The SYK Model by David Gross

oh good morning everyone and delighted to be here at the Asian winners school and to see all of you it will indeed as far as I can tell be an exciting school with an incredible set of lectures but this one probably won't be the best I can assure you I was hoping very much to prepare a chalk talk which is the best way over the last few days last week a incredible conference celebrating ten years to this wonderful Institute and it's been so full of great talks which I recommend that you watch they're all be online and there's really some wonderful talk and I didn't have time to prepare what I really would have liked to give which would be a pedagogical chalk talk instead I am Resort I'll have to resort by and large two slides that I've used in previous talks but there will be four and a half other hours of talks on the syk model given by my collaborator with a demure Rosenhaus whom I've been working on for the last two years on the syk model and anything that I do badly or leave out he will fill in I'm sure so let's let's proceed the syk model stands for such day of year and kitaev so it's there such day of and yeah our condensed matter theorists ooh over 20 years ago 25 years ago I think introduced a very interesting model of fermions living sort of at one space-time point I might call an atom or a or oh there's no space interacting in a very interesting way the motivation of severe substance was to find a soluble model which had a non-trivial quantum critical point the idea is that long-range behavior of quantum systems can have non Fermi liquid type behavior that might underlie things like high-temperature superconductors but very little is known about such critical points and any soluble example is instructive and there was quite a bit of work on this model in the condensed matter theory community by such dayavan and others the model by and large was forgotten and remarkably article quantum field theories never paid any attention to it alexa kitaev did and when he got interested in a black hole information paradox during two years where he which he spent at the kitv and thought that this might be a good example of a quantum mechanical system in one dimension just time which would be holographic in the 80s CFD sent to a two dimensional theory of quantum gravity and he simplified the model to some extent analyzed that particularly focusing on chaos or the chaotic behavior of this model and arguing that it behaved in a way that was characteristic of systems which are holographic to quantum gravity and that aroused a lot of attention since it's always nice to have a soluble model of holography and this appears to be one and it has been studied extensively by many people over the last few years and that is roughly what i will try to review so one of the most interesting aspects of this yes like a model which attracted me is that it's a new class of so-called large end theories and I'll briefly review the kinds of large and theories that have played a very important role in quantum field theory in condensed matter theory and in holography over the years and that's why Kay you know is a new class which has been overlooked and I and others have felt that a new class of large n theories is bound to be interesting not as a salary for its original purposes but who knows what studying in detail as I said it's a one-dimensional model so it's quantum mechanics there's no space it couples my Arana fermions real fermions half of Yurok fermions dimensions half of Dirac fermions and as you'll see the interaction is Q body or Q can be four six eight infinity coupling Q fermions or Q my or Ana's at a time and one of the unusual features probably the reason that quantum filter is didn't pick up on this model for years was that the interactions are random quench'd random numbers quenched just meaning that they're given once and for all by a set of numbers chosen from say a Gaussian distribution of couplings this is quenched random interactions are common they used in condensed matter to describe disordered systems where the couplings of say spins in a medium which is full of defects vary from point to point in a way which one can't control and in calculating observables one therefore only tries to calculate some kind of average over a reasonable probability distribution of such constants that's a bit unusual in in quantum field theory where we typically you know assume homogeneity and fixed couplings and we'll be discussing that a bit they have their interesting thing about this theory is that it it will have a critical point they point which exhibits scale invariance but actually even more than that the theory from the beginning is if you're morphism invariance if you're more fizz amande variant much like on them gravity like gravity that's broken by the interaction down to a conformal group in one dimension which is SL to our and it's soluble accidentally chaotic and that was the observation bike attire that led to the conjecture that it's holographic is some kind of gravity theory in two dimensions two dimensions is a particularly trivial but subtle case of quantum gravity where there are no dynamical gravitons and but still there is some dynamics in in the gravitational space-time and one can discuss black holes in this background in a particular background when you couple the gravity to a dilettante or to a scalar field and and in fact that ad s solution sorry with a in a background which as a negative cosmological constant which by general lore we believe might be holographic to a conformal field theory and one lesser dimension and that that appears to be the case at this model of quantum mechanics in one time no space is holographic say at finite temperature to a black hole in gravity in two dimensions with a negative cosmological constant are coupled to a scalar field [Music] so let me briefly say a word about large end theories in general as far as I said is the new class the old classes that have been used very efficiently over the last 50 years and autumn field theory and in condensed matter any body Theory our vector models and matrix models the simplest being vector models so in general what that means is that we have a class of quantum field theories with some field and an index that trend that runs from 1 to N and that this field forms a representation of a vector but the dynamics is invariant under some sson or su and symmetry group of the Agron Jian might be for example standard kinetic term in some number of dimensions with some interaction which it is only a function of fire I squared now in quantum field theory we are we would like to calculate the partition function the generating function with if we add sources of the oh that's worked in Euclidean space space time and so Lagrangian written here is positive definite and we have like to calculate the partition function maybe adding a source order to generate the correlation functions of the dynamical variables now I'm suppressing space-time labels and maybe even derivatives most of the features of large n emerge already in zero dimensions where one as I will here is just or there might have been a master me one is just doing an integral now in honor mechanics and many-body theory in quantum field theory we often just about the only thing we can easily do is expand such an integral in powers of H bar and what we're doing in that case of course is in a saddle point analysis of this multi-dimensional integral and we generate some of Fineman diagrams now so ignorin the source term we just have vacuum diagrams and which as you know are the exponential of connected ACK diagrams and let's take the simple case of set H bar equal to one and consider now just an integral of Phi I squared and some function of Phi I squared summed over in is over if we look at this integral having dropped the h-bar we note that phi phi i squared is a sum of n variables so as a function of n it kind of behaves like n and therefore for large n we might look for a saddle point even for finite H bar or finite coupling expansion and powers of H bar is actually an expansion in the couplings for example if we have here a coupling the simplest pi4 theory we can scale out I said taking Phi to Phi square to 1 over G Phi squared 1 over G squared Phi squared take out an overall factor of 1 over G squared so an expansion H R is like an expansion and coupling and fireman diagrams which summarized perturbation theory are really a weak coupling expansion so typically this will be a sum of of diagrams vacuum diagrams like this and then perhaps like this and so on this is just a one field going around a little PEZ no powers of the coupling but here there's a G squared and so on next order we get diagrams like this it's a border G fourth that's ordinary perturbation theory we evaluate that in this case by using as a propagator the propagator here has labels I&J for our fields a propagator is just in the case of zero dimensions just delta IJ and the vertex here is complicated because we can contract these indices in different ways it's of order G squared our perturbation theory is fine and people analyze perturbation theory in quantum field theory all the time use it to calculate lots of physical processes but the general problem in a perturbative analysis of quantum field theory many buddy theory is that in general these series diverge there might be signs but no matter how you choose the sign series like this as zero radius of convergence and this is always the case in any quantum field theory if you just expand in powers of the coupling already in quantum mechanics that is a makes the perturbative series totally Ilze fine without some kind of non perturbative information nonetheless we do calculations in QED and get answers that are correct to one part in a trillion because it's an asymptotic series so as long as n as long as we only consider the first in terms where n is much less than 1 over G which could be quite big or G squared of order 1 over hundred 37 we can get a very good answer but at some point it blows up so instead let's consider we consider four large end theories the behavior of this this generating function or the vacuum energy here as it goes to infinity and keep being G fixed in such a way that each term here is the same order as a function of n for example this first term as a propagator which is just delta j the there's no integral so this is best one but there n diagrams so this term is of order n just not surprising this is a vacuum energy of a system with n degrees of freedom or the energies proportional to n this term or two ways we can contract these indices and again we represent that by index lines so here this is 5i times Phi I by J times Phi J with summation Convention and we can contract I with I and that would correspond to this term where I goes here and J there or we can contract I with J with a Delta IJ and the index loop then looks like this they both have a factor of V squared but this one gives N squared and this one is N so clearly if we want to sorry I'm sorry this one gives it if we want to take end to infinity and keep a term of order n like the first term we should choose G squared to be proportional to 1 over N so G squared n equal to a constant in which case this term is down by one over n it's of order one whereas this term is of order and lambda so instead of getting two diagrams we only get one and it's easy to see that 2k for der in perturbation theory and powers of G squared or in powers of lambda so this is now lambda over N the only diagrams that will survive will be iterated bubbles we can just add bubbles anywhere here every time we add a bubble we get a new factor of N and a new factor of G squared or a new factor of lambda the general term here will be lambda to the K I'm some coefficient sorry times n that greatly reduces the number of grafts and it's easy to see that in general power series like this overall n sum of seed of the K lambda to the k @c to the K behaves like some 1 over lambda C is a K the terms grow geometrically and not factorial e and such series of course are converging for what less than lambda C so this is typical of luck Jan approaches and you can see that the advantage to begin with is that large in perturbation theories now converges in some range of couplings and you might go to beyond that by analytic continuation and so it gives you a well-defined expansion convergent expansion in powers of the rescaled coupling the other thing about large end expansions is that we all know that coupling constants in quantum field theory g squared tend to run they're not necessarily fixed in all regions of so that these expansions are totally often uncontrollable I mean the ordinary perturbative expansions where as what n is a number it's not a dynamical quantity and and finally large end for Fiat quantum field theories or quantum mechanical systems often even for rather small values of N the so as I said this gives an expansion n times some function of lambda and their Corrections of order one that's another function of lambda and so on and these expansions and bearers of 1 over N can be much better than the perturbative expansion itself it's easy to calculate the first term essentially a geometric series geometric series are easy to calculate and often exhibit the qualitative features of the exact Erie you don't lose necessarily physics oh whether you can extrapolate the large end behavior to the finite n is tricky for small values of it but for n large enough you capture the full physics often in small Corrections okay now such so if you try to summarize the graphs here you can see that the general structure is you draw a bubble and attach bubbles in all possible ways how do you sum such a thing well perhaps the easiest way is to cut this diagram and look at the perturbation expansion for the propagator which you can see is the sum of a free term if I cut this first interaction and then graphs like this now these are all so-called connected graphs right unparticle irreducible whereas here you can have graphs like this which what you can have here too so this is the full propagator graphs like this and so on and let's call that some the greens function as usual two-point function in general you can see that the son of graphs really says that the two-point function consists of free propagation and then all graphs that can't be cut into two so-called one-particle irreducible and so on iterated it one part of the reducible diagrams that can easily be summed by a geometric series and the one-particle irreducible graphs that can't be cut into two pieces by cutting one line consists of things like this where you where you are is would be simplest example where you have a bubble and then here you put in everything it consists of one particle going into one particle so you put the full greens function G so in a zero dimensional case these are just functions of lambda okay this is one and this is Sigma Sigma square and Sigma is just first term in G so that's also one and so on and it clearly is an algebraic equation which you can solve G equals one over and I really should put here a lambda 1 over 1 minus Sigma and Sigma is just G lambda G and you can all solve that equation that's a quadratic equation the branch cut singularity which is why its expansion in terms of lambda has a finite radius of convergence if you were to go to higher dimension you had had integrals here over say in momentum space or the momentum flowing in this line and you'd get a couple set of integral equations but it's still quadratic and they're easily solvable in any dimension of course the behavior of these functions in higher dimensions you know is tricky and have their feel to violate divergences and there can be multiple solutions a quadratic equation and you can do but but it's always soluble at least when you introduce off to deal with the ultraviolet divergence and large and vector theories have been used to study critical behavior long before one understood the general theory of critical behavior in the renormalization group have been extremely historically useful and remain it continued to be so that's Victor theories the second class of theories that so any questions about Victor argent expansions yes disconnected it's just how the lines come in to the vertex well I said the vertex with I wanna be careful i j KL has a variety of terms of delta i k delta jl austell fuh so we've joy Delta I J Delta K L and there's a third possibility and I write them like this or like this or like this so these are all index line or sometimes we put a vector on them if these are complex fields and there's an orientation and we're done just counting how the indices are contracted so among the various ways we can contract the indices these are the ones that get the two factors of N and this one only has one factor of n okay so I've separated a bit these lines but they're really at a four point vertex now yeah all right now well I don't know of any they're large and theories so the large and theories that are holographic are and there they have been studied in three-dimensions we're so large and theories of quantum field theories in three dimensions are a classic well study system especially for people interested in critical behavior because the world is to be dimensional and we look for phase transitions and second-order phase transitions which at the critical point are described by cons very few theories and those according to our understanding should be duals to some kind of quantum gravity holographic the ones that but it appears that they are dual to a kind of generalization of quantum gravity in AD Essbase which contains in addition to massless gravitons an infinite series of massless higher spin particles and are not very well understood whether those systems are maximum mechanical I don't think is known but they might be they're a little too trivial because they're essentially turns out integral or they're or free field theories in the sense so the dynamics is not that interesting but it could be interest there are other examples in higher dimensions much more interesting case of large in theories that has dominated a quantum field theory and string theory are large and matrix theories and large and Matrix theories are very different and they are the dynamical variables are matrices the dynamical variables are let's call then say em they have two indices and again we consider we consider dynamic since internet under some oh and acting on these indices so we might have hermitian matrices here the adjoint representation of a silicon or Avvo n and the Lagrangian is something again let's work just in three dimensions trace of M squared might be the the trace of M Squared is invariant say Amazon is a is a well let's take em to be complexed is a invariant under you the unitary transformations UN and this would be the kinetic term and we might have in general some function of an imagined now we have matrices a trace of a matrix and by n matrix is of order n so again this term we expect to be of order n and therefore you take the simplest term we brought but again put here something like G squared lambda over N m adjoint squared but anything here right so this again it's ev8 receives of order n there's sort of a lot for a large annual large and outside and I'm interested again let's say in zero dimension in calculating a the partition function where I integrate over all matrices hermitian matrices with some measure of e to the - L now in this case one can't play okay again the dynamical degrees of eating here are mij I can represent the propagator by to index Lauren's I and J opposite directions because this is just permission it's a product of fundamental times anti fundamental and the interaction here has the indices you can see contracted in a following way so I have a product of four matrices where I I identified the adjacent indices so this is the diagrammatic representation of the this kind of interaction and we can do the same kind of diagrammatic analysis of the perturbation expansion and powers of the coupling that we did here so for example the correction to the vacuum energy which is now of order n-squared of each index line is up over there n squared degrees of freedom and an N by n matrix so the first term is of order N squared and the second term is ax but again a bunch of different ways we can contract indices when we add an interaction and diagrammatically I can represent that as graphs like perhaps like this which is of order 1 2 3 n cubed times lambda so I should choose lambda to be of order 1 over N I should define it this way and then this graph will be of order N squared same as this but I'll learn for a large n I can throw away lots of other graphs like this well I want to throw away I want to throw away something what a throwaway well actually this order there's nothing I can throw away but in higher orders what you can easily see is that any graph that can be drawn like this will contribute something of order lambda to the K times N squared but graphs that have the lines crossing over in some way they can't be drawn on the plane non planar graphs will be down by powers of 1 over N and it's a famous result of a tuft that this can be written as a sum this has an overall N squared and a thumb over G of functions of lambda times 1 over N to the G minor who G where G goes from zero to infinity and that G labels the genus the number of handles of a two-dimensional sphere that is triangulated well that is triangulated by the bull graphed of this graph now I don't have any time to really explain how that works but this is a remarkable expansion one it tells you and if you take large n you're considering a very small set of graphs it's not a geometric series it's the sum of planar diagrams but it's much smaller than the sum of all possible diagrams and again the number of terms in this expansion of F or G equals zero or for any finite G is geometric it doesn't have an essential singularity and so there's some hope of being able to calculate this in each sequential term and furthermore the kind of diagrams that survive have something to do with triangulations of two-dimensional surfaces kinds of diagrams that survive in this vector model have looked sort of like if you draw lines to the center of these circles branched polymers whereas here the graphs that survived looked like various ways you can take a sphere and triangulate it a bunch of triangles now that made matrix models very interesting from the point of view of that many different points of view one was the ability to count such triangulations of a sphere but also in the limit where you approach critical point and terms here within or larger number of triangles dominates to try to represent the sum over two-dimensional geometries and thereby a connection perhaps to string theory and in fact this realization that the son of theater diagrams and or diagrams or the Riemann two-dimensional surfaces of arbitrary genus this kind of large n expansion that hoof made in the context of QCD or large in yang-mills theory and suggested strongly they already had [Music] reasons to believe that QCD with confining flux tubes might be represented by a theory of strings but this is another way of looking at it and there has been extensive analysis of many many matrix models of this type with lots of enormous amount back that Diwaniyah and that word bizarre have a wonderful look about this thick 700 pages of reprints of 20 years of of papers on mostly on matrix models and it's if you haven't studied this I strongly recommend you to the study this or a variety of reasons string theory one of them but but this is hard in general there is no obvious way of calculating these functions so for ordinary integrals like this with one matrix you can easily solve it by expressing this integral in terms of not n by n matrix matrices but just the eigenvalues since this is an invariant function this is just the sum over lambda I squared lambda I the eigenvalues of m and this is I really want this outside right what the choice squared and of some potential which depends only on the argument and so these integrals can be done in in just about any case but going beyond one matrix model that a very little is known in general and a theory like UCD has an infinite number of matrices age fields one at each point in space-time so in general large n matrix models although fascinating large n limit of supersymmetric a military we believe is what is exactly dueled to to be string theory in 80's background but in general except for special cases can't be solved in general now let me come to finally and I can see what happens when you don't have time to really prepare so matrix models planar diagrams let me come to syk whose Lagrangian Hamiltonian I'll write down in a moment but the there it turns out that the kinds of diagrams it survived are what are called melon diagrams and here you see what you do or continually add bubbles here you have all these circles these two index diagrams which give you a planar diagram on the plane here you have something which looks like vectors the lines are single value but the set of diagrams has not gotten by adding bubbles but rather taking each line and converting it to a bubble of melon like this so they have been called malonic diagrams and the point of view of duality well these diagrams were very interesting as I said in discussing clinical phenomena the critical points field theories which have such a behavior our perhaps dual to is kind of strange quantum theories in in which have many higher spin massless particles engage degrees and fit in which we don't understand very well these theories the critical points of such theories are presumably dual to critical string theory in 80s and these theories which will have a critical point are perhaps tool to some other theory of quantum gravity which we understand just a little bit of so far these are the ones that we've been study now for intensively for 20 years and and the dual theories string theory in 80s they are characterized by string theory which has in addition to light gravitons and assless particles many in fact they exponentially increasing density of of massive states and are complicate very complicated viewed that way but luckily we have a well-known theory of critical strings which we can analyze only in the dual bulk this theory as I said is simpler there in addition to the graviton their gravitational degrees of freedom of the dual theory have a infinite number of articles but they're all massless and degenerate and that call is very hard to understand such theories in general although even though they're quite trivial their dynamics and here one of the big open questions are certainly the one that I've focused on is what are the what is the bulk gravitational theory and the degrees of freedom beyond just those of gravity this was a question somebody other question yes yeah it's so by the way another way of saying all this is that in these larger theories you in effect are doing a quantum expand you're doing an expansion around a a trivial non interacting theory and n plays the role of H 1 over H bar and the leading term in effect describes a classical limit of this theory there are in quantum field theory there often eight there are many different kinds of classical limits H bar or one over N so in in the case of matrix models in string theory a large and expansion of services of yang-mills Theory gives you a sub kind of theory which leading order 1 over N is classical theory of something and in the case of supersymmetry getting those theory we know what that classical theory is it's the classical theory of strings critical strings in as a consider space times a sphere here we don't really know what the classical theory from a large n is and and what describes the this Tower of massive particles so anyway finally this is the Hamiltonian of the SK model it is a vector theory in the sense that the degrees of freedom which are my iran and fermions have an index of vector index there in the fundamental representation of their real my Arana fermions of Oh n the couplings Ji this term breaks that symmetry for random J's but we regard these couplings as random quenched couplings to choose them from an eight oh and invariant ensemble the theory on average will be a legendary so these are Gaussian random large and everything is Gaussian oh and symmetric and this theory indeed is all evil for a large end just by doing the kind of damn butt Grammatik sigh did accept summing the melon diagrams that survived where the result that one finds indeed in the infrared for large x or low energies a conformal and variant fixed point actually one of the groups into the subtleties of this theory is that that conformal endurance is broken you can't sit exactly at the conformal invariant point and accidentally countered yeah anything is cut for large and it doesn't matter anything is Gaussian that's called all large numbers so unless you're interested in the whatever in Corrections it doesn't matter I'll get to that accidentally chaotic in the sense that the Lepanto exponent of it tells you how things it doesn't matter large n and independent everything Oh Jay comes in J squared doesn't matter what the coupling is so again you analyze the Fineman diagrams as we were doing in other vector models this is the simplest funand diagram you have to go to order J squared so in this diagram I have a vertex coupling for these maryadas the indices are all different two of them were the same chi-squared is is zero cuz i can use with or a chi-square is one but e so the indices are anti symmetric and all indices you drop the diagonal term so all the indices are different here I comes in and you have goes into K L M with a coefficient in J I K L M and here you have that J and here you have another J well when you average over the ensemble or for large and it doesn't matter itself averages this had better be ji k L M itself that means that in this vertex again you have the same coupling and so I goes out that's a sign that you've preserved the symmetry I goes to I it doesn't mix with Jay and this correlation and then you get the sum over all values of KLM so that gives you a factor of and and and cubed so if you choose this to be Delta AI prime etcetera times J squared over n cubed you scale things in this way you'll get a J squared over and killed from this diagram and that's represented by this dashed line which enforces that all the indices here be the same as here and you can then analyze higher diagrams this one is of order J squared there's another way well here's an that's a melon which we just draw like this we go to higher orders we could do a diagram like this and then obviously I could put a melon here because we've said this is a water one this is of order now J squared the N is canceled this would be of order J squared if I contract indices this way other J squared but if I contract the indices in a different way like this this again is a change in the fourth diagram you'll see that i j k l has to be the same as jkm and so the number of indices I have here to play with is just JK LNM that n to the 4 / J squared n cubed squared and that gives me a suppression by 1 over N squared and so the final conclusion is the set of diagrams you have are gotten by taking this diagram and adding these bubbles on any line that you want and these are called generalized melon photographs it's very different it turns out than just these vector diagram and not however is complicated it has a matrix diagram and in fact and write down a closed set of kind of algebraic equations that summarize this some of these graphs so if you look at the full set of one particle irreducible graphs some of which I've drawn here you see that what we're doing is that every time we have an internal line we're dressing it in all possible ways so for this quartic interaction the one-particle reducible self energy is a product of these three propagators in the time domain this interaction J here in the in the Hamiltonian is independent of time so if I draw these diagrams in time this occurs at time tau this is at time tau Prime the propagator in Taos Bay in time-space cubed is the self energy and that's what's drawn here and then always the propagator is the inverse of this geometrical series the sum of the full propagator is the sum of irreducible geometric sum of irreducible diagrams roughly this thing which is one over Omega - Sigma of Omega so this is a simple geometrical right I mean algebraic especially but in frequency space if you see here the-- if you forget about time work in zero dimensions this is just no jew break equation although quarter quarter no longer quadratic but you can solve it okay closed set of equations for the two-point function that's characteristic of vector models not matrix models and having solved that every other term in the 1 over N expansion it can be calculated by a linear linear equation now these equations are a little unusual because this is time and this is frequency and so and generally you cannot solve these equations or nobody knows how to solve them that model is generalizable to Q term interactions where instead where Q equals four was the original model in which case here you'd have Q minus one and there are two cases which are solid evil one is cuticles two or you just have a mass term and it's kind of a free random mass a wire on a fermion with the ripped of mass that's kind of trivial or Q goes to infinity which simplifies life and we issued it solve this exactly otherwise the solution is only known for deep in the infrared where you can neglect this term so in the free theory which is just that of a free buyer honor article you have here a one dimensional theory the dimension of the field Chi is zero by power counting a propagator is just one over EDT or 1 over the frequency and at a term which is J these indices I'm sky and you see this the point of view of renormalization group is an operator with dimensions zero that's a relevant operator and it begins to be very important in the infrared and you get driven to the point where this is relevant where this dominates the dynamics and we have four fermions so that the four times in the dimension of the fermion must be equal to one you get dementia the fermion starts out of zero with a free theory and flows to infrared fixed point here where this term is irrelevant and now you can solve because in the time domain this just says that G times Sigma is equal to a delta function these expressions this is equations at the conformal fixed point are actually invariant under arbitrary time reprioritization you can see that this theory is invariant it has a 1 over D DT so there's no Ettrick here if you make an arbitrary memorization of time the DDT changes that's canceled by the integral ET and the same is of course true here if you take the transform Chi transform T to some function of T and at the same time transform I of T to F prime Phi of F of T f-prime to the 1/4 you in fact the product of four of these the F prime will transform DDT to the F of P so these equations have lots of solutions and you can make an arbitrary reprioritization if you choose any one of them you've broken at repair motorisation one-dimensional revenue down to the conformal group anyway so the solution in general for very short times in the UV is just given by the free theory where the propagator you can easily see in time and the time domain is just sine T in the infrared corresponds to zero dimension oh it's just the number whereas in the infrared the dimension goes to 1/4 and this should therefore behave like 1 over t to the 1/2 and this is the solution you have a flow from this to this that corresponds to Delta equals 1/4 and for a Q body reaction the dimension is 1 over D now you can take that solution and make an arbitrary paralyzation as I've just said and that's very useful because if you reaper ammeter eyes top and I've taken to be the real line to a circle by this transformation you immediately get the finite temperature correlate untrue and that's useful in analyzing this theory at finite temperature where the partition function retir ones from 0 to 1 over the temperature okay so that's the free theory very simple very straightforward you get a conformally invariant fixed point reprioritization time and variance has been broken down to the SL to are the two-dimensional conformal group and everything is hunky-dory and as I said in principle that solves the theory of a nonlinear point of view everything else you do can be calculated and be solved by linear analysis not that it's easy but it's linear and what you've done is flow to the point where this the dominant term and the fair man is acquired a dimension now the four point function is the simplest thing to calculate and you could easily see that that is equivalent to a linear problem I again okay that's the graphs that dominate oh so here you have a fermion coming in again because this quench'd random coupling must be the same as this one this index must be the same as this one so the indices work in this way preserving as Helen and the general diagram that survives for large n is just an iteration a ladder of such diagrams which means that you summit once you know properties of this kernel it's a linear integral equation for the full for point function and this was solved in a I Kataya of originally in schematic form and then more detail by polchinski and was enhanced and especially by nola scene and Stanford and the most interesting way Diana to analyze this thing is in terms of the ope so this theory like in all these matrix-vector models were really interested in the spectrum of gauge invariant or own invariant operators we have this theory of this is an example of such an operator but we can construct higher dimension operators which in the free theory might be these guys which are all in there and in considering the scattering amplitude I just analyze I just discussed we can try to sum these diamonds but what are we really interested in extracting there are two things we might be interested in extracting one is the so called out-of-time correlation function I'm not going to discuss which has to do with the behavior or large times of Chi I of T III of 0 KY j of t I J of 0 such a correlation function which is so called out of time order because you go from 0 to T and then back to 0 and T is what you use to analyze chaos it's not what we usually consider you usually consider time-honored onyx and this is a little trickier but then again is soluble by this linear technique okay point of view holography what we're really interested in are the gauge invariant or oh and invariant fields on the boundary in the quantum mechanical system which somehow are dual to all observables alright alright and so we'd like to use our understanding of this four point function as a function of the various times enter here to pick out when Towell is of order tell two terms in the operator product expansion when these times come very close and these times become very close this roughly product of these two operators can be expanded in terms of I ky behaves like a kite I'm sky behaves like some operator of dimension something times one over tau 1 minus tau 2 to the dimension of Oh n minus twice the dimension of KY so if we examine this or point function in the limit where these points come together we can pick out the ope coefficients in the set of operators that contribute to the sum and that's what was done by these authors and you kind of expect that there will be an infinite tower of operators which are roughly speaking these guys with dimensions which are roughly speaking n plus one at least in the free theory and at the conformal fixed-point should have dimensions equal to twice well n plus one plus twice the dimension of KY plus perhaps some well in general some anomalous dimension and so the analysis of that linear problem is what produced the so called spectrum of operators which in the free theory are these and they can be deduced from solving the linear problem which is picking that which is determined once we know the eigenvalues of the kernel and one reduce is such a formula for the various dimensions of those operators which as I said should roughly be well the ones that actually contribute because of these are hermitian operators are only even mentioned only author mentions otherwise in total derivatives they're not these are the primary operators in the theory and there you see equal to the dimensions of the twice a dimension the fermions plus the number of derivatives plus a correction which goes away or in the infinite limit of the spectrum now this is somewhat similar to matrix models in a DSC FD and su n gauge theory or again you have an infinite number of operators all single trace operators but there many more of them they're an exponential number of them as you increase a number of derivatives with a number of fields and but there there's a parameter that you can use a marginal parameter characterizes the conformal fixed point which is a coupling itself and when super-sub maximally sufism enter gauge theory in four dimensions is conformal for any value the coupling there for large coupling most of the operators develop infinite dimensions which correspond in the bulk the infinite masses which corresponds to the fact that the coupling in the boundary is dual to in a sense by class so in the limit where the coupling goes to infinity the splitting between the string States becomes infinite and the only left with a finite number of operators this is a critical factor in analyzing an idiocy of titre matrix models but it doesn't work here here you have a spectrum of operators that survive and there's no parameter anybody is found to be able to push an infinite number of them away so it means that all these operators will correspond to some degrees of freedom and the bulk generally massive except for the lowest dimension one which is normal to gravitational degrees of freedom and one eats to understand what the theory of those massive park excitations is now there is a enormous subtlety in this whole business because the lowest dimension operator which you might think corresponds to a massless particle of a bulk and the gravitational degrees of freedom is rather singular there are no graviton in two dimensions the dimensional quantum gravity is a little tricky it shows up in the bulk in the boundary are having that four point functionally actually blow up unless you include Corrections that we've dropped in the limit of infinite arms one over tau Corrections and this is indeed reminiscent of things that happen in the boundary which were analyzed over the years by many many people gravity in two dimensions but more recently by Marian Welch in ski it was a very detailed story of that and roughly speaking it corresponds to a one degree of freedom that is what left in the gravitational theory that dominates the the boundary theory in in the infrared event so this is the model a consistent model of 2d gravity with a cosmological constant to you're in and coupled to a peloton or scalar field which is adjusted to have a particular value and you you see the equations of motion for Phi are just the Einstein equations so there really have here an ad s2 background as an equation of motion but when you define the solution of this theory you have to fix a boundary condition and you have a this picture which is the beautiful picture taken for Moses lecture by maldacena shows that you have a freedom of finding the boundary condition on any Reaper amortizations of perhaps the trivial circle that you take towards the boundary and that gives you that degree of freedom [Music] in the park is the dominant surviving degree of freedom in the infrared on the boundary a function of tau the action for which you can derive from both the bulk and now the CNN Stanford it's from the boundary and action is given by the simplest action is invariant under a sell-through are the Schwarz Ian's this is where the maximal chaos comes from and has been shown in fact to be one loop exact it's a very simple agree of freedom corresponding to the gravitational degrees of freedom okay now I'm really run out of time I'm mostly but I have been mostly interested about not in these generalizations necessarily or large kill then it's with other generalizations but in understanding the bulk theory I should remark and I think Vlad is going to discuss this in more detail that this halt quenched random interaction and these melodic diagrams the melodic diagrams which will come from introducing a quenched random interaction have been reproduced already in the literature by people studying so-called tensor models you indeed might think that from after discussing large and vector matrix theories you might now discuss theories where the basic degrees of freedom have three indices or foreign disease or in general entering tensors and people were motivated to do this by trying for the same reason that the matrix model with two indices describes two dimensional geometry the thought was that well it's study three index objects they might be score this card we might be able to find finer diagrams that generate all three-dimensional challenges that's a tall order because three-dimensional geometries are not even there's no way of counting them or classifying them very easily and it fails by and large there is large and limits were analyzed for our example for theories and three indices with the indices now you need colors contracted in this way and they didn't generate they generated some subset of triangulation such a three-dimensional geometry but not in any sense full three-dimensional geometry but the techniques that were developed for those tens of models were or have been very useful as Witten first others because simplest Tenzer model with this way of contracting indices turns out as those people realized proved to be dominated by melon diagrams so that meant that one good take these tensor models as models of 3d geometry but reproduce the syk so for example this melon diagram that was the first one we had is just this way of contracting the indices and you sit and see you get one two three index lines so that again goes like n cubed just like we had here with a random couplings and it's been shown that just works to all orders and the expansion and if you take the leading draw all the diagrams and connect these index lines the dot that the diagrams that survive are the exactly in the mitotic diagrams this is however not a model with random quench couplings here the couplings are fixed and so it's much more like a quantum field theory with a hermitian hamiltonian and physically might make a lot more sense it in fact does differ from the syk model when you go beyond the large n limit one of our corrections are quite different and more complicated so if you're only interested in lard yep and well we've been largely working on is the classical limit of the Volk theory then you can take n to infinity and the two models of the same but igor klebanov and others have been working on this new class of tensorial models and they are much more complicated but often much richer since it's easy to generalize tensor models Thetas and tensors now to be bosons or fermions not just in dimension 1 but in any dimension and it's a new class as large an limits of quantum field theory very rich okay but the question that I've mostly been interested in which Vlad will describe in great detail our work on is the bulk gravity / string theory that needs to be discovered so here the goal is to assume that there is a holographic theory in the book which contains that already has been shown to degrade but there are an infinite number of massive modes of something this is would be analogous to starting with a DSC of T ideas they gain supersymmetric being Mills theory and discovering or constructing a critical string theory nobody's done that nobody knows how to do that you don't know how to root construct string theory from the bulk principle you know how to do it but no but it's I think it would have been extraordinarily difficult without knowing what that Polk string theory not having solved critical string theory previously and so we've been following this direct approach to try to take the degrees of freedom in the book which are exactly the he's filled with the type of dimensions and construct the three-point correlation functions of that conformal field theory and from theory once you have a complete set of operators for large end we do and you know they're three-point functions that's like informal endurance given by group theory with numbers but no these OBE coefficients you've solved the theory in general people a boots approach to so conformal field theories is to write down a set of equations that relates these OBE coefficients and try to find solutions or numerically or balance and so on here we actually have a quite a intricate quantum field conformal field theory with an infinite number of local operators and OB coefficients that we can calculate exactly and we've done that once you know that you can find the bulk theory that reproduces them and that's not very difficult at the if you want this so-called written diagram of a bulk theory whose boundary values have these ope confusions that's pretty straightforward and in principle once you've done that rather point functions you can using the operator product expansion construct the endpoint functions as these operators and that we've explicitly more or less done and then invincibly you can use that to try to describe construct the endpoint functions so at the cubic level you know infinite number of fields and you can translate the ope coefficients which are cut easily calculable that'll easily calculable in terms of summing finding diagrams like this a linear problem because witness is just that kernel we saw and construct the cubic couplings and in principle and over here are some of the answers ah let's see yeah that actually is the sir I was the three-point function which is a a number which depends on these dimensions and their operators and for that you can extract and study the three-point functions as a balk what's much harder are the endpoint functions both on the boundary which is pletely soluble and we've sort of outlined how to do that enough work who could calculate any endpoint function but translate it into the book is always tricky because in general a conformal field theory on the boundary yields are non-local well you don't expect it to be local there's no reason it should be local indeed string theory which is dual to H theory on the boundary is not a local theory a local Quantum's okay and it's a string theory so we have studied in detail before point function here are some properties of I'm not important yeah is the four-point function so four-point functional the boundary is quite killable and that's what we've done yes and then you in principle can start discussing the multi-point weapons in the bulk or what you discover already at the four point function level is nonlocality there is no simple quantum field theory in two dimensions and sister you know the gravitational degrees of freedom and an instant of power of massive particles it is described by a local column yoke Theory you don't expect there to be because after all string theory is not a local unfortunate thing here is that we don't know what string theory we don't have a good candidate so far for that bulk quote/unquote string theory or whatever non-local theory it exists there are lots of possibilities it's not too hard to get the spectrum you analyze the multi point couplings that we can calculate they look kind of stringy in some ways and I was trying to discuss but it's still a big mystery and I find it most fascinating because in my opinion the string framework as I like to describe it not a theory it's a way of constructing quantum gravity theories there but we don't one thing we're sure of is that we have no idea whether the methods we've used to construct string theory so far by any means exhaust the full set of string theories particular string theories and this very well might be a totally new class what's in the way that the sealy of gravity is a new class one that's very hard to understand and we're studying and the lab will fill in the details especially if the last part of this and take you up to the frontiers of our knowledge which as you will see are quite extensive that's good because it means there's a lot of stuff for all of you to work on if you want and to figure out thank you thank you very much yeah maybe we will have a couple of questions suspended David this is a no-brainer question but let me ask it anyway is it possible to organize these infinite number of fields that you just mentioned and write down some sort of a yes some sort of a string field theory yeah I mean I I'm sure what is the strength filthier string field theory is a is if it is all of our only known examples of a non-local quantum field theory if you want I'd like it's better described as a string theory so there are obvious ways of try to go about doing this which we and others have tried so far and you know we have very severe tests and we can calculate all the theory completely in the large n limit and have for all the endpoint function lots of ways of getting the spectrum but you know a strength here in two dimensions is a little strange string Theory's typically have the only physical degrees of freedom are the transfer loads there are no transfers loads in two dimensions so it's possible that one thing I've been exploring our longitudinal modes of strings which have been analyzed over the years in many different ways they're very tricky there are other approaches compactified strings from higher dimensions trouble is we don't see any sign of kulusuk wide modes but I don't know this is a wonderful challenge because in the world of string theory non-local theories that include gravity we were kind of lucky to find flat space critical or non-critical first and then critical strings very simple two-dimensional field theories arica mentally and were discovered therefore sort of accidentally as you know um little resonance model and so on so this was just a kind of lot dumb luck that physics often has of the existence of very simple integral limits that are not so far from the real world in some circumstances so hydrogen atom oh so now we have something which is soluble and we have to find what it's still - and as I said no she hadn't known about critical string theory you would have found it pretty hard to reconstruct string theory from the large n limit of two CD or we don't know how to do it for kisi D in fact there's another challenge large n limit of QC d is surely some kind of strength there and boy would it be great to have to know what that what it is what the classical theory of QCD strings is but this case is a little simpler and very explicit so I'm sure somebody will figure out how to do it jinke hi I have several questions first one is the usual one just what is the player oke to random hoping constant in the floor a cure and the second one is like in non-planar diagram in matrix mode what will be the role of the non malonic diagram in this holography well there whatever in Corrections as I said so you know the non malonic diagram the other terms in the whatever an expansion will correspond the interactions of this classical fury loopty-loop Corrections classical theory so of whatever it is and you can it's a straight totally straightforward task to do whatever in corrections to any large n limit once again a linear problem and the same of course is true in string theory or in any quantum theory once the large end limit in the bulk will be a classical theory of some non-local object some non-local classical theory but once you know the Lagrangian or the act you know then you know how to construct loop diagrams it's totally straightforward so you know you won't learn anything here the problem you won't I mean you'll learn something about one of red Corrections if you're interested here the problem is just to understand the classical it okay I think we should close now and on my side I would like to really thank David in the midst of this very punishing schedule that he had to given such a brilliant lecture and especially with the blackboard that brought back memories of from 25 years ago memories of running into lunchtime the two are two and a half hour long lectures that we used to have quantum field theory but thank you David again and I forgot to mention at the beginning the I said the Kavli foundation is supporting the winter school but this is the Kavli distinguished lecture that we are having as part of the winter school and so for that we have a small memento and as a token and I would like to invite Spencer to come and hand it over to David and conclude the this distinguished lecture

Channel: International Centre for Theoretical Sciences

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Id: Ms_55k8vXzo

Length: 103min 10sec (6190 seconds)

Published: Tue Jan 23 2018