The geometry of 3-manifolds

If you want to get an idea of the famous Poincaré conjecture for which Grigoriy Perelman was recently awarded a Millenium prize, I highly recommend watching this very entertaining lecture by Curtis McMullen at Harvard in 2006.

I had never realised that tori of genus greater than one are hyperbolic! It makes sense, if you think that a sphere (genus 0) has positive curvature, a torus (genus 1) is flat, that tori of genus 2 and more have negative curvature. These higher tori can be constructed by gluing together the edges of polygons.

I wonder whether anti-de Sitter space can also be viewed as a high genus space. Are the many black holes in the universe implying that it has a high "genus"? But the notion of genus isn't clear in higher dimensions...

So I started (very naively) to think about black holes on a latex surface, like the ones shown in the lecture. Consider a sphere and put a stone at the north pole. The sphere will bend, just like spacetime bends around the sun. Now imagine that you had a way to increase the mass of the stone, it would create a well at the north pole which would eventually touch the south pole from the inside. What happens then? The well could continue to become deeper and deeper but now it would create a spike out of the south pole. This is certainly misleading. More likely, once the stone approaches the south pole from the inside, its mass sucks it in, creating a well (as well).

Ultimately, what you'll end up with is a torus. The Planckian mass is at the centre of this torus, where there is really no spacetime: that's the black hole!

I had an unexpected confirmation of this picture in my kitchen. There was a frying pan full of a layer of greasy water and I noticed that it tended to leave discs where there was no water. By slowly pouring more water into the pan, the discs would shrink until there reach zero size, at which point a circular wave was emitted. I had just witnessed a topology change;) I could also reverse the process by taking some water out of the pan and stirring up the water. When the water had stabilised enough, discs were suddenly appearing and expanding quickly. That's my version of frying pan black holes: try it at home!

PS: the key is the circular wave.

"Holographic Superconductos in M-theory"

Last Monday we had Jerome Gauntlett telling us about exciting applications of the AdS/CFT correspondence to Condensed Matter systems.

Some quantum critical points (second order phase transitions) are at strong coupling, and hence intractable with standard CM technics, but the AdS/CFT comes to the rescue. This happens for example for "heavy fermions" (quasi-particles condensing to give Cooper pairs) or for high $T_c$ cuprates that have a superconducting phase. Holographic descriptions of such systems have been put forth by Gubser and by Hartnoll, Herzog, and Horowitz. The strongly coupled CFT in 3d has a holographic AdS dual, which has a $U(1)$ gauge symmetry that can be spontaneously broken by a charged field $\chi$. A finite temperature the description is in terms of a black hole in AdS, and it acquires charged hair at low temperature that breaks the $U(1)$.

Almost all the work on this topic has been taking a "bottom up" approach, but Jerome took the "top down" approach: constructing holographic superconductors from M-theory using consistent Kaluza-Klein truncation with Sasaki-Einstein spaces. Here consistent truncation means that you can keep only a finite set of the light KK fields (you cut the infinite KK towers) but then any solution of the low dimensional theory involving those remaining fields has to uplift to an exact solution of the original higher dimensional theory. So consider a stack of M2-branes at the tip of a Calabi-Yau cone on a Sasaki-Einstein seven-manifold for which the near-horizon limit is $AdS_4 \times SE_7$ with metric \[ d s^2 = \frac{1}{4} d s^2 (AdS_4) + d s^2 (SE_7) \] with \[ d s^2 (SE_7) = d s^2 (KE_6) + \eta \otimes \eta~,\] where $\eta$is the contact form. There is an associated Killing vector, which (although it does not always have to close) generates the required $U(1)$ symmetry. The four-form field strength is $G_4 = \frac{3}{8} \text{Vol}(AdS_4)$. This geometry is dual to $\mathcal{N} =3$ SCFTs in $d=3$ (or $\mathcal{N} = 8$ if $SE_7$ is replaced by a seven-sphere).
Now, had the membranes been replaced by anti-membranes, $G_4$ would have the opposite sign but more importantly the dual SCFTs would have $\mathcal{N}=0$. The corresponding geometry is called skew-whiffed $AdS_4 \times SE_7$. It is this geometry that can reproduce and generalize the HHH models. The consistent truncation found by Jerome has an extra neutral scalar field $h$ which corresponds to a deformation of the skew-whiffed CFT by an operator $\mathcal{O}_h$. At low temperature this system exhibits a superconducting phase.
Jerome concluded by suggesting that if superconductivity had not already been discovered, such a study would have pointed it out to us. M-theory might thus lead to a similarly radical discovery of qualitatively new phenomena...

David Blaine

I watched this TeD video where the magician David Blaine explains how he broke the Guiness record of apnea by staying a little more than 17 minutes under water.

To me, this guy is more than a magician, and more than an athlete, he's a pure artist. His determination to push the human body beyond its clinical limits is of course very spectacular and impressive, but it also says something about what it means to live. How much can one deprave the body from resources (food, oxygen) and still survive? There is also a sense of askesis very similar to what one can find in several religious traditions (fast, meditation, etc.)...

However, what struck me particularly was the fact that after a failed attempt he finally broke the record (live on TV) when he was convinced that he had no chance to succeed since his heartbeat was unusually high during the immersion. I think there might be some universal truth in this: great things are often achieved just at the moment when a long chain of intense efforts oriented to a precise goal ultimately gives way to the disappearance of all hope.

Nirmali

II. The twofold task in working out the question of Being. Method and design of our investigation

This is a summary of Heidegger's second introduction to Being and Time.

5. The ontological analytic of Dasein as laying bare of the horizon for an interpretation of the meaning of Being in general.

Dasein is a tricky thing to analyse in that it is in some respect obvious to us and in some other obscure. MH writes that "Dasein is ontically 'closest' to itself and ontologically farthest; but pre-ontologically it is surely not a stranger." As regard to its existence, we understand it immediately since "we are it". But "in spite of this, or rather for just this reason", it is the most difficult thing to really understand in its essence, because we are naturally inclined to consider it as obvious, self-evident. However, intuitively we have some sort of embryonic understanding of its essence since "Dasein is in such a way as to be something which understands something like Being". To perform an analytic of Dasein we must "choose such a way of access and such a kind of interpretation that this entity can show itself in itself and from itself".

MH announces that "we shall point to temporality as the meaning of the Being of that entity which we call "Dasein"," and that "we shall show that whenever Dasein tacitly understands and interprets something like Being, it does so with time as its standpoint". And in consequence, "time needs to be explicated primordially as the horizon for the understanding of Being, and in terms of temporality as the Being of Dasein, which understands Being." But the conception of time required will have to depart from the ordinary understanding of time (in particular time as a "criterion for naively discriminating various realms of entities", such as temporal, non-temporal, 'supra-temporal' eternal entities...).

"The central problematic of all ontology is rooted in the phenomenon of time, if rightly seen and rightly explained".

6. The task of destroying the history of ontology

Dasein's own past "is not something which follows along after Dasein, but something which already goes ahead of it" (Dasein "is its past"). This has the perverse effect that Dasein "falls prey to the tradition of which it has more or less explicitly taken hold. This tradition keeps him from providing its own guidance, whether inquiring or in choosing". And this also holds true for ontological understanding. Tradition "becomes master" and conceals itself. "It blocks our access to those primordial 'sources' from which the categories and concepts handed down to us have been in part quire genuinely drawn". The Greek ontology has thus "deteriorated to a tradition in which it gets reduced to something self-evident". Then "in the Middle Ages this uprooted Greek ontology became a fixed body of doctrine". Subsequent concepts such as "the ego cogito of Descartes, the subject, the "I", reason, spirit, person" (...) "all remain uninterrogated as to their Being". Just like in certain oriental philosophies, the primordial clarity of mind has deteriorated throughout history into rigidified clichés.

Our task is clear: "we are to destroy the traditional content of ancient ontology". AC/DC man! But nihilist enthusiasm is inappropriate: this destruction is "far from having the negative sense of shaking off the ontological tradition" (or "bury the past in nullity"). "We must, on the contrary, stake out the positive possibilities of that tradition, and this always means keeping it within its limits".

The point of departure for the task of destroying the problematic of Temporality will be the Kantian doctrine of time. After all, Kant is "the first and only person who has gone any stretch of the way towards investigating the dimension of Temporality". Still he didn't go very far because he "took over Descartes' position quite dogmatically". And Descartes, with his radical discovery of the 'cogito sum', left undetermined the meaning of the Being of the 'sum'. He "is 'dependent' upon medieval scholasticism". Furthermore, "Kant's basic ontological orientation remains that of the Greeks", and more precisely that of Aristotle.

In summary, the strategy is clear: in order to concretely address the question of Being with time as a horizon we must first destroy the associated tradition "until we arrive at those primordial experiences in which we achieved our first ways of determining the nature of Being".

7. The phenomenological method of investigation

"AdS/CFT from F-theory?"

At the lunch group meeting James told us about a fairly recent paper by Polchinski and Silverstein, entitled Dual Purpose Landscaping Tools: Small Extra Dimensions in AdS/CFT. Their goal is to construct vacua of string or M-theory with minimal supersymmetry of the form $AdS_d \times \text{small}$, where small means that the internal space is small compared to the AdS radius. The problem is that in the case of the standard Freund-Rubin backgrounds (whose dual CFTs are rather well understood in terms of the near-horizon limit of D3- or M2-branes at the tip of a Calabi-Yau cone), the radius of the internal space is of the same size as the AdS radius.

The new idea is to add some D7-branes. These contribute to the potential energy with an opposite sign to the curvature of the internal manifold, and so could be chosen such that the resulting cosmological constant is small.

This is where F-theory comes into play. The construction there is to glue IIB solutions using the $SL(2,\mathbb{Z})$ duality symmetry, and to allow for D7-branes. A D7-brane, which is codimension two and thus can be surrounded by a circle, is a unit magnetic source for the axion $C_0$ (a periodic RR scalar field): \[\int_{S^1} d C_0 =1~.\] The axion combines with the dilaton $\phi$in a complex field in the upper half-plane \[\tau = C_0 + i e^{-\phi} = C_0 +\frac{i}{g_s}~.\] $C_0$ has monodromy one around the D7-brane, meaning that going around the circle transforms it as $C_0 \to C_0 +1$.

Now splitting the 10d metric as a 4d Minkowski space-time times a 6d manifold $B$ and requiring $\mathcal{N}=1$ susy in 4d implies that $B$ is a Kaehler manifold and that $\tau$ is (anti-)holomorphic: $\bar\partial \tau =0$.

Mathematically, specifying the modular parameter $\tau$ in the fundamental region is equivalent to specifying an elliptic curve, and so this whole construction can be seen as an elliptic fibration $\pi: X\to B$, with $X$ a four complex dimensional manifold. The fibre over a point $p\in B$ on the base is an elliptic curve $E_{\tau}$ that is "biholomorphically" equivalent to a torus determined by a lattice $(1,\tau)$:
\[\pi^{-1} (p) \cong E_{\tau} \cong \mathbb{C}/(1,\tau)~.\]
What is very nice is that the D7-branes are located where the elliptic fibration degenerates, i.e. where the torus gets pinched off and becomes singular.
James made the point that this is not just a mathematical construction since one can T-dualise to IIA and the lift to M-theory, where the resulting solution is the product of a 3d Minkowski space-time and a CY four-fold. Polchinski and Silverstein's idea is to get $AdS_5 \times \text{small}$ solutions by looking at CY four-fold cones that are elliptically fibered...

Etymology of yo

Today, I think I made an etymological breakthrough: the interjection "yo", at least in some of its acceptations, is none other than the contraction of "you know". Word up!