More on my research
In this section can be found some topics of my earlier research.
An Intro: Gauge/Gravity Duality
The gauge/gravity correspondence is a duality relating quantum field theory (QFT) and gravity. More precisely, the correspondence relates the quantum physics of strongly correlated many body systems where computations are not easy tractable to the classical dynamics of gravity in one higher dimension. This is the reason that the duality is also sometimes referred as holographic. The idea is now 20 years old, but remains perhaps the most active part within the string theory. In its original formulation called also the Maldacena's proposal after the scientist who invented it, the correspondence related a four-dimensional Conformal Field Theory (CFT) to the geometry of an anti-de Sitter (AdS) space in five dimensions (AdS/CFT correspondence).
The correspondence can be realized in a surprising way. The fields describing the strongly coupled phenomena live in a space with one extra dimension in the dual theory which is a gravity theory. This extra dimension is related to the energy scale of the QFT making the connection between the two theories more concrete. To realize the importance of this idea, one may think that alternatively the holographic description serves as a geometrical feature of the quantum dynamics of the systems with a large number of degrees of freedom and provides exciting evidences that there are deep connections between quantum physics and gravity.
The initially discovered gauge/gravity duality was applied to the simplest toy model quantum field theory, the so called N=4 super Yang-Mills. This theory has the maximal symmetries allowed, very particular kind of degrees of freedom and describes phenomena at zero temperature. In simple words, it is very far from describing physics of the real world. Driven by the desire to understand even qualitative features, of strongly coupled phenomena experienced at the lab, the study of the correspondence has been extended to include very different domains, such as the analysis of the strong coupling dynamics of QCD and the electroweak theories, the physics of black holes and quantum gravity, relativistic hydrodynamics applications, physics of heavy ion collisions and condensed matter physics.
Perhaps the most active area of the applications of the gauge/gravity dualities related to quark-gluon plasma, a new state of very dense matter which behaves as a strongly coupled fluid and was created and observed in the experiments of CERN. There are many well known results that have been exported using methods of gauge/gravity correspondence to different branches of physics. Perhaps the most famous is the prediction that the quark-gluon plasma behaves as an ideal fluid. Its ratio of the shear viscosity over entropy density has been predicted to take values around 1/4π for isotropic dynamics. This prediction served also as a strong evidence that dynamics of the observed quark-gluon plasma in the colliders are strongly coupled, since the weakly coupled dynamics would imply a ratio of several orders of magnitude larger.
Surprisingly enough, recently it was found that the quantum theories with anisotropic dynamics, violate the famous shear viscosity over entropy density bound to obtain parametrically even lower value. These developments show that there is a continuous activity and development of the gauge/gravity duality and its applications.
Universal Properties of the Langevin Coefficients
In the context of gauge/gravity dualities the exact background of the QCD is not known so far. This is one of the important reasons to look for universal behaviors of certain observables among different theories, while these are at the same phase. To understand the meaning and nature of the universal relations, is useful to use an example, where we may have a ratio of two quantities with both of them being proportional to the black hole horizon of the background, and depending mildly or not at all to the other characteristics of the theory. Therefore, their ratio will give some universal behavior among different gauge/gravity dualities of finite temperature in their deconfined phase.
This is theoretically very important and interesting, but it may also lead to significant hints for the experimental data. If such universal relations hold for a wide range of theories they should hold for the QCD itself, at least to some extend. This is believed to be the case for the very successful prediction of the low value of the viscosity over entropy bound η/s, which has been justified by the experimental data, and it is one of the reasons we believe that the Quark Gluon Plasma is strongly coupled.
To study under which conditions these universal properties/relations on particular observables of the theory break down is of equal importance, and a very interesting question.
In a work with H. Soltanpanahi from Jagiellonian University, we have studied the universal properties of the Langevin coefficients of a moving quark in a QGP along the different theories. To establish the range of universality we have extended previous results in bibliography, in order to show that the longitudinal components of the Langevin coefficients are larger than the transverse ones even in theories with Lifshitz scalings. Practically, we have given the most generic relations in terms of the metric elements for this universal inequality to hold.
The natural question to ask after this development, is what is the exact range of this universal relation-which theories can violate it. It turns out that the only way to violate this universal relation is to consider plasmas which have a space anisotropy. As an application to our findings, we give a particular example of an anisotropic theory- the space dependent axion deformed N=4 sYM, that violates the universal inequality and the transverse Langevin component is larger than the longitudinal. Notice that the range of the universality of this inequality is the same to the viscosity over entropy bound, where it was also found that it is violated in the anisotropic theories.
Another enthusiastic finding of our work is that isotropic theories can not have negative excess noise in the case of quark motion. It turns out that the negative excess noise may be allowed in anisotropic theories. We have derived the generic condition in terms of the metric elements for that, and turns out to be quite strong and difficult to get satisfied. We have applied it to the known anisotropic theories and so far we have found strictly positive noise.
Non Integrability in AdS/CFT
A naive way to think integrability of a theory is as having as many conserved charges, ie. equations that need to be solved, as the number of the undetermined parameters. In practice the definition of integrability is more involved, but what is important is that when a theory is integrable, it is a strong evidence that the theory is solvable. Solving a theory means that we can compute all its observables: the spectrum of scaling dimensions, the scattering ampitudes, the surface operators, the correlation functions, the Wilson loops etc.
There are several ways to prove the integrability of a theory. In case of the sigma models a standard way to show integrability is to find a Lax Pair representation, from which an infinite number of conderved charges follow. The full set of necessary conditions in general, for the existence of a Lax Pair is not known. Therefore, is difficult in principle to prove integrability following this way. In certain cases there are other ways to prove integrability, for example using the Bethe equations.
An alterantive promising way which is more direct, is to attempt disprove integrability. In an integrable system, all the subsystems that can be derived as consistent truncations, should be also integrable. Therefore a necessary condition to prove non-integrability is that at least one consistent one-dimensional truncation of the full system of equations is non-integrable.
According to the Kolmogorov–Arnold–Moser (KAM) theorem, a non-integrable system is resonant in the corresponding phase space describing the motion in the angle variables. When weak non-linear perturbations are applied to an integrable Hamiltonian system whose motion is confined to invariant tori, then some deformed invariant tori may remain while others may be destroyed. The conditions for the regular and normal behavior of the perturbed solution are provided by the KAM theorem. Chaos is also a property of non-integrability and can occur only when the KAM theorem does not hold. Chaotic motion arises due to a noise applied to usually well behaved solutions, resulting to a motion that is highly sensitive to initial conditions. The appearance of chaos can also be viewed as a breakdown of integrability, but non-integrability does not always imply chaotic behavior.
The methods of non-integrability have been applied extensively to cosmological models. A common question to answer there is if the cosmological model is integral or chaotic as the time evolves. In our case we apply these technics in the context of AdS/CFT. We prove non-integrability in several theories, and therefore we exclude the possibility of the existence of the Lax pairs in these theories. For example, we show analytically that the imaginary beta deformed theories are non-integrable and as we switch on the deformation parameter the string solutions become chaotic, e.g. as in figure above. A striking fact for our method, is that we can even find simple algebraic conditions that when satisfied, non-integrability is ruled out for a large class of gauge/gravity dualities. For example, all the physical non-relativistic Lifshitz theories with trivial dilaton profile, are non-integrable, with the only exception being the z=1, AdS conformal space.
Probing Strongly coupled Anisotropic Plasma
The QGP in the early times after its creation appears to have anisotropies such as the elliptic flow and the rapid longitudinal expansion. In this work we make use of the gauge/gravity conjecture in order to calculate several observalbes in a dual finite temperature anisotropic N=4 sYM plasma.
To do so we have used the anispotropic gauge/gravity dualities, found recently. The way that the spatial anisotropy is introduced can be though as an effect of backreaction of additional branes. In case of the AdS, D7 branes can be introduced in a way that they wrap the 5-sphere and extend along the time and two more spatial directions, leaving untouched, say the z direction. The branes do not extend alond the radial direction and therefore do not add new degree of freedom in the theory as the flavor branes do. By increasing the number of the D7 branes, they start to backreact to the AdS geometry in a non-homogeneous way creating the anisotropy. In a more technical way this picture turns out to be equivalent as turning on an axion z-dependent field on the gravity side, and by an appropriate deformation of the background the IIB supergravity equations can be solved. In the field theory side this is equivalent, through the complexified coupling constant, of turning on a theta z-dependent term.
Therefore knowing the background ie. the theory, we proceed to calculate several observables, the jet quenching, the drag force and the static force. The target is to study how these quantities are modified compared to the isotropic theory, and also to compare them each other since they have different behavior in the parallel and transverse to anisotropy directions.
The static force measures the strength of the interaction between two probe quarks. In the gravity side can be calculated as a minimal surface ending on an orthogonal Wilson loop with appropriate boundary conditions and cancelation of the divergences that appear. A pair of heavy QQbar quarks introduced along the z direction and another pair is introduced in the transverse direction. The correct way to obtain physical results, as explained in the paper, is to take the derivatives of the potential in order to get the static forces. Doing that we show that the static force is decreased in presence of the anisotropy.
The drag force measures the dragging that a quark feels while moving on the plasma. In gravity side is calcucated by the momenum flowing from the boundary where the probe quark is to the black hole horizonof the background through a string. The drag force is found to be increased along the z anisotropic direction while in the transverse space, there exist a critical value of the velocity below which is reduced. It turns out that the drag force along the anisotropic direction is always greater than the one in the transverse space. The diffusion times follow the inverse relations although a complication appears when comparing with the isotropic case because the thermal mass receives corrections in the anisotropic background differently than the isotropic one.
The jet quenching measures the modification in the transverse momenum of a quark moving in the plasma per unit of length. Therefore, is a quantity which involves two directions, one the direction that the quark moves and one other that the momentum broadening happens. There are three inequivalent ways to place a system of two orthogonal vectors in our anisotropic space and therefore three different jet quenching parameters. The way to calculate this parameters in the gravity side is to consider a light-like minimal surface, and apply some approximations. We have found that the jet quenching is generally enhanced compared to the plasma and where higher enchancemnet is for a particle that moves along the z direction, then for a particle moving along the transverse direction having momentum broadening along the z direction and smaller enhancement is found for particle that moves and has momentum broadening along the transverse space. Notice that similarities have been found in weakly coupled calculations. If the comparisons are made in different scheme, like fixed energy density, the results may change for large anisotropies.
It is interesting all three observables have been written in the paper in a generic form in terms of metric elements, and can be applied to any background.
Non-singlet Baryons in Less Supersymmetric Backgrounds
This is a work done with Yolanda Lozano, Marco Picos from University of Oviedo and K Siampos from Ecole Polytechnique at that time. We have studied the properties of the holographic baryons and their dependence of supersymmetry and conformal invariance. The baryon in AdS/CFT represented by a brane located in the bulk and wraps part or the entire internal space. From this brane initiate k number of strings which go all the way to the boundary and therefore represent the heavy quarks of the baryon. To stabilize the existing brane-string configuration, N-k number of strings need to initiate from the brane with direction to the bulk.
It is known in AdSxS background that the number of the k strings that initiate from the brane has a lower bound, coming from the dynamics of the system and its stability. Motivated mainly by that fact, we have constructed baryon solutions in several gauge/gravity dualities: N=1 or 0 beta deformed theories, Sasaki-Einstein spaces, N=1 Maldacena-Nunez confining background. We have found that the k number of quarks in the boundary is not affected strongly on the reduction of supersymmetry. Technically, one can think that the wrapping of the branes in the internal spaces does not change significantly the system on-shell action. However, when the AdS part of the space gets modified ie. the space where the strings live, the lower bound for k can be reduced. Therefore confinement seems to affect stronger than susy the number of the quarks that can be placed on the boundary.
Giant Graviton Oscillators
This is a work done with Robert de Mello Koch, Matthias Dessein, Christopher Mathwin all from Witwatersrand University. It is an analytic paper, and parts of it can be used as a review of particular technics with Schur polynomials in N=4 supersymmetric Yang-Mills.
It is known that certain n-point functions of particular operators are given in terms of Schur polynomials. Certain Schur polynomials with N number of complex scalar fields Z have been identified with giant gravitons. Giant gravitons can be though as Dp branes wrapping part of the space with angular momentum, and are stable solutions. On the other hand Schur polynomials with order N^2 fields are identified with backreacted BPS geometries like the Lin-Lunin-Maldacena (LLM) ones. In this work we study the dilatation operator on restricted Schur polynomials labeled by Young diagrams with p number of columns (or rows). This setup corresponds to the p-giant graviton. The Shur polynomials are build by a number n and m of order N, for Z and Y complex fields respectively. In order to study their dilatation operator we need to consider the restricted Schur polynomials. The corresponding Young diagrams with the n+m boxes have irreducible representations of S_ (n+m). By considering the subgroup or it, S_n x S_m, the initial irreducible representations, in general, may correspond to several irreducible representations of Sn x Sm. It is essential to keep track of this particular multiplicity and for this reason we use a version of Schur-Weyl duality.
Then for an AdS p-graviton, which has a Young diagram with p rows, we obtain the dilatation operator acting on our operator which can be diagonized at least in some particular limits depending on how large are the number of boxes per column. Roughly speaking it can be diagonilized when the Young diagram goes smoothly from row to row without having a large number of boxes difference between them. In this limit the dilatation operator reduces to a decoupled set of harmonic oscillators. From the gravity side we have found that the number of stringy excitations that lead to zero total charge of the giant's worldvolume is equal to the number of possible operators we have constructed using our method revealing a natural intepretation of our results.
Flavor Corrections in the Static Potential in Holographic QCD
This is a work done with Nikos Irges from NTUA, Greece. In this paper we examine the modifications of particular Wilson loops when the backreaction of flavor branes is taken into account using a perturbative backreacted version of Sakai-Sugimoto model.
In the gravity there are two approximation levels that one can introduce the dynamical degrees of freedom. The flavor branes in the background are placed in a way that usually wrap the spheres and extend through the radial direction to touch the boundary of AdS and therefore introduce new degrees of freedom. In the probe approximation these branes enter in the backgorund but do not deform it. This is done by having a much larger number of N_c color branes than the number N_f of the flavor branes. Therefore in the case of AdS for example, one has the original background with some additional probe D7 branes. Using fluctuations of these D7 branes one can calculate the meson spectrum for example, ie. how the dynamics of the dynamical quarks depend on the gluons, but can not study the modifications of the glueballs due to dynamical quarks. Moreover in the probe approximation one can not find any screening on the static force since in the string tube there are no virtual qqbar pairs. This quenched approximation in the field theory is equivalent of ignoring the fermonic determinant. To take into account these effects one needs to take a large number of flavor branes, then these backreact on the background, and therefore their information starts to be encoded in it. Mathematically the backreacted backgrounds are difficult to be found due to the complicated systems of differential equations arise, although some of them have been found using certain approximations. For example smearing of the flavor branes instead of localization helps significantly, or perturbative backreacted backgrounds lead when possible to easier calculations. The realisation of the flavors in the lattice field theory is also difficult due to amount of the computational time required.
In our paper we have used a perturbatively backreacted version of the Sakai-Sugimoto background which is the D4-Witten background with additional D8-D8bar branes. It is confining and can describe chiral symmetry breaking. The pertubative backreacted background is known analytical in some particular regions. By using non-trivial properties of the minimal surfaces in the confining backgrounds and some peculiarities of the perturbative backreacted background we manage to find the static force and see the expected screening. Additionally we have calculated particular parts of static potential in the Witten D4 background, which turn out to be close to lattice field theory computations.
Gluon scattering amplitudes in Finite Temperature Gauge/Gravity Dualities.
This work was done with G. Georgiou from Queen Mary University and Democritus research institute. We have studied a special extension of the the Alday-Maldacena duality or otherwise called Wilson loop/scattering amplitude duality to the finite temperature limit.
The Alday-Maldacena conjecture gives a connection between the amplitudes in the wilson loops in the AdS space. The scattering amplitudes are given from the open strings ending on a regularization brane in the bulk with Neumann boundary conditions and the corresponding world-sheet can not be calculated easily. The trick to do the calculation is to do a T-duality on the hole configuration. This makes the strings ending on the brane located on the boundary of the space, with particular Dirichlet boundary conditions. The momentum conservation, now is translated to a closure of the loop. To find the minimal surface then is an easier exercise.
In our case we have extended the duality to theories in finite temperature, the N=4 sYM; the N=4 sYM in presence of chemical potential or presence on non-commutativity. To make the calculation analytically we choose a special amplitude which corresponds to a forward scattering of a low energy gluon off a high energy one. In the gravity side this is defined as a light-like Wilson loop which lives at the horizon of the T-dual black holes of the backgrounds. Increase of the chemical potential or non-commutativity leads to decrease of the corresponding minimal surface. We also examine the properties of the minimal surfaces that we expect to be generic for any scattering amplitude in the finite temperature limit.
Deformed Sasaki-Einstein gauge/gravity dualities and BPS and non-BPS string solutions.
The Sasaki-Einstein 5-manifolds can be realised as containing a base B with an axially squashed 2-sphere bundle over a round 2-sphere and the whole manifold having three U(1) global symmetries.
Therefore they can beta deformed choosing every time at least a U(1)xU(1) of the three global symmetries. The supersymmetry of the resulting theory may be equal or less to the initial one depending on which U(1) directions the TsT transformation is done. There is also the availability to perform multible beta deformations in order to completely break the supersymmetry.
The beta deformation can be realized in the field theory side, by redefining the ordinary multiplication of the fields to a version of star product, where its exponential determined by the U(1) chrges of the fields. In the gravity side the solution can be formally constructed by reducing the ten dimensional theory to eight dimension on the two torus. A group of SL(2,R)xSL(2,R) appears, acting on the Kahler modulus and the complex structure modulus of the internal torus. The symmetry acting on the Kahler modulus has as real part the B-field components and as imaginary the volume of the two torus, of the gerenrated solution. There are several other ways to generate the beta deformed backgrounds, but perhaps the most practical one is the TsT deformation, described briefly below, in the part I speak for the Wilson loops. In order to marginally deform the Sasaki-Einstein manifold without reducing further the supersymmetry, we need to avoid the transformation along the direction which affect the holomorphic (3,0) omega form which partly specify the Calabi-Yau structure on the metric cone and therefore leaving invariant the Killing spinors too. The U(1) angle that appears in the holomorphic 3-form is the azimuthal coordinate of the squashed 2-sphere and we do not include it in the deformation, leaving the supersymmetry of the deformed Sasaki-Einstein manifold unchanged.
Having constructed the deformed Sasaki-Einstein duality, we move on to study the BPS solutions. This is done by studying the massless geodesics in this background. We find that the BPS point-like strings move in the submanifolds where the two U(1) circles shrink to zero size. In the corresponding T^3 fibration description, the strings live on the edges of the polyhedron, where the T^3 fibration degenerates to T^1. Then we move to the non-BPS strings, derive their dispersion relations and compare them with the undeformed ones. At the end we comment on the range of the validity of our solutions and their dependence on the deformation parameter, which has many different characteristics to the string solutions on the beta deformed five sphere.
Semi-classical strings in Sasaki-Einstein manifolds, non-BPS string solutions in Sasaki-Einstein gauge/gravity duality
This work has been started while I was still a Phd student in Durham. I have found a wide class of non-BPS string solutions in Sasaki-Einstein manifolds Ypq and Lpqr and have examined the BPS solutions as well.
The Sasaki-Einstein dualites have supersymmetry N=1, and are formed when large number of D-branes are position at the tip of a Calabi-Yau cone. the metrics of the Ypq and Lpqr are now known. The manifolds have Ricci tensor that is proportional to the metric and have a Kahler form that satisfies certain conditions and a complex structure that is closed. The Sasaki-Einstein manifolds are manifolds whose metric cones are Ricci flat and Kahler. Certain coordinates in the metric have to satisfy some contrains, the Sasaki-Einstein contrains that are required by the smoothness of the space. These conditions are related to the parameters p,q,r.
In this work we have presented the methods to find string solutions in these manifolds. We have allowed the strings to extend and spin along the U(1) directions. The process is quite involved, due to complicated metric but several analytic solutions can be found. We have made a classification of the solutions of the particular types. It is interesting that the string solution analysis can be done in full generality for any p,q,r. We have found however that certain string solutions, although they solve the equations of motion they can not be accepted due to violation of the Sasaki-Einstein contraints. Moreover, certain solutions are accepted in class of Sasaki-Einstein manifolds, ie. for a range of p, q, r values but not for all of them. The energies of the strings turn out to depend on the p,q,r parameters too.
We have also found an interesting property that for certain strings, there exist values of p,q, ie, certain Ypq manifolds, where the strings are approacching the BPS strings. It would be very interesting to see how this result appears in the field theory side with the corresponding operators and may lead to non-trivial tests of the gauge/gravity dualities.
UV-divergences of Wilson loop for Gauge/Gravity duality
This work is done with Chong-Sun Chu from Durham University UK. We have found the most generic conditions to hold for the UV divergences of the WIlson loop, so to get cancelled using the Legendre transform.
The Wilson loop in AdS/CFT is represented by a world-sheet which extends from the boundary of the AdS space to the bulk. This distance is infinite and therefore the correponding surface contains divergences. This picture can be understood better in the static potential with the orthogonal Wilson loop. There the infinite contributions can be thought as coming form the self energy of the correponding quarks. In that case one can subtract the divergences by subtracting the contributions to two infinite strings that extend from the boundary to the bulk and represent the quark self energy. In the case of more involved Wilson loops, the cancelation of the divergences can be done with the Legendre transform. The reason is that the the minimal surface problem is not completely Dirichlet and Neumann boundary conditions appear too.
We have considered a class of generic backgrounds and we have found which are the conditions that the background should obey, in order to subtract the UV divergences on the corresponding Wilson Loop. By looking at our results, one can know directly whether or not the Wilson loop UV divergences cancel, in the corresponding theory. Suprisingly enough, it turns out that the UV cancelation does not depend on the supersymmetry of the theory at all. We have applied our generic results to several known theories at the time and we have also the appropriate boundary conditions of the minimal surfaces that cancel the Wilson Loop Uv divergences.
Another interesting part of this work is that with the derivation of the UV divergences cancelation conditions, one can make predictions of the form of the Wilson loop operator in theories that it is not known. This is due to the fact that the coupling of the fields in the operator is determined by the cancelation conditions derived in our paper.
Near BPS Wilson loop in beta deformed theories
This work is done with Chong-Sun Chu from Durham University UK. We have studied the analogous of the 1/4 BPS Wilson loop in N=4 sYM, in the beta deformed theories with N=1 sYM.
The 1/4 BPS Wilson loop in the maximally supersymmetric theory is a circle in AdS and a circle in the S5, away from the equator where it becomes 1/2 BPS. Its expectation value has been found using several independent approaches. The motivation for this work was to examine the corresponding Wilson loop in a theory with less supersymmetry. The beta deformed theories, can be obtained by a Leigh-Strassler deformation, while it turns out that the process is equivalent to modifying the superpotential by a star product containing the U(1) charges of the scalar fields. In the gravity side one can do an SL(2,R) transformation on the Kahler constant, or perform a TsT transformation which consists of a T-duality in one U(1) angle of the sphere, a shift and mixing with another U(1) angle where the deformation parameter beta enters, and finally a T-duality in the initial U(1) angle. Another method to obtain this background is by the use of the Buscher rules. The deformation explained above can be done for any theory that has at least two U(1) global symmeties. In the case of the AdSxS the supersymmetry gets reduced by to N=1. However, by performing series of TsT transformation using the third U(1) isometry of the sphere one can go to N=0. The reduced supersymmetric gauge/gravity dualities have been used to examine the gauge/gravity correspondence in a setup with reduced supersymmetry widely. A reason for that is that the background is not very involved and similar to the undeformed sphere.
In our case we have considered the analogous of the 1/4 BPS Wilson loop in the beta deformed theories. It turns out, that one has to allow to the worldsheet to move to more directions in the bulk in order to get minimized consistently. This is quite common feature of classical solutions in the beta deformed theories. We have found that the expectation value of our Wilson loop remains undeformed, and the beta dependece is cancelled non-trivially in the expectation value. These results agree with works for Wilson loops in beta deformed theories in higher representations. Moreover, we have found that the UV divergences of our Wilson loop cancel. This suggest that at least in large N the Wilson loop operator should be similar to the undeformed case and has several further applications. For example, as lready one can predict from the nature of the TsT deformation, one expects that the Wilson loop/scattering amplitude conjecture should give the same results in the N=4 sYM and in the beta deformed theory.