10.5 The Petersen algebra and the Desargues configuration

We want to end this section by considering an example which is more complicated, but very interesting from the algebraic point of view. There exist ten geometric configurations of the form , i.e., with exactly ten points and ten lines. In [61], these were associated to supergravity solutions with ten components of the electric field turned on. This result was re-analyzed by some of the present authors in [96] where it was found that many of these configurations have a dual description in terms of Dynkin diagrams of rank 10 Lorentzian Kac–Moody subalgebras of . One would therefore expect that solutions of the sigma models for these algebras should correspond to new solutions of eleven-dimensional supergravity. However, since these algebras are infinite-dimensional, the corresponding sigma models are difficult to solve without further truncation. Nevertheless, one may argue that explicit solutions should exist, since the algebras in question are all non-hyperbolic, so we know that the supergravity dynamics is non-chaotic.

We shall here consider one of the -configurations in some detail, referring the reader to [96] for a discussion of the other cases. The configuration we will treat is the well known Desargues configuration, displayed in Figure 54. The Desargues configuration is associated with the 17th century French mathematician Gérard Desargues to illustrate the following “Desargues theorem” (adapted from [145]):

Let the three lines defined by and be concurrent, i.e., be intersecting at one point, say . Then the three intersection points and are colinear.

Another way to say this is that the two triangles and in Figure 54 are in perspective from the point and in perspective from the line .

As we will see, a new fascinating feature emerges for this case, namely that the Dynkin diagram dual to this configuration also corresponds in itself to a geometric configuration. In fact, the Dynkin diagram dual to the Desargues configuration turns out to be the famous Petersen graph, denoted , which is displayed in Figure 55.

To construct the Dynkin diagram we first observe that each line in the configuration is disconnected from three other lines, e.g., have no nodes in common with the lines , , . This implies that all nodes in the Dynkin diagram will be connected to three other nodes. Proceeding as in Section 10.2.2 leads to the Dynkin diagram in Figure 55, which we identify as the Petersen graph. The corresponding Cartan matrix is

which is of Lorentzian signature with
The Petersen graph was invented by the Danish mathematician Julius Petersen in the end of the 19th century. It has several embeddings on the plane, but perhaps the most famous one is as a star inside a pentagon as depicted in Figure 55. One of its distinguishing features from the point of view of graph theory is that it contains a Hamiltonian path but no Hamiltonian cycle.

Because the algebra is Lorentzian (with a metric that coincides with the metric induced from the embedding in ), it does not need to be enlarged by any further generator to be compatible with the Hamiltonian constraint.
It is interesting to examine the symmetries of the various embeddings of the Petersen graph in the plane and the connection to the Desargues configurations. The embedding in Figure 55 clearly exhibits a -symmetry, while the Desargues configuration in Figure 54 has only a -symmetry. Moreover, the embedding of the Petersen graph shown in Figure 56 reveals yet another symmetry, namely an permutation symmetry about the central point, labeled “10”. In fact, the external automorphism group of the Petersen graph is , so what we see in the various embeddings are simply subgroups of made manifest. It is not clear how these symmetries are realized in the Desargues configuration that seems to exhibit much less symmetry.