"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Now, had the membranes been replaced by anti-membranes, $G_4$ would have the opposite sign but more importantly the dual SCFTs would have $\mathcal{N}=0$.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...

"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...

"Gauge/gravity in three dimensions"

Today's Theoretical Particle Physics seminar was given by Diego Rodriguez-Gomez (Queen Mary, U. of London), based mainly on 0809.3237 and 0903.3231.

Given the broad audience, he started by a small review of the AdS/CFT correspondence as a specially clean incarnation of 't Hooft's and Susskind's holographic principle. Whereas the 4d case has been relatively well understood for some time, the 3d case began to reveal itself only recently. In fact, until roughly two years ago, it was thought that the IR fixed point at the end of the RG flow of the maximally supersymmetric $\mathcal{N}=8$ SYM in 3d had no Lagrangian description. The reason for this belief was that the dual description has a non-constant dilaton blowing up at small radius: $e^{\phi} =(R/r)^{5/4}$. The BLG theory (anticipated by Schwarz) was an $\mathcal{N}=8$ Chern-Simons-like theory but it had an $SU(2)$ gauge group instead of the large $N$ needed in the AdS/CFT context. The resolution came from relaxing the requirement of maximal susy down to $\mathcal{N}=6$, which allowed arbitrary $N$ [ABJM].

Diego was interested in reducing the supersymmetry to $\mathcal{N}=2$ by putting the M2-branes at the tip of a Calabi-Yau four-fold. He focused on a specific example: the cone over $Q^{1,1,1}$, which is similar to the conifold in 6d and has the advantage to have been extensively studied (the metric is explicitly known). The simplest proposal (inspired by crystal models) is that the dual field theory is a quiver with four gauge groups, six fields, and a sextic superpotential $W$ with two terms. As in the ABJM case, all the global symmetries are not manifest, but appear by studying how scalar fields get identified. This results in a mesonic moduli space that is an $\mathcal{N}=2$ orbifold $Q^{1,1,1}/\mathbb{Z}_2$. Here are the toric diagram and the associated quiver:
Diego and his collaborators were able to show that the chiral operator spectrum is matching the supergravity harmonics, at least at large CS level $k$. This is a non-trivial check of the non-Abelian superpotential $W= C_1 A_1 B_1 C_2 A_2B_2 - C_1 A_1 B_2 C_2 A_2B_1$.

Remaining mysteries include the question of whether theories with $\mathcal{N}<3$ susy are conformal, which would require to have an equivalent in 3d of a-maximization; the inverse algorithm (from the CY4 to the quiver) and the connection to type IIA string theory; the small $k$ limit and monopoles operators, which is the genuinely M-theoretic limit, since at large $k$ the theory reduces to IIA.

"Topology and Relativity"

OK, so for this Friday's Theoretical Physics Seminar we had Maulik Parikh from IUCAA in India talking about two of his current projects paired for the occasion under the title "Topology and Relativity".

Topology and special relativity

This is a collaboration with Brian Greene and Jana Levin, to appear.

The starting point is the twin paradox on a cylinder. For memory, the twin paradox in flat space-time concerns the age difference of two twins, one of which has been sent to the outer space and back. There is in fact no paradox, since the Principle of Relativity states that all inertial observers are equivalent, whereas the space twin accelerated and decelerated on his Odyssey.

In the case where space is a circle and time a line, and so on a space-time cylinder, both twins, gracefully christened A and B, could be inertial and still go their different ways and meet again. Who, then, asketh Maulik, is younger?
Well, the catch is that the periodic identification of the space coordinate picks a globally preferred frame (the one with winding number zero I suppose). So the first lesson is that

A nontrivial topology breaks global Lorentz symmetry.

The preferred frame could be determined by experiments by sending photons in different directions. In particular, Einstein clock synchronization would only be possible for preferred observers. There is also a discontinuity in the time coordinate, as the inhabitants of Kiribati, lying (not anymore since 1995) on the International Date Line, know very well (after all, as far as time zones are concerned, the Earth's worldvolume is essentially a cylinder).

Suppose there is a compact extra dimension (of mm size according to the ADD scenario). Can we tell the velocity of our (3+1)-brane around it? Obama says "Yes we can!", for which Maulik thinks he should get the Physics Nobel Prize as well...

If the LHC fires gravitons in the extra dimensions, then measuring their return time could make it possible to determine the motion of the brane. This is in fact not realistic, since graviton interactions are suppressed by the Planck mass.

Another effect of an compact extra dimension would be a modification of Newton's law. Since a source placed at a point along the extra coordinate would be repeated infinitely at interval $L$, the standard potential
\[
V(r) = -\frac{GM}{\pi r^2}
\]would be replaced by
\[
V(r) = -\frac{GM}{\pi } \sum_{n=-\infty}^{\infty} \frac{1}{r^2 +(nL)^2}\\
\simeq -\frac{GM}{r} (1+ 2e^{-2\pi r/L})~.
\]For a moving brane, one would have to replace $L$ by $\gamma L$ where $\gamma$ is the relativistic factor. This opens the amusing possiblity of a magnified extra dimension, if our brane were to move ultra-relativistically.

Topology and the Fate of the Universe

The second part of Maulik's talk was on a disconnected topic, base on his paper Enhanced Instability of de Sitter Space in Einstein-Gauss-Bonnet Gravity. The geometry of the early universe is well-described by de Sitter space, which is perturbatively stable. However, Bousso and Hawking showed that dS space can be destabilized by non-perturbative effects (such as instantons, black hole tunneling, etc.). The probability of a gravitational instanton is
\[
\Gamma \sim \frac{\exp(-I_E[\text{instanton}])}{\exp(-I_E[\text{background}])}~.
\]where $I_E$ is the Euclidean action. Now instead of taking the usual Hilbert-Einstein action, Maulik considered the Einstein-Gauss Bonnet action (which appears for instance in the low energy effective action of heterotic string theory). The novelty is the in our dimension the Gauss-Bonnet action is a topological invariant
\[
I_{GB} = -\frac{\Lambda V_4}{8\pi G} - \frac{2\pi \alpha}{G} \chi~,
\]where $\Lambda$ is the cosmological constant, $V_4$ is the volume, $\alpha$ the coupling constant, and last but not least $\chi$ is the Euler number.

The Gauss-Bonnet topological term can enhance the instability of primordial de Sitter space.

In the case of the extremal Nariai black hole with topology $S^2\times S^2$, using the Hilbert-Einstein action as Bousso and Hawking did leads to an instanton probability $\Gamma = \exp[(-\pi L^2/3G)$, which means this is only relevant for a length scale $L$ close to the Planck scale. Disappointing.

In contrast, with its additional topological term, the EGB action leads to an enhanced production of Nariai black holes (with $\chi =2+2=4$) by a factor of $\exp(4\pi\alpha/G)$.

Maulik also mentioned a bound on the maximal curvature of empty dS space.

Other applications include the effect of the topological term on the probabilities of Calabi-Yau manifolds in the string landscape. For example, the quintic has $\chi=-200$, which means it could be suppressed by roughly $\exp(-2000)$.

"Mirror symmetry, Langlands duality, and the Hitchin system"

Today, at the Geometry and Analysis Seminar organized by Nigel Hitchin, Tamas Hausel talked about his paper with Michael Thaddeus Mirror symmetry, Langlands duality, and the Hitchin system. The room was packed. He started by giving some background informations about the three concepts in his title.

Mirror symmetry

The basic idea is that the symplectic geometry of a d-dimensional Calabi-Yau manifold $X$ can be related to the complex geometry of another CY manifold $Y$. There is a topological test of this relation, referred to as topological mirror symmetry, which equates (mirror pairs of) Hodge numbers of the two CYs:
\[ h^{p,q} (X) = h^{d-p,q}(Y)~. \]

Note that any hyperkaehler manifold satisfies $h^{p,q} (X) = h^{d-p,q}(X)$, so in a certain sense TMS is already built-in. Tamas mentioned two important developments in the history of mirror symmetry: homological mirror symmetry proposed by Kontsevich in 1994, which reads
\[
\mathcal{D}^b (\text{Fuk}(X,\omega)) \cong \mathcal{D}^b (\text{Coh}(Y,I))~,
\] where $\omega$ is the symplectic form and $I$ the complex structure. Another breakthrough was the geometric construction of $Y$ from $X$ elaborated by Strominger, Yau and Zaslow in 1996.

Langlands duality

The aim of the Langlands program is to describe $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ via representation theory.
To each reductive group $G$ is associated a Langlands dual $^LG$. The Langlands conjecture leads for instance to class field theory, in the case $G=GL_1$; in the case $G=GL_2$, it leads to the Taniyama-Shimura conjecture (which is famous because it implies Fermat's last theorem). An important progress towards the proof of the conjecture was made by Ngo in 2008 with his proof of the fundamental lemma for the function field $\mathbb{F}_q(X)$.
There is a geometric version of the conjecture, obtained by replacing $\mathbb{F}_q(X)$ by $\mathbb{C}(X)$ for $X/\mathbb{C}$ (Laumon 1987, Beilinson & Drinfeld 1995):
\[
\{ G\text{-local systems on $X$} \}
\]\[
\leftrightarrow \{ \text{Hecke eigensheaves on Bun$_{^LG}(X)$}\}~.\]
Hitchin systems

Recall that a Hamiltonian system $(X^{2d}, \omega)$ has an energy functional $H: X\to \mathbb{R}$ and an Hamiltonian vector field $X_H$ such that $\text{d}H=\omega(X_H,\cdot)$. A function $f:X\to \mathbb{R}$ is a first integral if $X_H f = \omega(X_H, X_f) =0$ (involution). The system is completely integrable if there is $d$ first integrals. The generic fibre is then a torus (examples: Euler and Kovalevskaya tops, spherical pendulum).
An algebraic version is obtaiend by replacinf $\mathbb{R}$ by $\mathbb{C}$, and many examples can be formulated as Hitchin systems (1987).

Now I cannot say I completely followed the rest of the talk in all its glory, but I'll try to restate what I understood. Tamas was considering different moduli spaces, which are all smooth non-compact varieties: $\mathcal{M}_{\text{Dol}}^d (G)$ is the moduli space of rank $n$ and degree $d$ Higgs bundles $(E,\phi)$, $\mathcal{M}_{\text{DR}}^d (G)$ is the moduli space of flat $G$-connections on a genus $g$ curve, $\mathcal{M}_{\text{B}}^d (G)$ is yet another thing -- but they're all equivalent by a non-Abelian Hodge theorem. The Hitchin map $\chi(\phi)$ is completely integrable and its fibre $\chi^{-1}(a)$ is a torsor.

Inspired by the SYZ conjecture, Hausel and Thaddeus noticed in 2003 that $\chi^{-1}_{SL_n}(a)$ and $\chi^{-1}_{PGL_n}(a)$ are torsors for dual Abelian varieties. (I think this means they are related by T-duality on the toroidal Hitchin fibres, but he said "fibrewise Fourier-Mukai tranform" instead :)

A confirmation that their conjecture are more or less sane came from the 2006 work of Kapustin and Witten on S-duality (electric-magnetic duality) in $\mathcal{N}=4$ super-Yang-Mills in four dimensions, which Tamas qualified as "a major work" with many fertile ideas. (In fact T-duality in the Hitchin moduli space corresponds to S-duality in the gauge theory, see Witten's Strings on the Beach! talk in 2005.)

Using stringy Hodge numbers (also known as "orbifold cohomology") Tamas made a conjecture with a TMS test, but he also made another one using mixed Hodge numbers.
His final questions were: "Why two conjectures? Why same Hodge numbers instead of mirrored ones? Why Geometric Langlands and not classical Langlands?

He ended up by mentioning a curious hard Lefschetz conjecture for weight and perverse filtrations that left the crowd speechless.

"Adding Flavor to AdS4/CFT3"

I thought I would try to write short summaries of interesting talks I attend here in Oxford. The first one of Hilary term is

Adding Flavor to AdS4/CFT3 by Andy O'Bannon from the Max Planck Institut in Munich, based on 0909.3845.

The motivation is that the AdS/CFT correspondence only really becomes useful for applications (quark-gluon plasma at RHIC, condensed matter systems, often 3-dimensional) when it involves not only fields in the adjoint representation of the gauge group---strings starting and ending on the same brane---but also fields in the fundamental representation. For this one needs to add new branes so that strings can stretch between different branes.

Such procedure is well-understood in the AdS5/CFT4 context: the supergravity action acquires a new term describing the new Dp-branes (be it D5 or D7), $\large S_{10d} = S_{IIB} + S_{Dp}$, and this is dual to super-Yang-Mills with flavors in 4d, $S_{4d} = S_{\mathcal{N}=4} + S_{\text{flavor}}$. The story in M-theory is less understood. The "membrane minirevolution" (as Lubos calls it) led to a duality between $N_c$ M2-branes with a $AdS_4 \times S^7/\mathbb{Z}_k$ horizon, and a (2+1)dimensional Chern-Simons theory with N=6 supersymmetries with fields in the bifundamental of $U(N_c)_k \times U(N_c)_{-k}$. This is the famous ABJM theory (see Klebanov & Torri for a recent review). The goal of the talk is to understand what happens on the field theory side when on add M5-branes. What is the $S_{\text{flavor}}$ dual to $S_{M5}$?

Since the whole heuristic argument is based on being able to use the intuition of a string stretched between different branes being in the fundamental, and since there is no string in M-theory, the strategy is to start with type IIB supergravity with $N_c$ D3-branes, add Dp-branes and NS5-branes to get some flavor, and then T-dualise to IIA and lift to eleven-dimensional supergravity, the low-energy limit of M-theory. Here is roughly how it goes.

The D3-branes are first considered as hanging along one direction between two NS5-branes, as so (thanks to Cyril for allowing me to use his drawing device:):
Now perform a dimensional reduction on this compact interval and you get a (2+1)d SYM with N=4 and gauge group $U(N_c)$. If you replace one NS5 by a (1,k)5 = NS5 + k D5, then you get (after considering bounday terms...) a Chern-Simons theory with level k. Pretty close already!

Now consider two stacks of D3-branes stretched between the (1,k)5 and the NS5
(the (1,k)5 has to be tilted to preserve N=3 superymmetries, with and angle $\large \tan\theta = k$):
What you get now is a CS theory with N=3 and fields in the bifundamental of $U(N_c)_k \times U(N_c)_{-k}$, which is starting to realy look like the ABJM theory. The superymmetry can be enhanced because of Kaluza-Klein monopoles (which correspond on the field theory side to take the low energy limit by integrating out masses greater than $g_{YM}^2 k$) but Andy passed over this important subtlety, and so do I.

All that is left to do is to T-dualise this whole brane construction and lift to eleven dimensions to produce M2-branes and Kaluza-Klein monopoles (which are described purely geometrically...):
\[D3 \to D2 \to M2 \]

\[NS5 \to KK \to KK \]

\[(1,k)5 \to KK + D6 \to KK'\]

The KK monopoles interesect at a $\mathbb{C}^4/\mathbb{Z}_k$ orbifold singularity, and placing $N_c \to \infty$ M2-branes at this singularity produces a near-horizon geometry $AdS_4 \times S^7/\mathbb{Z}_k$, which is dual to the ABJM theory.

So now to understand flavors in AdS4/CFT3 you can add some Dp-branes in the IIB background and repeat this translation procedure to M-theory. Andy went through two examples, one with D5-branes, the other with D-branes, and showed that they in fact both lead to the same CS theory with flavor and $SU(4) \times U(1)$ isometry.

He finished by mentioning an application: fractional quantum Hall effect.

Fun with the G-string

If you like parodies of academic articles (and I know you do), check out how the early fathers of string theory used to have fun: The Super G-String. I don't know who is behind it (it is hosted on Warren Siegel's webpage) but the fake authors are V. Gates ("wie geht's?", a reference to Jim Gates), Empty Kangaroo, M. Roachcock (cockroach), and W.C. Gall (let me know if you crack this one---my attempt is the French word calvitie...).

I came across this paper because it is mentioned in the introduction of Aspinwall's lectures on D-Branes on Calabi-Yau Manifolds, where he writes in a footnote about D-branes that "The first reference to such objects that the author is aware of is, oddly enough, section 4 of [1]."

And indeed, on section 4 of the parody, there is an allusion to what we would call nowadays braneworld model-building, with the open string ending on a four-dimensional submanifold. So humour can be serious sometimes! After all the researcher's work isn't that different in its essence from a ludic fantasy...

Compactification and singularity

On the train from Lausanne to Geneva Airport I recalled the way Polchinski introduces his chapter on toroidal compactification :

"In general relativity, the geometry of spacetime is dynamical. The three spatial dimensions we see are expanding and were once highly curved. It is a logical possibility that there are additional dimensions that remain small."

J. Polchinski, String Theory, volume I, CUP 1998 (p. 231)

My personal version of this argument is to say that in GR space can be curved and hence it is a "logical possibility" that there are dimensions so curved that they actually close on themselves. I find that it is a very seducing way to introduce the notion of extra dimensions that are hidden because of their compactness, since it follows from the well-established curvature of space-time.

Polchinski then goes on to explain Kaluza-Klein theory with a periodic dimension without further explanations. This leaves the following question open :

What kind of massive object curves some dimensions so as to make them compact while leaving other dimensions non-compact ?

I've been drawing a lot since then and realized a few things that amazed me for five seconds before revealing their complete triviality... Nevertheless, here are some of my reflections.

At first the idea that an object curves only some dimensions and not all of them seems bizarre. Imagine a toy universe that has only two dimensions so that one can envision curvature as the bending of an elastic membrane with a ball on it. This video which is part of a documentary about Hawking has a nice illustration of this after 3:00.

Now if some dimensions are to be so strongly curved that they become compact, this must be the effect of a very massive object. So let us give the ball a tremendous mass; the way I picture what happens is that the weight of the ball stretches the membrane in the form of an extremely long tube. So neglecting the extremities, the membrane has now the topology of a cylinder, which is the direct product of a non-compact manifold, a real line, with a compact one, a circle. What seems a bit weird is that the radial direction, which is curved by the ball, remains non-compact, whereas the angular direction, which is not curved, is compact ! But there is a fallacy here : to have an angular coordinate (with a finite range) does not mean there is a compact dimension. The plane is not any compacter in polar coordinates than in Cartesian coordinates...

I'd like to understand spaces that have a non-compact part and a compact part. For simplicity I'll take direct products of real lines and spheres. The simplest case is R x S^1. This is pretty clearly a cylinder if you think of it as an S^1 fibration over R, but you could also think of it as a R fibration over S^1, in which case one can see that is is also a cylinder by taking into account that the fibers must not intersect. In fact, only the topology matters here and the cylinder can be continuously deformed at will. The formal definition of this direct product is R x S^1 = {(x, theta) | x \in (-\infty, \infty), theta \in [0, 2\pi]}. So you just need to specify a couple of coordinates, one for the real line (x), and one for the circle (theta).

An interesting deformation of the cylinder is achieved through a conformal mapping, where one defines a coordinate z = exp(x + i theta). This can be thought of as a radial coordinate r = exp(x), and an angular coordinate theta. The radial coordinate ranges from 0 to infinity but at r=0 the circle degenerates, which does not correspond to the cylinder bounded by circles at both its extremities at infinity. In consequence, the origin has to be removed. So one ends up with the manifold R^2 (in fact the complex plane...) with the origin removed : R x S^1 ~ R^2\{0}. R^2 is non-compact but by removing a point you make it isomorphic to a space that is the direct product of a compact space with a non-compact space. There is thus an equivalence between manifolds with a singularity and partially compact manifolds !

Coming back to our latex membrane, we understand now that in order to obtain a compact dimension the ball must be so heavy that it produces a black hole singularity.

Now consider R x S^2. It is hard to understand as a fibration : is it some kind of solid cylinder ? I guess that a particle can move arbitrarily inside the cylinder but as it approaches the surface it is forced to move tangentially to it (?) It is much easier to think about it after a conformal mapping, which leads to R^3\{0}. Again, the origin is removed for the angular coordinates to be well-defined everywhere. (Notice that the manifold is not complex anymore, given that it is odd-dimensional...)

Another simple case is R^2 x S^1, which can be reduced to the first case by rewriting it as R x R x S^1 ~ R x R^2\{0}. We get a manifold R^3 with a line singularity along the third axis. This is nice because it means we can get more than just one non-compact dimension (the radial direction), namely the dimensions of the singularity itself.

We are ready to generalize : after conformal mapping, the direct product R^n x S^d gives a (n-1)-dimensional singularity at the origin of a (d+1)-dimensional space.

String theory requiring 10 dimensions for its consistency while we only observe 4 non-compact dimensions, we might try the direct product R^4 x S^6. However, according to our reasoning, this is isomorphic to R^3 x R^7\{0}, so that the fourth non-compact dimension differs from the others in that its boundaries at infinity are six-spheres... (how bad is that ?)

If this has to be avoided, one must rather consider R^5 x S^5 ~ R^4 x R^6\{0}. This is pretty close to what appears in the context of the AdS/CFT correspondence ! There, the singularity would be black 3-branes and there would be a warping between R^4 and the radial coordinate to get AdS_5.

Remark :

The bending latex representation is misleading since it implies that the membrane is curved towards a third dimension. I don't think this is the proper way to view it, because otherwise the curvature of four-dimensional space-time would by itself imply the existence of extra dimensions... Rather, one should maybe picture a curved space by a density plot. The darker the color, the larger the curvature. The surface of the black hole would be a very dark circle. 

From the density plot the surface of the black hole is at a fixed radius, but from the curved membrane representation it forms a very long tube. It is folly to suppose the coordinate along the tube is the coordinate U = r/alpha' with r->0 defined by Maldacena.

Party String Theory

And of course you go to a party and of course someone asks you what you do and of course you say string theory and of course what is it ? and of course you don't know where to start. // Classic.

So it gets you thinking about how to present string theory and so you find one way and so you feel warm inside and so you post a post. // Classic.

A good way to introduce string theory is as a conceptual framework that unifies the two current pillars of theoretical physics : general relativity and quantum mechanics. To give a feel of what string theory is about, it suffices to focus the explanations on just one specific (but pretty central) aspect of each theory : black holes for GR, the uncertainty principle for QM.

1. The central result of Einstein's general relativity is that a very massive object -- or equivalently a very energetic object, remember E=mc^2 -- has the effect of curving the neighbouring space-time (which in fact implies gravity...). But there is a bound to this phenomenon, since if an object is too massive, it will curve space-time so much that it will end up tearing it apart and creating a black hole. (As an illustration, imagine space-time had only two dimensions, and think about it as a stretched piece of cloth ; putting heavy objects on it will curve it, and if an object is heavy enough it will make a hole in it.)

2. At the heart of quantum physics lies Heisenberg's uncertainty principle, according to which a particle cannot have both a precise position and a precise momentum (mass times speed). There is a tension between the precisions of those two properties : if the position of a particle is determined with great accuracy, unavoidably its momentum becomes very uncertain -- we know precisely where it is but not where it is going nor at which speed it is moving. And inversely, if the momentum is precise, it is the position which becomes uncertain, as if the particle were dissolved into some sort of a fuzzy cloud...

Now if you ask the philosophical question of knowing what is the ultimate building block of reality, and that you undertake to break objects down into little pieces, then in littler pieces, and then yet littler and so on, there will be a point where the piece you consider is so small that by the uncertainty principle its momentum (and so its energy) is so fuzzy that it can produce a black hole. From this point, the division process cannot be carried any further.

String theory aims at describing what is happening at this ultimate scale, where tiny black holes appear and disappear randomly by quantum fluctuations, and where the very notions of space and time lose their pertinence...

[Poetic suspension]

What we call "reality" is nothing but the surface of a chaos of black holes...

[Poetic suspension again]

And hence holography, AdS/CFT, and the like. Good.

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On a related note, people are usually amazed when I tell them that I switched from philosophy and art to physics. The most coherent explanation in my bag of tricks has to do with my interest in perception and the essence of reality, but I think I just found a new twist on it, as I was staring at the moon (beautifully speckled tonight). It goes something like this :

At some point in my early life it occurred to me that creation was the purest source of joy (this expression is in fact taken from my application letter to Oxford -- how naive I was...). So that might explain why I was interested in arts and why I even wanted to become a painter. But I was aware (I think) that the product of an artist's creation is not a work of art (although it is, don't get me wrong !), but rather a new perspective on existence.

[Which example could I find to illustrate that ??? I reckon that if you don't already know it, it will be quite impossible to convince you... For example when the Gothic cathedrals started to be built with thinner walls and with more light, it was not (even though maybe ingenuously it was) to make a "pretty" building, but to be in phase with the evolution of the societies from a feudal to a seigniorial organisation. (About this fascinating topic I recommend the brilliant book L'Art féodal et son enjeu social by André Scobeltzine.) But it is not a real good example, because the notion of artist was ill-defined at the time... Anyways !]

OK so the point is that what I am doing currently, namely theoretical physics, is also a creative venture. And what the theoretical physicist creates [notice my emphasis on the "theoretical" bit ? ; )] is not an article, not even a physical theory (although it is, don't get me wrong !), but -- reality itself !

Just like the philosopher in fact... And like the artist as well in fact... I'm afraid I would have utterly lost my party interlocutors by this time in fact... Perhaps that explanation ought to be substantiated more substantially in fact...


Addendum : Well my basic intuition arises roughly from the following type of considerations. Before Galileo the earth was the centre of the universe. It's not just that everybody wrongly thought it was. Go back to Timbuktu : if nobody sees the Dalmatian in the picture then there is no Dalmatian in the picture. Peace.

Spider-scientist

I was absorbed in abyssal thoughts about my research when a little black spider landed on my left shoulder from totally nowhere!

I have been quite amazed with spiders recently: how do they manage to build their webs hanging between two unrelated objects? That don't make no sense to me! (Also, the other day when I went running - in fact it was this morning - I was thinking about the fact that spiders always seem to know where they can build their webs and where they don't, for example in a place with a high circulation; but then I thought it certainly is absurd to think about it that way...)

And so, plötzlich, it struck me that a researcher has something to learn from a spider, in fact. Just like the spider, he is flying through the air from one idea to the other, blown by the wind of his imagination, and like the spider, his ultimate goal is the construction of a large web of concepts (able to catch some juicy preys... for the SOUL nigga!).

[As an aside, I would put forth the comment that when Witten was saying string theory is a piece of the XXIst century mathematics (did he venture to say IIIrd millenium?) fallen by chance into the XXth century, that would translate in my metaphor into a spider web with positive slope, built with the help of a favourable wind.]

But that only works if the researcher keeps track of his own thought process. If the line breaks, it won't be possible to relate the distant concepts, and everything is lost. Which, as you may imagine, is just what happened to me that day when the little spider paid me an unexpected visit... She was clearly philosophical enough to know that sometimes when you want to get a message through you have to violate its spirit.

First Live Citation

On Monday appeared my paper written with He and Lukas, entitled "An Abundance of Heterotic Vacua" [I tried to put forth something more irreverent like "Get Your Fill of Heterotic Vacua," playing on the full/empty contradiction, or "Heterotic Bundles All Over the Place," to convey the image of some kind of orgy, but some of my collaborators were not so hot about it...]. Given that Ron Donagi was giving a talk on Tuesday at Strings 08 about "Heterotic Standard Models," the eventuality that he would mention our paper was non-negligible. Indeed, his last slide contained a reference that slightly differed from the other by its typography, indicating its late addition. It was just cryptically reading: "[GHL]." Ron undertook to enumerate the authors: "He, Lukas, and... [I was getting excited]... errmmm... I can't remember, never mind."

Great! Thanks Ron! Two thumbs up! :-P
So close to my first live citation, and yet so far...

However I got my revenge today, during Hermann Verlinde's talk on "Holographic Gauge Mediation." I arrived late to that talk because I had already heard it at Eurostrings, so I wasn't following with great attention. I suddenly woke up when I saw my name on the screen: "cf. Gabella, Gherghetta, Giedt." My first paper from my Master project in Minneapolis!

That was a pleasant feeling to have the impression to participate to a collective research effort. (My joy was soon to be a bit tempered by the fact that he then kept referring to us as "some phenomenologists" (in which he wasn't completely wrong, I have to confess...)).

Linde's speech at Strings 08 banquet

The task of entertaining the audience at the end of the Strings 08 banquet at UniMail was accomplished with spirit by Andrei Linde. He started by saying that he was going to make a discourse in the Georgian-Russian fashion, but maybe it was not appropriate... He said that after our visit at the United Nations on Monday, he was seeing two goals of utmost importance: 1). world peace, and 2). experimental evidence for string theory.

Given that we live in an expanding universe, in a few billion years the galaxies will be beyond each other's horizon, and so there will be no intergalactic war any more. First problem solved.

He addressed the second problem with a succession of anecdotes from various famous physicists. One of them was about a Russian scientist (something in Z... I forgot) that once told physicists which were discouraged by the absence of experimental evidences in favour of baryon asymmetry that the fact that parallel lines do not intersect was actually an evidence (he quickly explained what he meant but I couldn't get it). "That is another type of evidence," Andrei said.

Then he talked about Murray Gell-Mann who asked one of his students to measure the height of a tower with a barometer. One week later the student came back and said that he had found three ways of doing it: The first option was to go to the top of the tower, attach a rope to the barometer, slide it down, and then measure the length of the rope. "Myeaaah. What is the second way?" Murray asked. The second option was to go to the top of the tower, throw the barometer down, and count the time it takes to reach the ground. "That is not really what I expected..." The student's third way was to go to the superintendent and tell him: "This barometer is worth about 30 €. It is yours if you tell me the height of the tower."

Andrei's point was that in order to get evidence for string theory, we need to use the barometer in a clever way (measuring the pressure difference at the top and bottom of the tower). There
is no straightforward way to do it (like the first two solutions of the naive student), and all the superintendents left the universe about 13.7 billion years ago (that is, at its creation).

And this clever way is to think about dolphins (not fishes: dolphins!). Dolphins live in water because it is the place where they can live, just as we live on the ground because we can live here. The tiny value of the cosmological constant can be explained in this way, but only at the condition that there exist a huge number of possible universes with different values of it. (Huge as in 10^500, which journalists, as they cannot typeset exponents, write as 10,500... "Anyway: big number.") Yes: anthropic reasoning. [I thought about David Gross at that point, who certainly won't miss on Friday to repeat his warning of last year's conference in Madrid: "Do not give up!"]

Andrei ended his discourse by saying that he could have proposed a toast for the LHC, the biggest machine ever built -- but that would have been too banal. Then for string theory -- not surprising enough. He had to be surprising (Georgian-Russian style!). He finally drank a toast to "You, the people that are making string theory, because we enjoy working together and learning." I found this apparition of a human, relational factor indeed quite surprising in such a context (although it would have been completely trivial in many other contexts).

In summary, Andrei's discourse operated two shifts:

• A shift from the dream of a unique and unambiguous explanation of everything to the acknowledgement of the relevance of environmental determination.

• A shift from a machine or a theory to mutual enjoyment.