Editing Cartan matrix (section)

== Cartan matrices in M-theory ==
In [[M-theory]], one may consider a geometry with [[Cycle graph|two-cycles]] which intersects with each other at a finite number of points, in the limit where the area of the two-cycles goes to zero. At this limit, there appears a [[gauge group|local symmetry group]]. The matrix of [[intersection number]]s of a basis of the two-cycles is conjectured to be the Cartan matrix of the [[Lie algebra]] of this local symmetry group.<ref>{{cite journal|last=Sen|first=Ashoke|title=A Note on Enhanced Gauge Symmetries in M- and String Theory|journal=Journal of High Energy Physics|volume=1997|issue=9|pages=001|year=1997|doi=10.1088/1126-6708/1997/09/001|arxiv=hep-th/9707123|s2cid=15444381}}</ref>

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

One may [[Compactification (physics)|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-brane]]s. There is a [[U(1)]] local symmetry group for each D-brane, resembling the [[Degrees of freedom (physics and chemistry)|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 root (root system)|simple root]]s, which are related to the two-cycles in the basis that is chosen.

Generators in the [[Cartan subalgebra]] are represented by open strings which are stretched between a D-brane and itself.