having made this observation, one may then revisit the conventional $N=2$ and $N=4$ cases (1) and (2) above. When we looked at the seven qubit subsector $E_{7}(C)\supset SL(2,C)\times SO(12,C)$, we gave an $N=4$ supergravity interpretation with symmetry $SL(2,R)\times SO(6,6)$ [4], but we could equally have given an interpretation in terms of $N=2$ supergravity coupled to $11$ vector multiplets with symmetry $SL(2,R)\times SO(10,2)$.

Moreover, $SO(l-1,2)$ is contained in $SO(l+1,C)$ and $SO(6,m)$ is contained in $SO(12+m,C)$ so we can give a qubit interpretation to more vector multiplets for both $N=2$ and $N=4$, at least in the case of $SO(4n,C)$ which contains $[SL(2,C)]^{2n}$.

2 Five-dimensional supergravity

In five dimensions we might consider:

1. $N=2$ supergravity coupled to $l+1$ vector multiplets where the symmetry is $SO(1,1,Z)\times SO(l,1,Z)$ and the black holes carry charges belonging to the $(l+1)$ representation (all electric) .

2. $N=4$ supergravity coupled to $m$ vector multiplets where the symmetry is $SO(1,1,Z)\times SO(m,5,Z)$ where the black holes carry charges belonging to the $(m+5)$ representation (all electric).

3. $N=8$ supergravity where the symmetry is the non-compact exceptional group $E_{6(6)}(Z)$ and the black holes carry charges belonging to the fundamental $27$-dimensional representation (all electric).

The electrically charged objects are point-like and the magnetic duals are one-dimensional, or string-like, transforming according to the contrgredient representation. In all three cases above there exist cubic invariants akin to the determinant which yield the corresponding black hole or black string entropy.

In this section we briefly describe the salient properties of maximal $N=8$ case, following [16]. We have 27 abelian gauge fields which transform in the fundamental representation of $E_{6(6)}$. The first invariant of $E_{6(6)}$ is the cubic invariant [13, 17, 16, 18, 19]

[TABLE]

where $q_{ij}$ is the charge vector transforming as a $27$ which can be represented as traceless $Sp(8)$ matrix. The entropy of a black hole with charges $q_{ij}$ is then given by

[TABLE]

We will see that a configuration with $J_{3}\not=0$ preserves 1/8 of the supersymmetries. If $J_{3}=0$ and ${\partial J_{3}\over\partial q^{i}}\not=0$ then it preserves 1/4 of the supersymmetries, and finally if ${\partial J_{3}\over\partial q^{i}}=0$ (and the charge vector $q^{i}$ is non-zero), the configuration preserves 1/2 of the supersymmetries. We will show this by choosing a particular basis for the charges, the general result following by U-duality.

In five dimensions the compact group $H$ is $USp(8)$. We choose our conventions so that $USp(2)=SU(2)$. In the commutator of the supersymmetry generators we have a central charge matrix $Z_{ab}$ which can be brought to a normal form by a $USp(8)$ transformation. In the normal form the central charge matrix can be written as

[TABLE]

we can order $s_{i}$ so that $s_{1}\geq s_{2}\geq|s_{3}|$. The cubic invariant, in this basis, becomes

[TABLE]

Even though the eigenvalues $s_{i}$ might depend on the moduli, the invariant (2.4) only depends on the quantized values of the charges. We can write a generic charge configuration as $UeU^{t}$, where $e$ is the normal frame as above, and the invariant will then be (2.4). There are three distinct possibilities

[TABLE]

[TABLE]

[TABLE]

Taking the case of type II on $T^{5}$ we can choose the rotation in such a way that, for example, $s_{1}$ corresponds to solitonic five-brane charge, $s_{2}$ to fundamental string winding charge along some direction and $s_{3}$ to Kaluza-Klein momentum along the same direction. We can see that in this specific example the three possibilities in (2.5) break 1/8, 1/4 and 1/2 supersymmetries. The respective orbits are

[TABLE]

[TABLE]

[TABLE]

This also shows that one can generically choose a basis for the charges so that all others are related by U-duality. The basis chosen here is the S-dual of the $D$-brane basis usually chosen for describing black holes in type II B on $T^{5}$ . All others are related by U-duality to this particular choice. Note that, in contrast to the four-dimensional case where flipping the sign of $J_{4}$ (1.14) interchanges BPS and non-BPS black holes, the sign of the $J_{3}$ (2.4) is not important since it changes under a CPT transformation. There is no non-BPS orbit in five dimensions.

In five dimensions there are also string-like configurations which are the magnetic duals of the configurations considered here. They transform in the contragredient $27^{\prime}$ representation and the solutions preserving 1/2, 1/4, 1/8 supersymmetries are characterized in an analogous way. We could also have configurations where we have both point-like and string-like ch the point-like charge is uniformly distributed along the string, it is more natural to consider this configuration as a point-like object in $D=4$ by dimensional reduction.

It is useful to decompose the U-duality group into the T-duality group and the S-duality group. The decomposition reads $E_{6}\to SO(5,5)\times SO(1,1)$, leading to

[TABLE]

The last term in (2.7) corresponds to the NS five-brane charge. The ${\bf 16}$ correspond to the D-brane charges and the ${\bf 10}$ correspond to the 5 directions of KK momentum and the 5 directions of fundamental string winding, which are the charges that explicitly appear in string perturbation theory. The cubic invariant has the decomposition

[TABLE]

This is saying that in order to have a non-zero area black hole we must have three NS charges (more precisely some “perturbative” charges and a solitonic five-brane); or we can have two D-brane charges and one NS charge. In particular, it is not possible to have a black hole with a non-zero horizon area with purely D-brane charges.

Notice that the non-compact nature of the groups is crucial in this classification.

3 $D=5$ black holes and qutrits

So far, all the quantum information analogies involve four-dimensional black holes and qubits. In order to find an analogy with five-dimensional black holes we invoke three state systems called qutrits.

3.1 $N=2$ black holes and the bipartite entanglement of two qutrits

The two qutrit system (Alice and Bob) is described by the state

[TABLE]

where ${A}=0,1,2$, so the Hilbert space has dimension $3^{2}=9$. The ${a}_{AB}$ transforms as a $(3,3)$ under $SL(3)_{A}\times SL(3)_{B}$. The bipartite entanglement is measured by the concurrence [21]

[TABLE]

The determinant is invariant under $SL(3,C)_{A}\times SL(3,C)_{B}$ and under a duality that interchanges $A$ and $B$.

The black hole interpretation is provided by $N=2$ supergravity coupled to 8 vector multiplets with symmetry $SL(3,C)$ where the black hole charges transform as a $9$. The entropy is given by

[TABLE]

3.2 $N=8$ black holes and the bipartite entanglement of three qutrits

As we have seen in section (2) in the case of $D=5,N=8$ supergravity, the black hole charges belong to the $27$ of $E_{6(6)}$ and the entropy is given by (2.2).

The qutrit interpretation now relies on the decomposition

[TABLE]

under which

[TABLE]

suggesting the bipartite entanglement of three qutrits (Alice, Bob, Charlie). However, the larger symmetry requires that they undergo at most bipartite entanglement of a very specific kind, where each person has bipartite entanglement with the other two:

[TABLE]

where $A=0,1,2$, so the Hilbert space has dimension $3.3^{2}=27$. The three states transforms as a pair of triplets under two of the $SL(3)$’s and singlets under the remaining one. Individually, therefore, the bipartite entanglement of each of the three states is given by the determinant (3.1). Taken together however, we see from (3.4) that they transform as a complex $27$ of $E_{6}(C)$. The entanglement diagram is a triangle with vertices ABC representing the qutrits and the lines AB, BC and CA representing the entanglements. See Fig. 3. The N=2 truncation of section 3.1 is represented by just the line AB with endpoints A and B.

Note that:

1. Any pair of states has an individual in common

2. Each individual is excluded from one out of the three states

The entanglement measure will be given by the concurrence

[TABLE]

$J_{3}$ being the singlet in $27\times 27\times 27$:

[TABLE]

where the products

[TABLE]

[TABLE]

[TABLE]

exclude one individual (Charlie, Alice, and Bob respectively), and the product

[TABLE]

excludes none.

3.3 Magic supergravities in $D=5$

Just as in four dimensions, one might also consider the “magic” supergravities [22, 23, 24]. These correspond to the R, C, H, O (real, complex, quaternionic and octonionic) $N=2,D=5$ supergravity coupled to $5,8,14$ and $26$ vector multiplets with symmetries $SL(3,R),SL(3,C),SU^{*}(6)$ and $E_{6(-26)}$ respectively. Once again, in all cases there are cubic invariants whose square root yields the corresponding black hole entropy [20].

Here we demonstrate that the black-hole/qubit correspondence continue to hold for these $D=5$ magic supergravities, as well as $D=4$ . Once again, the crucial observation is that, although the black hole charges $a_{AB}$ are real (integer) numbers and the entropy (2.2) is invariant under $E_{6(6)}(Z)$, the coefficients $a_{AB}$ that appear in the wave function (3.5) are complex. So the 2-tangle (3.6) is invariant under $E_{6}(C)$ which contains both $E_{6(6)}(Z)$ and $E_{6(-26)}(Z)$ as subgroups. To find a supergravity correspondence therefore, we could equally well have chosen the magic octonionic $N=2$ supergravity rather than the conventional $N=8$ supergravity. The fact that

[TABLE]

but

[TABLE]

is irrelevant. All that matters is that

[TABLE]

The same argument holds for the magic real, complex and quaternionic $N=2$ supergravities which are, in any case truncations of $N=8$ (in contrast to the octonionic). In fact, the example of section 3.1 corresponds to the complex case.

Having made this observation, one may then revisit the conventional $N=2$ and $N=4$ cases (1) and (2) of section (2). $SO(l,1)$ is contained in $SO(l+1,C)$ and $SO(m,5)$ is contained in $SO(5+m,C)$, so we can give a qutrit interpretation to more vector multiplets for both $N=2$ and $N=4$, at least in the case of $SO(6n,C)$ which contains $[SL(3,C)]^{n}$.

4 Conclusions

We note that the 27-dimensional Hilbert space given in (3.4) and (3.5) is not a subspace of the $3^{3}$-dimensional three qutrit Hilbert space given by $(3,3,3)$, but rather a direct sum of three $3^{2}$-dimensional Hilbert spaces. It is, however, a subspace of the $7^{3}$-dimensional three 7-dit Hilbert space given by $(7,7,7)$. Consider the decomposition

[TABLE]

under which

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

This contains the subspace that describes the bipartite entanglement of three qutrits, namely

[TABLE]

So the triangle entanglement we have described fits within conventional quantum information theory.

Our analogy between black holes and quantum information remains, for the moment, just that. We know of no physics connecting them.

Nevertheless, just as the exceptional group $E_{7}$ describes the tripartite entanglement of seven qubits [4, 5], we have seen is this paper that the exceptional group $E_{6}$ describes the bipartite entanglement of three qutrits. In the $E_{7}$ case, the quartic Cartan invariant provides both the measure of entanglement and the entropy of the four-dimensional $N=8$ black hole, whereas in the $E_{6}$ case, the cubic Cartan invariant provides both the measure of entanglement and the entropy of the five-dimensional $N=8$ black hole.

Moreover, we have seen that similar analogies exist not only for the $N=4$ and $N=2$ truncations, but also for the magic $N=2$ supergravities in both four and five dimensions (In the four-dimensional case, this had previously been conjectured by Levay[4, 5]). Murat Gunaydin has suggested (private communication) that the appearance of octonions implies a connection to quaternionic and/or octonionic quantum mechanics. This was not apparent (at least to us) in the four-dimensional $N=8$ case [4], but the appearance in the five dimensional magic $N=2$ case of $SL(3,R)$, $SL(3,C)$, $SL(3,H)$ and $SL(3,O)$ is more suggestive.

5 Acknowledgements

MJD has enjoyed useful conversations with Leron Borsten, Hajar Ebrahim, Chris Hull, Martin Plenio and Tony Sudbery. This work was supported in part by the National Science Foundation under grant number PHY-0245337 and PHY-0555605. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The work of S.F. has been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104 Constituents, fundamental forces and symmetries of the universe , in association with INFN Frascati National Laboratories and by the D.O.E grant DE-FG03-91ER40662, Task C. The work of MJD is supported in part by PPARC under rolling grant PPA/G/O/2002/00474, PP/D50744X/1.