Umut Gursoy et al. “Thermal Transport and Drag Force in Improved Holographic QCD“. Umut with collaborators have shown that bulk viscosity of strongly coupled quark-gluon plasma does not exceed shear viscosity, although grows in the vicinity of the phase transition. Also, if you want to know what exactly people mean by “improved holographic QCD”, a good minireview of it is contained in the beginning of the paper.

Gavin Salam, “Towards Jetography” – everything you need to know about jets in QCD.

Charlotte Gils et al., “Topology driven quantum phase transitions in time-reversal invariant anyonic quantum liquids“. A new type of phase transition in anyonic quantum liquids is discovered – it is driven by quantum fluctuations of topology. I think it is amazingly cool result and the paper definitely deserves to be called the best paper on cond-mat on the day of publication.

C. Germani et al., “Relativistic Quantum Gravity at a Lifshitz Point“. I was not able to go through the paper thoroughly yet, but it seems that they were able to relate Horava gravity to relativistic vector-tensor theory in some particular gauge – in other words, one can derive a kind of full Einstein-Hilbert action from the non-relativistic Horava gravity. If this is indeed so (I still have my doubts), then result is definitely a strong one. Did anybody check out this paper? If yes, what’s your opinion?

Post from: NEQNET: Non-equilibrium Phenomena

Other interesting things in ArXiv (11 Jun 2009)

1. Quantum information

T. Tilma el al., “Is entanglement a critical resource for quantum metrology?”

Can we beat the shot-noise limit (and get to the Heisenberg limit) in quantum metrology by playing with entanglement of a quantum state? As it turns out, typically the answer is positive, but depending on what exactly you measure there might exist regimes such that entanglement actually prevents achievement of the Heisenberg limit.

2. Quantum field theory, string theory

J. Ellis et al., “The probable fate of the Standard Model“. If you perform RG analysis of the SM, three scenarios may realize depending on the value of the mass of the Higgs. First of all, if Higgs mass is very large, Higgs self-coupling blows up – and we need some non-perturbative physics (which we are currently not quite aware of) to control it. On the other hand, if Higgs mass is very small, the potential of SM is unstable. Finally, SM may survive up to the Planck scale if the mass of the Higgs boson is not too small or too large. Authors try to figure out which scenario will be most probably realized in reality using info that we currently have about constraints on Higgs mass.

Two nice string-theoretic papers, V. Kumar, W. Taylor, “String universality in six dimensions” and T. Erler, M. Schnabl “A Simple Analytic Solution for Tachyon Condensation” are well discussed in the recent Lubos Motl’s post and I don’t have anything to add to what he has said already.

3. Astrophysics and cosmology

W. Buchmueller et al., “Probing Gravitino Dark Matter with PAMELA and Fermi“. The main conclusion of the paper is that fluxes observed by PAMELA and Fermi LAT require astrophysical sources for gravitino dark matter.

QuAD collaboration. “Improved measurements of the temperature and polarization of the CMB from QuAD“. A white paper by QuAD collaboration. QuAD and ACBAR will allow to improve constraints on several cosmological parameters (such as running index) compared to WMAP alone.

D.L. Band et al., “Prospects for GRB Science with the Fermi Large Area Telescope“. Another white paper, for Fermi this time. Large Area Telescope will allow for detection of gamma ray bursts in the > 100 MeV band (the most data we had so far are for 10 keV – 1 MeV band).

EPIC collaboration. Study of the Experimental Probe of Inflationary Cosmology (EPIC)-Intemediate Mission for NASA’s Einstein Inflation Probe. This Monday is definitely the day for white papers – this one is huge, more than 150 pages. Actually, it is a EPIC mission concept report for NASA. EPIC will allow precision measurements of CMB B-modes of polarization.

Bruce Bassett et al., “Fisher Matrix Preloaded – Fisher4Cast“. Authors havedeveloped a very nice software package for working with Fisher matrices.

4. Condensed matter physics

M. Kastner, “Monte Carlo methods in statistical physics: Mathematical foundations and strategies“. Very nice pedagogical introduction into Monte Carlo.

Post from: NEQNET: Non-equilibrium Phenomena

This and that in ArXiv on Monday

Regarding analog models I don’t have too much to report – since I am located here at relatively close vicinity to Grigory Volovik (he works in Espoo, while I work in Helsinki), I think I know the agenda quite well, and my overall impression that no so many exciting things happen on the field was confirmed on the workshop. Basically, it proves to be relatively easy to construct models of relativistic chiral fermions and vectors from non-relativistic condensed matter systems (for example, He-3). However, it seems to be impossible to construct relativistic dynamical gravity (that is, effective theory with Einstein-Hilbert action) starting from these systems – recent attempt by Horava seemed to be promising, but the ultimate answer is still the same. What we can do at most is to model a “relativistic” field theory on a curved background (such as Painleve-Gullstrand BH), but this background is static and backreaction of our field theoretic degrees of freedom on it is zero. That’s what activities on the field of analog models of gravity revolve around for almost decade.

So, let me turn to modified gravity and Nima’s talk. Since nobody in the physics blogosphere seems to really discuss the content of his talk (see Mark Trodden’s report – he attended the workshop, too), let me proudly do it for you

As you may already know, the title of the talk is “Don’t modify gravity – understand it“. Nima started by saying that he spent too much time inventing models of modified gravity and now wants to officially confess his sins.

Why? First (but not the most important as you’ll see below), because modified gravity is boring – in all (or most all) it can be reduced to usual GR + scalar field. More accurately, he has introduced the following classification: all modified gravity models can be divided into two classes -

1. boring, with subclasses a) very boring (and not excluded) and b) moderately boring (and excluded by experiments) – because of the name of the class he did not want to talk about those models at all
2. exciting. This class, according to Nima, includes only deeply flawed models such as DGP (where aforementioned scalar field possesses “Galilean invariance”) and Higgs phases of gravity (where scalar fields are essentially Goldstone modes of spontaneously broken spacetime symmetries).

So, why exciting models are deeply flawed from Nima’s point of view? The reason is the fact that, according to well-known theorem quantum gravity (based on usual GR) can not have local observables. The physical reason for that is simple. Quantum mechanics in principle allows us to measure positions of quantum particles with infinite precision (not both position and momentum though). However, measuring position with infinite precision assumes that we have an infinitely heavy apparatus to measure it. Quantum gravity in turn does not allow us to have an infinitely heavy apparatus (sufficiently heavy one would trivially turn into black hole).

A correct language for describing gravitational degrees of freedom should look more like holography. Basically, holography means non-local degrees of freedom, and non-local degrees of freedom mean holography.

Yet, non-locality of gravity will only be noticeable only if we take into account non-perturbative effects, suppressed by

,

where is Newton constant, that is, gravity is non-local but in a very subtle way.

Now, why exciting models are deeply flawed according to Nima? Well, Higgs phases of gravity violate “non-local” part in the statement above – they are manifestly local. On the other hand, DGP violates “subtle” part of the statement above, since it allows for superluminal propagation.

Basically, a general effective scalar field theory featuring CP violation looks like

.

DGP is a rather special CP violating theory, where the last term before the dots is canceled due to a special symmetry, and this allows theory to feature superluminal propagation.

He concluded by explaining what questions should one study to understand non-local nature of gravity better. Basically, since non-locality is suppressed by a factor , it should become important again in situations where some kind of enhancement is present – such as questions related to BH information paradox and eternal inflation (in the latter case, enhancement comes from the fact that you are never able to measure more than models in dS universe, where is de Sitter entropy).

I’ll try to cover remaining talks of the second day tomorrow.

Post from: NEQNET: Non-equilibrium Phenomena

Workshop on tests of gravity in Case Western – day 2 and Arkani-Hamed’s talk

Our recent paper Real-time gauge/gravity duality offers a prescription for the computation of real-time correlation functions using the gauge/gravity duality (i.e. the AdS/CFT correspondence and its generalizations). In this post I would like to explain the motivation for the prescription as well as some general ideas; technical details and additional references can be found in the paper.

Let us begin by considering quantum field theory in Lorentzian signature spacetimes. Such ‘real-time’ quantum field theory is somewhat more complicated than its corresponding Wick-rotated counterpart: one finds lightcone singularities, insertions, operator orderings, non-uniqueness of classical propagators and the need to carefully specify initial and final state or density matrices. On the other hand, all this extra structure precisely allows for the dynamics in our everyday lives and we therefore better know how to deal with these real-time subtleties!

Fortunately, in quantum field theory the situation is at least conceptually well under control. Most of the complications of the previous paragraph are in fact related and can be brought together rather elegantly. Since we will need some aspects of real-time quantum field theory below, let us give some details on how this is done.

Real-time quantum field theory
Consider a real-time quantum field theory path integral. It allows us to compute time-ordered correlation functions, which have specific analyticity properties (often specified in terms of insertions of factors of ). As we will now review, these analyticity properties are intimately related to the proper specification of the initial and final state in the path integral.

Let us for example suppose that the initial and final state are the vacuum state of the theory. In quantum mechanics the wave function of the vacuum can be obtained by computing

(1)

for some position eigenstate and an arbitrary state . Indeed, inserting a complete set of energy eigenstates in (1) and taking the limit projects upon the vacuum state (which we take to have zero energy) and we find the vacuum wave function up to an overall factor that does not depend on .

Since the factor in (1) is just the time evolution operator over an imaginary time interval of length , the amplitude (1) can be computed by a Euclidean path integral over a segment in imaginary time of length . The above arguments also hold in quantum field theory, and we may conclude that vacuum-to-vacuum amplitudes like the familiar real-time partition function,

are computed by path integrating not only along the real time axis with a source for an operator , but rather by adding two semi-infinite Euclidean segments on both the initial and final end of the Lorentzian segment as well. The complete contour is sketched in figure 1.

Fig.1. Contour in the complex time plane corresponding to vacuum amplitudes. The vertical (Euclidean) segments are extended to infinity, and the crosses indicate possible operator insertions.

As we mentioned above, the ‘physical’ effect of these wave function insertions in a Lorentzian path integral is that they lead precisely to the familiar factors of in correlation functions. A quick (but nonrigorous) way to see this relation is the following: consider a mode of the form in the Fourier transform of the two-point function . At a certain large time we turn to the Euclidean segment and substitute with , so our mode becomes . We now only find regularity as if : this is precisely compatible with Feynman’s -prescription corresponding to ‘positive frequencies to the future’ (and similarly we obtain ‘negative frequencies to the past’). For a proper derivation of these facts, see for example Weinberg or the appendix of our paper.

Translation to gauge/gravity duality
Let us now turn to string theory and in particular the gauge/gravity duality. According to the standard dictionary, we should regard spacetimes as semiclassical approximations to some ’stringy path integral’ with boundary conditions for the fields specified by the field theory sources. However, in Lorentzian signature a quantum field theory partition function also depends on initial and final states (in the above example we have just seen that this dependence is nontrivial even for the vacuum state). Correspondingly, one expects that such states need to be specified in the bulk as well. So what would be the precise map between bulk and boundary states?

In our proposal (using an idea of Maldacena), we very literally translate the aforementioned Euclidean path integral approach to the bulk theory. We thus propose to take into account the entire contour for the field theory path integral and not just the Lorentzian segments. This means that we have to ‘fill in’ the entire contour with a (d+1)-dimensional asymptotically AdS spacetime, which then consists of various ’segments’: those that end on the Euclidean segments of the contour generally have a positive definite bulk metric, whereas those ending on the Lorentzian segments have a Lorentzian metric. These segments are then glued together along bulk hypersurfaces that should end precisely on the corners of the contour. We can for example `fill’ the contour of figure 1 (times a circle in d = 2), with segments of three-dimensional Euclidean and Lorentzian AdS, leading to the bulk space of figure 2 (where we actually compactified the semi-infinite Euclidean segments to hemispheres).

Figure 2. The bulk space(time) corresponding to the contour of figure 1, with the operators now inserted on the bundary of the bulk space.

Nicely enough, the construction fits in naturally with the Hartle-Hawking proposal for a ‘wave function of the universe’. The hypersurfaces where the metric jumps should then be seen as the moment where we begin and end a ‘time’ evolution in quantum gravity, and the Euclidean segments then provide exactly a wave function of metrics for these initial and final surfaces. The particular extension of the corners in the boundary contour to bulk hypersurfaces is actually irrelevant for physical observables, which makes the setup very ‘holographic’ as well.

We mentioned before that taking account of the initial and final states in a field theory path integral leads to the correct insertions in correlation functions. On the other hand, in the gauge/gravity duality we compute these correlation functions as functional derivatives of an on-shell supergravity action with respect to its (radial) boundary data. Does our construction then indeed lead to the correct insertions if we compute the correlation functions in this way? This indeed seems to be the case, since all examples we worked out agree with field theory expectations. The (again nonrigorous) explanation is that the Euclidean segments account for the proper insertions in the bulk-boundary propagators, which then map one-to-one into the same insertions in the boundary correlation functions.

Future directions
Let us finally mention some applications and directions for future research. First of all, the prescription allowed us to rederive and generalize the popular recipe by Son and Starinets (see also this paper by Herzog and Son) for thermal two-point functions obtained from black hole backgrounds. We expect further applications to arise in the study of nonequilibrium systems in the bulk and boundary theories, notably the collapse of matter to black holes and the associated thermalization in the boundary theory. The prescription may be particularly useful in studying the holographic encoding of global properties in the bulk, for example the presence of bulk horizons and the regions beyond the horizon. It would also be interesting to further extend the analogy with the Hartle-Hawking wave function and to study aspects of Euclidean versus Lorentzian string theory beyond the supergravity approximation.

Post from: NEQNET: Non-equilibrium Phenomena

Real-time gauge/gravity duality

This post is about my recent paper with Alexander Gorsky and Alexander Monin about a correlator of a Wilson and a ‘t Hooft loop. Before I proceed, I should explain what these objects are and why they are important to be studied. QCD possesses a consistent description in terms of “dual variables” – charges and monopoles. Reader familiar with the systematics of particle-like solutions in different theories would stop me at this very moment by pointing out there are no monopoles in QCD. True, there are no monopoles in the sense of e.g. Georgi-Glashow model. However, effectively there is such a thing as monopole, which is widely observed on lattice as a non-zero Abelian flux through a closed lattice surface. A lot is known on “thermodynamics” and “phenomenology” of these quasiparticles. They do not exist in the sense of theory spectrum. Still, they are an important tool of describing QCD. The QCD phase transition, which is an element of common lore, can easily be understood in terms of monopoles (Fig.1).

This diagram distinguishes between a phase of condensed, strongly correlated or dominating monopoles in each particular phase of QCD. In the transition area both monopoles and charges are determining the dynamics of the theory. In general, most interesting thermodynamic properties of the theory are supposed to take place in the region of the phase transition. Therefore, after we know something about monopoles and charges per se, one would ask himself what the dynamics of their interaction is.

A typical measure of charge or monopole mutual properties are Wilson lines. In particular, they allow one to introduce quark-quark potential by studying a correlator of two straight lines, and many other useful things non-perturbatively. A reasonable guess then would be to use Wilson lines as a measure of quark-monopole interaction. Therefore, a natural idea for someone interested in QCD properties next to phase transition point would be to consider a correlator of corresponding Wilson loops. actually the loop carrying magnetic charge is also known quite well in the literature as ‘t Hooft loop.

One could actually have done this on a lattice (actually, this was how this project started due to a discussion with our lattice-oriented colleagues). This however hasn’t been accomplished so far. Perturbative calculation is out of question, for a small coupling of the charge will automatically make monopole charge large. A resummation technique a la Erickson-Semenoff-Szabo-Zarembo may apply, but that would be a long story in its own turn. So the only way to calculate something well-defined would be to use duality approach.

There have already been some posts discussing some aspects of AdS/CFT here at NEQNET, as well as more particular models of AdS/QCD, so I will just briefly remind the reader a simplest formulation of duality: maximally supersymmetric gauge theory on a 4-dimensional boundary is equivalent to IIB string theory in the bulk, the equivalence requiring identification of currents in the boundary with bulk SUGRA fields’ limit towards boundary. When we calculate a Wilson or ‘t Hooft loop on the boundary, its AdS counterpart is a string two-dimensional surface in bulk, whose one-dimensional boundary is the contour of the loop. This recipe will be used by us as well. To find a correlator of a Wilson and a ‘t Hooft loop we must consider a configuration in AdS with two different boundaries, carrying electric and magnetic charges correspondingly.

Charge conservation would not allow to connect directly an electric string to a magnetic one. Thus an intermediate dyonic string world-surface emerges. Happily, there exists a vertex “charge, monopole, dyon”, or, more generically, vertex, which relates more generic dyons with each other. This vertex will allow the connection between electric, magnetic and dyonic string surfaces.

The question on whether monopole and charge degrees of freedom are mutually strongly correlated or not acquires a simple and beautiful explanation. Dependent on conditions which I discuss below, either configuration shown in Fig. 2

may be realized. The configuration on the left takes the advantage of the dyon vertex. The configuration on the left consists of free, non-correlated charge and monopole. Actually they are not absolutely non-interacting, since they are capable of exchanging SUGRA modes. Still, this is already at PT level and is not of foremost interest to us.

The choice between the two configurations is done dynamically, dependent on which of the two actions is the lower. Contribution by the other configuration to the partition function is then exponentially suppressed. The parameters on which the action depends are temperature, internal radius and external radius. I show in Fig. 3 the phase transition between the connected and disconnected phases dependent on radii ratio. Denoting them in the Figure as “interacting” and “inert” I mean the absence of non-perturbative correlation between the particles; they may still interact via exchange of supergravity modes.

The meaning of this transition is simple and physically quite clear: only when monopoles and charges are sufficiently close, they are non-perturbatively mutually strongly correlated. At low temperatures, this limiting ratio of radii is approximately 0.6. When , the loops are already uncorrelated. However, at high temperatures the critical ratio falls like , therefore, monopoles and charges may become correlated.

We have shown some temperature results, although we never said where the temperature came from. For that purpose the pure AdS space is generalized towards either a t-direction compactified version of the latter or a black-hole-in-AdS background. The transition between the two is known as Hawking-Page transition. Although we are far away from normal QCD with fundamental fermions, this phase transition maybe in some or other way related to QCD phase transition at zero chemical potential.

The results for the phase transition may lead us to a speculation on thermal behaviour of QCD, remembering again the caution that should be taken when extrapolating SUSY results upon a non-supersymmetric theory. We however remember here the standard lore that at finite temperature QCD and SUSY are closer in their dynamics, thus we hope that these results are actually more useful in a non-SUSY context than what one might think from the first sight.

The other issue we discuss in our paper, related to the existence of the vertex and different configurations of monopole/charge/dyon worldsheets is the generalization of the entanglement entropy.

This notion has been introduced long time ago but attracted a lot of attention recently because of its effective derivation in the holographic picture. Roughly speaking if we have a set of regions
divided by boundaries than the entanglement entropy is defined as the entropy seen by an observer in a region who does not communicate with the other regions. In the simplest case one has two regions A and B and introduce the vacuum density matrix . Then the reduced density matrix defines the entanglement entropy . The entanglement entropy is generically UV divergent but the UV divergent part of the entropy does not depend on the size of the region hence the finite -dependent contribution to the entanglement entropy can be, for instance, safely defined as the difference of the entropies at two different and .

The multicomponent regions has been investigated as well and the following generalization has been suggested, inspired by the one-dimensional case

where is the entropy of the single component and and are the right and left boundaries of the -th component. An important question concerns the property of the strong subadditivity which has been proven in holographic picture by Hirata. Another interesting feature of the system to study is the extensive mutual information

where

.

It was argued in literature that the extensivity does not generically hold which is triggered by nonvanishing tripartite information function

The holographic calculation of the entanglement entropy is practically identical to the Wilson loop calculation hence our mixed correlators suggest the natural generalization of the entanglement entropy when the charges are attributed to each boundary. That is, the entropy function for each interval takes values in lattice and has the following structure

for the -th interval. In the conformal case the calculation of the generalized entropy corresponds to the calculation of the partition function with nontrivial boundary conditions. One can define the generalized entropy by summing over the all boundary charges or introducing a kind of boundary chemical potentials for different charges.

Our recipe for the holographic calculation of the generalized entanglement entropy is very transparent. One has just to calculate of the area of the composite minimal surface as the function of the geometrical characteristics. Since all boundaries generically have electric and magnetic charges, the corresponding boundary contour has to be a boundary of the string worldsheet. Such connected composite surfaces may exist or not depending on the geometry of the boundary regions. Similar to canonical entanglement entropy, charged entanglement entropy is UV divergent but the UV divergent part is independent of the geometrical factors.

A natural question concerns the properties of the generalized entropy. The first one to be mentioned is the strong subadditivity which can be simply tested in the holographic picture. A comparison of the corresponding area indicates that for the simplest (0,1)-(1,0) correlator this property is satisfied, however, the analysis of the multiple loop correlators deserves a special consideration. The most interesting question related to the generalized entanglement entropy concerns its modular properties. Indeed, when we have a correlator of multiple dyonic loops, it takes values in with some integer and it would be very interesting to investigate the action of the -duality group on it, which could be related to the deconfinement phase transition.

Certainly there are other questions to come after this work. It would be interesting to recognize the phase transition in terms of the summation of the perturbative series in the spirit of Zarembo. However in the case under consideration the perturbation analysis is more involved since the interactions between electric and magnetic objects have to be summed up.

One of the most interesting questions concerns the action of the S-duality group on the generic correlators of the dyonic loops. A generic correlator of , dyonic loops has to possess interesting properties under the action of group. In particular, it would be interesting to investigate the modular properties of phase transition points of dyonic loop correlator.

Calculation of a correlator of several nonlocal observables has a lot in common with the calculation of the entanglement entropy. Our calculation suggests the natural generalization of the entanglement entropy notion to the case when the boundaries of the regions are charged under the -duality group. That is, generically the generalized entanglement entropy for the region with boundaries takes values in the group tensor product . Since the entanglement entropy at strong coupling is similar to the Bekenstein-Hawking black hole entropy the generalized entanglement entropy can be considered as an analogue of charged black hole entropy. We plan to discuss these issues elsewhere.

Post from: NEQNET: Non-equilibrium Phenomena

Correlator of Wilson and t’Hooft loops at strong coupling in N=4 SYM theory

This post is based on arXiv:0904.3744, in collaboration with Giuseppe Nardelli. Check the links for references and introductory reviews on the subject.

A question. The prototype of instanton in local scalar theories is the classical Euclidean solution (a hyperbolic tangent) for a double-well potential, . The study of instantonic solutions is an essential tool to understand the vacuum structure of the corresponding Lorentzian theory with the potential upside down. The same problem is neither trivial nor of mere academic interest as far as nonlocal theories (i.e., with an infinite number of derivatives) are concerned.

A simple example of a nonlocal scalar with a static double-well potential is provided by the tachyon of open string field theory (OSFT). Nice reviews on OSFT were written by Ohmori, Sen, and Fuchs & Kroyter. The effective lowest-level Euclidean equation of motion for the tachyon of the polynomial superstring field theory is

This equation is highly nonlocal and it is not obvious how to solve it. For instance, an expansion of the operator would not do, since theories with higher-order derivatives are physically inequivalent to nonlocal theories (on general grounds, the former have ghosts, the latter have not). These “perturbative” solutions have a limited range of validity and by no means include all possible solutions. For this reason, equation (1) has never been solved nonperturbatively, although it admits an oscillatory solution corresponding to a brane with marginal deformations. This class of solutions plays a major role in string field theory, as they describe the initial or final stage of tachyon condensation.

An answer. Eventually, we have been able to find an approximate solution of equation (1), namely, the error function

The global accuracy of this solution is between 0.9% and 1.5%, depending on some details, and it can be estimated via two duplication formulae for incomplete gamma functions. To show that is a solution, we used a method developed in a series of papers. The idea is to promote to an auxiliary direction and impose to obey the diffusion equation

Then equation (1) is localized, since nonlocal operators act as translations along the extra direction:

(In general, powers of the scalar field do not obey the diffusion equation, but in this case they approximately do with very good accuracy.) In particular, solutions can be explicitly constructed. This type of theories is ghost-free and characterized by a well-defined Cauchy problem. The error function is a kink:

For comparison we show also the usual kink of the local theory.

One can compute the probability of the “quantum mechanical” instanton to tunneling between the two vacua (minima of the effective potential) and the result is very close to the corresponding local system (), despite the fact that the local equation and its solution are radically different.

Note that the error function is also solution of the equation

with different values of the parameters. This equation has been often used in the literature as a simpler substitute of equation (1).

Inverse problem. A different way to recast the above results is to start with the following inverse problem:

What is the simplest nonlocal system which generalizes the double-well instanton and has the error function as a soluton?

The answer is:

The tachyonic effective action of open string field theory at lowest truncation level!

The mass and nonlocal exponent appear as separate inputs in the effective equation of the OSFT tachyon, although both are determined by conformal invariance. Obviously, a time rescaling can change their ratio, which is precisely the job done by the parameter in our model. However, it turns out that their product is fixed once is chosen. The remarkable fact is that the value of the parameter in the simplified equation of motion is very close to the one dictated by string theory, once the mass is fixed to the OSFT tachyon mass . In particular,

Branes. Regarding the above equations of motion as living on Minkowski and changing to a spatial coordinate, the solution becomes a spatial Minkowski kink, that is to say, a soliton. In fact, the energy of this configuration is peaked around , and the latter can be interpreted as a lower-dimensional brane according to Sen and Horava. More precisely, this solution represents a unstable (non-BPS) -brane in a -dimensional target spacetime decaying into a stable (BPS) -brane.

To support this claim, we must check that the ratio of the brane tensions is the one prescribed by Sen’s descent relations. At its local maximum the effective tachyon potential equals the tension of the non-BPS -brane, which is

where is the open string coupling. When , the brane coincides with the target spacetime of Type I/IIA theory. On the other hand, the tension of the stable -brane is

The prefactor takes into account reduction of dimensionality of the brane () and the fact that the tension of an unstable -brane is times the tension of a BPS -brane. To proceed, we first revert to the original effective action of string field theory and then fix the normalization of the solution. The truncation level of the action affects the value the non-BPS brane tension and possibly the ratio

For the approximate potential in equation (2),

Considering that was regarded as the approximate solution of the lowest-level approximate effective action, the agreement is impressive. We can conclude that the error function is a nonperturbative OSFT tachyonic profile. This does not correspond to a marginal deformation, so it is not clear, at least to me, how to obtain a similar result with the modern techniques recently developed in the full theory.

The parameters of the system with the nonlocal potential, equation (1), are not so close to OSFT as those of the simplified system or, if they are, the global accuracy of the solution is lower (around 3 to 4%). Nonetheless, the brane tension ratio is about the same, if not better. The evaluation of the action on the solution cannot be done numerically unless one implements a careful numerical procedure which takes into account the nonlocal operators (a truncation of the latter would not be reliable). However, we can use the duplication formulae again. One gets

We conjecture that the exact numerical result is extremely close to the theoretical value.

A puzzle. To summarize, the effective equation of the string tachyon with similar values of the coupling constants, as well as the brane descent relation in Sen’s tachyon condensation, have been obtained starting from an apparently different framework. It would be desirable to explain this open problem. The fact that string field theory may be viewed as a diffusing system was already pointed out in arXiv:0708.0366 and arXiv:0802.4395, where tachyon solutions of OSFT and boundary string field theory were mapped onto each other. In a forthcoming study we will argue that the diffusion equation naturally implements some large gauge symmetries of OSFT at the level of the effective dynamics. In the meanwhile, we can discuss together on NEQNET.

Post from: NEQNET: Non-equilibrium Phenomena

String theory and the diffusion equation

I am very happy to find myself writing a blog about a recent paper written by Jose Juan Blanco-Pillado, Alex Vilenkin and myself, and titled “Quantum tunneling in flux compactifications“. In this paper we studied bubble nucleation rates in a 6-dimensional Einstein-Maxwell theory. The two extra dimensions are compactified into a 2-sphere, and their radius is stabilized by a magnetic flux through that sphere. We picked this toy model because it is simple enough to allow a quantitative analysis, yet it also includes some of the essential features of string theory compactifications (a related paper by Sean Carroll, Matthew Johnson and Lisa Randall was posted on the same day as ours!).

But why do we care about bubble nucleations, extra dimensional theories and compactification? To help set the mood, let’s come down to earth and think about a vanilla galaxy. Mmm, nice. How about the Hubble deep field? Mmmm – very nice! We can see thousands of galaxies effortlessly suspended in space billions of light years away. The universe we see is undeniably beautiful and incomprehensibly vast. But still, we can’t help but notice that our laws of physics are telling us that there may be more to the story – infinitely more!

For over two decades eternal inflation has been telling us that our entire observable universe is just a small part of an infinitely large universe (that was spawned some 13.7 billion years ago in the Big Bang), which itself is only one out of an infinite number of other universes (each the product of their own “local” big bang). Here’s the thing: once inflation starts it never ends! To illustrate this, let’s think about a simple model with a scalar field potential that has two metastable de Sitter minima separated by a barrier. Let vacuum A have a larger cosmological constant than vacuum B. If the universe starts in vacuum A, bubbles of vacuum B can nucleate and begin to expand at a speed approaching that of light within A. However, vacuum A is itself expanding, always leaving room for new bubbles to form. Furthermore, since B also has a positive vacuum energy, it can itself become a parent vacuum to type A bubbles. This simple recycling universe is an example of a “multiverse” which gets populated by the two possible vacua in the theory. This idea generalizes to theories with many different vacua – all possible types of bubbles are nucleated one within the other in an everlasting effervescent cosmic extravaganza!

More recently String theory has echoed the same sentiment, suggesting the existence of a multitude of vacua characterized by different values of the low-energy constants of Nature. String theory (currently our best candidate for a quantum theory of gravity) demands that we consider  or  dimensional spacetimes instead of the mundane  that we’re used to! The idea of extra dimensions predates string theory, going back roughly a century to Kaluza-Klein theory, which attempted to unify gravity and electromagnetism by considering a  gravity theory which precipitates electromagnetism in a reduced  perspective. At any rate, it does seem as though we live in a  4d world, so where do all the extra dimensions go? This is where the idea of compactification comes in. Physicists have been able to show that if we start with a higher dimensional world, some of the extra dimensions can be “compactified” so that we don’t “experience” them directly (although what’s going on in those compactified dimensions does influence our effective  reality).

It turns out that there are many ways to compactify extra dimensions. In string theory, the role of scalar fields is played by the moduli that characterize the sizes and other geometric aspects of these extra dimensions. String theory vacua also involve additional objects, such as fluxes and branes. There are so many ways to combine these ingredients (we’re talking numbers in the googols here!) to produce different vacua, that we land up with a “string landscape” of possible vacuum solutions. When the string landscape is combined with inflationary cosmology, the picture of an eternally inflating “multiverse”, populated by all possible types of vacua comes into sharper focus. The calculation of bubble nucleation rates is an essential part of the irresistible task of quantitatively understanding the multiverse and it’s evolution.

Plot of the effective potential, in units, as a function of the modulus field . We show the potential for 3 different values of the flux quantum .

So now that we have motivated why we study bubble nucleation rates, let’s get back to the  Einstein-Maxwell model we investigated in the paper. Our action included a  cosmological constant term and we assumed that a 2-form magnetic flux permeates the extra dimensional  sphere. For our metric ansatz we assumed a maximally symmetric  [/tex] Riemannian manifold, and compactified the extra dimensions on a 2-sphere. While the model can be studied directly in  it is easier to see what’s going on when we dimensionally reduce our model so that we have an effective  potential as shown in the figure. Each minimum in the figure corresponds to a metastable vacuum with a given quanta of the Maxwell field flux  . The set of minima with different values of  , constitute a “small” landscape.

We set out to describe “flux tunneling” from a configuration with a given value of  to a neighboring minimum with flux quantum  (upward jumps may also be allowed if the initial vacuum has positive vacuum energy ). We showed that this process of vacuum decay occurs through the nucleation of magnetically charged 2-branes, which look like expanding spherical bubbles in the large 3 spatial dimensions and are localized in the extra 2 dimensions. The vacuum inside the bubble has its extra-dimensional magnetic flux reduced by one unit compared to that of the vacuum outside. We estimated the instanton action corresponding to this flux tunneling nucleation event, and used it to calculate transition rates.

While the effective  potential has stable vacua under small perturbations in the compactification radius (modulus field  ) for any given value of the flux  , we see that it tends to zero for large values of the radius/modulus field  . This in turn means that positive-energy vacua should be able to decay by tunneling through the barrier, leading effectively to decompactification of space. This seems to be a generic situation for four dimensional effective potentials for moduli fields that represent the size of internal manifolds and that are stabilized at non-negative values of the  cosmological constant. We estimated the decay rate of the above vacua towards decompactification and compared it with the flux tunneling decay rates.

We found that for light and extremal branes (extremal branes have a simple relation between their tension and charge), flux tunneling proceeds far more rapidly than decompactification tunneling, while for superheavy branes the two tunneling rates are comparable.

There is another flux compactification sector of our  theory. The existence of this branch of the landscape is more easily understood in the dual picture, where we have a four-form field flux that one could turn on in the four sphere. One can then find solutions of this model with two large spacetime dimensions (having de Sitter, Minkowski, or anti-deSitter geometry) and with the remaining 4 dimensions compactified on a  . We can study tunneling processes between different values of the flux number on the 4-sphere or go to the Maxwell description where the 4-form flux along the internal dimensions gets dualized to an electric field along the large spatial dimension. It is easy to see then that one can understand the tunneling between vacua in this sector as the Schwinger decay of this electric field.

We are currently investigating whether or not there is an instanton that interpolates between the two sectors in this model. This would probably be a more complicated instanton than the ones we have already studied, as it should involve a topology change to be able to interpolate between the different compactification schemes. This is an important point, since the existence of this type of instanton is necessary in order for the multiverse to explore all the sectors of the landscape.

Another interesting area for ongoing research involves bubble collisions. It is usually assumed that when two bubbles of the same vacuum collide, their domain walls annihilate in the vicinity of the collision point, with great energy release, and the two bubbles merge. At late times after the collision, the resulting configuration has the form of two expanding spheres which are joined along a circle of ever expanding radius. In the case of bubbles with different vacua, a similar configuration is formed, but now the colliding walls merge to produce a new wall that separates the two vacua inside the bubbles.

In contrast, branes separating flux vacua in different bubbles are generally localized at different points in the internal manifold and will therefore miss one another in the colliding bubbles. So the branes will not merge or annihilate, and the bubbles will simply propagate into one another, forming a new vacuum in the overlap region. This new type of behavior could have important phenomenological consequences for the observable signatures of bubble collisions.

To summarize: we set out to study bubble nucleation rates in a toy string theory landscape – the  Einstein-Maxwell model. We showed that vacuum decay can occur via the nucleation of magnetically charged 2-branes. From the  viewpoint, these branes look like expanding bubbles which have their magnetic flux on the inside reduced by one unit compared to that on the outside. We calculated the instanton action for this flux tunneling and compared it to the decompactification decay channel.

We also emphasized that the expanding bubbles resulting from flux tunneling are bounded by co-dimension  branes, which are generally localized at different points in the internal dimensions. We expect, therefore, that in bubble collisions, the branes will generally miss one another and the bubbles will continue expanding into each other’s interior, forming a new vacuum in the overlap region. This may have interesting observational implications, which we hope to explore in the future.

References to the literature can be found in our paper: Jose Juan Blanco-Pillado, Delia Schwartz-Perlov and Alex Vilenkin, “Quantum tunneling in flux compactifications” arXiv:0904.3106v1 [hep-th].

Some suggested further reading includes:

(1) the popular book by Alex Vilenkin, Many Worlds in One – the search for other universes.
(2) a Scientific American magazine article by Raphael Bousso and Joseph Polchinski, The string theory landscape, September 2004.

Post from: NEQNET: Non-equilibrium Phenomena

Quantum tunneling in flux compactifications

We all know that sound travels at about 343 m/s in air, and much faster than that in many solids. But just how much faster could sound travel if given the chance?  Could there be a medium in which the speed of sound can approach the speed of light?  Or might there be some more stringent fundamental bound on the speed of sound?

As it happens, the answer to the last question is no.  There is a medium in which the speed of sound  () can approach the speed of light :  a gas of pions (each of mass ) at a finite isospin chemical potential .  When , but is small compared to , it is possible to calculate the speed of sound using using chiral perturbation theory, and it is not hard to show that it approaches the speed of light in the chiral limit of going to zero. (see Son and Stephanov.)

So clearly there’s no general speed limit on the speed of sound for all consistent field theories.  But let’s lower our ambitions a bit.  Might there still be some broad class of theories that doesn’t include the counterexample above where there IS an interesting speed limit for the speed of sound?

As it turns out, the answer to that question is yes!  But conditionally: we have to lower our ambitions still further.  Working with Tom Cohen, we have been able to show is that in an a certain class of strongly coupled systems, must approach from BELOW at high temperatures.  That is, at least in this class of theories, and at least at high temperatures, there is indeed an interesting speed limit for sound.  We do not yet know whether the speed limit applies away from the high temperature limit – that’s a subject for future work.

(We’re not the only ones who have worked on this: Paul Hohler and Misha Stephanov independently got the same results we did using different methods, and our papers appeared simultaneously. )

To calculate the speed of sound, we used the gauge/gravity duality. The gauge/gravity duality is a marvellous tool:  it lets us calculate observables in some strongly coupled gauge theories just by doing classical calculations in theories with gravity. But as with most good things in life, there is a catch: there are no known gravity duals for any of the gauge theories currently used to describe nature.  This means that the duality can’t be used to make quantitative predictions for nature:  at best, one can hope to learn some interesting qualitative lessons about the behavior of strongly-coupled systems.

In our work, we considered 3+1 dimensional systems that have gravity duals with a single scalar field .   These systems can be thought of as strongly-coupled large N conformal field theories deformed by the addition of a single relevant operator , which is dual to the bulk scalar .    These single-scalar gravity theories are the simplest non-conformal gravity duals.  Different choices of potentials for the bulk scalar field correspond to different dual gauge theories.

In a 4D conformal field theory, the speed of sound is simply a constant:  , as can be seen from the general fact that ( and are the pressure and energy density of a system), and the fact that the trace of the stress-energy tensor vanishes in a CFT, so that .  In a non-conformal field theory, the speed of sound depends on the temperature and other properties of the theory.   The theories we  decided to work with are relevant deformations of a CFT, so they should look like CFTs in the far UV.   This means that at high temperatures, the should approach 1/3.  What is not so obvious is whether approaches 1/3 from above or below:  that is, is the 1/3 an upper bound?

To answer this question, we developed a high-temperature expansion for the geometry and the profile of the bulk scalar.  At very high temperatures, the geometry should look like an Schwarzschild black hole, with a vanishing scalar field – that is, the system should become approximately conformal.  Thus, our high-temperature expansion is an expansion around an Schwarzschild geometry.  It turns out that the leading correction to the geometry is almost completely insensitive to the details of the scalar potential: one only needs to know the UV scaling dimension of of .  Note that , where the upper bound is due to the restriction to relevant operators, and we use the lower bound to avoid a subtlety involving the BF bound for stability of scalars in spaces (this does not affect our conclusions).

Once we found a way to calculate the high-temperature geometry, calculating the speed of sound was easy.  In systems at zero chemical potential, the speed of sound can also be written as  , where is the entropy.  The entropy and temperature can be read off from the geometry, and we find that in the high temperature limit

where

Clearly, the correction away from 1/3 is always negative for in the allowed range.  So for all systems in the class that we considered, is the ultimate speed limit for sound, at least at high temperatures!  It would be very interesting to see whether (and under what circumstances!) this speed limit still holds for lower temperatures.

Well, at this point you might say that this is all very well, but this was in the context of a pretty limited class of theories.  That’s a fair point.   As it happens, our result holds in systems with gravity duals with more than one scalar field as well, as will be discussed in a paper we are now finalizing.

But in fact, to our knowledge,  in all 4D theories with gravity duals, at least when the systems in question are energetically stable (i.e., in a state of lowest free energy).  This is true both in fairly ad-hoc models like the ones we worked with in our paper, and in more sophisticated top-down models using various brane constructions.

So what are the next steps?  First, it makes sense to look for counterexamples, to try to figure out the right domain of validity of the sound bound.   Just how broad (or narrow!) is the class of theories to which it applies?

In a similar vein, it would be great to come up with some general argument that would show whether this kind of speed limit is a general property of holographic theories, or of some interesting subclass of them.  Maybe such a speed limit can tell us something interesting about holography?  If is indeed bounded by 1/3 in all theories with gravity duals, the next obvious thing to look at would be 1/N and finite coupling corrections to the speed of sound, to see how robust the results are away from the supergravity limit where N and the ‘t Hooft coupling are infinite…

For more on all of this, see the two papers below, and the references in them.

1. A. C., T. D. C., and A. N., http://arxiv.org/abs/0905.0903

2. P. M. Hohler and M. Stephanov, http://arxiv.org/abs/0905.0900

Post from: NEQNET: Non-equilibrium Phenomena

A bound on the speed of sound from holography?

In this post I will describe recent work on fermionic Schwinger-Keldysh propagators from AdS/CFT. For further details and references see ArXiv: 0904.4869.

A formulation of the AdS/CFT correspondence relates correlators of a quantum field theory at strong coupling to the boundary behaviour of bulk classical supergravity fields in an asymptotically AdS background. For spinor bulk fields and fermionic dual operators, the prescription is embedded in the relation

,

where  and  is the scaling dimension of  , related to the mass  of the bulk spinor. The boundary lies at  . The early prescription yields Euclidean correlators. In many circumstances standard Feynman diagrams and S-matrices calculations are not adapted. In non-equilibrium settings, interactions cannot be discarded or switched adiabatically or the system might be unstable. All in all, it’s generally not possible to find an asymptotic state and use the LSZ reduction formula. The initial state is known though, so that  matrix elements still provide valuable data. On top of that, some systems such as the quark-gluon plasma and condensed matter models are at strong coupling. It would be of interest to find a way of obtaining real-time correlators from AdS/CFT.

But let us first review the Schwinger-Keldysh formalism for real-time propagators in quantum field theory. The Schwinger-Keldysh prescription provides a way to study real-time Green functions by considering a contour in the complexified time plane. Fields “live” on this contour. In some sense, quantum dynamics does the doubling of the degrees of freedom required for describing non-equilibrium states. Alternatively, one could view this forward-backward closed time contour as a way to introduce a fake doubled Fock space. Initial states are defined on the product of the true Fock space and the twin Fock space. A state-vector prescription has been forged for systems which are described by a density matrix (that’s the case of systems at finite temperature, out of equilibrium, etc.). The action underlying a microscopic description of the system, along with the partition functions are now split according to contributions from the four parts of the time contour, with sources  and fields  .

Schwinger-Keldysh contour

This way, contour-ordered Green functions are mapped into a matrix

.

In the operator formalism

,
,
,
.

with the convention that upper signs stand for fermions and lower ones stand for fields obeying the Bose-Einstein statistics.

and  denote the time-ordering and anti-time-ordering operators.
Those Schwinger-Keldysh correlators are related to the more familiar retarded and advanced Green functions through

,
.

with the fancy notation  referring to either a commutator or an anticommutator.

Inserting a complete set of states gives the key relations, e.g., for bosons

,
,
,
.

For fermions similar formula hold, with hyperbolic tangents, some sign changes and the Fermi-Dirac distribution starring instead.

When  ,  and  . Since  , the relation

holds as required, whatever the quantum statistics of the field.

Now, are those Green functions directly related to some measurable quantities? From its definition, the lower Green function  is related to the average density of particles in the system or to some current density. One might also define a spectral density from the lower and upper Green functions. When particles in the system are interacting, one can check that  encodes the average transition probability when an extra particle of momentum  is added. Besides, switching to retarded-advanced variables, so-called symmetric or Keldysh Green function appear as the sum of  and  . They are proportional to the imaginary part of the retarded propagator. Their interpretation is as correlators for stochastic forces experienced by the system.

Real-time correlators in AdS/CFT

Recall that the original prescription in AdS/CFT for finding gauge theory correlators from classical bulk supergravity fields is in Euclidean signature. However it might actually not be possible to perform an analytic continuation to find Minkowski correlators. This would require some knowledge of the Matsubara frequencies. Yet, in many cases the bulk wave equations can only be solved in some restricted frequency limit. Actually, there exists however a prescription, put forward by Son and Starinets in hep-th/0205051, for computing Minkowski signature retarded Green function in AdS/CFT. It involves a choice of in-going boundary conditions. The drawback is that this cannot be obviously generalized to higher point Green functions. So it might still be of interest to try and compute real-time correlators a la Schwinger-Keldysh in an AdS/CFT setting.

A hint on how to achieve this relies on the observation that Penrose diagrams of asymptotically AdS spacetimes with a black hole exhibit two boundaries on which dual gauge theory fields live. On the other hand, fields and Fock spaces are also doubled in the Schwinger-Keldysh formalism.

Kruskal diagram for the AdS-Schwarzschild black hole

This analogy was used by Herzog and Son, cf. hep-th/0212072. They show how the  matrix of two-point correlation functions for a scalar field and its fictitious partner field is reproduced from the AdS dual supergravity action.

The main idea is to study fields on the Kruskal diagram. Kruskal coordinates (usually labelled U and V) are suited to the study of space–times with horizons. The original, asymptotic observer coordinates (such as those which appear in the familiar Schwarzschild metric) behave badly at the horizon. Yet, a space-ship which would happen to cross a black hole horizon would not see anything particular, until, much later, its crew dies in awful circumstances experiencing unbearable tidal forces as they near the true singularity. This physical singularity cannot be removed by any choice of coordinates. But the fake horizon singularity of the early Schwarzschild metric does not appear any more once you switch to Kruskal coordinates. They also provide an extension to extra regions than the initial Schwarzschild metric. Those are the left and lower quadrants in the Penrose diagram pictured below. One can check that the near horizon behaviour of Fourier modes of the solutions to the scalar equation of motion can be expressed in terms of Kruskal coordinates. While initially the field equation was restricted in the right quadrant of the Penrose diagram, it can be extended to the whole diagram. Fields in the L quadrant can be viewed as the doubled fields of those of the R region, with the Schwinger-Keldysh formalism in mind. Besides, when one extends the mode functions to the complex Kruskal coordinates U and V planes, it turns out that positive–frequency solutions to the wave equation are analytic in the lower U and V complex planes. A solution is composed of only negative-frequency modes provided it is analytic in the upper U and V planes.

Only two linear combinations can be built from the modes in each quadrants which meet the above criterium on holomorphicity. These are

where  is an out-going solution to the wave equation which vanishes in the L quadrant. Similarly,  is in-going and vanishes in the R region.

From the analyticity requirement, the in-going and out-going cross-connecting functions  and  are constrained to be

The prescription devised by Herzog and Son then consists in expanding a supergravity field in a basis of outgoing  and in-going  modes in the full Kruskal plane. The coefficients multiplying those basis functions are determined from the boundary behaviour of the field in the L and R quadrants.

For the scalar field they consider in their paper, the Bose-Einstein distribution then naturally appears. To compute real-time correlators, one finally just has to insert the mode expansion for the supergravity field back into the boundary supergravity action. The Son and Starinets prescription expressing retarded and advanced Green functions in terms of in-going and out-going solutions is called up in the process, which finally yields the Schwinger-Keldysh propagators. They obey the relations reviewed at the beginning of this post.

Given the recent interest in fermionic operators in AdS/CFT, with possible applications to condensed matter physics, how does the Herzog and Son recipe generalize to fermions? The near-horizon behaviour of solutions  and  to the Dirac equation in a AdS-Schwarzschild black hole background is such that additional  or  factors appear as compared to the scalar field case. Given the importance of the solutions being analytic in a complexified extension of the Kruskal plane, 0904.4869 shortly reviews the theory of spinors and twistors in curved and complexified space-times mainly developed by Penrose. In particular, it appears that one can choose a basis for expanding dotted spinors  , which embodies those extra square root factors with a spinor dyad  . By dotted spinor it is meant (in the  case; there’s a generalization to other dimensions) the negative-chirality Weyl spinor component of a supergravity Dirac spinor  .

Those square root factors provide the key ingredients for generating the Fermi-Dirac distribution. Let us see how this works. As for the scalar field case, the conditions that positive-frequency solutions are analytic in the lower U and V complex planes and negative-energy modes are analytic in their upper counterparts leads to the following linear combinations

with

They provide a basis for a spinor field defined over the full Kruskal plane of the AdS-Schwarzschild geometry

.

A point worth noting is that one does not have to expand the  field. Earlier work has established that the leading-order part in an expansion of this field near the boundary must be fixed. One must fix the “position” and leave the “momentum” free to vary in a set of canonically conjugate pairs given by  , the leading order part of  near the boundary and  \psi_0 , the leading-order component of  .

The coefficients  ,  are determined by requiring that  approaches  and  on their respective boundaries. This entails

,
.

with the Fermi-Dirac distribution making its way through the algebra.

A key ingredient comes from the effect on spinor fields of time reversal from going to the R-quadrant to the L one. It must also be taken into account when considering the L quadrant part of the boundary action. When the dust settles, taking functional derivatives of the boundary action with respect to boundary R and L spinors yields the Schwinger-Keldysh correlators for a gauge dual fermionic operator. A review on how retarded fermionic propagators are defined in AdS/CFT and how the leading-order spinor components of  and  are related is best left to the references.

Open questions

In this post, we have focused on the quadratic part of the supergravity action. It would be interesting to compute higher point real-time correlators from higher order components of the action. It might also be of interest to extend to fermions the work of Skenderis and van Rees. As explained in 0902.4010 their work generalizes the Herzog, Son prescription and accounts for some subtle issues. Recently, there has been a sustained interest in fermions from theories with gravity duals. An open problem is to apply the approach exposed in this post to geometries dual to non-relativistic conformal field theories.

Post from: NEQNET: Non-equilibrium Phenomena

Fermionic Schwinger-Keldysh propagators from AdS/CFT

Post from: NEQNET: Non-equilibrium Phenomena

Witten interviewed by Ira Flatow on Big Ideas: video of the day

Recently Chern-Simons theories have attracted much attention as they are thought to describe the world volume theory of the elusive M2 brane. In this short article I will attempt to outline some of the recent developments in this field and describe how brane tilings can be used to find and investigate a large class of these theories.

A Cartoon of an M2 brane and its transverse geometry

The recent interest in the topic of M2 branes was triggered by the discovery of highly supersymmetric Chern-Simons theories. Any theory that describes the world volume physics of an M2 brane should be maximally supersymmetric in 3 dimensions. Such a theory should therefore admit N=8 SUSY. Previously, it was thought that Chern-Simons theories in three dimensions can have at most N=3 SUSY, however Bagger and Lambert [1] recently found a Chern-Simons theory which admits the full N=8 symmetry and fits perfectly as a theory for M2 branes in flat space. Aharony, Bergman, Jafferis, and Maldacena later found the the world volume theory of a stack M2 branes transverse to a geometry [2].

A 2+1 dimensional Chern-Simons (CS) theory is a quantum field theory that has a non-dynamical gauge field . The action for such a theory has the form

Where we take to be a non-abelian gauge field transforming in the adjoint representation of the gauge group U(N).

In order for the theory to make sense, it must be well behaved under gauge transformations. While it is relatively easy to show invariance in the abelian case, the non-abelian case is a little more subtle. In this case

Where N is a integer related to the winding number of the gauge transformation performed. When quantizing the theory using Feynman’s path integral formalism, we insist upon being gauge invariant. This leads to the condition that . This integer k is called the Chern-Simons level for the gauge field . Typically every gauge group in the Chern-Simons theory has a level associated to it.

The Chern-Simons theory above is not at all supersymmetric. However it is possible to make the gauge field a component of an N=2 vector multiplet. Doing this necessarily introduces two scalar fields (one of them auxiliary) and a 2-component dirac spinor to the theory. Supersymmetry invariance naturally enhances the action to include many new terms, including the possibility of superpotential terms. Exactly which superpotential terms, and Chern-Simons levels we pick affects how much more additional supersymmetry the theory has. By carefully consideration of these factors, it is possible to find a theory that can admit the full N=8 SUSY (16 supercharges).

A Typical Quiver Diagram

For certain special Chern-Simons theories we can pictorally represent the Lagrangian by using a directed graph called a quiver diagram. These theories are called quiver gauge theories. Each node of the quiver corresponds to a gauge group and each edge corresponds to a chiral field. Each chiral field transforms under a bi-fundamental representation of the two gauge groups that the edge connects. To define the quiver gauge theory’s lagrangian, superpotential data must also be supplied. In this discussion we shall insist that each field should appear exactly twice in the superpotential – once in a positive and once in a negative term. This is known as the toric condition. It should be noted that, due to gauge invariance, each superpotential term corresponds to a closed loop in the quiver, although the converse is not always true. To completely specify the theory living in the world volume of a stack of M2 branes, we must also specify Chern-Simons levels for all of the gauge groups in the theory.

A Typical Brane Tiling

A brane tiling can be viewed as an extension of a quiver to include superpotential data. The tiling is a bi-partite graph that lives on , or alternatively one can view it as forming a tiling of the plane. The tiling’s dual graph is the periodic quiver and so the tiling’s faces correspond to gauge groups and its edges correspond to chiral bi-fundamental fields. A given term in the superpotential is composed of all of the fields (edges) that the corresponding node connects to. As the tiling is bi-partite we can colour its nodes, say, black and white with different coloured nodes corresponding to superpotential terms of different signs. The tiling’s superpotential then naturally satisfies the toric condition as each edge connects precisely one white and one black node. As is mentioned above, Chern-Simons levels must also be specified to fully define the theory. Faces in the tiling are given integers equal to these Chern-Simons levels.

Once a brane tiling and a set of Chern-Simons terms are chosen then it is interesting to analyse the Moduli Space of Vacua of the theory.  For CS theories with superpotentials satisfying the toric condition, this space is generically a toric Calabi-Yau 4-fold.  The first and in some ways most simple object we look at is the space of vacua when only the superpotential’s F-terms are taken into account. This space is known as the Master Space of the theory and can be given by the quotient where can be regarded as a charge matrix associated with the F-terms. The space is a space of special liner combinations of fields called perfect matchings. A perfect matching is a collection of fields such that each term in the superpotential contains exactly one of these fields. These perfect matchings can be found easily from the tiling [3] and , which encodes the relationships between these perfect matchings, is also very easy to find.  The columns of  correspond to perfect matchings and rows of  correspond to charges.

It is also possible to take into account the D-term conditions. The moduli space of vacua, taking into account both F-terms as well as D-terms is called the mesonic moduli space and can be identified with the space transverse to an M2-brane in M-theory.  Rows of the charge matrix correspond to baryonic symmetries of the theory and tell us how perfect matchings (corresponding to columns) are charged under these symmetries. The computation of   is straightforward and is described in [3].  It turns out that the total number of charges (from both and ) is equal to which means that the (complex) dimension of the mesonic moduli space is equal to 4. This is expected as there should be 8 real (or 4 complex) dimensions transverse to an M2 brane in 11 dimensional M-theory.

The mesonic moduli space, being a toric Calabi-Yau 4-fold, can be specified by a convex collection of lattice points in called a toric diagram, which is a kind of fingerprint for the manifold. This toric diagram can be found directly from the and matrices that define the theory’s mesonic moduli space. The total charge matrix can constructed by gluing together the two charge matrices:

The kernel of this total charge matrix contains the toric data of the CY 4 of interest. Specifically, a collection of 4-vectors are generated which form a convex zonotope in . We should note that the toric points live in a 3-dimensional hypersurface in this 4-dimensional space, which is a manifestation of the Calabi-Yau condition on the mesonic moduli space. This collection of toric points defines the aforementioned toric diagram. The process of finding this toric data for a given Chern-Simons theory is known as the forward algorithm and is a very efficient way of finding large numbers of Chern-Simons theories that live on M2 branes. Given a tiling and a set of Chern-Simons levels, we can easily compute details of the mesonic moduli and so find out which M2 brane geometry the Chern-Simons theory corresponds to. As a special case, we find the well known ABJM theory [2].

One interesting phenomena worthwhile mentioning is that of toric duality. This is when two different CS theories, corresponding to two different tilings with different CS levels have the same mesonic moduli space and so describe the same M2 brane physics. This phenomena is well known and is called toric duality. In Phases of M2-brane Theories [3], these dualities are tested by comparing both the spectrum of gauge invariant operators and also scaling dimensions.

References:

[1]  J. Bagger and N. Lambert, arXiv:0712.3738 [hep-th].

[2] O. Aharony, O. Bergman, D. L. Ja eris and J. Maldacena, arXiv:0806.1218 [hep-th].

[3] J. Davey, A. Hanany, N. Mekareeya and G. Torri arXiv:0903.3234 [hep-th].

Post from: NEQNET: Non-equilibrium Phenomena

381. M2 branes and Chern-Simons theories

As was mentioned before, the twistor space corresponding to 4-dimensional real Minkowski spacetime is a complex projective space – that is, by definition we introduce complex coordinates such that and the points and are identified for arbitrary .

It is possible to describe lines in by a pair , for any . The set of these straight lines therefore depends on four complex parameters. Conformal structure in twistor space is defined by the condition that the distance between any straight line crossing a given line and the line itself is zero (therefore the line belongs to the light cone with origin somewhere in ).

Let us consider a real hypersurface (called Hermitian quadric) given by the equation

.

It divides the twistor space into two parts: the form is positive in one part and negative in the other. The set of all straight lines belonging to this Hermitian quadric depends only on 4 real parameters and therefore represents conformal compactification of Minkowski space. Cones of all lines from the set that cross the given line are just light cones. If we want to get the usual Minkowski space, we need to choose a particular line (for example, , ) and remove all the lines belonging to the set from the twistor space.

That’s how 4-dimensional flat space with Minkowski signature is embedded into the twistor space … Euclidean space () can be embedded into as follows. Consider a set of straight lines connecting points and . All these lines either do not intersect or coincide. This way we can divide into classes of non-intersecting lines or introduce fibration in other words.

Finally, let us talk a bit about various symmetries associated with twistor space and various embeddings discussed above. First of all, the group of projective transformations of space is . It has a subgroup which conserves the quadric above. It therefore induces the group of conformal transformations of Minkowski space. If we want to keep the line defined above fixed, the corresponding subgroup of the group is nothing but Poincare group of Minkowski space. Finally, if we fix one more line not crossing , we will get Lorentz group.

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339. Twistors: getting more formal

As it turns out, twistors also help dealing with non-linear differential equations. For example, Ward and Sir Michael Atiyah have used the language of twistors to construct self-dual solutions of the Yang-Mills equations – instantons. I guess I don’t need to explain how much instantons are important for the quantum Yang-Mills theory – they are non-trivial topological configurations of the Y.-M. field that extremize the Y.-M. action and (which is much more important) give contribution into the overall Y.-M. partition function of non-zero measure.

Instantons – solutions of the self-duality equation – defined in the 4-dimensional Euclidean space turn out to correspond to complex vector bundles in twistor space. This fact allowed Atiyah, Hitchin, Drinfeld and Manin to find all possible instanton solutions for the Y.-M.

Another direction of research where twistor formalism proved to be useful was finding new solutions of the Einstein equations. The point is that instead of Minkowski space we can of course consider general curved spacetime. While 4-dimensional Minkowski spacetime corresponds to the set of straight lines in the 6-dimensional complex space, it is natural to expect that generic curved spacetime should naturally correspond to the set of curved lines in the same complex space. Of course, as usual, reality is more complicated than the first naive guess about it. Not all curved manifolds can be realized as sets of curves in the twistor space, but only those which satisfy to vacuum Einstein equations and additional conformal condition of autoduality (autodual part of the Weyl tensor is equal to zero). Such manifolds indeed correspond to sets of curves in a curved twistor space, and the language of twistors allowed us to find many self-dual solutions of the Einstein equations.

Next time, let me discuss a couple of examples which will hopefully make all this twistor ideology clear.

Some basic literature

1. Yu. Manin, Gauge field theory and complex geometry. A very strong book about complex geometry, holomorphic bundle theory and some physical applications.
2. R. Penrose, W. Rindler, Spinors and Space-Time, Vol. 1. In this volume authors discuss twistor space corresponding to Minkowski space and free massless fields.
3. R. Penrose, W. Rindler, Spinors and Space-Time, Vol. 2. Here, the authors describe twistor description of curved spacetimes and also discuss many physical applications.

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337. Twistors and non-linear differential equations. Curved spacetime

Since everybody currently seems to be a bit crazy about twistors – see for example, the Witten’s paper “Perturbative gauge theory as a string theory in twistor space” and the buzz it started – I decided that the time has come for me to learn what it is and write minireview post about it.

So, what is twistor? Twistor is a straight line in the complex projective space , which is use to realize 4-dimensional Minkowski space, correspondingly, twistor space is a linear space of all such lines. Twistors were first introduced by Roger Penrose (on the left) in the late 1960s.

The set of all twistors – lines in the complex projective space – depends on 4 complex parameters. Minkowski space is realized if we choose a subset in the twistor space that depends on 4 real parameters. The idea to use complex geometry in order to work with real spacetime is highly non-trivial and powerful as we will see. The (twistor) space of all lines in can be interpreted as a complexified and conformally compactified Minkowski spacetime. If you consider two causally connected events (for example, connected by a ray of light) in Minkowski space, they correspond to lines in the twistor space which are crossing each other. Euclidean 4-dimensional space can also be naturally realized as a subset of twistor space, so Wick rotation is a very natural operation in the twistor language.

Congruence of null lines

The fundamental idea of Penrose is that the primary physical structure is not 4-dimensional Minkowski space, but complex twistor 3-dimensional space: twistor equivalents of various physical quantities should be described easier than the 4-dimensional quantities defined in Minkowski space. According to Penrose, many physical field equations just follow from the analyticity conditions in the twistor space. After you deal with a quantity in the twistor space, you reduce it to its equivalent in Minkowski space by an integration over twistors. In other words, you introduce some kind of integral transformation from the complex 3-dimensional twistor space to the real 4-dimensional Minkowski space.

This program is easily realized in the simplest case of free massless fields of various spins: scalar field, spinor, vector field and linearized Einstein equations. These massless fields propagating in 4-dimensional flat spacetime correspond to solutions of Cauchy-Riemann conditions in the twistor space. This fact is not of a physical interest, though, since what you really get is new representations of solutions of linear differential equations.

Next time I am going to discuss a bit how twistors help dealing with solutions of non-linear differential equations and present some examples.

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335. What is twistor

Hope you’ll enjoy it

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324. Video of the day: nice visualizations of elementary particles