Sunday, September 19, 2010

Theory Update 1

For years now this blog has posted M Theory lessons, but my original reason for using this title (which I leave to the reader's imagination) has somewhat diminished in importance and so instead of Lesson $360$, today we begin with our theory update number $1$. It is just a lazy choice of post title, and I am not about to change direction.

In the two dimensional path table, or monomial table, we can consider maps (arrows) that embed one group of paths into a larger group, either to the right or above it. Observe that such a map always exists for objects that are adjacent, either horizontally or vertically, and these maps can be composed along a line. So instead of a single map between path sets, we can define a category with $1$-morphisms between two objects, given by the pair of arrows that end in a rectangle corner. Moreover, given a pair of such $1$-morphisms, we can similarly define a $2$-morphism using a larger rectangle, as shown by the medium arrows in the diagram.

In a typical $2$-category, these $2$-morphisms have two types of composition, vertical and horizontal. This is usually drawn as a globule picture, but the diagram illustrates the point. Now given two $2$-morphisms there is a $3$-morphism, given by the paths around the big arrows in the diagram. Observe how a vertical composition of $2$-arrows results in a different kind of $3$-morphism to the horizontal composition.

For example, consider the maps discussed in the last lesson. There are two sets of maps of interest into the qutrit tetractys: one from the three qubit path set and one from the two qutrit path set. The former selects $8$ noncommuting paths and imbeds them into the $27$ paths, whereas the latter embeds $9$ paths. The former is associated to the Jordan algebra over the bioctonions and the latter to the $57$ dimensional nonlinear version of $E_8$. In general, we have maps $n^{d} \rightarrow m^{t}$ between the $d$ $n$-dit space and the $t$ $m$-dit space.

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