Off-topic, Moriond 2013 (right sidebar has link to live broadcast): ATLAS' diphoton rate is \(1.64\pm 0.30\) times the Standard Model, 2.1 or 2.3 sigma too much...Two bright Gentlemen whom I know from Harvard, Joshua Lapan and Alex Maloney, and two bright Ladies who are actually at Harvard right now, Alejandra Castro and Maria Rodriguez (yes, their affiliation footnotes are marked as ABBA on the title page, and yes, "A" are females while "B" are males), published a wonderful paper
Black Hole Monodromy and Conformal Field Theoryof the kind that makes one happy, somewhat humble, and somewhat proud to have helped to make similar developments possible. Almost exactly 10 years ago, your humble correspondent and Andy Neitzke wrote a paper (with numerous colorful pictures; I've been almost always using TkPaint to create them: a cool mini-tool) that introduced the "monodromy method" (monodromies around singular point in the complexified \(r\) plane of black hole solutions that mix up the linear space of solutions to the Klein-Gordon and other equations on that background) into the black hole research.
Our goal was to rederive the real parts of asymptotic frequencies of the highly damped, quasinormal (ringing) modes of the black holes, a constant that was known to be proportional to \(\ln 3\), a mysterious constant mostly numerologically guessed by Shahar Hod years earlier and proven by Taylor expansions and recursive relationships for the coefficients by your humble correspondent a month earlier. Wow, that paper looks amazingly rich in comments about loop quantum gravity to me today – at that time, I had to believe that there was a chance that its methods could actually hide a toad of truth. Andy Neitzke extended the monodromy method to an analysis of greybody factors three months after our paper.
For nearly a month at the end of 2002 and at the beginning of 2003, we intensely struggled with some subtle and shocking analytic properties of the Bessel functions (functions we previously wanted to remain bizarre abstract boring references in courses for students, not actual parts of our practical lives!) such as the Stokes phenomenon and with the right formulation of the proof we knew had to exist and we could always see its "foggy skeleton" but hard work was still needed to become sure that "monodromy" was the right word and to figure out what the relevant monodromies were and how they had to be teamed up to create the argument.
Of course, other people were involved in the research of the quasinormal modes and other people after us have used the monodromy method in many ways. Interestingly enough, Alex Maloney and Matt Schwartz, another young man from Harvard, were very helpful when I was writing the December 2002 paper because they created the first Mathematica program in my life (or at least the part I could remember) with cycles and conditions, to analyze the behavior of the recursive maps (I told them what the short program should do). At least my knowledge of Mathematica has slightly improved over the last 10 years! ;-)
On TRF, you may read Quasinormal story on quasinormal modes and some other articles about the topic.
Now, in 2013, Alejandra, Josh, Alex, and Maria use similar methods to clarify some mysterious facts that emerged in the quantum black hole research in recent years. Using the stringy methods, the black hole entropy was microscopically calculated for many types of black holes including objects such as the Kerr black hole in \(d=4\). The dual description exploits a conformal field theory (CFT), by a generalization of the AdS/CFT correspondence. And this two-dimensional CFT incorporates left-moving and right-moving excitations that are different in general (an asymmetry you know from the heterotic strings).
The black hole entropy has separate contributions from the left-movers and the right-movers and the interesting fact is that if you revert the sign of the smaller among these two contributions, the resulting difference still has many thermodynamic properties we associate with the entropy. In fact, it describes the area of the inner horizon. I would say that it has to be so because the radii of both horizons solve an algebraic (quadratic) equation and they differ by a different choice of the sign of the square root. But I have never written it before haha.
More generally, it's interesting that the parameters describing the inner horizon appear at many places when you calculate things like scattering on the black hole background. Now, these four authors probably have not only an explanation why it is so. By having found it, they may also explain why the thermodynamics of seemingly different black holes may be analyzed by similar maths. And they have a new, more conceptual, and independent of approximations, way to unmask the origin of the hidden conformal symmetry of Kerr black holes that Alejandra, Alex, and Andy Strominger clarified using low-energy approximate methods in 2010. (There are dozens of TRF blog entries that have something to do with the Kerr black hole entropy and related research by Andy Strominger et al. in recent years.)
And they may calculate lots of quantities about the scattering purely from the monodromies – in some sense, by "group theoretical methods". One could say that these monodromies may be beginning to play an equally important role as the monodromies in the \(\NNN=2\) gauge theories studied by Seiberg and Witten in the early 1990s.
I don't understand all the details yet; the preprint only appeared yesterday. However, let me sketch some basic tools and steps. They study the Klein-Gordon equation on the background of a black hole; as I mentioned, this may cover many apparently different black holes in many spacetime dimensions. It's a second-order linear equation (the time derivatives matter here) so for each frequency, it will typically have two linearly independent solutions. In this two-dimensional space, one may choose a basis but there is no universally preferred basis.
In fact, the space is still two-dimensional near the singular points in the \(r\)-plane, the complexification of some radial ("tortoise"...) coordinate describing the black hole geometry. And around these points, the two-dimensional space gets mixed up by a monodromy transformation in \(GL(2,\CC)\). The monodromies around the inner horizon, outer horizon, and the point at infinity satisfy conditions such as\[
M_- M_+ M_\infty = {\bf 1}
\] which are the key to their calculations. They play with the eigenstates and eigenvalues of these monodromies and they reformulate the scattering problem – a transformation of incoming and outgoing waves between the initial and final state – as something similar to a monodromy, too. There are various technical subtleties they must be careful about – some of those were known to me and Andy Neitzke, most of them were probably not.
But an outcome is that some bilinear function of the scattering amplitudes may be extracted from the monodromies regardless of the type of the black hole. The fact that so many seemingly different black holes carry entropies that admits a two-dimensional CFT approximation is interpreted as the consequence of their having two singular points at finite values of \(r\) and one singular point at \(r=\infty\). The monodromies around these points know about all the basic things in black hole thermodynamics, including the question how the central charges etc. should be divided between the left-movers and the right-movers!
For black holes with a greater number of singular points, the situation is harder and the relevant quantities aren't analytically calculable from the monodromies (right now) but an interesting generalization could exist for those black holes, too. I won't add too many comments here because this is meant to be just an advertisement for the paper and you may read it yourself. The same four authors are preparing a longer paper on similar topics, "Black Hole Scattering from Monodromy".
I still suspect, just like I did at the end of this 2005 blog entry, that the analytic continuation will play an increasingly important role in our understanding of quantum gravity. The Euclideanized black hole solutions according to Hawking and Gibbons have previously allowed an elegant calculation of the black hole entropy and temperature. The requirement was that the solution had no deficit/excess angle at the Euclideanized horizon.
However, the irregularities – e.g. monodromies – that actually do exist might turn out to be even more useful. They could know about lots of things. In this paper, the monodromies are acting on the low-energy fields only. But maybe one should think about the action of the monodromies on the space of all possible microstates, too. A black hole itself may be identified with a mathematical structure defined by purely group-theoretical constraints of a sort and all thermodynamic features of the black hole may be encoded in them.
And of course, I can't forget to recall that analytic continuation is also likely to play an important role in the right explanation of the black hole information loss, firewalls, quantum xeroxing, and other non-existent anomalies, as we discussed in the context of the Papadodimas-Raju paper.
Off-topic, soccer
Tomorrow, my hometown may very well look like this:
That's how the fans of Fenerbahce Instanbul reacted in May 2012 when they lost to another Turkish team. The UEFA home matches over there are closed for fans because UEFA managed to notice that the latter are wild animals. (The ban of fans didn't help; they were shooting parachuting suicide bombers by ballistic missiles into the stadium from outside when they played BATE Borisov.) However, Pilsen won't enjoy this kind of protection. At least 800+ Turkish fans will be present on the stadium and many more may be around. The reward for our having defeated Naples, currently the 2nd team of the top Italian league, 5-to-0 aggregate may be somewhat bittersweet.
Fenerbahce is by far the most popular Turkish soccer club, counting 1/3 of the Turkish population as its fans. They include this jerk in the middle. Note that the very fact that they play the Europa League is a playful act of political correctness. A fast look at the map of Instanbul reveals that Fenerbahce is an Asian club.
Well, the match carries the highest risk rating. Our police has prepared 300 heavily dressed cops, cops on horses, a helicopter, ready teams of traffic cops, immigration cops, even some German and Austrian cops, public order cops, many other subspecies of cops, and the anti-conflict teams. Let's pray that these wild Asians won't destroy our elegant city. ;-)
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