 Okay, but as we are lucky that Thibault is actually here. And as Thibault was complaining that there are not enough calculations on the backward blackboard and just ideas and he wants to see the calculation part. So I have an end in anticipation that he may be here nevertheless today. So actually the lecture is going to be in two parts. The first half hour, I will do very fast. I will highlight points which I would have done in greater detail to conclude the series of three lectures. But the other part, one hour is going just to be about anomalies. I'm going to discuss anomalies, how you have global anomalies, how you have local anomalies and do specific calculations which I think are instructive. And in everything else I told you there are open problems also there, but the area is much better studied and much better known than here. So what were the things that we passed through in the two formal lectures? We discussed that semi-classical geometry captures two averages which are very small and are of the order of e to the minus s. One of them was, well both of them were related actually to noise. One of them was related just to the behavior, the long time behavior of a two point correlation function. And we discussed chaotic case and integrable or semi-integrable cases. We saw a large variety of possibilities for the value of the noise as a function of the entropy. And we saw that semi-classical geometry gives correctly these very small numbers. We saw however that semi-classical geometry fails to give more exclusive information in particular how the correlation functions behave at any given time and not just averaging there. This appeared once in the about ten years ago in the context of black hole information paradox and reappeared in the case of the ER equal EPR paradox where once again it seemed that geometry may not play a role but actually we have seen here the geometry can capture a very small quantity which in this case was related to the EPR structure of various objects in particular eternal black hole. And then deviations from that and I won't go into any more. Then I said that I would like to study more about singularities. Of course also the black hole has a singularity inside it and of course also the firewall is a singularity. But I wanted also to discuss other type of singularities. Now we know that in string theory singularities which are time like actually are resolved from time to time. There are examples in string theory where such singularities are resolved. And in a way and now I don't know how the camera can do it but you should close your eyes and I will tell you because I don't want you to see as I flip through everything and I'll tell you when you can open them when I reach my target. So my advice is close your eyes if you insist on keeping them open your problem. So I wanted to reach here actually that's true I want ones to reach there. And we know that in string theory any concept that we know as unambiguous when one probes nature by point particles has an example where it becomes ambiguous or if you wish symmetric that's a nicer word when you probe nature by probes which are extended. The most the earliest and most famous one is just simple distance. You would think that if we have one dimension which is compactified on radius r and one on radius 1 over r that the two are totally different. If you do quantum mechanics of particles on these objects the spectrum is completely different. When you do string theory that is not the case and as you know that is a tip of the iceberg of a class of symmetries known as t dualities and there is a whole groups very long structure a lot of it penetrated into mathematics under mirror symmetry and there are a lot of ambiguities of that nature an infinite number of them discrete infinity of them. Then we have examples of theories where the target space has different topology and nevertheless string theory would give the same answer. We have that for small cases and we have it when the when the things are small I won't go through all the examples we don't have here at the time but just remember that if I write down a WZW model on level one on a certain group manifold then I can describe it either by the group manifold whose dimension would be the dimension of the group or I can define it on the carton sub algebra which dimension is the rank of the group. So you have here an example of A the topology being different because the topology of u1 to the r is different than the topology of the different groups and the dimension is different because it's dimension g in one case and r in the other case. So we have this concept of topology ambiguous the number of dimensions we know is ambiguous large or small in the first lecture I spent time about how four dimensions with a certain set of degrees of freedom can describe ten dimensions with another set of degrees of freedom. Then you can go on and do commutativity we know that we have examples where for a certain degree set of degrees of freedom we can consider the system on a manifold which is commutative and for another set of degrees of freedom the manifold is not commutative. You can look at associativity there are examples in the presence of fluxes where the manifold may be associative or not and you would get the same answers and then you go to singularities so the simplest case in string theory when it's small you take a c equal 1 model you look at a circle of a certain radius or you look at an orbit fold of a different radius the orbit circle is just there okay here I don't have but you can use your imagination just you just let the target space contain a piece in it which is a circle where you have identified the angles theta and minus theta so by doing that you have objects which on a circle were open strings if you attach them to these points they become closed strings and they produce a twisted sector you can study the whole model but the main thing is that this object has of course two fixed points at zero and that pie so this is actually the line with these two fixed points and this object has a time like singularity in quantum mechanics however you know that when it has a certain radius it is exactly equivalent to a certain circle with another radius so that is already one example then there are other examples which are more sophisticated which I won't enter in where you you go into certain points to generate extra massless particles and then you have a different type of topology and different type of singularity structure so actually I don't know of any concept in mathematics which has not become ambiguous or symmetric when one study when there are at least examples of where it has become ambiguous or symmetric in the presence of extended objects as probes and I think that's very interesting and I think it's a great pity that we don't have a unified picture of that but each case we solve on its own while I would imagine that eventually mathematicians would see or we would teach them or they would teach us how to have a unified picture which explains how come all these ambiguities come out now as part of these ambiguities I would like now to look into cosmology and I would make here the claim that one can look on a cosmology which seems to have no singularities time goes from minus infinity to plus infinity and you just lie on your how do you call it hammock and that's it nothing ever happens to you and on the other hand you have another dual frame in which you have a drastic end of time and in which there is a catastrophe everything explodes so you have two different pictures of the same phenomena in the bulk so this is a very interesting thing and we study that in with José Barbón in quite a lot of detail to try and understand how come it is possible to have both these pictures now we know that in general when we do singularities and when we study singularities in a normal theory the fact that they appear in our theory is a reflection of our ignorance we don't know that there is quantum mechanics we think about classical physics and we miss that there is quantum mechanics we miss massless excitations but when we increase our knowledge these singularities are not there anymore general relativity is a trickier object because it has in it horizons and under the skirts of horizons maybe some things can be hidden they are it goes as you know under their general principles which are not again proven from anywhere but you impose very general censorship issues in hope that such things are all such singularities are always protected by horizons but does it have to happen does it not happen this is not obvious and there are a set of examples where this actually does not happen namely you can show that it using again ADS CFT as a tool that there is no need that the singularity be there so the way I distribute the half hours I want some things to remain with you so one thing I want to remain with you is this picture of the multiverse so you have here various islands in this case the islands where we can live and one minimum could be better than the other and we try somehow to jump from one to the other and in this case I also captured the idea that okay you also want to live on the other minimum now in general when you have a field theory which has two minima with no gravity then we know that there is a mechanism by which you go from this minimum to this minimum and this was analyzed by many people and by the Russian group by by the US US groups and the mechanism actually comes a little bit even the intuition from statistical mechanics is okay let's say you are sitting in this you have a wave functional which is concentrated around this state you try and stay there now this is not an eigenstate of the Hamiltonian you must if you just write here the wave functional with no components here it will not be a wave functional as an eigenstate of the Hamiltonian and then the idea is that suddenly a bubble of the true vacuum will appear and the question is will this vacuum will the revolution succeed or will the revolution fail so this is an attempt to make a revolution if it will the bubble will increase the revolution will be successful if it will shrink it will fail now how does the analysis go we know that if you do you create such a bubble of the right phase in the wrong phase and I'm not talking no gravity involved then you are in a situation where you gain energy proportional to the volume of the system but you lose energy proportional to the derivatives that you have to pay for if you go from one phase to the other phase and here I took what is called a sin wall approximation where the where it's very drastic and all the all the derivatives are concentrated here you can also make them more blurred and do analysis on what happens in any case one effect goes like the volume and one effect goes like the surface so this means that there will be a revolution which will succeed it could be that if the bubble which forms is too small it will contract again but eventually there will be a bubble for which no matter what the parameters of the problem are let's say some 5-4 theory so you have lambda 2 and lambda 4 no matter what the parameters are this is going to there will be a fluctuation which is large enough and it will take over and everybody will be the whole system will move to the other vacuum so this situation of tunneling from one vacuum to the other is what happens in field theory and it's not unexpected because both of these actually live in the same Hilbert space it was never a doubt that the state here is just some excited state of what exists here now what Coleman and the Lucia studied what happens when you add gravity and the when at the time the cosmological constant was supposed to be zero so the question was what happened when the vacuum energy was larger than zero now actually what is interesting for us is what happens when you tunnel between two vacua both of which have let's say a negative cosmological constant because we are using ADS as a tool in that case what you see is not what you get and the main reason is let's say you are sitting here or even here and you ask am I stable or not stable so in ADS there is a major feature as you know well that the surface and the area and the volume scale in the same manner so once you know what the parameters of the model are you can't have a victory by having it a very large going to a very large scale the scale the issue of if this will work or not work is just determined by the parameters so you may sit in a vacuum here but in that vacuum you will nevertheless not go here because the parameters are such that it doesn't allow it there is also what is called the Brighton-Mueller Friedman bound even if you sit at a point here and you think you have a tachyon or an unstable direction if the tachyon doesn't have a large enough mass squared you will also be stable but what is important for the discussion here what happens is that generically unless you do very fine tuning and this was shown in the paper of Kolmande Lucia with calculations that if you go from here to here you crunch it's not that you will now go to a new vacuum no matter how many victims you had but at least you settle in a new vacuum no you crunch so crunch means that if you do the semi-classical calculation till the region of validity of your calculation it seems like you're forming a curvature singularity and afterwards you're outside the validity of the calculation and you don't know what happens so ADS is a very good place to try and study what happens when you have singularities inside and what does the holographic theory tell you about it because the holographic this is the words I said because the holographic theory contains the total non-perturbative thing here I mentioned that in flat space a large enough bubble will always beat the surface in ADS no it depends on the parameters of the problem sometimes it will and sometimes it will not and then we went again I advise you to close your eyes we've we made a setup in which one can study this and one particular case is you can take the ADS space and make it a setter foliation of it and when you do that you have inside here somewhere a crunch in the Laurentian part I don't have again time to go into this diagram in detail it just you start with the Euclidean part then you built here the Hartle-Hawking wave function and then you move from here to the Laurentian part by doing the analytic continuation and again I invite you to look at the papers for all of that the bottom line in the time I'm dedicating to it is this system which it has ADS in it and has generically a singularity in here it consists of gravity matter and matter has an OD comma one symmetry in the Laurentian part and OD plus one symmetry in the Euclidean part so if you have that symmetry then generically you have a crunch solution inside you demonstrate by looking at this coupled system of equations then you go to the boundary and you have some type of theory on the boundary and this theory on the boundary to study is easier than to study black holes because you find out that the singularity reaches the boundary or if you wish the UV degrees of freedom of the field theory in a finite time so there is a representation of the problem on the field theory side and you try to see what is a representation and the lesson are the following you can have a theory on the boundary which has a relevant operator if the relevant operator has a positive sign so let's think of a mass squared let's say that you have m squared phi squared this theory now when you go back into the bulk will have a singularity in it but the singularity will not be a crunch but would be that as you know what is the IR limit of a theory which has a mass you go just to the vacuum state if you have some interesting topology okay you will get more than just a vacuum state but generically you just end up with a vacuum state which means that in the bulk you have drained out a piece of geometry and you create what is called the bubble of nothing so this is the situation when you add a positive relevant operator when you add a negative relevant operator you may succeed that in this new vacuum you still have many degrees of freedom if you have there many degrees of freedom you have an untrivial geometry also here and it is this geometry which in the bulk has a singularity I'm not showing here again the details I'm telling you the just the fact so you have here on the sitter space but no gravity just an expanding universe you find yourself in a situation where you have a well defined theory so the boundary theory tells you there is no problem everything goes well the bulk has in it a singularity how do you see in the on the boundary that there is a problem you change coordinates and I think I gave a talk about this several years ago here and you can expose that singularity but if you use this conformal boundary this is a bread and butter spontaneous oh and let's call it or some other spontaneous global symmetry theory and it looks perfectly well on the boundary it looks it has a singularity in the bulk so that means that you can try and reconstruct details of the what happens in the bulk and we took a quantum mechanical example in a paper and I think the thing to remember about that you would imagine that the term r phi squared which appears every time you have the conformal field theory and then you perturb it but you have the term r phi squared you would imagine that in quantum mechanics it disappears but actually to our surprise we didn't see it notice before it does not because the curvature goes to zero at the same rate as a conformal coupling which multiplies it goes to infinity and the product is finite usually in four dimensions you are used to have a one six multiplying the r phi squared but there is a general dimension structure of that and when d goes to one that is quantum mechanics the coupling explodes the curvature goes to zero and the product is finite so you can do play with this and see how you do diagnostics of a theory which on the boundary has no problems while in there which means it doesn't have problems while in the bulk it does and this okay then I would say in one word you can also have a theory where the theory has a operator for example margin and irrelevant which is unstable if it's unstable also the bulk is not curable so you will have a problem on the boundary and you will have a problem on the bulk there is an interesting number theory problem which arises if you allow this thing to oscillate a butterfly effect like the capizza problem of the inverted pendulum and actually that has very interesting structure in number theory I'm not I won't discuss it what is the point which I wanted to discuss for further reference the the point for further reference okay here I just told you the facts the for further reference the point I wanted to discuss is this okay I told you about t duality you all know about the stories that if you have a circle it's r or one over r now tell me how did we actually how do our brains know because after all the universe could be in one direction at a circle we have no evidence that it's not we don't have any evidence that it is so let me announce now that the discovery was just made in some new experiment that you never heard about and they just discovered that there is a circle so how do we decide if it's r or one over r how do how do we know to say that okay in my opinion the the reason is the way we are built by looking at local observables this is our structure so for us the momentum modes and not the winding modes are the natural probes of the system so we would we our distinction of r and one over r goes by momentum modes we think through localized wave packets or local plane or plane waves winding modes are not built into us now our feeling is that the same thing happens when you have the two dual descriptions of this singularity in the bulk that is if you have probes which are local then you will not know let me do here again i will jump on all this i will just use this if you have probes which are local you will and you live in the sitter you will not know that there is a crunch you can go on your hammock forever if you build probes which are very large and increase in size together as the Hubble constant as the Hubble constant grows and your world volume even though it's not gravitationally coupled but it just grows if you use such large observables we believe that you would have doomed day people saying they would know about the end so this would be the analogy and to meet an open problem to identify these observables what are the winding like modes which would tell us that indeed a catastrophe is going to happen in the coordinate systems in which you see a catastrophe the ostriches which would not see it would be objects which would shrink much faster than the Hubble constant on the scale so they would never notice because they are shrinking they would not notice the imminent catastrophe so to me that's an important question to understand is it true that indeed the boundary theory says that the singularity can be cured without anything extra happening as it would seem or not and i think this is one of the keys to look at it okay now in the last okay five minutes i will i will of that part i discovered the issue of complexity i defined complexity last time i'm glad in private to give people again tell them what complexity is and it was supposed to be a measure if it's what happens what is the structure of the singularity and there was a conjecture which was born by looking at black holes for long times that if complexity increases then everything is okay no singularities are formed in the process however if complexity decreases then for a black hole you would expect the horizon to be like a firewall and this is what i told you last time so we asked ourselves what happens for various type of singularities and what we found the general structure again please close your eyes if you don't want to get a what we found that in the cases we studied of singularities and we had three type of singularities that the complexity decreases towards the singularity of course all the calculations can cannot continue when you're very when you're a string scale because then your your tools are not there but till you reach scales which are string scales and curvatures which have the order of string scales appropriate one over then you find that the complexity decreases and one case is for example the Kassner which universe you in that case that's the most trivial example because you just take the world volume and shrink it to t equals zero so here when i say decrease increase it's toward t equals zero so you have to go back from positivity to t equals zero this is the metric of the problem you this is a metric on the boundary you can extend it because it's rich rich flat to the bulk this is one extension which has the same boundary you do the analysis of calculating the complexity i told you last time you use the idea that you have an extremal volume surface never mind the how the final result of of this thing is that as you approach t equals zero the system the complexity decreases and this is the volume of the cft times the number of degrees of freedom times the cutoff you put because you don't want to reach the singularity itself so you're approaching it but you're doing the calculations before the singularity and this reminds us of when trying to think about it reminded us of the structures in models which Tibor used to play with and those are the structures of which we don't know again these works of Berezynsky leaf sheets and who did i forget Chalatnikov and Chalatnikov these these works we never knew if they are generic or not but they had the property that the system as you went towards the singularity developed much larger time derivatives and spatial derivatives so if you have a lattice picture the system looked like in the strong coupling on the lattice each problem lives on its own a quantum mechanics of the lattice and this as a state is a very simple state so its complexity is indeed very low so so somehow the fact that we got it for Kazner may be less surprising but we also got it for the cases i described before which were the singularities sitting inside ads in all these cases we found the singularities to decrease the reason that the singularity decreases if you wish were the a serum namely in the computations themselves and i will this will be the the last thing again please turn shut your eyes okay this let me show the results of the calculation so the results were that if you looked at the extremal volume you found that the contributions you get from the part which is the uv part are of what i've written here what you get from the ir part these decrease with time these increase with time when you sum them up together the reason that the sum decreases is because the number of degrees of freedom in the infrared is smaller than the number of degrees of freedom in the uv and the process has in it a constant flow of degrees of freedom from the uv to the ir so this is the final answer that you get and it decreases so with the with the prescription that the volume is what is supposed to be the complexity we founds for three type of or four types of singularities they all had one common feature the complexity decreased which somehow gives you the picture of a system which is decoupled into many different space time points so maybe one step into getting a feeling of what singularities look like using the tools which actually are tools on the boundary field theory and with this i actually even fulfilled my promise and i'm glad in private to discuss more but this concludes the this part of of the singularity i would would maybe just say again to me was very surprising that we can live with a a theory which in the bulk has a singularity this is not what i would have expected usually we wait for the night on the white horse to come and save us quantum mechanics something string theory here it seems that you can live without it however as i told you there is the issue of what probes we use and it could well be that with probes which we are not used we have not yet developed in these theories which are long extended probes one would actually see the singularity even when you are on your hammock thinking you will never see the singularity okay now i go to the i want to discuss anomalies so when tibaut asked me originally asked me to take essentially my bless pascal lectures so this was part of this was part of one of my bless pascal lectures and it was more for to educate and for an advanced audience so anomalies kinkiness whatever plays plays a role in our physics in our physics understanding and in string theory as you know there is the central charge minus 26 and the central charge minus 15 which come out of an anomaly play a very important role in how we construct string theory and the open question i want to discuss will be related to that in gate theories it took many more decades to appreciate the fact that you can't write down any gate theory you want there are some restrictions on what you can do and one of the restrictions goes under the name of gauge anomalies and i would like to discuss simple examples of this and give an understanding of what happens because physicists hate authority so if somebody tells you okay you can't quantize the theory they say why not i want to quantize it so what punishment will you suffer in the situation where we'll ignore these anomalies now in general how do you get to anomalies so you have a certain our way we are maybe maybe this will change the not the show here so this here we'll start here so one set of very interesting lectures could have been should we really continue and write lagrangians and actions this has been a very successful way of doing physics but as you know we are very ungrateful and everything that succeeds we have to would say ah maybe we should have done maybe we should have done it differently and ignore it so one point of view which is you have heard probably many times recently is maybe one doesn't need lagrangians or not that you don't need them but there are circumstances where even if you want them they aren't there you have no idea how to write them down no the usual way we write first the classical lagrangian we identify its symmetries okay now we can be pleased by it having symmetry or we may not care if it has symmetry but it seems that it in order to understand at least the basic interactions which we observe in nature assuming a symmetry which many call now redundant which is gate symmetry was a very useful way to do computations and to confront them with experiment so we have spent a lot of time in writing down lagrangians of systems which have gauge which are gauge invariant and they contain spin one particles and they contain spin a half particles and they contain spin zero particles now the symmetry played a very important role especially when one tried to take into account the full quantum mechanical structure of the theory that is durinormalization when one does renormalization one finds out that's this what is called the quantum lagrangian which is a true one cannot preserve the symmetries of the classical lagrangian now sometimes maybe it cannot preserve at all sometimes you may be in a situation where this lagrangian has several symmetries and by preserving one you can do it but you cannot preserve others and then you are faced with a choice do i don't maybe i don't give a damn about all of them maybe i want to preserve one and not the other and what will be the consequences of doing that so i want to exemplify these in simple cases i will start with a quantum mechanical case now in quantum mechanical case you can say you talked about renormalization you talked about the infinities how i mean how will you see any type of anomaly in quantum mechanics so actually when one discovered anomalies they gauge anomalies they came in two parts one wrote down some partition function and the question was this could be some regularized partition function the original one had a symmetry a full symmetry and you ask what happens to that when you apply to it a gauge transformation so how is z of a of g related to the original z of a you had now the gauge group as a gauge group is of course a local symmetry but itself has a local structure and the global structure and i will i will show here in the example in quantum mechanics we will actually discuss the gates the local gauge symmetry but its global part global not in the sense of that it doesn't change with space-time coordinate but it's part of the global structure of the group because the group of gauge transformation is some group and this group has local structure let's say near the identity but it can also have some non-trivial topology and you can ask how does this object behave under large gauge transformations and this is anomaly i want to discuss now the system i'm going to take will first have a quantized gauge and then the gauge will not be the gauge field first will be not quantized and then will be quantized so it's a very given the time again i'll take a very simple system i'll make it a billion you can discuss the same thing for non- a billion systems and the reference for this is work done together with schmuel elitzu itzhak frischmann and adam schweemer some of them say it's too trivial to be worthwhile even to mention i will i will mention it so the Lagrangian way is going to we are going to take we'll have a complex we'll have a fermion so it will be a fermion psi of t and it will be coupled so there is only time because it's quantum mechanics with the t component of a to psi of t this will be the Lagrangian because i am in the appropriate number of dimensions i don't need to worry about gamma five or gamma zero and so on just they're all defined to be ones now this system is invariant under certain symmetries it has a gauge symmetry and it has a global symmetry and the gauge symmetry will construct so it has a global part and the local part so what is the gauge symmetry so there is a u1 local gauge symmetry and that is if i take psi of t and i multiply it as one is used to by e to the lambda t that would be i take psi of t to and multiply it by that phase i take psi bar and i multiply it by this phase and shift a of t into a of t plus dt of lambda t this Lagrangian as written is invariant under that symmetry it is also invariant under a global symmetry this is a symmetry i will present to clash with and that symmetry is take and interchange psi and psi bar and take a and move it into minus a that is what you would expect from charge conjugation so let's call it charge conjugation this is a global symmetry i do that symmetry at one point i take psi to psi bar and a to minus a now in order to give this interesting structure i'm because i want to build the case where there is an anomaly so i have to give some structure and the structure is i'm going to put t on some let's on some region from zero to capital t and i'm going to impose periodic boundary conditions on the gauge field here so i have t goes to the t goes from zero to t where from the point of view of the gauge field i want to impose periodic boundary conditions okay so now i won't show you but it's a trivial exercise you can do for yourself just interchange psi and psi bar put a to minus a and then try to get back your original Lagrangian remember the anticommutation relations of the psi's and psi bar and remember the psi and psi bar the anticommutation relation is one you will retrieve back this model after doing this so therefore you know that you have two symmetries at the classical level okay now just note for a moment that when you look at the system you write down which is also actually modern today you write down all line integrals all things that that sit in that system so for example let us consider the Wilson loop so i can write a loop or in this language some people sometimes call it the polyakov loop because it goes in the temporal direction let's look at this object from zero to t a of t dt now let us check its gauge invariance properties so if i substitute in here i substitute this i have the first term is the term so now let's do the substitution so i'm going to get the original term which is exponent of i adt from zero to t and i'm going to get a second term sorry it would be a product and the second term because i have a total derivative here is going to be i times lambda at t minus lambda at zero now here you will see the the structure of the gauge group coming into play i could say that i wanted the gauge group be such that at the edges both lambda of t and lambda of zero should be zero namely i'm interested only in local gauge transformations which near the they start and they end at the same point for example the identity i don't allow them to change in that case for those transformations and the class of all lambdas which obey this this will be this will be the local part of the local gauge invariance the Wilson loop which is the quantity here is invariant under them another quantity however which would be in another set of gauge transformations which would be invariant under them would be the large gauge transformations and i could allow myself that lambda of t minus lambda of zero not be the same so but b 2 pi n this would also keep the theory invariant now w and this action here is invariant under both so i have the full gauge transformations here okay and however there is only one lesson to learn usually and i will do it in the second example when we go to two dimensions usually when you quantize the theory or a nice way to quantize the theory is by going to the gauge a not equal zero there you can have gauss's law and you have the Hamiltonian and then you diagonalize your theory by submitting your way functional to gauss's law and by this you reduce the number of degrees of freedom from d to d minus one with a not and by applying gauss's law i will repeat it on the states you get d minus two degrees of freedom okay but in this system is it possible to always go to the gauge a not equal zero the answer is not because one of the things which defines the system is the value of the Wilson loop you have now what is possible you can take any configuration a of t and turn it into a time independent gauge turn into a time independent field so any field that i give you as a background or afterwards i'll integrate over a of t can always be reduced to a constant and the value of the constant this can always be taken by a legal gauge transformation by a local one it will be here one over t if you wish the average of a of t dt from zero to t okay another place where this average comes in but for a different reason and this this object d a bar dt by definition vanishes because i've taken out all the t dependence in it now how will a bar change under the residual gauge transformations i have which are the gauge transformations which have this property so i can take for example a local gauge transformation which could start from zero let's say and end at 2 pi n how would a of t change under this gauge transformation so this as you see is part of the local part of the global this is part of the global part of the local gauge transformation because at time zero and at time t it has different values cannot be deformed to the identity so under this you will see that a bar goes to a bar plus 2 pi n over t so unless a but unless you are in a situation where already this object here has been arranged to be a 2 pi n you cannot take and go back to zero so you are in a situation where you have a remnant of the gauge group which you have to maintain as a symmetry and has to remain a symmetry of your partition function and that is the one where a bar gets shifted in such a way the wilson loop and this will have different values depending on with what background i put i cannot go to a bar equals zero unless i'm in a very special configuration okay so this is these are the constraints we have and now we have to go ahead and calculate the partition function now there are two ways to do it and this is part of the question i'd like afterwards to ask you there is one part which is very complicated on some scale and that is at least for someone learning quantum mechanics would be complicated and let's just do the function do the integral over psi and psi bar and see how the thing depends on a using the Lagrangian i will show how to do that but then the same thing can be done with the Hamiltonian and the Hamiltonian gives it the answer in two lines where the Lagrangian as you will see if one likes long calculations it is a long calculation so now let's do the functional integral over psi and psi bar if these were bosonic degrees of freedom we have a Gaussian the Lagrangian is Gaussian in the field psi we would have a determinant to the power minus a half let's say if it would be one real field because these are drassmann variables you get the opposite sign so what you would have here when you calculate the functional integral you would get the z of a bar remember i am i've i used gauge freedom as much as i could i fixed this down to the integer part the appropriate i kept the sorry the opposite i kept the fractional part of the value of the background field which i cannot modify anymore by gauge transformations it's gauge invariant and i look how the partition functions depends on that and this was the result by integrating d psi d psi bar and we had an exponent of i s over h bar and we just went ahead and did it so this result is the determinant of the linear derivative minus dt plus i a bar okay and if you wanted some infinity in quantum mechanics you got it here because this object here needs regularization it's an infinite product this determinant so you need to regularize it so the question is comes now how to regularize it and now we keep in mind that there were two symmetries which at this stage we treat the same we don't prefer one to the other we would like to maintain that z a bar gauged under the the global part of the local gauge be equal to the original z of a that is under the two pi n over t i want the symmetry and i also want that my regular partition function bz of minus a and the question is can i keep these two things together this was a global symmetry this is part of the global part of a local symmetry a global again in the from the point of view of the topology not in the sense that it's a global transformation and can we do it okay that's a question you will see in the Hamiltonian calculation it's very easy immediately you see what is possible and what is not possible here you need to work so there is one way in which one does the calculation in higher dimensions and that's called pauli-villars what you do is you take the system at hand you add to the system at hand a mass term the mass term by definition obeys no matter global local part of the of the symmetry of the gauge group it's always there and if you look at this object you have changed the theory but then what you do is hello you take the mass to infinity and when you take the mass to infinity you can do that which means physically you have decoupled that particle now we will see in the what will in the short part of the second part of the second part is that you can't always decouple massive particles when there is an anomaly present it's it the revenge will come at you but here there is no anomaly because i took care that all the symmetries the gauge symmetries be preserved so i can just take m to infinity and i can see what happens so now in order to be able to do that one adds one actually needs to add a few pauli-villars i will just i will write to you how this goes on and see if there is some combination which will give you a finite result whatever that finite regularized result is you're sure it's going to be invariant under all all types of gauge transformations and you are at its mercy if it's also invariant under a goes to minus a this in this way you you do it you don't prove that there is no other regularization which would preserve both but on the other hand you respect the old tradition of using pauli-villars which is an indicator that that's the case as i told you when we go to the Hamiltonian formulation you will see immediately what is possible and what is not the the one you won't have to go through the phase space of all possible regularizations okay so what do you have in the case here so the partition function z of alpha which is regularized using pauli-villars that you actually have here two masses you need in order to get the finite result so you take both to infinity and what you have here is determinant of minus t plus i alpha bar multiplied by the determinant of minus tt plus plus m1 which will be taken to infinity to the power alpha one and if this tells you if you wish alpha one here tells you how many species of pauli-villars particles you want to take but this is a mathematical regularization so it doesn't have to be an integer but in principle it would tell you the the number of species because the determinant would go up by the number of species then you would have here determinant of minus dt plus i a bar plus m2 all this to the power alpha 2 now if i add no i i i need the truth is i don't remember now but as far as i remember this is the correct thing and i would have to justify to you why how to do that why to do that this turns out to be finite when 1 plus alpha 1 plus alpha 2 vanishes and when alpha m1 plus alpha 2 m2 is zero so you see that if i have here an i in the conditions it would fall now one can show it in detail and what you get here is this is for you now just one see so there is a limit of m1 goes to infinity the limit of m2 goes to infinity and then you have a product for n going from minus n1 to plus n2 where n1 go to infinity and n2 go to infinity i mean nobody tried to play here double scalings i don't know if there are several ways you would do it at this stage it looks like it doesn't matter how they play maybe there would be a little more structure if one would relate the various ways of going to infinity in any case you get this product off so you have a free field so you know you have two pi these are fermions so when you do functional integrals not we have something going from zero to t so for the gauge field i emphasized it's periodic but if you want to preserve the quantization for the fermions you have to do anti periodic so that's why you have here the n plus a half plus a bar this comes from the first thing and then you have this multiplied by two pi over t n plus a half plus a bar and here is what you asked so maybe i just omitted the i there and here and then here m okay the i remember it's idt which is so again this we have to go through i don't remember so you take this product and this now becomes products of gamma functions so if you define now so now this becomes z pauli delars of a bar becomes to be the limit of mi goes to infinity limits of ni goes to infinity where mi was defined to be mi t over two pi and a which will come in a moment is a bar t over two pi and you get this limit and you have here the gamma function of n two plus three halves plus a divided by the gamma function of minus n one plus a half plus a and then you have here ratios of n two plus three half plus a plus i m one and then you have something similar to the power gamma two should be n two plus three half plus a plus i m two this power gamma two and here you will have the gamma of minus n one plus a half plus a okay as i said nothing is event known here about the double scaling limit if they are needed or not in any case this now goes down and becomes the limit of mi goes to infinity of cosine pi a times your cosine of a plus i m two to the power alpha two and from here you get the conditions which i've written up there and the final once you take them into account what do you get you get one plus exponent of i a bar t what a long way to get such a simple result but that's the way and yes please it was the type or on the work upstairs in the infinite problem there is alpha one and alpha two powers which are yes it was just yes here you mean here yeah it should be to the power of the one yeah okay you you correct it okay yeah at least here they were not so this is the answer that one gets this answer is has two properties it is invariant under the transformations that that remained from a bar so remind you what did a bar i need to do a bar we were checking is this invariant under a bar goes to a bar plus two pi n over t it is invariant under that however it is not invariant in general for a goes to minus a so we saw that that pauli-villard's regularization gives you a regularization which is gauge invariant but doesn't work for a equal a bar you're not invariant under a goes to minus a bar okay by the way if one adheres to the usual one half id by dt both ways this is the anti-symmetry between cyber psi dot and cyber dot psi no not you will see if you will see where the you will see where the ordering and tricks of ordering come in but not not at the Lagrangian level i think this is a standard dirac lagrangian it's i this it's i d slash okay we can discuss playing that game well this one uh i chose that one it has the right it has the symmetries i want it so i can do that and then we can see what happens if we want to change it so one thing is you can ask how do i make this invariant under a goes to minus a okay that's a legitimate question you multiply by the face and you get the same no actually to do that i need to multiply by term which is not gauge invariant i need to add to it a term which is if i would multiply that by an exponent of minus i a bar t this sorry a bar capital t over two this would do the job but this term is not gauge invariant in general under the transformations we have discussed before it will depend of if n is even or odd so i can i can do this okay i can try and this is actually looks like a churn simons term if you wish i add because in this model this is the churn simons like term i add this term i can correct and now have z a equals z minus a but i have lost for a general value of n i have lost the symmetry so in that case i have okay that again it doesn't say that this is the only way to do it maybe there are other ways but this particular way left us with one symmetry and not with the other you can play around but i think this is the easiest way to see it that in order to be able to make it invariant that is turn it into a cosine you had to multiply it by a function which itself is not in general invariant under the global part of the local gauge transformations not the most general ones there are some for which it is let's say half for which it is and half for which it's not so this is the result that one obtained if one starts again given the time i won't now show you that but you can start in a way which ab initio is natural you are guaranteed at each step that you are going to have charge conjugation you don't care about gauge invariance but you do a charge conjugation regularization it's a little different and then you indeed get the result that you would get by multiplying these so you would get that you are as you are guaranteed at every step you will also in the limit we charge in conjugation invariant but you are not invariant under the general a where was it not under this general transformation because of this factor two which enters in okay so this was an example of how two ways one which guarantees gauge invariance which is the pauli villas which now whether the file comes back to me I think the eye is what is special about pauli villas actually you put it there they're not natural fermions because you put the eye that was what pauli that was a trick of pauli and villas I'm sorry the file dropped after a while and on the other hand if you do I didn't show you the natural regularization which obeys charge conjugation you are going to find out that you obey charge conjugation guaranteed but you will miss gauge invariance okay so the first question it's always terrible solomon's choice whom to choose to sacrifice and maybe sacrifice both I mean who cares about them actually or do I choose one or not the other this decision cannot be made immediately it has to be made once a is quantized if a is not quantized it's left to the arbitrary will of you you can keep one you can keep the other there are other ways to regularize it and then you will have neither now let me show you before we make the solomon's decision which in this case is again you choose one and not the other is how would this look in the Hamiltonian formulation now again I should emphasize that this type of anomaly for decades was missed and actually was discovered by Whitten in the mid 80s maybe or something because for decade nobody realized that this extra imposition is and it actually gives you a limit on the number of on the oddness or evenness of neutrinos for example that you are allowed to have in your in your system so it has consequences it's here just the quantum mechanical example but it has more serious consequences okay here tiball I will do what you want I will take ambiguity in ordering into account so we know what the classical Hamiltonian is it's a system which is first time derivative so you know that when you do the Legendre transform you will actually not see the conjugate momentum because it's a first time derivative so it drops and all you are going to see is back the term you had itself so the Hamiltonian of the system is going to be I a bar okay but how in what order so now that I'm going to quantum mechanics I know that I can have this I can write it like this which is a little bit what you want it to do and write it like that you can okay but well it depends as you know on how we decide what how we do that so let's not quarrel about that now even though I agreed it out of all numbers two pies are less important eyes are crucial this I would agree so under the condition that alpha minus beta equals one I can have any type of combination they're all legal they all have the same classical limit so the problem here which was the problem of regularization I had a determinant it's an infinite product I have to define it transmutes into a much simpler problem I just have the issue of ordering there's nothing else there's nothing infinite here but I do have the ambiguity in how I want to order this and the results will depend on how I order this because it multiplies this will multiply the a bar and we saw that the gauge dependence will depend on that so to look at such a Hamiltonian in the general case okay even in the regularization you can already see that in order to have once charge conjugation symmetry for example that would require that beta equals minus a half in order to have the symmetry that you change psi and psi psi dagger and psi and a with minus a now in general you can now go ahead and calculate the partition function and the partition function this operators psi dagger psi the number operator is two eigenvalues zero or one so when writing down the partition function either the fermion state either you have a fermion or you don't and this you just sum over these two options and when you do that that's a very simple thing you're going to get the exponent of if I removed alpha and remain with beta I'm going to get here exponent of minus a bar t times beta times one which I get when the state is empty and then I have when the state is full I would have an exponent of minus a bar t so this is my partition function so here we only ambiguity I had was the ordering and I have to see is there a value of beta for which both symmetries can be preserved we I showed you or told you that when you do this transformations you will get better equal minus a half which tells you already that it's not going to be the case that you can keep both so here unlike the Lagrangian where I really had to prove to you that there is no regularization which exists which can preserve both here you see depends what better you are going to take you can arrange to get both sorry both violated or you can have one of them for example if you took beta equal zero you would have one of them if you took beta equal n you could have the gauge transformation if you if you take beta equal minus a half you got charge conjugation but you did not get the the other symmetry okay so this shows you an example where the Hamiltonian is so much simpler than the Lagrangian and I wonder I don't remember I think Landau suggested to bury the Lagrangian with it with the honors it deserved or the Hamilton one of all the Hamilton one of them was suggested to be buried with the honors it deserves I think it's the Lagrangian but it could be the Hamiltonian in any case you see that in this particular example it's really much easier to look at the Hamiltonian formulation I'm wondering if there are other systems where you have very complicated calculations doing a Lagrangian if things could simplify when you look at the sub-sector just through the Hamiltonian okay now let's make so here we saw that for gauge invariance beta should be n we had a case where n equals zero but other integer n's would work as well and the minus a half doesn't work now what about the quantum mechanics so in the quantum mechanical case the way this manifests itself is what is Gauss's law now Gauss's law on the states is obtained by the derivative with respect to the variable which is not cyclic or sorry which is cyclic which is a not or a bar here so Gauss's law would tell you that when acting on the state let me call it the operator g which is psi dagger psi plus beta so when g acts on a state psi sorry not minus beta I should get zero that's a that's a claim that's Gauss's law now in general we started with the example of beta equal zero and that was okay but if I take beta equal minus a half there is no way I can add fermions and the basis condition if I take beta equal n it means my system wouldn't have one fermion but I would have n and it could work but for beta equal minus a half there is nothing I can do which means my Hilbert space is empty and this is a manifestation in a Hamiltonian formulation this is true in higher dimensions the way that Gauss's law comes the way you see that in the quantum theory you cannot give up the symmetry of global transformations is that you will end with an empty Hilbert space in the Lagrangian formulation you would integrate what I've written which was was beta equals zero this was a partition function if you integrate it over dA which a bar which you should if it's a functional integral it would vanish to zero so there are two ways to see it in a Lagrangian formulation where you have a global anomaly you integrate over the gauge field when it's quantum you get zero so the theory you don't have a partition function you can't define anything it's cleaner to mean the Hamiltonian my Hilbert space is empty so I put too much the system is over constrained too bad so I will have to sacrifice with great regret charge conjugation so the system even though classically it looked like it had charge conjugation the consistent quantum theory cannot have charge conjugation this is the example for the global anomaly now can I give an example of the local anomaly and this I will do in the five minutes which remains I will because I can tell you the consequences so this I cannot do in quantum we don't know how to do in quantum mechanics but in a work which they did again with Adam Schwimmer Ian Halliday and Mike Chanowitz we tried to take a model which can be solved exactly even though it has an anomaly and that is the Schwinger model in two dimensions just go ahead and solve it and see what it gives now the problem the open problem I have in mind which I think has not yet been solved and I wish it I would have solved it but if not someone else solve it is do non-critical string theory but don't do the trick of making it critical by adding the appropriate coupling leave it non-critical as it is and quantize it and solve it and see what theory this gives because all the way up to now where people did what they call non-critical string series but actually completing canceling the anomaly by adding a piece to the system and then solving it which is interesting gives very nice results but it doesn't face the problem of what happens if you disregard the rules the authority and you just do it and bad things should happen so let's see how how in general one would look and then look at the two-dimensional case so what is the general structure that you would expect so just like you can do with the Wheeler-DeWitt equation you can do or was done much earlier counting degrees of freedom in the cage of gate series so you know you have a gauge object a mu which has d degrees of freedom you can go and forget now the global structure issues which I discussed to a gauge or a not equal zero so the problem became d minus one you write down a Hamiltonian but you find that the generator of gauge transformations which keep a not equal zero which means the time independent gauge transformations that generator is just Gauss's law and let's say the Hamiltonian is e squared plus b squared g for the free case would just be del dot e in general would also have minus some charge density e and e and b don't commute because e and a conjugate variables and b is curl of a but nevertheless when you do the calculation because this is the generator of the residual gauge transformations of the system this will vanish you do the calculation and you find that h and g vanish which means that you have the super selection rule of freedom to diagonalize the two simultaneously and the way you get the correct number of degrees of freedom you say okay I diagonalize h and I look at all eigen functions of h however I pick only those eigen values oh sorry only those eigen functions which are annihilated let me call it Gauss's law is better by Gauss's law this will be the sector where for example Lorentz invariance will be resuscitated because here I made an explicit breaking of Lorentz invariance but because the theory is gauge invariant then if I work at the subset of space of states which are annihilated by g I look at the spectrum it is a dispersion a relativistic dispersion relation I have resuscitated it and the number of degrees of freedom went to what it should be d minus two and you can do the same with the willard-the-witt equation and go from in the gravity case from the number of degrees of freedom embodied in g mu nu which is d times d plus one over two reduce it by the same type of operations go to the synchronous gauge impose fourth or d type of Gauss's laws and you will get the same counting so this works very nice is there any loophole in the argument I gave yes there is one loophole I have assumed that I can simultaneously diagonalize all the g's which means I've assumed the g of x and g of y vanish for every x and y if they don't then I can't I stop there because I cannot diagonalize them at the same time and the point of anomalies which appear not the topological ones I've discussed but the simpler ones which were known for many decades the ones which are related to gauge transformations which are near the identity is that this doesn't vanish when you have theories where this doesn't vanish you have an anomaly now where does it come from it doesn't come from the part del dot e it comes from the part rho now it's true that in non-Abelian gauge theories like in the Willow-DeWitt equation you don't get the constraints commute but you get on this side another constraint so on the subspace where the constraints vanish it does commute and therefore it works well so if you have a pure gauge theory let's say just qcd without quarks there is no problem there are indeed there is a i and there is a j and there will be here a gk and there will be some fi jk's whatever appropriate structure constants but on the subspace where the g vanishes they commute so there's no problem however the row part this is where the chiral matter comes in doesn't have that property there you have derivatives depending on what dimensions you are but you have delta functions or derivatives of delta functions depending again how many derivatives appear depend on then your dimensionality and in particular in two dimensions where you can solve the problem when you add chiral matter chiral matter means that er is different than e left you're going to get here some let's say something which is er squared minus el squared multiplied by a derivative of a delta function so that will be a situation where you have an anomaly so in general a lesson to take away independent of the details of the model the way that the anomaly manifests in a Hamiltonian formulation if it is a local part of the local gauge group it's the fact that Gauss's laws don't commute there is a c number of course not not commute there is a c number on the right hand side on the other hand when you have a global part of the local gauge group as a with an anomaly the Hilbert space is empty you cannot you have over constrained your system you will find no state which obeys the laws so you have to have a theory where this anomaly is cancelled and you cancel it in string theory when we have the conformal anomaly we we speak very roughly we speak about 26 dimensions we keep about 10 dimensions we can speak about other ways of of doing that that's the way we cancel the anomaly in string theory the way you cancel the anomaly in su5 is by having a 5 bar under 10 the way you cancel it in with gravitational anomalies is by having relations between leptons and quarks there are many and crucial ways in model building where the anomalies come in so to conclude because i didn't this would have been lecture number five which would not happen so if i would solve this model exactly it can be done in two dimensions i would solve it in the gauge a naught equals zero so the first thing is should i expect a Lorentz invariant dispersion relation i see no reason if the theory has an anomaly if e left is different than er there is no reason why the gauge a naught equals zero would be Lorentz invariants i picked a direction and the answer is not Lorentz invariant you get an explicit dispersion relation you have in it e related to k to the force n to k squared so it's not a relativistic dispersion relation but you can solve the theory exactly then just like here you had both of you suggested to multiply by this there is a way here to add a west zoomino term to cancel the anomaly but adding this west zoomino term is always equivalent to having ab initio another set of fermions which cancel the anomalies and integrating them out and the integrating out gives the west zoomino term there there is an ambiguity and here i'm talking really to the super experts there there is an ambiguity of gauge invariant terms the west zoomino term cancelling the anomaly is unique but in the integral there can also be gauge invariant terms what do they represent there are infinite number of ways to cancel an anomaly if e right is different than e left i can have an infinite number of distribution of charges which i add to my theory each of which will end up exactly cancelling the anomaly okay for example let's say that er minus e left was one okay so i need to add an e left which is one but i can add e left which is three and then i can add an e right which is two or i can add two e rights which are one there are an infinite number of ways to cancel the anomaly by adding extra stuff which is not anomalous okay you just pick up the single part which is and that's what the gauge invariant terms reflect and it's an interesting thing to know and then the final surprise i had on the subject is they are in conformal field series what are known cos as coset models coset models are models which are very interesting because in in string theory they can replace geometry by algebra and you can do exact calculations using the algebra instead of living with a semi-classical picture which is geometry so they have had and i think will have and are having lots of uses now if you take a model which has a west zomino witten model g it has a subgroup h and you write a suga-wara construction as a different of the g's and the hs this model itself is not a wzw model but still is conformal and its conformal charge is the difference of these two and we have shown in the past with koukout bar dachi that this model can be written as a gauged as a strong coupling limit of a gauged west zomino witten model and you can see exactly how this model constructs now the funny part is that if we have a group g over h i for cosets so that's a strong coupling limit that's a deep infrared of the problem we found that actually and you can gauge i thought for years that i cannot gauge an anomalous symmetry in this cases you can gauge the anomalous symmetry and actually you find the relativistic dispersion relation which was to me a surprise so again these models you take by writing down some cft gauging that cft but not adding a term f squared the term f squared in two dimensions has conformal weight so if conformal invariance is not spontaneously broken it will not be created and indeed it's not created but you could think also of the model with the f squared and then just take the coupling which is dimensionful like the mass to infinity it's the same consequence and in that case as i told you somehow this anomalous schwinger model remains with a relativistic spectrum which to me is a little bit of a puzzle how that how that could have happened so i think by giving puzzles which are concrete and in the beginning telling you about an area which i find really very interesting and i believe will develop well there will be interesting results but in which things are not as well defined as we would like and one needs much better definitions which is a quantum information theory related to gauge series and to gravity i hope i get i gave you some feel of that and i hope by this i finished my duties here on that and the the next day louis michel professor would have to do the same