Topology change and new phases of N=4 SYM theory

Last friday (06/01/07), I attended the last of the Winter 2007 Caltech High Energy Seminars. The talk was at 1pm, given by KITP post-doc Sean Hartnoll. Hartnoll discussed topological phase changes of N=4 SYM theory (hep-th/0703100). Using both numerical and analytical techniques, Hartnoll found that at weak coupling, the six-sphere eigenvalue distribution transitions to a five-sphere distribution.

An interesting interpretation of this second order phase transition involves the fate of the large AdS black hole spacetime at weak coupling. Given that there are no further phase transitions as a function of coupling, the AdS black hole is described by the five-sphere eigenvalue distribution.

Hartnoll also explained how the five-sphere distribution generalizes the two-dimensional quantum Hall effect. During the question and answer session I mentioned the four and eight-dimensional quantum Hall effects to Hartnoll. He agreed that it would be interesting if his techniques could be applied in such dimensions.

Mapping the geometry of the F4 group

While browsing the arxiv, I came upon a cool new paper on the group F4. The paper contains a plethora of mathematical treasures, including: the exceptional Jordan algebra, OP^2 and the generalized F4 Euler angle construction I mentioned to Kea and Carl earlier this year. Pierre Ramond explored these issues in the context of M-theory back in 1998 (hep-th/9808190). More recently, the physical context has been extended to extremal black holes in N=2 Maxwell-Einstein supergravities (hep-th/0512296).

Mapping the geometry of the F4 group

Abstract:

In this paper we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3 x 3 hermitian matrices with octonionic entries. We use a parametrization which generalizes the Euler angles for SU(2) and is based on the fibration of F4 via a Spin(9) subgroup as a fiber. This technique allows us to determine an explicit expression for the Haar invariant measure on the F4 group manifold. Apart from shedding light on the structure of F4 and its coset manifold OP2=F4/Spin(9), the octonionic projective plane, these results are a prerequisite for the study of E6, of which F4 is a (maximal) subgroup.

Strings in Two Minutes or Less

The eight finalists for the Discover Magazine "String Theory in Two Minutes of Less" contest have been selected and can be found at discovermagazine.com. Even though Brian Greene is the official contest judge, there is a viewer's choice award that net surfers can participate in. I casted my vote for the 'String Ducky' entry, as it is pretty accurate and easy to follow. Which is your choice?

Hawking in Space

Already a giant in theoretical physics, Stephen Hawking has just completed eight rounds of zero-G weightlessness in preparation for his journey into space. The zero-G flight took place on a modified Boeing 727 plane named "G-Force One", boarded at NASA's Shuttle Landing Facility in Florida. As to why Hawking is going through so much trouble for space flight, he commented:

"Many people have asked me why I am taking this flight. I am doing it for many reasons. First of all, I believe that life on earth is at an ever-increasing risk of being wiped out by a disaster such as sudden global warming, nuclear war, a genetically engineered virus or other danger. I think the human race has no future if it doesn’t go into space. I therefore want to encourage public interest in space.

"I have long wanted to go into space, and the zero-gravity flight is the first step toward space travel. I also want to demonstrate to the public that everyone can participate in this type of weightless experience.

"Finally, I am doing this flight to raise money for a number of charities, including ALS, Easter Seals, Starlight Starbright and the X Prize Foundation."

The planned space trip has been so inspiring that Steve Kohler, president of Space Florida, has renamed the zero-gravity research center in Hawking's honor.

Full Cosmic Log (MSNBC) ...

(photo by Steve Boxall)

E6 and Entanglement of Three Qutrits

Michael Duff and Sergio Ferrara posted a follow-up to their papers Black hole entropy and quantum information and E_7 and the tripartite entanglement of seven qubits entitled E_6 and the bipartite entanglement of three qutrits. In order to relate quantum information to 5D black holes, they invoked the use of qutrits, the ternary logic generalizations of qubits. This is a very natural choice and has its roots in the structure of Jordan algebras of degree three. Jordan algebras of degree three underlie the N=2 5D magical supergravities where the charge space of N=2 5D BPS black holes is equated with the Jordan algebra as a vector space over R. In the octonionic case, the Jordan algebra of degree three is called the exceptional Jordan algebra, and the black hole charge space has real dimension 27.

At the end of the paper, Duff and Ferrara state that the "analogy between black holes and quantum information remains, for the moment, just that" and that they "know of no physics connecting them." They mention a comment by Murat Gunaydin (via private communication) suggesting that "the appearance of octonions and split-octonions implies a connection to quaternionic and/or octonionic quantum mechanics." Murat's comment seem quite accurate, but the details of the quaternionic/octonionic quantum mechanics are quite subtle. In 1934, Pascual Jordan, John von Neumann and Eugene Wigner investigated quaternionic/octonionic quantum mechanics and found problems in their Hilbert space formulations. Bischoff in 1993, eventually gave a Hilbert space formulation using the regular representation of the Jordan algebras of degree three. Support for this approach was given by Dray and Manogue's Exceptional Jordan eigenvalue problem. My most recent paper combines the work of Bischoff, Dray and Manogue and elucidates the role of eigenmatrices in the description of N=2 extremal black holes with symmetric moduli space E6(-26)/F4.

The interplay between extremal black holes, entangled qutrits and octonionic quantum mechanics is quite elegant. I tend to take the view that quantum information is the most fundamental, however. Jordan algebras, Freudenthal triple systems and octonions may ultimately be just convenient representations for the description of the quantum logic of nature. As the year progresses, we'll see how this story unfolds, and if we're lucky it will reveal secrets about the nature of spacetime itself.

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)

Ternary Logic

Earlier this week, Kyle (a fellow physics grad student) got me thinking about the set of single variable functions from a finite set A={a,b,c} to a finite set B={0,1,2}. Such functions involve triples of elements of A x B, and take for example the form f_012={(a,0), (b,1), (c,2)}. I ended up using shorthand for such functions, writing f_012 as 012 for instance, to reveal the nice single variable ternary function structure. Ultimately, I came up with the above diagram to show how multiple copies of the parity cube vertices arise in this set of functions. I guess one can also look at it as a Qutrit function diagram. Kea and Carl may see further applications in particle physics. ;)

AIM Team Maps E8

Congratulations to the Atlas team for their fine work in computing the Kazhdan-Lusztig-Vogan polynomials for the large block of the split real form of E₈. AIM's popular overview can be found on their E8 page. For the wonderful details see the Atlas overview.

3D Discrete Dynamical Systems

Shown above are 3D scatter plots of several trajectories for a saddle point origin. The program is written in C, based on an old computer graphics project. The L triplet denotes the eigenvalues used to generate the system.

Standard Model Particle Masses

For all of you that suddenly awaken at night (in a cold sweat) because you forgot your SM particle masses, make sure to check out the PDG page for a refresher.

Twistors' Revenge

One of the recent (Jan 23) jewels on the arxiv is Andrew Neitzke, Boris Pioline and Stefan Vandoren's "Twistors and Black Holes" paper. It seems quaternionic-Kahler manifolds are all the rage now.

Twistors and Black Holes

Abstract:
"Motivated by black hole physics in N=2, D=4 supergravity, we study the geometry of quaternionic-Kahler manifolds M obtained by the c-map construction from projective special Kahler manifolds M_s. Improving on earlier treatments, we compute the Kahler potentials on the twistor space Z and Swann space S in the complex coordinates adapted to the Heisenberg symmetries. The results bear a simple relation to the Hesse potential \Sigma of the special Kahler manifold M_s, and hence to the Bekenstein-Hawking entropy for BPS black holes. We explicitly construct the ``covariant c-map'' and the ``twistor map'', which relate real coordinates on M x CP^1 (resp. M x R^4/Z_2) to complex coordinates on Z (resp. S). As applications, we solve for the general BPS geodesic motion on M, and provide explicit integral formulae for the quaternionic Penrose transform relating elements of H^1(Z,O(-k)) to massless fields on M annihilated by first or second order differential operators. Finally, we compute the exact radial wave function (in the supergravity approximation) for BPS black holes with fixed electric and magnetic charges."

NASA Announces Dark Matter Discovery

Astronomers will announce how dark and normal matter have been forced apart in an extraordinarily energetic collision at 1 p.m. EDT Monday, Aug. 21.