A few days ago Lubos reported on an intriguing new paper by Volker Braun describing how to construct a Calabi-Yau manifold with 6 real dimensions and minimal Hodge numbers using the 24-cell. Such manifolds can be applied to the compactification of superstring theory down to our familiar 4 dimensional spacetime. The predictions for physics based on this particular manifold would be unrealistic but its discovery is an important step towards understanding the fuller range of possibilities. It is also of considerable mathematical interest in its own right.
I’m not going to say anything more about that paper but I do want to say something about the 24-cell and its curious relation to 4 qubits as well as a surprising relationship between the invariants of 4 qubits and the platonic solids. I found out about these things after talking to Mike Duff and his coworkers on the qubit/black-hole correspondence (Borsten, Dahanayke, Duff, Marrani, Rubens). I think this is a bit more specialized than the kind of stuff I usually report on here, but some of our regular commenters expressed an interest and I am always happy to oblige. For anyone who does not understand any of the mathematical terms here they are all explained in wikipedia.
The 24-cell is a very special regular polytope in 4-dimensional space. It has the special property of being self dual in the same sense as the tetrahedron is self-dual in 3 dimensions. It can also be tessellated to fill 4 dimensional space just as a cube can tessellate to fill 3d space. in fact the 24-cell is the only regular polytope in more than 2 dimensions that has both of these properties. The only comparable shapes in this sense are the triangle, square and hexagon in two dimensions.
The vertices of the 24-cell can be plotted in 4D co-ordinates at the 24 points given by
It’s dual can be plotted at the points,
As many readers of viXra Log are undoubtedly aware, there are many connected mysteries surrounding the number 24 in mathematics and the 24-cell is one of the more enigmatic. 24 is famous as the dimension of the Leech lattice which is connected to the significance of the number in the theory of finite simple groups, especially examples such as the Mathieu groups, the Conway groups and the Fisher groups. The existence of the Leech lattice can be explained in terms of the 24-bit Golay code which can in turn be constucted using special properties of quadratic residues in Z24. Alternatively the Leech lattice is a reduction of an alternating lattice in 25+1 dimensions using a null vector relying on the fact that the sum of the first 24 square numbers is 70^2. This closely connects together one set of circumstances where the number 24 appears in mathematics.
Then there is a second interlinked set of places where the number 24 shows up in number theory and the theory of special functions. This includes the Ramanujan discriminant function, a modular form that is the 24th power of the Dedekind eta function. This can be connected to the fact that the value of zeta(-1) is 1/12. It has implications in bosonic string theory where it is linked to the critical dimension where the two dimensional worldsheet vibrates in the remaining 24 dimensions.
These two sets of places where the number 24 appears, one in group theory and the other in number theory, do not seem to have a causal link. You cannot reason that one implies the other. Yet you can combine the two by compactifying bosonic string theory over the Leech lattice. This was a realisation that led to the famous proof of the monstrous moonshine conjectures and a Fields medal for Richard Borcherds. This much will be familiar to anyone who follows related discussions on the internet and especially if they have read John Baez’s lecture on the number 24. As far as I know there is nothing else that clarifies the mystery of this connection. For example there is nothing that directly links the Golay Code to the Ramanujan Discriminant function except Moonshine.
What about the 24 cell, where does that fit in? According to Baez it relates to the number theory side via elliptic curves, but in a way that involves a group of order 24. Perhaps this is a clue to a missing direct link between between number theory and group theory. The connection described by Baez points to the fact that the moduli space of elliptic curves is given by modding out the group SL(2,3). The vertices of the 24-cell when plotted as quaternions (Hurwitz quaternions) also form a group, and it is the same one, also known as the ditetrahedral group because it is the double cover of the rotation group of the tetrahedron. This seems very nice but actually there are only 15 groups of order 24 and only seven that are not direct products of smaller groups, so saying that two structures form the same group of order 24 is only a small factor better than saying that they have the same size. What we really need to find is a more direct way in which the 24-cell relates to elliptic curves.
This is where the 4-qubit system comes in. The wavefunction of 4 qubits is represented by a 2x2x2x2 hypermatrix of 16 complex numbers. Local transformation on these qubits take the form of SL(2,C) transformations applied to each qubit independently so the overall symmetry group of the system is SL(2,C) 4 . To understand the entanglement possibilities for 4 qubits the first step is to find the polynomial invariants under this group. This is a non-trivial computation but it can be shown that there are 4 independent invariants of degree 2, 4, 4 and 6 in the 16 components of the hypermatrix (see e.g. http://arxiv.org/abs/quant-ph/0212069 for a construction.) However, there is a special invariant that is a combination of these known as the hyperdeterminant which is of degree 24. The hyperdeterminant is a discriminant for the hypermatrix that is zero iff the quadriliear form constructed from the hypermatrix has singular points where all derivatives vanish. You don’t have to understand the details, just notice that this is another structure where the number 24 has special significance.
It turns out that the 4 qubit hypermatrix is related in a fundamental way to elliptic curves with the hyperdeterminant being related to the Ramanujan discriminant modular form mentioned above. I have described this relationship in detail at http://arxiv.org/abs/1010.4219 so I won’t repeat it here. This makes a direct link between the number 24 that appears as the degree of the hyperdeterminant and its appearance in the theory of modular forms linked to bosonic string theory. After discussing this with Mike Duff I was also able to link the 4 qubit system directly to bosonic strings and I used this in my FQXi essay.
The classification of 4-qubit entanglement is a tricky business. The SL(2,C) 4 transformation group has 12 independent paprameters so it should be possible to use these transformations to reduce any state with its 16 components to representative states parameterised by just 16-12=4 variables. A clean solution was provided by Verstraete et al in http://arxiv.org/abs/quant-ph/0109033 . They found nine perameterised classes of states where the largest class known as Gabcd has 4 parameters and includes all states whose hyperdeterminant is non-zero. For present purposes I am only interested in this class. It takes a form that can be written in qubit terms as
Φ = x (|0000> +|1111>) + y (|0011>+|1100>) + z (|0110> + |1001>) + t (|1010> + |0101>)
For this class of states, we can work out any of the invariants including the hyperdeterminant which is going to be a polynomial of degree 24 in the four variables x, y, z and t. This has the potential to be a complicated expression, after all the full hyperdeterminant in 16 variables is an expression with 2894276 terms. In practice for the reduced state the hyperdeterminant simplifies and when you work it out you will notice that the result factorised into 24 simple factors
Det(Φ) = x2y2z2t2(x+y+z+t)2(x+y+z-t)2(x+y-z+t)2(x+y-z-t)2(x-y+z+t)2(x-y+z-t)2(x-y-z+t)2(x-y-z-t)2
These factors correspond in an obvious way to the Hurwitz quaternions and therefore the vertices of the 24-cell. This provides a direct link between the number of vertices in the 24-cell and the degree of the hyperdeterminant for 4-qubits which in turn is linked to the exponents in modular forms and the critical dimension of bosonic string theory, just as we wanted.
Is there a better way to understand why the hyperdeterminant factorises so conveniently? Yes there is. Although all the 12 dimensions of the group SL(2,C) 4 were used to reduce the hypermatrix to the class Gabcd , there remains a discrete subgroup that maps states of Gabcd , (in the form above) onto themselves, so this discrete subgroup provides a group of linear transformations on the 4D space parameterised by x, y, z and t. This subgroup turns out to be the Weyl group of D4 whose system of root vectors is the 24-cell. The polynomial invariants of this reflection group as functions of the four parameters x, y, z and t are also of degree 2, 4, 4 and 6 and correspond to the 4 qubit invariants. The Weyl group is the symmetry group of the root system so it just permutes the 24 factors in the hyperdeterminant making it an obvious invariant. Notice that D4 as a Lie-algebra is SO(8) or its split form SO(4,4) which is the group used to construct the 4-qubit/black-hole correspondence. This was what Borsten et al used to classify the entanglement of four qubits using a classification of nilpotent orbits that had already been worked out for black holes in M-theory. Their answer matches the Verstraete classification nicely.
We can go one step further and extend the group of transformations to include permutations of the 4 qubits. This gives a larger discrete group acting on Gabcd ,which can be identified as the Weyl group of F4. The corresponding root system is now the 48 vertices of a 24-cell combined with its dual. The polynomial invariants of this group are of degree 2,6,8 and 12 and they correspond to the primitive invariants of the hypercube that is symmetric under the permutations of the qubits as well as the usual SL(2,C) 4 transformations.
This leads to one last curious correspondence that I want to point out. The degrees of the primitive invariants of the hypercube (2,4,4,6) are not trivial to work out, but can you see what they are related to? Think of a tetrahedron with 4 vertices, 6 edges and 4 faces. The remaining number 2 corresponds to the inside and outside of the tetrahedron which can be regarded as the two three dimensional parts which cover space in combination with the vertices, edges and faces. Remember that the 24-cell as a group is the double cover of the rotation group of the tetrahedron so there is a connection. For the invariants that are symmetric under permutations, the larger root system of 48 vectors from the two 24-cells combined also forms a group when the root vectors are regarded as unit quaternions. The full set of unit quaternions form the group SU(2) which is the double cover of SO(3) so any finite subgroup must be the double cover of some rotation group in 3D. In this case it is the rotation group of the cube or octahedron. This corresponds to the fact that the symmetric invariants for four qubits are of degrees (2,6,8,12) because the cube has 6 faces, 8 vertices and 12 edges (or you can use the dual octahedron).
So despite the fact that the invariants of the 4 qubit system are non-trivial to construct, their polynomial degrees correspond to the geometric elements of three of the platonic solids. What about the other two regular solids, the dodecahedron and icosahedron? There is another reflection group H4 whose root system corresponds to these solids and it therefore has invariants of degree (2,12,20,30) . Since this group acts on the same 4D sapce you can use it to construct four invariants of the 4 qubit system with these degrees, but there is no lie algebra corresponding to H4 and its significance is not so obvious. However, these three cases are part of a system of mysterious “trinities” as noted by the mathematician Vladimir Arnold. This means that there must be a lot more going on that we don’t really understand yet.