## Thursday, November 26, 2009

## Thursday, November 5, 2009

### 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

## Wednesday, October 28, 2009

### "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...

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...

## Monday, October 26, 2009

### 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!

## Friday, October 23, 2009

### "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.

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.

## Thursday, October 22, 2009

### I. The necessity, structure, and priority of the question of Being

This is a summary of the first section of the Introduction of Martin Heidegger's major work,

*Being and Time*(1926), where he is aiming at reawakening an understanding of the meaning of the question of Being.I. THE NECESSITY, STRUCTURE, AND PRIORITY OF THE QUESTION OF BEING

1. The Necessity for Explicitly Restating the Question of Being

Since Plato and Aristotle the question of Being has been trivialized, and hence considered as superfluous. MH lists three prejudices, according to which this question is unnecessary. First, the assertion that 'Being' is the most universal concept does not imply that it is also the clearest: "It is rather the darkest of all." Secondly, even though 'Being' cannot be defined -- since a definition is typically of the form "something

*is*this and that", it presupposes an understanding of the word "is" -- this does not eliminate the question of its meaning. Thirdly, the "self-evident" character of the concept of 'Being' remains,*a priori*, an enigma. So not only does the question of Being lack a clear answer, but "the question itself is obscure and without direction".2. The Formal Structure of the Question of Being

The structure of any question splits into: that which is asked about [

*sein Gefragtes*], the starting point of the curiosity; that which is interrogated [*sein Befragtes*], the things one turns to in order to find an answer; and that which is to be found out by asking [*das Erfragte*], the answer itself.In the case of the question of Being, the

*Gefragtes*is... Being [damn! I knew it], or more explicitly "that which determines entities as entities". But "the Being of entities 'is' not itself an entity", in the sense explained above that Being cannot be defined. Rather, Being "must be exhibited in a way of its own" (quite intriguing ain't it?). [I'll hereinafter push the acronym perversity so far as to reduce Being to B.]The

*Befragtes*are the entities themselves, that is "everything we talk about, everything we have in view, everything towards which we comport ourselves in any way" (a rather broad concept, mind you). [As an aside, I find such periphrases very revealing of the superiority of the German language to forge concepts.] But in sight of the unlimited possible choice, can one single out a specific entity that will be particularly useful to discern the meaning of B? Well we can, and it is ourselves, the inquirers. MH calls this special entity the "*Dasein*": "This entity which each of us is himself and which includes inquiring as one of the possibilities of its Being". What singles us out, among the infinities of entities, is the very fact that we are able to grow an interest in that which determines us as entities, i.e. our B.There is an apparent danger of 'circular reasoning': are we supposed to come to grip with the meaning of B by inquiring into entities that are inquiring into their own B by inquiring into entities that are inquiring into their own B by inquiring--you got it--? MH discards this "always sterile" argument, since (if I understand correctly) it is fine to 'presuppose' B "provisionally". No circular reasoning then, but "a remarkable 'relatedness backward and forward' " between the

*Gefragtes*(B) and the*Befragtes*(us,*Dasein*). [This is somewhat reminiscent of self-consistent systems in condensed matter theory.]3. Ontological Priority of the Question of Being

The message of this subsection is basically that "Basically, all ontology, no matter how rich and firmly compacted a system of categories it has at its disposal, remains blind and perverted from its ownmost aim, it if has not firstly clarified the meaning of B, and conceived this clarification as its fundamental task". What MH means (supposedly) is that among all the scientific investigations, the question of B is the most primordial. But I suspect that he doesn't mean "priority" in the sense that it has to come first. Indeed, in any of the fields of scientific research (or "areas of subject-matter"), the "basic concepts" are (provisionally, "beforehand") "worked out after a fashion in our pre-scientific ways of experiencing...". So the subject-matter of interest is primitively understood, may I say, intuitively, and it is over this original intuition that a science, that is a "system of categories", develops. But:

"The 'real' movement of the sciences takes place when their basic concepts undergo a more or less radical revision which is transparent to itself. The level which a science has reached is determined by how far it iscapableof a crisis in its basic concepts." (p.9)

[Remark: I have a similar approach to evaluate the richness of a personality.] And MH to enumerate a few instances in science of what he perceives as "freshly awakened tendencies to put research on new foundations": the foundational crisis in

*mathematics*, in spite of it being "seemingly the most rigorous and most firmly constructed of the sciences" (a clear presentiment of Goedel's incompleteness theorem of 1931); the "problem of matter" in the context of the relativity theory of*physics*(I don't understand his point at all, even at the seventeenth reading...); the "new kind of B" defined in*biology*; and the realisation of the "inadequate" foundation of*theology*(he's probably being polite).Ontological (concerned primarily with B) inquiries, such as Kant's

*Critique of the Pure Reason*, are "more primordial" than the ontical (concerned primarily with entities) inquiry of the positive sciences. But they still lack an understanding of 'what we really mean by this expression "Being" '.4. The Ontical Priority of the Question of Being

I find this last section more difficult to understand, perhaps because MH "anticipate[s] later analyses". Here we see listed and swiftly defined a series of key Heideggerian mottos and concepts. For instance, about Dasein, he is writing that "Being is an

*issue*for it" and that"Understanding of Being is itself a definite characteristic of Dasein's Being. Dasein is ontically distinctive in that itisontological."

The distinction between "existentiell" and "existential" is also briefly described. As far as I can tell, the former means what you would naively expect, namely it characterises an understanding of existence, and more precisely Dasein's own existence.

"Dasein always understands itself in terms of its existence--in terms of a possibility of itself: to be itself or not itself."

(Here one can foresee the concept of authenticity.) MH underlines that Dasein is responsible for its existence, either actively or passively (for example, I suppose, if it lets external circumstances like childhood trauma dictate its conduct).

"Only the particular Dasein decides its existence, whether it does so by taking hold or by neglecting."

"Existential" on the other side characterises an understanding of the context of the "structure of existence".

"By "existentiality" we understand the state of Being that is constitutive for those entities that exist."

Coming back to the ideas of the previous section, MH writes that "Sciences are ways of Being in which Dasein comports itself towards entities which it needs not be itself", but his message is that Dasein "must first be interrogated ontologically." And just like the basic concepts of sciences are first worked out in a pre-scientific way, the question of B has to be worked out in a pre-ontological way:

"the question of Being is nothing other that the radicalization of an essential tendency-of-Being which belongs to Dasein itself--the pre-ontological understanding of Being."

This is the end of the first introduction of

*Being and Time*.I mean come on! How amazing is that? The guy is slowly cracking the very kernel of the most fundamental question you can imagine, the question of Being. Heidegger was saying in an interview that an entirely new form of thought is now called for, in our modern time. It is simpler than the old way of thinking, more natural, but it is also more difficult, in that it requires a much greater care with the use of language. This perspective should be motivation enough to overcome the disgust inspired by the bestiary of Heideggerian concepts. So let's keep calm and carry on!

## Friday, October 16, 2009

### "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 $\exp(-2000)$.

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 $\exp(-2000)$.

## Monday, October 12, 2009

### "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:

\[ h^{p,q} (X) = h^{d-p,q}(Y)~. \]

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.

\[

\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.

## Thursday, October 8, 2009

### "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...):

\[D3 \to D2 \to M2 \]

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.

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.

## Sunday, October 4, 2009

### Merleau-Ponty: L'Œil et l'Esprit

*L'Œil et l'Esprit*, écrit en 1960, est le dernier texte de Maurice Merleau-Ponty (1908-1961). J'en ai souligné quelques passages et je tente maintenant d'en restituer l'argument. [Désolé pour l'état squelettique de cette note..]

**Section I**

MMP (c'était aussi le surnom d'un charismatique professeur à la barbe blanche de

*Quantentheorie*à la*Humboldt Universität*:) prend le problème du désenchantement de la science moderne comme point de départ."La science manipules les choses et renonce à les habiter." (p.9, première phrase du folio)

"Il faut que la pensée de science -- pensée de survol, pensée de l'objet en général -- se replace dans un « il y a » préalable, dans le site, sur le sol du monde sensible et du monde ouvré tels qu'ils sont dans notre vie, pour notre corps, non pas ce corps possible"…

"Dans cette historicité primordiale, la pensée allègre et improvisatrice de la science apprendra à s'appesantir sur les chose mêmes et sur soi-même, redeviendra philosophie…" (p.12-13)

La solution doit venir de la peinture, le vrai sujet qui intéresse MMP.

"Or l'art et notamment la peinture puisent à cette nappe de sens brut dont l'activisme ne veut rien savoir." (p.13)

Note comme la ponctuation, son absence en fait, indique bien comme MMP ne compte pas s'appesantir sur les autres arts que la peinture, qu'il discrédite tous en deux phrases expédiées. Lui, c'est la peinture son truc.

"Le peintre est seul à avoir droit de regard sur toutes choses sans aucun devoir d'appréciation." … "Quelle est donc cette science secrète qu'il a ou qu'il cherche?"

Le décor est planté.

**Section II**

MMP essaie de cerner "cet extraordinaire empiétement" du monde tel que nous le voyons et du monde dans lequel nous nous déplaçons (je ne suis pas certain de bien le suivre ici; oppose-t-il la perception visuelle à la perception tactile?).

"C'est en prêtant son corps au monde que le peintre change le monde en peinture." …

"Le monde visible et celui de mes projets moteurs sont des parties totales du même Être." (p.16-17)

"L'énigme tient en ceci que mon corps est à la fois voyant et visible." (p.18)

"… l'indivision de sentant et du senti." (p.20)

C'est intéressant, et j'aimerais y acquiescer (et faire le lien avec la théorie quantique, éventuellement…), mais je peine à trouver la substance de sa réflexion. Ses formules sonnent creux:

"le dessin et le tableau"… "sont le dedans du dehors et le dehors du dedans"… (p.23)

*See what I mean?*

Ce qui a le plus excité mon crayon sont des citations:

"«

**la nature est à l'intérieur**», dit Cézanne." (p.22)"Max Ernst"… "« … le rôle du peintre et de cerner et de projeter ce qui se voit en lui. »" (p.30)

Même si je ne suis pas emballé par l'argumentation de MMP, je trouve ces citations très inspirantes. Après tout, ce n'est pas réellement les sortir de leur contexte, MMP s'en est déjà chargé.

(Il m'arrive d'ailleurs souvent d'aimer une citation au fi de son utilisation particulière. Par exemple, bien que je ne me soie pas à vrai dire délecté de

*La Porte Etroite*(1909) d'André Gide (1869-1951), j'ai été profondément touché par une citation paradoxale du Christ, elle-même tirée de Pascal: "Qui veut sauver sa vie la perdra.")Ce que j'aime dans la citation de Cézanne, c'est sa force et sa beauté brute. Elle est faite de mots simples, tirés du langage quotidien, et pourtant elle frappe l'entendement et fait résonner les hautes sphères spirituelles. La nature, c'est ce qu'il y a devant moi, là, ce que je vois par-delà ma fenêtre. Et en moi sont mes sentiments et mes idées, immatérielles et intimes. Et bien les deux ne s'opposent pas, on ne peut pas poser l'une comme fondamentale (tels les cartésiens l'esprit, ou les positivistes la réalité physique) et y prendre appui pour explorer l'autre, non. Les deux sont une seule et même chose, et il ne fait sens de les considérer qu'en conjonction.

"Les parfums, les couleurs et les sons se répondent." Certes, certes, bravo Charles, voilà qui devrait faire mollir les âmes tendres, mais ce n'est pas tout! La nature est à l'intérieur. Comprends bien ça.

Quand Cézanne peint la montagne Sainte-Victoire (plagié-je?), il se fout pas mal du paysage, tout comme il se fout pas mal du panier de pommes qu'il peint. Ce qui l'intéresse c'est l'intérieur, ou plutôt le fait hallucinant que la montagne, toute géologique qu'elle puisse être, elle est à l'intérieur. A l'intérieur de lui, quand il la voit.

Et c'est là que MMP nous dit à point

"la peinture ne célèbre jamais d'autre énigme que celle de la visibilité." (p.26)

**Section III**

"Comme tout serait plus limpide dans notre philosophie si l'on pouvait exorcise ces spectres, en faire des illusions ou des perception sans objet, en marge d'un monde sans équivoque! La

*Dioptrique*de Descartes est cette tentative. C'est le bréviaire d'une pensée qui ne veut plus**hanter**le visible et décide de la reconstruire selon le modèle qu'elle s'en donne." (p.36)On croirait presque entendre le sarcasme des physiciens quantiques à l'égard de ceux qui (comme Einstein) ne veulent pas abandonner leur rêves déterministes...

Le mot clé ici est "hanter". Si nous étions immortels nous pourrions nous en passer. A défaut, nous hantons, et c'est tant mieux.

Après avoir discrédité "la chose même" et "l'espace en soi", "l'en soi par excellence" cartésiens, MMP concède

"Descartes avait raison de délivrer l'espace. Son tort était de l'ériger en un être tout positif, au-delà de tout point de vue, de toute latence, de toute profondeur, sans aucune épaisseur vraie." (p.48)

J'aime bien cette idée qu'il est nécessaire de passer par une étape de pensée rigide, positiviste, réaliste, avant de pouvoir dans un deuxième temps s'en affranchir et accepter l'incertitude. MMP nous dit par exemple que

"

**l'espace n'a pas trois dimensions**"en quoi, de nouveau, je serais friand de pouvoir le suivre mais je ne vois pas bien comment.

"les dimensions sont prélevées par les diverses métriques sur une dimensionnalité, un Être polymorphe, qui les justifie toutes sans être complètement exprimé par aucune."

Mais, ai-je presque envie de dire, cet "Être polymorphe" est-il Calabi-Yau?

Si je comprends bien, la dimensionnalité serait une propriété du mode de perception plus qu'une propriété intrinsèque des choses. Une qualité secondaire, et non une qualité primaire. Hm. Intéressant.

"Quelque chose dans l'espace échappe à nos tentatives de survol." (p.50)

"En vérité il est absurde de soumettre à l'entendement pur le mélange de l'entendement et du corps." (p.55)

La science moderne, nous dit MMP, est tellement dégénérée, plus cartésienne que Descartes etc, qu'il nous en faudrait une autre, entièrement nouvelle, pour pouvoir envisager une cohabitation amicale avec la philosophie.

"si nous retrouvons un équilibre entre la science et la philosophie, entre nos modèles et l'obscurité du « il y a », il faudra que ce soit un nouvel équilibre. Notre science a rejeté aussi bien les justifications que les restrictions de champ que lui imposait Descartes."

"La pensée opérationnelle"... "est fondamentalement hostile à la philosophie comme pensée au contact"... (p.57)

"Notre science et notre philosophie sont deux suites fidèles et infidèles du cartésianisme, deux monstres nés de son démembrement."

Et voici la voie à suivre selon MMP:

"Nous

*sommes*le composé d'âme et de corps"... (p.58)"L'espace n'est pas celui dont parle la

*Dioptrique*, réseau de relations entre objets, tel que le verrait un tiers témoin de ma vision, ou un géomètre qui la reconstruit et la survole, c'est un espace compté à partir de moi comme point ou degré zéro de la spatialité. Je ne le vois pas selon son enveloppe extérieure, je le vis du dedans, j'y suis englobé."... "La lumière est retrouvée comme action à distance, et non plus réduite à l'action de contact"... (p.58-59)"Toutes les recherches que l'on croyait closes se rouvrent. Qu'est-ce que la profondeur, qu'est-ce que la lumière"... " non pas pour l'esprit qui se retranche du corps, mais pour celui dont Descartes a dit qu'il y était répandu"...

**Section IV**

MMP nous dit qu'il a le sentiment

"d'une discordance profonde, d'une mutation dans les rapports de l'homme et de l'Être, quand il confronte massivement un univers de pensée classique avec les recherches de la peinture moderne." (p.63)

"Quatre siècles après les « solutions » de la Renaissance et trois siècles après Descartes, la profondeur est toujours neuve"... (p.64)

"De la profondeur ainsi comprise, on ne peut plus dire qu'elle est « troisièmes dimension »." (p.65)

Et de nouveau une très belle contribution de Cézanne au débat:

"Quand Cézanne cherche la profondeur, c'est cette déflagration de l'Être qu'il cherche, et elle est dans tous les modes de l'espace, dans la forme aussi bien. Cézanne sait déjà ce que le cubisme redira : que la forme externe, l'enveloppe, est seconde, dérivée, qu'elle n'est pas ce qui fait qu'une chose prend forme, qu'il faut la briser cette couille d'espace, rompre le compotier --- et peindre, à la place, quoi?"

[Le lecteur attentif aura remarqué une coquille dans le mot "coquille" -- justement...]

"C'est donc ensemble qu'il faut chercher l'espace et le contenu."

La couleur est "« l'endroit où notre cerveau et l'univers se rejoignent »"

(comme la théorie des cordes;)

"C'est cette animation interne, ce rayonnement du visible que le peintre cherche sous les noms de profondeur, d'espace, de couleur."

"L'effort de la peinture moderne n'a pas tant consisté à choisir entre la ligne et la couleur, ou même entre la figuration des choses et la création de signes, qu'à multiplier les systèmes d'équivalences,"...

Ce qui me fait penser aux dualités de la physique théorique...

Il y a aussi un passage sur le cinéma et la photo.

"Rodin a ici un mot profond : « C'est l'artiste qui est véridique et c'est la photo qui est menteuse, car, en réalité, le temps ne s'arrête pas. »" (p.80)

"La peinture ne cherche pas le dehors du mouvement, mais ses chiffres secrets."

La vision est "le moyen qui m'est donné d'être absent de moi-même, d'assister de dedans à la fission de l'Être, au terme de laquelle seulement je me ferme sur moi."

..."par elle nous touchons le soleil, les étoiles, nous sommes en même temps partout"...

"la «simultanéité » ''' mystère que les psychologues manient comme un enfant des explosifs."

Parle-t-il de la synchronicité de Jung?

... "le propre du visible est d'avoir une doublure d'invisible"...

**Section V**

... "profondeur, couleur, forme, ligne, mouvement, contour, physionomie sont des rameaux de l'Être"...

A propos du "vrai peintre":

"Même quand elle a l'air partielle, sa recherche est toujours totale."

A rapprocher de la façon dont Tim Gowers explique dans sa conférence

*The Importance of Mathematics*au*Clay Mathematics Institut*e en 2000 qu'il serait faux de discréditer les domaines de recherches mathématiques qui paraissent les moins applicables parce que les mathématiques forment un réseau intriqués de connexions, et qu'on ne peut en couper une partie sans par la même occasion en affaiblir la totalité organique..... "la trouvaille est ce qui appelle d'autre recherches. L'idée d'une peinture universelle, d'une totalisation de la peinture, d'une peinture toute réalisée est dépourvue de sens." (p.90)

(tout comme l'idée d'une

*Theory of Everything*...)"Le plus haut point de la raison est-il de constater ce glissement du sol sous nos pas, de nommer pompeusement interrogation un état de stupeur continuée, recherche un cheminement en cercle, Être ce qui n'est jamais tout à fait?

Mais cette déception est celle du faux imaginaire, qui réclame une positivité qui comble exactement son vide. C'est le regret de n'être pas tout."

... "chaque création change, altère, éclaire, approfondit, confirme, exalte, recrée ou crée d'avance toutes les autres."

## Sunday, July 19, 2009

### Zen Zoe

If you wander around in one of Oxford's many parks, you might meet an old lady peacefully drawing on a huge sheet of paper. Well, it's Zoe Peterssen. The rest of the story has been told many times, for instance here---needless to say that I was quite disappointed to see that many people had experienced just the same intimacy with Zoe, and that I wasn't "special" at all, she opened up to pretty much everybody. Fuck you for that Zoe bitch!

Under such circumstances, I'll just restrain to a couple of remarks.

Firstly, about her technique. She's drawing on huge sheets of paper, I told that already. What's surprising then, is that the next step is to photocopy those sheets repeatedly, reducing the size by half every time, until the tree is small enough to hold on a little card that she gracefully gives to wanderers that she likes (who are welcomed to then also give her something, a few pounds will do). So from say 2 meters to 3 centimetres, she must photocopy a drawing six times. What a labour! This shrinking process is actually akin to bonsai growing.

This comparison came to me because when I said that I was doing particle physics she mentioned a book she'd read about it,

*The Tao of Physics*by Fritjof Capa (1975). I'd like to re-read this book.She also mentioned bows and told the story of a man who wanted to learn archery and spent many years at it until finally he shot an arrow without feeling any shot, to which his Zen master said that he now knew how to shoot an arrow. I'd like to know how to shot an arrow.

## Wednesday, June 24, 2009

### Masterclass with Mitsuko Uchida

Yesterday I went to a master class given by Mitsuko Uchida to four performance students of the Music Faculty. The students presented pieces by Schubert, Schumann, Bach, and Beethoven. It was fascinating to see how much emotion and contrast she was able to percieve in those pieces and how well she could communicate it. She used a variety of ways to do it: obviously by playing herself on the piano, and by singing and screaming, but also with her body, directing with her hands and dancing, jumping around really, but also through more indirect ways. She used metaphors, she talked about a "dry sunny day, when it hasn't rained yet and you can smell the warm sand" and so on. She also resorted to musical analysis, for instance she explained how Bach was moving between the tonic, the dominant, and the sub-dominant---in fact, she apologized for having to sound so technical but it was just a way for her to express how she felt the piece should be played, what was at stake, what psychological tensions and what resolutions.

The precious moments when Mitsuko was playing for more than two seconds revealed the depth of her concentration and total involvement into the music. It was amazing---also certainly completely unavoidable to any great performer---how she was never playing lightly, but always mobilizing her full life's experience, all her pains, all her joys, all her emotions and sensations, her entire mind and body focused on the musicality.

She had something of a yoga teacher trying to encourage her students to stretch their bodies. She has probed the extremities of the various human emotions. One advice that struck me particularly and that she repeated to almost every performer of the day (at least the girls) was not to play "too beautifully". She encouraged them not to iron the peculiarities, the oddities of the compositions, but on the contrary to make the strangeness emerge, accentuate the dissonance, show how bizarre reality is. (It is something I think about a lot these days, bringing out the surreal...)

So you had for example this nice little American blonde playing as beautifully as she could, and hirsute Mitsuko shouting to her to play it uglier. At some point she told her "Play with your biggest pain!"

She also got mad when one of the students told her that the reason she was playing in a certain way was because of the influence of some famous pianist (although I think she misunderstood what the student meant). She said that every interpret makes mistake and so it is pointless to copy them, one should create her/his own mistakes :)

Mitsuko has recently become a 'Dame' and

today she was receiving and honorary doctorate at Oxford's

*Encaenia*Ceremony (the word apparently meaning "festival of renewal"). Since she lives in London I was looking forward to her next concert, but according to her agenda, she will have sooner toured the planet several times than perform again in the UK. Farewell Mitsuko!## Sunday, June 7, 2009

### La religion comme mise en abîme de soi-même

Je suis tombé sur la reproduction d'un tableau très étonnant:

Ce qui a d'abord attiré mon attention est que je n'avais jamais vu un autoportrait de femme aussi ancien. Sofonisba Anguissola [Cremona, c. 1532-Palermo, 1625] l'a peint en 1556 -- seulement huit ans après le probable premier autoportrait féminin par Caterina von Hemessen, lequel est également le premier autoportrait (tous sexes confondus!) à représenter l'artiste devant son trépied.

J'ai d'abord hésité quant au sexe de Sofonisba, étant donné que ce prénom est en soi déjà tellement étrange qu'il pourrait tout aussi bien être celui d'un homme, que son corset rigide ne laisse pas entrevoir sa morphologie, et que son visage pourrait être celui d'un jeune apprenti aux traits efféminés. Mon réflexe en cas de telles incertitudes est de lire quelques phrase du texte accompagnateur jusqu'au premier pronom "il/elle". Ca m'a pris un peu plus de temps que d'habitude: l'image issue du catalogue de l'exposition El retrato del Renacimiento au Musée du Prado était accompagnée d'un texte en espagnol, langue moins portée sur les pronoms sexués que le Français (mais heureusement plus encline que lui à spécifier le sexe d'une profession: "la artista", "la pintora", etc.). Bref, Sofonisba est bel et bien une femme.

Je me suis mis ensuite à admirer sa simple féminité sans artifices. Bien qu'elle ait embrassé une carrière habituellement réservée aux hommes, elle a su conserver une sensibilité de femme, enfantine et délicate. La grâce de ses mains et de son visage émerge de l'habit sévère qui cache ses formes.

Je note en passant (franchement, excusations pour toutes ces digressions!) qu'il était inacceptable à l'époque pour les artistes femmes de s'exercer au dessin anatomique, qu'il était pourtant nécessaire de maîtriser pour pouvoir prétendre composer des oeuvres de larges envergures impliquant de nombreux personnages. Ca explique en partie qu'elles se soient rabattues sur les portraits, et en particulier les autoportraits.

(Une tache vive de couleur magenta sur sa palette m'a fait sourire, parce que j'ai une théorie selon laquelle les adolescentes d'Oxford portent des vêtements de cette couleur le soir pour indiquer leur disponibilité sexuelle...)

Mais mon enthousiasme a vraiment pété le plafond lorsque je me suis posé la question "mais que peint-elle?" Naïvement, le tableau nous montre Sofonisba peignant une scène religieuse , sans doute à usage dévotionel, représentant la Vierge et le Christ (ou, peut-être, à cause de l'attitude sensuelle des personnages, Aphrodite et Cupidon?). Mais à la réflexion, ce n'était pas une scène religieuse qu'était en train de peindre Sofonisba lorsqu'elle peignait son autoportrait (et pour cause), mais sa propre image: je crie à la supercherie!

Si Sofonisba avait voulu peindre fidèlement ce qu'elle voyait, c'est-à-dire sa propre réflexion dans le miroir, ce qu'elle aurait peint sur son canevas aurait été une seconde reproduction d'elle-même en train d'en peindre une troisième et ainsi de suite jusqu'à l'infini. Ce procédé d'incrustation répétée ad libitum dans une image d'une réplique d'elle-même au format réduit est une mise en abîme. Mais Sofonisba a choisi de remplacer ce puits sans fond s'ouvrant sur elle-même par une image pieuse. J'y vois un enseignement sur la religion:

La religion se substitue à la limite infinie

de l'auto-représentation.

Je trouve ça très intéressant. (On pourrait presque pousser jusqu'à y voir une manifestation du principe holographique...)

Une interprétation alternative serait que Sofonisba, par le remplacement de sa propre reproduction par l'image religieuse, essaie d'exprimer qu'elle s'identifie à la Vierge -- un peu à la manière dont un bon chrétien est censé s'identifier au Christ et tâcher d'imiter sa vie. La croix que forment son pinceau et son bâton d'appui (comment appelle-t-on cet outil?) confirment cette lecture. Mais il faut admettre que c'est forcément plus compliqué, puisque le tableau que peint Sofonisba ne représente pas seulement la Vierge, mais aussi son enfant (voire même Aphrodite et Cupidon). S'il y a identification, ce n'est donc pas avec un seul, mais deux personnages bibliques (mythologiques). Elle s'identifierait avec quelque chose de plus abstrait, du genre de l'amour maternel (ou de la sensualité amoureuse).

## Monday, April 6, 2009

### Alien mathematics

Unlike what you might be expecting from the title, this post is not about Großendieck's fundamental reploughing of algebraic geometry, but about the following hypothetical question:

Imagine we discover another civilisation living in some corner of some galaxy, where we know that the physics is essentially the same than on earth. Would they have the same mathematics than us? And if yes, would they have the same mathematics history than us?

David Gross was proposing this thought exercise at the end of a public talk by Robbert Dijkgraaf at KITP, and I figured the best way to attack it was by taking a hot bath. In fact, the bath was so hot that I could feel my spirits evaporate and I felt that were I to think about this question in such circumstances, I would develop biased conceptions that would definitely screw all my chances to reach any interesting conclusions... Nevertheless I did it, and it's a mess.

(Note that Dijkgraaf wasn't too inspired by the question -- I guess this is the difference between a Nobelised and a non-Nobelised physicist: the former becomes philosophically oriented (remember Josephson))

Probably it is sensible to start by the second part of the question, assuming the answer to the first is positive. Then it is obvious that it would be quite challenging to defend the opinion that their math history is exactly the same as ours. They would have needed to have e.g. an Evariste Galois killed at 20 in an obscure duel, etc. But might they have had the same structure in the development of their mathematics?

What would Kant say about this question? I believe that it would be something along the line of "Their mathematics would resemble ours inasmuch as we are able to perceive them, and them us. (After all they could be made of "dark matter", in which case it would be strange if they had the same concepts as ours...)

## Sunday, March 29, 2009

### Mehr Licht/Mehr Nicht

"Mehr Licht!" Tels sont les derniers mots traditionnellement attribués à Goethe sur son lit de mort. Mais les mauvaises langues ont prétendu qu'il fallait en réalité entendre "Mehr nicht!" -- je n'en puis plus. Cette anecdote m'est revenue en tête hier, et j'ai pensé qu'on pouvait y voir plus qu'une moquerie: et si les deux interprétations n'étaient pas inconciliables? Ma thèse est celle de la coincidence, dans l'esprit du créateur, de l'instant de l'illumination et de celui de l'épuisement. Après tout, comment pourrait-il en être autrement, étant donne que pour percer la coque opaque de l'inconnu par un éclair de lucidité, il faut avoir atteint la frontière lugubre du périmètre de sécurité de l'esprit? Prenons l'example de l'athlète qui établit un nouveau record: le mehr Licht autant que le mehr nicht l'habitent, et se confondent en lui.

Dans The Crucible d'Arthur Miller, il y a un vieil homme honnête qui est exécuté par l'Inquisition par écrasement: on lui pose des pierres de plus en plus lourdes sur la poitrine jusqu'à ce que mort s'en suive... Ses dernières paroles furent : "More weight!"

Ici aussi on peut le comprendre de plusieurs manières: la manière comique (qui n'est pas ma favorite, va sans dire) selon laquelle il se moquait de ses bourreaux, mais aussi la manière, disons, existentielle qui appelle à plus de poids dans les décisions et les attitudes de vie de ces concitoyens. Une vie plus lourde de sens.

## Monday, February 23, 2009

### 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...

## Saturday, January 3, 2009

### Failblog

There is a YouTube channel called failblog which is dedicated to the biggest possible failures. At first I laughed my ass off by watching some of those videos, but then I started to think that there might also be something very interesting there. After all this is a precious collection of what is know as "actes manqués" in psychology (parapraxis), which are the equivalent in action of what a Freudian slip is in conversation. Where it becomes interesting is that such failures are, according to standard psychanlaytic theory, expressions of subconscious pulsions, and as such it is possible to make sense out of them.

Let us watch a few examples and you'll see what I mean.

First a very awkward frisk by a policeman who is probably repressing his homosexual tendencies. It's called "Frisk Fail" :

What is very striking is that the gesture of the policeman does not look at all like an inadvertent slip but rather like an intentional pass. The policeman is overwhelmed by a subconscious desire and loses touch with admissible behaviour. When he realises what he did and his conscious self takes control again, he certainly cannot believe it and apologizes repeatedly.

This interpretation is corroborated by the discussion that they are having, which resembles some kind of seduction game :

Cop -- Spread your legs for me please... I see you're shaking, I make you nervous/That makes me nervous.

Guy -- It's cold outside.

Cop (laughing, at ease) -- Yeah? It's cold? ...

The policeman is trying to relieve the tension in this delicate situation, full of sexual implications, but when the guy acknowledges his sensitivity to the cold like a shy girl adorably shaking in her first evening dress he's getting so loose that repressed emotions take over and push him to make the fatal move.

Now let's look at this other failure, also of sexual origin I'm afraid (most of them probably are---but not all !---which misled Freud to think that sexual repression alone could account for the entirety of human behaviour).

The failure here is unbelievable : how could she fail to see the pole just in front of her ? There is apparently something big going on here ! But there is not far to look for to find it (some YouTubers also suggested it in the comments of this video in fact). She is obsessed by her sexual frustration in her little village in Alaska. When she's faced to a symbol of her object of desire (a phallus), she fails to recognise it. (Notice the similarity with the cop video...)

Now the "Best Man Fail" :

The obvious thing to say here is that the best man is probably secretly in love with the bride, to the point that he does not even admit it to himself. His subconscious then forces him to do what he should have had the guts to do in full possession of his senses : throw himself at her feet to tell her his love and interrupt the wedding...

But what I find more interesting here is that there is a controversy in the comments of this video (like for many other videos on failblog) about its authenticity [I always find that very disappointing, not being able to trust something as a piece of reality, but I have to live with it you know; after all everything is fiction, etc.]. Someone is writing that the best man is acting like a bad actor in a Bollywood movie when he shouts "Nooooo!", and another that everyone is behaving totally unnaturally. And it is indeed quite weird how they all repeat "Oh my god!" one after the other... However, I do not think that this video is a fake, but that it is an illustration of how we Americans are preconditioned to behave in a stereotypical manner in our everyday life.

Here comes a video that amazed me deeply, "Proposal Fail" (you might enjoy it more without the sound first) :

This girl, unless her great emotion and the pressure of the crowd, is still able to swim against the tide, and to do it with the grace of an angel. I was moved to the soul by her lovely body language : the sparkle in her eyes and the little lateral movement of her mouth just before she starts speaking, her hand gently laid on her unfortunate lover's raised arm while she speaks, the other protecting her genitalia, her quiet precipitation and awe once she's spoken...

Alas, alas ! This girl was not for real : the proposal was a prank, as guessed by the reporters, and confirmed by a commentator who remembers reading it in the paper the next day.

(See here for a genuine reaction -- note the larsen effect at the crucial moment...)

I thougt I'd also show an example that is not related to any kind of Freudian bullshit :

Actually I failed. Notice his reaction : "Oh ffffffff.. Shit. Oh my god." This guy manages to repress his impulse to say "fuck", but only to replace it by another similarly vulgar word, namely "shit". The "oh my god" comes only after, once the first emotion has passed. No doubt Freud would have concluded that his sexuality is repressed, and expresses itself in a scatological way...

Sorry for this tasteless post.

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