M Theory Lesson 13
John Baez's TWF244 is really cool. He talks about a recent paper by Leinster on the Euler characteristic of a finite category. Consider the following. Order the objects in the category $1, 2, 3, \cdots ,n$. The integral adjacency matrix for the category sets $a_{ij}$ to be the number of arrows from $i$ to $j$.
For example, in the category
we would have a 3x3 matrix with all entries 1, since we must not forget the identity arrows. If the inverse to this matrix $A$ existed, the Euler characteristic would be the sum of entries in $A^{-1}$. Let's fill in a few more arrows. Imagine there were $k$ arrows from $1$ to $2$, $2$ to $3$ and $1$ to $3$ which forms a basic composition triangle. Similarly, imagine the dual triangle had $m$ arrows at each edge. Then the adjacency matrix would be a circulant based on $1,k$ and $m$. Such triangles look like idempotent equations $k^2 = k$, but here $k$ is an ordinal and we should wonder about composing one of the $k$ arrows from $1$ to $2$ with another one from $2$ to $3$, because the number of such compositions would naively be more than $k$. If $k^2 = k$ were true, and $k$ was ordinal, then it must be zero or one, which gives a particularly simple kind of category otherwise known as a poset.
Anyway, to cut a long story short, this characteristic works nicely for all sorts of things, such as orbifolds. In M Theory we counted the number of particle generations using an orbifold Euler characteristic, which might be a rational number in general. So we can think of this as a cardinality of a category! This is wonderful, because the physical result follows from the universality of $\chi$.
Moerdijk looked at Lie groupoids as a foundation for orbifolds, which seems like a logical thing to do from the Symmetry point of view. But remember that we encountered the orbifold Euler characteristic in the work of Mulase et al on ribbon matrix models.
For example, in the category
we would have a 3x3 matrix with all entries 1, since we must not forget the identity arrows. If the inverse to this matrix $A$ existed, the Euler characteristic would be the sum of entries in $A^{-1}$. Let's fill in a few more arrows. Imagine there were $k$ arrows from $1$ to $2$, $2$ to $3$ and $1$ to $3$ which forms a basic composition triangle. Similarly, imagine the dual triangle had $m$ arrows at each edge. Then the adjacency matrix would be a circulant based on $1,k$ and $m$. Such triangles look like idempotent equations $k^2 = k$, but here $k$ is an ordinal and we should wonder about composing one of the $k$ arrows from $1$ to $2$ with another one from $2$ to $3$, because the number of such compositions would naively be more than $k$. If $k^2 = k$ were true, and $k$ was ordinal, then it must be zero or one, which gives a particularly simple kind of category otherwise known as a poset.
Anyway, to cut a long story short, this characteristic works nicely for all sorts of things, such as orbifolds. In M Theory we counted the number of particle generations using an orbifold Euler characteristic, which might be a rational number in general. So we can think of this as a cardinality of a category! This is wonderful, because the physical result follows from the universality of $\chi$.
Moerdijk looked at Lie groupoids as a foundation for orbifolds, which seems like a logical thing to do from the Symmetry point of view. But remember that we encountered the orbifold Euler characteristic in the work of Mulase et al on ribbon matrix models.
posted by Kea | 8:21 PM
6 Comments:
These diagrams look similiar to the Eightfold Way. That makes one wonder whether non-linear operads will be key to extending the Standard Model. That would be refreshing, since physics has gone nowhere in 30 years.
Yeah. And the ribbon diagrams are just what 't Hooft ordered for QCD back in 1974. Seems so simple, doesn't it?
Hi Kea
Thanks for the link to TWF 244. As usual John Baez has a great and interesting blog.
I am fascinated with item 5 which discusses Euler’s bridges. From my perspective this is a problem ideal for the strategy analysis of game theory.
Geert Jan Olsder [mathematics] Delft University in 2005 presented a variant of this problem as a train routing schedule using game theory.
MAX PLUS IN HET (TREIN)VERKEER ...[Text in English]
webserv.nhl.nl/~kamminga/wintersymposium/Olsder2005.pdf
If you like, on Amazon.com you can view inside this great book: Dynamic Noncooperative Game Theory (Classics in Applied Mathematics) (Paperback)
by Tamer Basar, Geert Jan Olsder.
Basar is also still active as a Professor of Electrical and Computer Engineering. Engineers might be considered “applied physicists“.
Doug, we appreciate your enthusiasm for Game Theory, but I have to say, I'm not sure when I'm going to find time to read any of these references, once I slog through a few papers in particle physics, math phys, maths, cosmology, philosophy ..... sigh. So little time.
Thanks for the Leinster reference Kea. I'm pleased to know that whatever the fundamental objects of M-theory are, we can still talk about their Euler characteristic!
Why, yes, kneemo! This is important. We can't go to all this trouble to create a 'true BI' theory and then throw it all out by cheating on the actual number crunching at the end. The numbers must fall out of a universal construction.
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