"AdS/CFT from F-theory?"

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

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

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

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

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

Etymology of yo

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

"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.

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 is capable of 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 it is ontological."

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!

"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.