Cartan Matrix - Cartan Matrices in M-theory

Cartan Matrices in M-theory

In M-theory, one may consider a geometry with two-cycles which intersects with each other at a finite number of points, at the limit where the area of the two-cycles go to zero. At this limit, there appears a local symmetry group. The matrix of intersection numbers of a basis of the two-cycles is conjectured to be the Cartan matrix of the Lie algebra of this local symmetry group .

This can be explained as follows. In M-theory one has solitons which are two-dimensional surfaces called membranes or 2-branes. A 2-brane has a tension and thus tends to shrink, but it may wrap around a two-cycles which prevents it from shrinking to zero.

One may compactify one dimension which is shared by all two-cycles and their intersecting points, and then take the limit where this dimension shrinks to zero, thus getting a dimensional reduction over this dimension. Then one gets type IIA string theory as a limit of M-theory, with 2-branes wrapping a two-cycles now described by an open string stretched between D-branes. There is a U(1) local symmetry group for each D-brane, resembling the degree of freedom of moving it without changing its orientation. The limit where the two-cycles have zero area is the limit where these D-branes are on top of each other, so that one gets an enhanced local symmetry group.

Now, an open string stretched between two D-branes represents a Lie algebra generator, and the commutator of two such generator is a third one, represented by an open string which one gets by gluing together the edges of two open strings. The latter relation between different open strings is dependent on the way 2-branes may intersect in the original M-theory, i.e. in the intersection numbers of two-cycles. Thus the Lie algebra depends entirely on these intersection numbers. The precise relation to the Cartan matrix is because the latter describes the commutators of the simple roots, which are related to the two-cycles in the basis that is chosen.

Note that generators in the Cartan subalgebra are represented by open strings which are stretched between a D-brane and itself.