Monday, June 04, 2007

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.

5 comments:

Kea said...

Good to see you're having fun over there. Interesting stuff.

Doug said...

Hi Kneemo,

I understand only a little of the Hartnoll paper and have a few questions.

RE: Figure 1: Conjectured phase diagram as a function of temperature and coupling.

Could temperature and coupling be strategic degrees of freedom as opposed to spatial?

Am I correct in interpreting that section #6 suggests that the difference between S6 and S5 is due to perturbation?

“Saddles” in mathematical game theory may be evidence of equilibrium.

Doug said...

With my interest of finding similarities among theories and existence, consider:

1 - I found this NASA artist concept ‘Running Rings Around the Galaxy’ that resembles a Rössler Attractor,

References:

Running Rings Around the Galaxy
http://www.spitzer.caltech.edu/Media/happenings/20070530/

Rössler Attractor
2 representations
... one at top -> [looks similar to running rings?]
... one at bottom with equations
[from the Open Computing Facility, UC Berkeley]
http://www.ocf.berkeley.edu/~trose/rossler.html

WIKI: system of three non-linear ordinary differential equation
“The banding evident in the Rössler attractor is similar to a Cantor set rotated about its midpoint. Additionally, the half-twist in the Rössler attractor makes it similar to a Möbius strip.”
http://en.wikipedia.org/wiki/R%C3%B6ssler_map

2 - Take a look at these animations from
The Nonlinear Dynamics Group at Drexel University
See directions ... need the latest version of the Macromedia Flash player
http://lagrange.physics.drexel.edu/flash/

kneemo said...

Could temperature and coupling be strategic degrees of freedom as opposed to spatial?

Relating temperature to strategic degrees might work. Let's think about it in terms of black holes. Assume that strategic degrees of freedom (at the level of cellular automata) are equated with the density of states of a black hole quantum system. Then the entropy is proportional to the logarithm of the density of states, i.e. the logarithm of the strategic degrees of freedom. In this way we can relate the temperature to strategic degrees of freedom, as well as horizon area.

Doug said...

Petri Nets were developed for graphing cellular automata and are used by engineers in the powerful Max-Plus Algebras.
Nodes are used in lieu of vertices and arcs rather than edges.
Petri nets also use transitions and tokens. The latter might be able to represent energy quanta.
“Petri Nets is a formal and graphical appealing language which is appropriate for modelling systems with concurrency and resource sharing. Petri Nets has been under development since the beginning of the 60'ies, where Carl Adam Petri defined the language. It was the first time a general theory for discrete parallel systems was formulated. The language is a generalisation of automata theory such that the concept of concurrently occurring events can be expressed.”
http://www.informatik.uni-hamburg.de/TGI/PetriNets/

Other examples are in “MAX PLUS IN HET (TREIN)VERKEER’ [English], Geert Jan Olsder Delft University, on the web.
http://webserv.nhl.nl/~kamminga/wintersymposium/Olsder2005.pdf

The web paper above serves as the basic example for this textbook found in many university mathematics libraries.
‘Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications’ (Princeton Series in Applied Mathematics) by Bernd Heidergott, Geert Jan Olsder, and Jacob van der Woude (Hardcover - Nov 7, 2005)