## Wednesday, March 28, 2007

### Jordan Algebras and Extremal Black Holes

Back in 2003, Pierre Ramond wrote a classic paper entitled Exceptional Groups and Physics where he considered the mysterious relationship between M-theory and the exceptional Lie groups. On pages 8 and 9, he discussed the exceptional Jordan algebra (EJA) and its automorphism group F4, arguing that the SO(9) subgroup of F4 should be interpreted as the light-cone little group in eleven dimensions. He concluded with the statement:

"If the SO(9) subgroup of [the] EJA automorphism group F4 can indeed be identified with the light-cone little group in eleven space-time dimensions, it will suggest the EJA as the charge space of a very special system."

In February 2005, Murat Gunaydin showed the EJA is actually the charge space of an extremal black hole in N=2, d=5 Maxwell-Einstein supergravity. Come summer 2005, Andrew Neitzke, Boris Pioline and Andrew Waldron joined forces with Murat Gunaydin and formulated a method for counting microstates of four-dimensional BPS black holes in N >= 2 Maxwell-Einstein supergravities. Gunaydin gave a December 9th talk on the approach at the KITP and by December 22 the work culminated in a paper entitled BPS black holes, quantum attractor flows and automorphic forms.

It turns out there are more Jordan algebraic goodies that add to the story, so I put together a paper and posted it here. :)

(Note: The image above is a depiction of the Bajoran wormhole from the Star Trek: Deep Space Nine series)

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## 3 comments:

For a possible connection between the powers of 3 in the pure constants and black holes, see Lubos Motl's recent blog entry.

Nice catch Carl!

The ln(3) was supposed to be related to the use of cubic graphs in spin networks. Some problems came up, however, and the earlier LQG black hole entropy calculations became suspect. The story goes on of course, and so does the idea that LQG and M-theory may one day be united in some larger background independent theory. This is what Kea's M(otivic)-theory is aiming to accomplish via categories and operads.

Motl's april 3rd post on F-theoretical GUTs is insightful as well. Try to give the Bourjaily papers a read and connect it to the discussion we had on PF with Alejandro Rivera concerning SU(5), SO(10) and E6 GUT models.

Another comment on the "tripled Pauli statistics".

I bet that you can rewrite the weired function as a usual thermodynamic partition function if you add some extra exponentials to the denominator. These would not appear in the usual formulation because those degrees of freedom are frozen out.

That is, the 3 comes from ignoring the extra Pauli degrees of freedom that have energies on the order of the Planck mass. Another way of saying this is that the black hole partition function is a low energy approximation only. To put it into standard thermodynamic form, you have to add some high energy degrees of freedom that are frozen out. Then it acts like a triply degenerate Pauli partition function.

I'll try and write up something about this and put it up on Physics Forums over the next day or two. Right now I'm in San Antonio, where the latest news is that the law has been changed regarding what sorts of weapons blind people are allowed to use while hunting.

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