So welcome everyone to the belated spring school on super string theory and related topics. My name is Bobby Acharya I'm a staff member here at the ICTP in the high energy group. I'm here with my colleagues Pavel Kutrov also from ICTP and other fellow organizer Francesco Bernini from CISA neighboring Institute. To start the meeting, I'm going to give a brief introduction to the ICTP, which has been hosting this school, essentially since the 1980s. The ICTP is the International Center for theoretical physics located here in Trieste, Italy, on the northern end of the Adriatic coast. So the Institute was founded in 1964 by Abdus Salam. And part of its aim is to enhance international cooperation through science. In addition to a world class research program, the ICTP has a mission for developing and fostering science in the in developing countries. We're governed by three organizations of the Italian government, UNESCO and the International Atomic Energy Agency. Most of our budget actually comes from Italy with a significant contribution from the Atomic Energy Agency. And we are administered by UNESCO and it's through UNESCO that we implement our mission for fostering science. I'll say a few words about the structure of the ICTP. Broadly speaking, we can divide it into three legs, research, education and outreach. So I'll go through these very briefly in turn. So the research at the ICTP falls into six main groups. So there's our group, which is the high energy cosmology and astroparticle physics group. We cover, broadly speaking, most of the areas in fundamental theoretical physics, also a bit of experimental high energy physics. We have a condensed matter and statistical physics group. So historically the high energy and condensed matter groups were the first two groups to be formed. We also have a mathematics group, science, technology and innovation unit. We have research studies, earth systems physics, which could be things like climate modeling or earthquake models and many other related topics. More recently, we started a group on quantitative life sciences. And we're also expanding in the areas of sustainable energy and high performance computing. So some significant contributions from the ICTP over the years. That was the Nobel Prize of Abdus Salam in 1979 for the standard model. We shared in the Nobel Peace Prize. Filippo Georgie, he's the head of our earth systems physics group, was one of the co-chairs of the IPCC that received the Nobel Prize for climate change. We have, as I mentioned, an experimental physics group that works on the Atlas experiment that contributed to the Higgs discovery. One of our high energy physics members, Alexei Smirnoff, was recently retired, has contributed fundamental research to the theory of neutrino oscillations. And our collaborators in our partner institute in Brazil worked on the LIGO experiment which received the Nobel Prize for gravitational wave detection. So education at ICTP, there's various decrees and diplomas that we coordinate. So we're not a university, so we don't give degree certificates, but we have various programs running jointly with collaborating institutes, most of whom are in Trieste. That's the University of Trieste, CESA, but we also have a master's degree in complex systems which involves universities further than Trieste. Our most successful educational program is our postgraduate diploma program, which takes students from developing countries in which there's no comparable diploma program. So at the moment we have a diploma in high energy physics, mathematics, condensed matter, and earth systems physics, so reflecting our research areas. And in each program we take roughly 10 students in each for one year, for a one year intense program. So we've had more than a thousand graduates since the program started and as you can see many of them go on to research careers. We organize, help organize and organize various training activities. So that's something like typically around 60 or so conferences and workshops each year, and that's not just at the ICTP, but we also organize activities in developing countries. We have a significant number of visitors from all over the world every year. And in addition to the sort of main ICTP activities we also support scientists through additional hosted activities. This is just to give you an idea of where scientists that come to the ICTP come from. And these are statistics that go back around 50 years or so. And we've had visitors from literally 188 countries making ICTP one of the most international science institutes globally. And we pride ourselves on supporting scientists throughout their entire careers with through the programs that I mentioned, which are starting at those masters level through to PhDs. But then we follow our alumni through their careers. Some of them become what are called associates of the ICTP. Once they go back to their home countries and join or develop research groups there. And there's a number of other programs that help continue to support our alumni and associates throughout their careers. And once they become faculty members in their home countries they send students back to us and so we have a nice cycle of career trajectory. Now come to the outreach part. And so with the ICTP has had a long tradition of scientific capacity building in developing countries. We have an office of external activities that supports and helps coordinate networks, meetings, other types of scientific activities all over the world. In the last decade or so we've developed partner institutes, most notably in Brazil, China, Mexico and Rwanda. So for example our institute, our partner institute in Brazil, the South American Institute of Fundamental Research was one of the co-hosts of the strings conference earlier this year which I'm sure some of you will have attended. Our partner institute in Rwanda has a master's, a two year physics master's program, mostly aimed at students from Africa. And the institute in Rwanda, although albeit smaller and junior at the moment compared to ICTP 3S there, sort of is mirroring the model of the sort of structure that we have here that I've outlined here. Around 2013 we started the Physics Without Frontiers program to engage undergraduates and master's students in particular in areas of conflict or rather difficult areas of the world to be studying science. We also have a focus in Physics Without Frontiers on issues like gender in science. And in recent years we've had projects in Afghanistan, Iraq, Gaza, Syria just to give you an idea. Then we have the Science Dissemination Unit which is part of the Science and Innovation Research Department and they develop all kinds of innovative ways of disseminating science and engaging with the public. And you can read more about more of these things on our website. And this plot is supposed to show you the sort of reach of our scientific outreach just last year. So these are all, every dot represents a different activity and meeting or outreach event that took place last year. We also do more traditional sort of outreach and since 2012 we've had the Salam Distinguished Lecture Series featuring physicists giving a series of lectures typically in January. And some of our, this is just to highlight some of our former alumni, say two of whom started off life as diploma students and now have distinguished positions elsewhere in their home. In the case of Narayan, he has a research group in Nepal that we continue to support and some of you will recognize Freddie over there. So this is just a testimonial from Stephen Hawking some years ago testifying the impact that the ICTP has had over the years. And there's a similar one from David Gross. And yeah, you can read more details about the ICTP website. So now we are ready to sort of open the school. The school is not only supported by funds from the ICTP but also by from the INFN, the Italian sort of National Institute for Nuclear Physics. So we're grateful to them for their support. We have, as you will have seen from the program, three lecture courses taking place. And from Kazuya, Irene and Alex. And I'm now going to let my co-organizer, one of my co-organizers, Pavel say a few words about the format and also the Slack channel. Pavel. Okay, so yeah, first, since we're short on time, so let me just briefly advertise again that we have created this Slack workspace. So you should all in principle have received a link to it, how to join, but I will also later a bit later put the link in the chat in Zoom here. The idea that you can ask there whenever you have time, like outside of the dedicated Zoom sessions, you can ask the questions to speakers or to other participants and people can reply to these questions when they have time, not necessarily immediately. And also, yeah, so also you can find some lecture notes there made by our lecturers, which can help you to follow the lectures. And so there are three dedicated channels where you can ask questions related to those topics of the courses and also separate channel where you can ask questions, not necessarily any other questions. Yeah, that's basically it. So, Bobby, do you want to tell yourself how the questions during the Zoom will be handled? Yeah, so I think if you've got questions, just raise your hand on the Zoom. The question sessions are not going to be recorded. So feel free to ask whatever you want, we really encourage discussion and questions as much as possible.  No, yeah, that's it. Yeah, so yeah, after each lecture there will be a dedicated kind of discussion session and it will be not recorded and neither it will be streamed on YouTube, so you can feel free to ask questions in a formal manner. Yeah, I think that's it. But okay. So without further ado, then I think we can start. Kazuya, are you able to share your screen? Okay, so let me share. So yeah, it's a pleasure to invite Kazuya Yonekora from Tohoku University to give his first lecture on non-perturbative fermion anomalies. Okay, so thank you very much for the introduction. And first of all, I'd like to thank the organizers for putting together this school in this difficult time. So my lecture is going to be about non-perturbative fermion anomalies. By the way, I'm Kazuya Yonekora from Tohoku University. So I'm asked to talk about anomalies and symmetries and so on. And I focus on just fermions because they are very basic and fermions appear in many places in physics. It appears in string theory and high energy physics and also condensed metaphysics. But I should perhaps also mention that there are many other developments in the area of anomalies and symmetries. So there are some generalized symmetries such as higher form symmetries and non-inventive symmetries and so on. So this area is developing very fast. But I try to be elementary as much as possible. But I needed to assume some background knowledge. So I assume that you know some basics of QFT. And in particular, I assume that you have learned about perturbed anomalies in the standard textbooks. So for example, perturbed anomaly is described by this kind of triangle diagrams in textbooks. I don't use any technical details of those perturbed anomalies. But at least I assume that you know the concept of anomalies in quantum fields theory. And then I will talk about non-perturbed version of anomalies, which is called global anomalies. And also another thing I assume is some basics of geometry and topology. I use some fiber bundles and their connections. So these are the language to describe gauge fields. And also I assume that you know some differential forms and manifolds. So these are the basic backgrounds. And the difference of my lectures is this paper. This is my paper with Edward Wittman. So if you want to know more details of my lecture, then please consult this paper. So it contains much more details of my lectures. Okay, so in my lectures, I talk about anomalies of symmetries. And there are two types of symmetries. The gauge symmetry and global symmetries. So let me comment on these symmetries. So first gauge symmetries. So I think you have learned, for example, part of the anomaly cancellation in the standard model. So this is a very basic textbook material. So in the standard model, the anomalies are canceled very miraculously. So if you have never checked this anomaly cancellation by yourself, then I recommend to do it. This exercise. So this is a beautiful cancellation. And also anomaly cancellation was one of the important historical developments in the study of string theory. So in type 2B string theory, there is again some miraculous cancellation of anomalies of fermions and also some chiral performance fields. And in heterotic string theories and type 1 string theories, there is a green-schwarz mechanism by which we can cancel part of the anomalies. So in any consistent theory, part of the anomalies must be canceled. Any question? Maybe no? So okay. So for the gauge symmetries to be consistent, part of the anomalies must be canceled. So in principle, we also have to check that there is no anomaly at the nonpartavatable level. So these textbooks treat only part of anomalies. At the nonpartavatable level, it is more nontrivial to check the anomaly cancellation. And it is even difficult in the standard model. So in the case of standard model, it is possible to show the nonpartavatable anomaly cancellation, but it's not so easy. It requires some mathematical machinery to show the anomaly cancellation in the standard model. And as far as I know, so there is no complete answer in string theory. So no one has demonstrated complete nonpartavatable anomaly cancellation in string theories. There are some developments in this direction, but still there are many things to be done in string theory. However, maybe you ask this question. So we are very confident that string theory is consistent. So there are lots of studies on string theories, and all these studies show that string theory is really a consistent theory. So we strongly believe that string theories exist. So why do we care anomaly cancellation? So anomalies should be obviously cancelled if the string theory is consistent. And we believe that the string theory is consistent. One of my answers to that question is the following. The study of anomalies in string theory or maybe also in quantum field theory is very much related to non-trivial topological structures. And I don't have time to explain the details of this point, but let me mention a few facts. So as an example, let's consider direct quantization of fluxes. So string theory contains many P-form fields, such as Ramon-Ramon fields. And so in text books, it is argued that they must satisfy some direct quantization condition. So direct quantization condition over P-form field is given by this kind of equation. So this equation means that if we integrate a P-form field over some P-cycles, then that value must take values in integers up to some constant. So this is a standard statement for the direct quantization condition. And it is often assumed. However, in string theory, this is not always the case. And sometimes this condition is modified by additional term. So this integration of the flux does not take values in integers, but it takes values in integers plus some constant, which I denote by A. And it turns out that this contribution is related to anomalies of the world volume theory of D-blends. So I mean, so direct quantization condition, this one, is usually discussed by requiring the consistency of the coupling of the D-blends to Ramon-Ramon fields, for example. But D-blends contain some world volume degrees of freedom. And they sometimes have some anomalies. And then the direct quantization condition is modified in this way. So this means that if we just naively impose this direct quantization condition, the theory is inconsistent. So I believe that the string theory itself is consistent theory. But it can happen that some compactification of string theories can be inconsistent because of the anomaly. So we have to be careful about the topologies or compactification of string theory. And these topologies are determined by this existence of anomalies. So by this reason, even if we believe that string theory is consistent, we still have to study anomalies of degrees of freedom, which appears in string theory. So for example, I believe there is a question from Igor, maybe Igor can unmute himself. Hello, thank you. Is correct to say that the anomaly part is computing the cohomology class of the manifold where the field lives? In a sense, yes. I don't explain it in my lecture, but it turns out that these anomalies are described by what is called generalized cohomology. So in that sense, the answer is yes, but it needs generalized concept of cohomology, not just the ordinary cohomology. I see. Thank you. Okay, thank you. Okay, so this is one of the motivations to study anomalies in string theory. And of course, if we consider some field theory, which is beyond the standard model, then you have to check that the theory has no anomalies. So even if we are just interested in quantum field theory, rather than string theory, we still need to understand anomalies as much as possible if we are interested in some new physics. Okay, next let me talk about global symmetries. So in this case, the anomalies of global symmetries are called two-fifth anomalies. So this two-root anomaly is a anomaly of global symmetries. And the existence of anomalies of global symmetries does not imply any inconsistency, but they are very helpful. They are very useful to study the dynamics of quantum field theory. So for example, let's consider this kind of RG flow, denormalization group flow, from some UV theory to IR theory. Then, so originally two-fifth argued that the two-foot anomaly is conserved under the denormalization group flows. So if this UV theory has some global symmetry, and then if that global symmetry has some anomaly, then that anomaly must be conserved in this IR theory. So the anomaly of the UV theory must be the same as the anomaly of the IR theory. So this is a very powerful constraint on strong dynamics, for example. So for example, in the original case of two-fifth, this UV theory is the QCD in the massless scope limit. So we consider massless QCD. And then in the IR, the gauge dynamics become strongly coupled. And so because of the stronger coupling, it is very difficult to compute what happens in the IR dynamics. So however, even though we cannot solve the IR dynamics completely, we know that the anomaly must be conserved in this RG flow. And by using that constraint, two-fifth argued, in the case of QCD, that the chiral symmetry must be spontaneously broken for some color numbers and flavor numbers. So in the massless limit of QCD, there is an SUNF times SNF chiral symmetry, and this symmetry has some anomalies. So they have perturbative anomalies. And by using this chiral symmetry, by using the anomaly of this chiral symmetry, two-fifth argued that these symmetries must be spontaneously broken in the low energy limit if the confinement happens. So in that way, so this two-fifth anomaly is very helpful to study the dynamics of quantum field theory. Okay, and also, I should also mention about condensed matter physics. In the condensed matter physics, the motivation comes from what is called topological phases of matter. So these topological phases of matter includes, for example, integer quantum hole effect, and topological insulators, and so on. And in the language of high energy physics, we can understand these topological phases of matter in terms of anomalies in the field theory limit. So in this condensed matter context, the topological phase corresponds to the anomaly of some degrees of freedom which appear on the boundary. So the situation is something like this. So I'm sorry that this figure is not so beautiful. My handwriting is very bad. So here, by this figure, I mean some material in condensed matter. So this material has a bulk, and it also has some boundary. If the bulk is some topological phase, then there is some degrees of freedom, some massless degrees of freedom, or some topological degrees of freedom on this boundary. So they have anomalies, and the anomalies of these boundary degrees of freedom is completely determined by the property of this bulk topological phase. So that is the basic relation. For example, in the case of integer hole effect, in the bulk topological phase, effectively, we have some Chan-Simon's term in the bulk, and then on the boundary, we have some chiral fermion. And then by the, what is called the anomaly inflow mechanism. So the anomaly of this chiral fermion is canceled by the bulk Chan-Simon's term. So in this way, so bulk topological phase and anomalies are very much related, and it is believed that there is some one-to-one correspondence. So this picture is motivated from condensed matter physics, but it turned out that this is also essential for the nonpart of the formulation of the concept of anomaly itself. This means that even if you are not interested in condensed matter physics, we still need this picture for the understanding, for, in particular, for the nonpart of the understanding of anomalies. And so this picture is very important. For example, if we try to see, so I mentioned that there is some modification of direct condensation condition by some anomaly. And this is shown by some, I mean, some introducing some, this kind of bulk. And then we can show the shift was a direct condensation condition. So this picture is very important. Okay, so this is the condensed matter physics motivation and also this is important for anomaly itself. Okay, and finally, let me make one remark about this lecture. In this lecture, I will not distinguish gauge and global symmetries. So I just treat gauge fields as a background field. So in the case of gauge symmetry, we are interested in pass integral like this. So we integrate over gauge fields. So this capital A is a gauge field. And we also integrate over fermions. So this psi is a fermion. So we are interested in this kind of pass integral in gauge theories. We can first perform the pass integral over fermions. And then let me denote the result by this deep side. So this is the partition function of fermions in this background field A. So in this definition, I only perform the pass integral over this fermion. And then for gauge theories, we also perform pass integral over the gauge field. But in my lecture, I only consider this quantity, the partition function of fermions in the background field. So gauge field is always treated as a background field. And in the case of global symmetries, so it's very helpful to introduce some background fields corresponding to some global symmetries. So I mean, so in the case in perturbation theory, we can introduce such a background source field. So for example, if we have some continuous global symmetries, then there is some corresponding current operators. And we can couple these current operators to some background fields, background source fields. And then we define generating, sorry, we define generating functional. So this contains the information of correlation functions of the current operators. So by taking the functional derivatives of this generating functional, we can obtain correlation functions of current operators. So that is explained in standard text books. We can also consider these background fields feature topologically non-trivial. Then this partition function contains more information. And those information is very helpful in the study of quantum field theory. So even for global symmetries, I introduce some background gauge field. So then I can discuss gauge symmetry and global symmetry simultaneously. So I don't distinguish these two cases. Okay, so any questions so far? Hello. Yes. Hi, just to clarify, sir, needed whenever you mention about this boundary of this topological, whatever topological phase topological object that picture. This is some kind of real boundary of some kind of material, right? Yes. Okay. Okay, thanks. Thanks a lot. Yeah, yeah. On this matter physics, we really have some, we really have some material. And this is the area boundary of that material. Okay. Yes. Thanks. There might be another question from Igor. Igor, you cannot just unmute yourself. Oh, hello again. We began under the assumption that string theory is a mathematically consistent theory. But we also know that it's a quantum theory of gravity. And there are many evidences that consistent theory of quantum gravity cannot have global symmetries. So how can we identify gauge symmetries and global symmetries? In the context of mean theory, I always consider gauge symmetry. So, yes, so there's no global symmetry in string theory. So we believe that. So, so here I just mean gauge symmetry. For example, type one string theory contains gauge group SO32. Yes. Then we must check that the anomaly of this gauge symmetry group must be cancelled. Okay. Yeah, that was shown by anomaly cancellation in type one string theory, or catalytic string theory is shown by Green-Schwarz at the part of the level. But as far as I know, no one knows the nonpart of anomaly cancellation. Okay. Thank you. Yes. Okay. So, if there is no other questions, then let me begin the technical part of my lecture. So first, I want to talk about the relation between topology and anomaly. So I only discussed some elementary examples in this section. And so I assume that you know part of the anomalies. So I only focus on global anomalies. So global anomaly means some nonpart of the anomaly. Even if there is no part of the anomaly. And then so we can consider various kind of symmetries and we can in particular consider, for example, discrete symmetries. So discrete symmetries does not have any part of the anomalies, but they can have some global anomalies. So in some cases, those anomalies of discrete symmetries are very helpful. And also, we must be careful about topologies for symmetry groups. So usually in some basic textbooks, so they only care about the algebra of symmetries rather than the D groups. But in the study of global anomalies, the global structure of symmetry groups is very important. And I'd like to demonstrate this point by some elementary example. So I consider some topology of symmetry group and the basic example is the following. So we can consider SU2 symmetry group. And we can also consider SO3 symmetry group and O3 symmetry group. So often in the physics literature, these groups are not carefully distinguished. But for the study of global anomalies, we must distinguish these three groups. So they have different global structures. And the algebras are the same. So the algebras are just described by three generators. So we are very familiar with them in the study of angular momentum. So they have three generators and they satisfy some algebra like this. Yeah, something like this. So this is a very standard algebra of angular momentum. So because of this reason, because they have the same real algebra, they have the same part of the anomalies. But at the nonpart of the level, we must distinguish them. So first let me talk about the difference. So first let me discuss the difference between SU2 and SO3. So there are two related facts about these groups. So first, the allowed representation of these groups are different. In SU2, we can have spin odd representation. Sorry, spin. Half integer spins are possible. In SU2, in SO3, we only have integer spin representation. I will make more comments later on this point. But first let me mention another fact about the difference between SU2 and SO3. So another fact is that some fiber bundles are possible only in the case of SO3 and not in SU2. I will give examples soon. So first let me explain this first point. So these two points are related to each other. And first let me mention this point. So for example, we can take U1 subgroup of SO3. So let's say that this is just a rotation around the z-axis. So this U1 inside SO3 may be realized by this kind of group element. So we take this generator, J3, and then we take the exponential of this J3. And then this represents the rotation around the z-axis by the angle phi. So we can take this U1 subgroup. Then the exponential of 2 pi i J3 is, so this is identically equal to 1 in SO3. So this means that 2 pi rotation is just identity. So this is just the usual fact in three dimensions. So this means that this J3 cannot take values in half integers. It cannot be one half plus integers. So in SO3, half integer representations are excluded and only integer representations are possible in SO3. So this is the basic difference between SO3 and SU2. And next let me consider some fiber bundle. In the case of SO3, we can consider the following non-trivial fiber bundle on SU2. On SU2, I take some polar coordinates theta and phi, just the standard polar coordinates. And then I take the gauge field A, the gauge field of SO3 to be like this one. So maybe I should write a figure. So this is SU2. So SU2 has some north pole and south pole and says some equator. And then on the northern hemisphere, I take the gauge field, which is given by this formula. So I take the gauge field like this in the region theta is less than 2 pi. And in the southern hemisphere, I take the gauge field to be like this. Okay, so if you are familiar with some Dirac monopole configuration, then I'm just taking Dirac monopole configuration by taking the... So I take the U1 subgroup inside of this SO3. So this J3 is the generator of the U1 subgroup. And then in this U1 subgroup, this configuration is just a configuration of Dirac monopole. And this configuration is consistent because we can glue the gauge field in the northern hemisphere and southern hemisphere in the following way. The gauge field in the southern hemisphere is given by gauge transformation of the gauge field in the northern hemisphere. So these two gauge fields are related in this way. And here, the gauge transformation parameter, this G, is given by exponential of i, j, 3, 5. Okay. So we can glue these two gauge fields by this transition function. And so this is consistent for SO3. Well, in other words, this function G, this gauge transformation function G, is a single-valued SO3 group. So in the case of SO3, this function is single-valued. So this gauge transformation is consistent. But in the case of SU2, this is not consistent. In the case of SU2, this J3 can take half integer values. Then this function G is not single-valued. So this configuration is not possible in the case of SU2. So this is only possible in SO3. So in this way, so which representations are possible and which fiber bundles are possible are closely related to each other. So for the purpose of my talk, SO3 has more interesting topology. In fact, we can see this function G as a map from S1. So this S1 is the equator here. So we have S1 here. So this G is a function from S1 to SO3. And the topological characterization of this function is given by the homotopy group, the first homotopy group. So this G gives... So topologically, this map is classified by the fundamental group. So the first homotopy group, pi1 of SO3. So this pi1 of SO3 is known to be G2. So this example is known to be a non-trivial element of this G2. So this transition function has a non-trivial topology. And that is the reason that this bundle constructed here is topologically non-trivial. I don't explain the details here, but anyway, so this is a kind of well-known fact in mathematics. Sorry, Katsuya. There is a question in chat. Yes. It asks, isn't G a map from S1 to the group U1? Yes, so... Yes, that's right. So this map factors like this. So this is a map from S1 to U1. So this U1 is a subgroup inside SO3. And then this... So because this U1 is a subgroup of SO3, we can further map this to SO3. And because we are interested in this SO3, I just wrote the directory's map from this S1 to SO3. Ah, okay. But maybe I should mention that, yeah. So in the case of U1, this homotopy group of U1 is given by the set of integers. So this means that the non-trivial bundles on this S2, non-trivial bundles on S2 are classified by integers. And these integers are precisely the magnetic flux of Dirac monopoly configuration. So this integer specifies the magnetic flux. And so this fact about SO3 means that we can cut... Sorry, we can have a kind of magnetic flux for this SO3 gauge group. But this magnetic flux is conserved. It is topologically stable, only more to G2. So if we combine two magnetic fluxes of SO3, then the combined flux is trivial. So that is the meaning of this homotopy group. Kazuya. Yes. We should think about entering the discussion session soon. Ah, okay. Yes, okay. Maybe I stop here. Let's go to the discussion session. Thank you very much for this very nice opening lecture. So we're going to stop the recording here and enter the discussion session. But let's thank Kazuya before we do that. Harvard and Madrid, it seems. Yeah, please take it away, Irena. Okay, thank you, Bobby. And thank you for the invitations to give these lectures here. And thank you to all the students for coming here to listen. So, yeah, my name is Irena Valenzuela, and I'm going to tell you about the swan plan during this week. So I'm not expecting you to know about it. I mean, I will try to be as pedagogical as possible and start from the beginning. So please, the most important thing is that if you have any question at any time, don't hesitate to interrupt me. Okay, you can just stop, ask any question.  That's not the point we are here to learn. Okay. So, okay, so these are here some references that I put out the swan plan. I said that my goal is try to be as slow as possible. So of course, we will not be able to cover everything that has been done. So if you want to know more, these are nice references. I will mainly follow my this. Yeah, this reference here, which comes from another course that I gave last year. So also the things that I'm talking here, I mean, if you don't understand something that you want to know more details, you can just check this paper and you will see everything explaining a bit more detail, but it's the same same ideas and same files. Okay. And also, yeah, as I said, if you want to know more or especially know the references, you can check this review, on Ferran party or the another desi lectures from Google. Okay, so let's start. Let's try to define what seems to zoom. Okay. Okay, so the idea of the swan plan. Okay, we'll see that we are going to break with this intuition of effective theories. Okay, so let me start putting us all in the same page and defining what are typically effective theories. effective field theory. So effective field theory is something that we use all the time in physics has been very useful to progress in high energy physics because we can provide a description of the theory that is valid up to some energy scale which we call the cutoff and describes the physics around that scale without worrying too much about what happens in the UV at higher energies. Now if we have some UV theory, some theory that is valid in the UV we can always integrate out the degrees of freedom up to some high energy scale up to some cutoff and obtain some infrared description. So this is integrating out and here we get, as you have probably learned many times, you get some randomizable part plus an infinite tower of non-randomizable operators that are valid up to the cutoff of the theory which I call lambda. Okay this is very nice. The problem or the thing is that it's not guaranteed whether the reverse process can always be done. So that given some effective field theory that you construct from a bottom-up perspective following the rules of quantum field theory, like for example you require that it's unitary or anomaly free and so on, it's not guaranteed that this will have a UV completion so that you can go back. Okay so this is something that already happens in QFT but in the presence of gravity it's even less clear. Okay whether any EFT can be UV completed in quantum gravity and the fact that this is not always possible is what is underlying the swampland program that we will discuss. Okay so not the UV of t can be UV completed in a quantum gravity field. Okay so it's not compatible with a quantum gravity description at higher energies and the goal of the swampland is to understand what are the constraints that effective field theory must satisfy to allow for this UV completion. Okay so of course there are some things that we know like anomalies right you need to satisfy gravitational anomalies now to be able to complete it in quantum gravity but the point what we are learning from state theory and also thinking of black holes is that this is not enough like you need to require more things to guarantee that you reach some unitary quantum gravity description. Okay any question about this? Okay so this is the typical picture that we put about the swampland in the sense that if you imagine here like the horizontal axis all effective field theories that you could construct following the rules of quantum field theory and not all of them will have a UV completion as we are saying only a subset which we call the landscape so this is consistent with quantum gravity it can be UV completed and this is not consistent and the theories that are not consistent in the sense that they don't satisfy these additional constraints we say that they belong to the swampland okay and the name and the name was chosen precisely in opposition to the landscape which are the theories that the effective theory that can arise as low energy descriptions from state theory for example. So one of the most important things of how to distinguish a swampland constraint is that it should disappear if the mass Planck if the Planck mass goes to infinity right because it's a gravitational thing so if you decouple gravity the constraints should completely disappear and that's why like in this cone if the UV completion is infinitely far away like everything is allowed okay and the constraints become stronger and stronger as you increase the energies typically okay so for example you have some effective field theory that lives here yeah it's here in the landscape maybe at this energy scale is fine it's an effective theory that makes sense but if you want this effective theory to be valid up to a higher energy scale there is a moment in which it's going to found to find like this boundary of the cone this is a bit qualitative but to get you an intuition I mean at some point we'll probably hit this boundary which means that the effective theory will not be valid anymore unless you modify it and sometimes these swampland constraints are telling you what is the scale at which this happens okay so what is the cutoff at which you have to modify that effective theory so that it's still you know within the the cone that is compatible with quantum gravity so that the cutoff in which new fishes have to appear and you have to modify the theory seems to be below the plank mass okay so this is one of the the lessons that we will learn from the swampland program that this cutoff this quantum gravity cutoff which quantum gravity is telling you that you have to modify the theory is is below the plank mass which is the naive cutoff that we put for quantum gravity okay as I said if there is any question just interrupt whatever comes to your mind okay so if this is true and if there are these new constraints that we need to satisfy I think it's amazing news especially for phenomenology because basically we are giving new constraints that effective theory can satisfy so they can be used as new guiding tools to progress in high energy physics and we are they are capturing the quantum gravity imprint at low energies okay so we are saying that quantum gravity can matter at low energies even if the plank scale naively is very high and seems that gravity is very weak and quantum gravitational effects are not important still to guarantee a quantum gravity consistency of higher energy there are no trivial constraints and this can also solve like bring new insights into the natural issues that we observe because they are kind of breaking this expectation of separation of the scales between the uv physics and the infrared okay so you might wonder like how is this possible that um these constraints are important no in the infrared like we always learn that when you integrate out all the heavy fields in the end maybe you generalize some non-random noise operators but this is going to be suppressed so the effects will not be so important so why it's so important to consider all these corrections well the point is that in the presence of gravity this intuition can fail okay so we can really have a mixing between the uv and the ir physics so this Gulsonian effective theory approach can fail and an intuitive way to see this I mean very intuitive is thinking of black hole physics so if you imagine okay we have some black holes okay so let's think of black holes if we imagine that we scatter particles at very high energies right like what we do at the LHTC uh higher energies right typically imply smaller distances right so by scattering at higher and higher energies we improve smaller distances however there is a moment in which if we continue going to higher energies we are going to start generating a black hole right and a black hole will be larger and larger okay so that putting more energy into the system does not give rise to proving smaller distances but not proving larger distances okay so we break this notion that higher energies go to smaller distances and so on okay so this is a very intuitive and void in argument but mainly the point is that by studying three theory compactifications and also black hole evaporation in more detail you can derive these new swan plan constraints okay that we will study in these lectures and what is more interesting is when these constraints go against the effective field theory expectations okay when constructing billion standard models and so on so the examples that we will see is that some features which are completely fine from the perspective of quantum field theory like global symmetries or small gauge couplings right so we have a perturbative description of a gauge theory or for example large field ranges okay so we can have models of large inflation and so on these things are problematic in quantum gravity okay sometimes we will see that this implies that they cannot happen and we have to break the global symmetries sometimes it means that if you want to engineer small discovery or a large field range the cutoff the quantum gravity cutoff is going to go it's going to be very low it's going to go to zero so new features have to appear soon okay okay any question i'm going to continue asking for questions unless someone is brave enough to ask sorry rena there is a question in chat oh okay sorry i didn't see it should i read it for you or you can read it i i can read it so the question is if suitable modifications bring a new team back in consistent region can i sharply distinguish landscape for example yeah the thing is that the notion of landscape of somplan is depends on the energy scale that's the point okay so some of the constraints i mean depends which one okay some of the constraints since the plan scale is finally far away some of the i mean some of the effective theories will always be ruled out in the empire so there are some effective theories which cannot be there and some somplan constraints are sharp in that sense like for example you cannot have global symmetries but others are related to this cutoff so that they tell you okay if you want to engineer this feature in the eft the cutoff has to be at this energy scale which means that if you are above that energy scale that theory is inconsistent but if you are below it's still consistent for some energy range so this distinction sometimes will depend on the energy scale at which you want the effective theory to be valid okay i'll also add a question but maybe just ask to maybe clarify a bit more what do you mean with the small gauge couplings or problem or maybe we'll come back to it later in the lectures yeah yeah so these are the main three uh well consequences of the main three conjectures that we'll see the small gauge couplings because it will be constrained for example by the good gravity conjecture and that we'll see also that if the gauge couplings become very small which in principle should be fine in quantum theory right from the quantum gravity perspective that's too close to restoring a global symmetry like if this coupling is going to zero it's becoming very very small yes you have a very approximate global symmetry and that's problematic and will mean that the cutoff of the theory has to go linearly with the gauge coupling so if the gauge coupling is very small the cutoff also becomes very small we'll see that does that mean if i just have say some some let's say some classical gravitational background and i want to put some some gauge theory on top of it like i mean like getting trouble so there's essentially no perturbative gauge theory on top of some i mean as i said it's going to be a relation of the cutoff so there is always some some perturbative gauge theory below a certain cutoff but this cutoff is something that you cannot extract from quantum field theory right it's a quantum gravitational effect that is telling you that the cutoff is also going to zero so it depends what yeah what energy skill you want to describe this effective theory then will be a problem or not so there's a correlation between the cutoff and the gauge coupling and yes let me remark is a good point all i'm i'm talking about here will be about effective theories that are weakly coupled to instant gravity okay so we always would take an effective theory in quantum field theory and we couple the theory to a semi-classical gravity description understand gravity description and then we wonder whether that effective theory will continue to be valid at quantum level right when we consider quantum gravitational effects and what are the constants any other question okay i see more question good very good people so it's right to say that the theory is falling in swampland if not we made consistent with quantum gravity so if you say a theory is in the swampland is because i mean when you say when you give an effective theory an effective theory is also a it also comes together with the definition of the cutoff like what is the energy scale at which that effective theory is valid so if you are saying that effective theory at that energy scale is in the swampland then it's not consistent with quantum gravity it's kind of part of the definition of the effect okay any other question okay good so this is this was the introduction so let me okay so this is a map i have a question too yes sure sorry what is a large field range what do you mean by large field range yeah so we'll see this more detail but okay i mean if you have some scalar field right the scalar field can have can take different vacuum expectation values uh in principle that value is not constrained could be whatever right from a quantum theory perspective it could even be very large it could even take a transplankian expectation value as long as the there is no energy that apparently comes above the cutoff right like i mean the the field i mean the field the scalar fields can take transplankian values as long as the energy doesn't go along the cutoff and that's in principle completely fine and that's what we use for example to construct large field inflationary models but from the quantum gravity perspective we'll see that there is also a sort of cutoff in field space like how large these vacuum expectation values can be okay so thanks okay so this is a map of the most important transplank conductors um what i'm going to do so let me put a bit of order here and then i will tell you what to what what we'll do in detail is to study the ones that they are in black okay so the options of global symmetry the gravity and the distance connection because they are really at the core of the sombran program i mean these are the most important ones and many of the others can be derived from relations to these ones and also are the ones that have been more studied and are better understood and they are in solid ground okay so it's very important to have these three connections very clear because from then we can start constructing other things okay and we can start giving also more constraints and the potential and so on but that only makes sense if these three conjectures are like very very clear okay so just to put a bit of order the first row here everything can be related and actually this year we are unifying these conjectures into a single statement okay so everything is about constraining what is the notion of global symmetry that is not allowed in quantum gravity okay so this is about constraining the kinematics and basically it's really everything is correlated to the absence of global symmetries that we study in detail now the next row is they are giving constraints on the field spectra okay so these ones are the ones that tells you that if you want to engineer small gauge couplings or large field ranges the cutoff is going to go to zero and you you will have new physics okay so we'll see this in detail basically we will get powers of states that are becoming light such that this quantum gravity cutoff becomes below and blank and then I'm sorry can I because you insist it's all the questions and welcome for the first line but no global symmetry so it's a conjecture right but then if there I guess a conjecture has some need to it but is there a is there a distinct distinction this conjecture for for discrete and continuous symmetries or so we'll see that the conjecture is in general both for continuous and discrete although there are different words given evidence for this and depending what work then applies to both continuous and discrete or only to continuous and so on so yeah I will tell you later the evidence then you can tell me like exactly if you want to know like okay where these symmetries are yeah okay and the last part are the like newer conjectures okay that have appeared recently and that are still on progress let's see so it's like here from the gravity and this last conjecture if we apply the scalar potentials actually these are like constraints on the potentials in certain setups we can see that they imply stronger constraints which tells you some features of the scalar potential or the vacua that are consistent with quantum gravity and these are conjectures that can have much stronger phenomenological implications but are also still in the process of understanding what is the exact statement that is realized okay so this is evolving very quickly and that's why when you hear about the swamplan it's important also to keep in mind what swamplan statement you are talking about okay it's not the same if you are talking about no global symmetries that we already have a lot of evidence and also proofs in certain setups you see like ADS-CFT or perturbative string theory and so on that if you talk about the sitter or other things because it's still in progress I mean you have to I mean there is something there but it's not completely clear what is the sharp statement that is realized okay and part of the most of the research is precisely trying to define properly what is the precise constraint that should come from quantum gravity okay so okay so let's focus then on the left and the black ones the last day depending how we go on time we will talk a bit about the the last ones on the bottom okay I can tell you what they are about and what we know and what we really know about them and what is the evidence that we have and what is missing like what are the open questions and if you have if there is someone in particular that you are more interested in you can also tell me by slack or something okay like I have heard about this statement can you talk a bit about it and we'll discuss it in the last day okay so as I said the plan of the lectures after this very long introduction is to talk about no global symmetry which is the most important one so we'll be talking the rest of the day about this and also part of tomorrow and then we will discuss the presence of all these towers of states like these new states that appear when we try to engineer like weak weak gauge theories and large field ranges and so on and how this gives rise to this with gravity and distance conjecture okay from the string theory perspective and then we will see how this particulate with gravity conjecture can be also independently derived from black hole physics and as you said the last I mean at the at the very end we'll see how much time we'll have to discuss the constraints on the scalar potentials and the string landscape okay so any question before I start with no global symmetries okay very good so the absence of global symmetries if the most I think is one of the most important and some plan statements is actually very old and I mean we'll see for example you can already find an explanation of it in the book of colchiski about string theory but in order to discuss it I want to explain carefully what would we mean with global symmetry okay how to distinguish a global symmetry from a gauge symmetry and so on because it this doesn't mean that there are no symmetries whatsoever in quantity okay it's just that this the symmetries if we have an exact symmetry it has to be gates in the sense that has to be associated to to some gauge degree of free it's like the global global part of the gauge group so the the intuition to distinguish between global versus gauge is that global charges as I put here cannot be measured from very far away okay so this is kind of a handwoven intuition but it's precisely what is behind or these black hole arguments I mean why would I not have global symmetries okay it's kind of the idea that everything it's kind of a holographic perspective of the world or so that everything can be measured from very far away so to give an I mean I think it's easier sometimes to give examples so let's give some example and then we'll give a more proper formal definition of this so the typical example is for example if you just have some action like some scalar field okay this has this is invariant right under a shift symmetry okay so I can transform phi with phi plus lambda and this has some global charge which by nether since I have a Lagrangian description this is just the integral of a current which in this case you can compute by using nether theorem that is star d phi okay so that this current is conserved okay now this is an example of a global symmetry okay that should not be allowed in quantum gravity because this is conserved now there are two options you can either break it okay so we can add a scalar potential for example so we break the symmetry or you can gauge it okay so the usual procedure as you have learned many times is that we can gauge it by coupling to a gauge field right gauge field A so that we add an extra coupling of the current to this gauge field and let me put a kinetic term for the gauge field as well okay so this is the typical stuccover coupling okay we can just I mean this is the right this way if you wish that the gauge field is heating up right the scalar and the global symmetry now is gauge because this Lagrangian is only invariant under the combined transformation of the action and the gauge field right so when phi goes to pi plus lambda and I mean we can now this lambda can be local actually the local parameter so a1 goes to a1 minus d lambda and this is a gauge transformation right so if lambda x at infinity goes to zero this is what we call gauge right local gauge transformations and these are redundancies of the theory but those transformations that are global not that are go to a constant at infinity are global transformations and they really act in the Hilbert space of physical states okay so this is a symmetry not a theory this is a global symmetry but this global symmetry which is fine in quantum gravity so we don't call this global but we call it gauged okay or sometimes people call it like the global part of the gauge group okay so this is fine fine for quantum gravity okay while the previous one without the gauge field should not really be allowed okay this is a I mean just in a very simple example what is the distinction sorry Irene I have a question yes so when you say that the symmetry must be either broken or gauged by broken you mean explicitly broken or would the spontaneously broken be okay no I mean explicitly breaking I mean in the end spontaneous breaking is like you still have a symmetry right it's just that this particular vacuum but some state that doesn't that breaks the symmetry but it's still an exact symmetry in your theory and your description so that's not allowed no it's not enough with spontaneous breaking okay thank you okay so this is like the okay so this global part is like the electric charge no like the electric charge that we always talk about if you write the current I mean the question of motion for the gauge field right now this is going to get modified okay so that you get this electric charge and this is something that we can measure from very far away okay that's why there is this intuition or distinguish between global and okay so I'm going to generalize this and give you a more formal definition of global symmetry but before sorry I want to give you another question yes sure so uh so I would like to understand better what is the intrinsic distinction between uh say standard global symmetry and the example that you gave here of global symmetry comes from boundary conditions is it related to super selecting sectors so that is uh for me the easiest way so we'll see but I think the most precise way to distinguish them is when you then write it in terms of what are the charge operators that are that are charged under the symmetry I mean whether they are catching variant or not so whether I mean we'll see that's I mean now we will formulate this or in terms of topological operators and charge operators and the difference between this gauge global symmetry and the previous one is that the operators that are charged under symmetry are gauge or not gauge invariant so whether you have genuine operators that are charged and this is very sharp because you can just right like classify all the operators that you have and check you have any operator that is genuine so it's gauge invariant and it's charged under some topological operator I see so so we can say that so this symmetry is okay because there are no gauge invariant operators that are charged under this symmetry exactly we'll see a couple of slides but yeah that's the sharpest thing okay so before going to this sharper definition I thought can be useful to give you as I see there's another there are more questions yes why charges of global symmetries can't be measured far away well intuition is because I mean how you measure right some symmetry because you have some dynamical degree of freedom and for example when it's gauge is it becomes the electric charge so you can measure it because you have the gauge field right and you can make an experiment to measure the the gauge charge from very far away because then you have a agos low right so agos low is something that appears only when the symmetry is gated and even if it's discrete and you would say okay but then I don't have for like a dynamical gauge degree of freedom what's going on still you can have like an a hat or no both effect and you can measure from far away so you need to have some dynamical like some degree of freedom associated to it and if it's global it's completely global like we didn't have the gauge field there is no experiment that you can think of that you can measure that charge from far away there is no goes low there is nothing like that this is kind of the intuition and we'll see also that it becomes I mean right now we'll see that it becomes very clear also when thinking of black holes because the black holes and the the horizon depends on what is the gauge charge but that's not I mean the selective charge but it's blind completely to global charges so from very far away you cannot see if the black hole is charged under a global symmetry or not we'll see we'll see okay let me go on slide we'll see why it's important that can be measured from far away okay so let me exactly so before getting to more formal definitions I want to give you an intuition of why what is the problem with that okay now I just gave you a distinction okay so this is the difference between two types of symmetries that we can have and the intuition is that one can be measured from far away the other one cannot but why is that why is that a problem okay so why this global symmetry should not be allowed in quantum gravity okay so here I put like what is the evidence that we have for this conjecture um there are some proofs that we have okay in stream perturbation theory that you cannot read by in pochinsky book because any any global symmetry of the worship give rise to a gauge symmetry in spacetime okay and you cannot have global symmetries in spacetime and because I mean global symmetry something that are conservative that's a matter of the duality frame so all the global symmetries in the worship are gauged in the spacetime so we can only have gauge symmetries in spacetime and the same happens in adcft thinking of the like global symmetries in the cft are gauged in ads the thing that this this proof in perturbative theory works for continuous symmetries when you have a current okay but recently by harrow and augury they generalize the this to talk about more general symmetries even if they are discreet um as long as they are speedable in the boundary so to explain all these things I don't know you need no many more things also about entanglement which reconstruction so on so I think I will not explain it here um but if you have you want to know more about these derivations after the lecture I know the half an hour later or whatever like you can stay and you can ask me about it and I can tell you more detail what the proofs are about okay instead what I want to do now is to give you what are the black hole heuristic arguments okay of why these global symmetries are not allowed so you have at least some intuition of what goes wrong if you cannot measure the global charge from far away and also let me remark that recently um there are also many more works um trying to to make more concrete this absence of global symmetry from black hole physics because all the all the recent developments regarding the unitary and like the derivation of the page curve right for the unitary black hole evaporation are also useful to show the absence of global symmetries and people have shown that all these wormholes that are breaking the global symmetries or more generally and whenever you have some topology changing process in quantum gravity this is going to break global symmetries so we'll see that having a global symmetry is in one-to-one correspondence with having some topological operator okay something that is invariant and there are different morphins because has to commute with the stress energy tensor and that's fine in quantum theory because in quantum theory we fix the topologies but in quantum gravity topology something that can fluctuate can change and this gives you an intuition why global symmetries are not allowed because then we cannot have topological operators okay so let's do this so let's study these heuristic arguments okay and then we will define sharply what are exactly how to identify the symmetries that are not allowed and how all the recent progress in quantum field theory generalizing the notion of symmetry could also can have a very profound impact here okay so just the heuristic argument so why is it a problem if i cannot measure from far away well if you think of a black hole this is the metric for example for a sparsil black hole that is neutral so the by the know her theorem and so on i mean we know that the metric the horizon where the horizon is not sensitive to the global charge you will have for imagine that we have different black holes that are charged and there are global symmetries so we can construct a black hole for all possible charges and global charges all these black holes even if they have a different global charge they have the same metric okay so from far away they look the same and now this the fact that you have infinitely many ways like you have infinitely many possible charges for example and there are some you want and so infinitely many ways to to describe the same horizon the same object from far away gets translated to some like infinite uncertainty that's and i mean it should give relates to some infinite entropy associated to the black hole right like if we think that this the entropy of the black hole is associated to how many different ways no like how many microstates we have if we have some black hole charge and the global symmetry this would give rise to infinite entropy and would be against the expectation of back and same bound whatever we have some always a finite entropy for the black holes now you can make these habits on a small precise but more quantitative if you think of the constructing black holes for every possible of the every possible value of the charge and letting the black holes evaporate okay since Hawking evaporation is blind to this global charge in the end you are going to end up with an infinite number of very long-lived almost stable remnants of size and plank this is the size and this is what is called the travel with remnants in the sense that we can count how many of these black holes we have which is going to be a sum of e to the s now this is number of states where s is the entropy and now we have to sum over all possible charges global charges and the masses and this goes to infinity so having an infinite number of states in a weekly couple descriptions of instant gravity doesn't make sense because everything couples to gravity and is going to normalize the plank mass and the I mean it's not going to be instant gravity anymore so this is the the idea that exactly this effective theory weekly couple instant gravity doesn't make sense okay and we'll see when talking about the gravity connection how this travel with remnants is solved if you have a gauge charge instead because if the cement is gauged then this metric now will have a care black hole and the metric is sensitive to the value of the charge so we'll have different different black holes for each possible value of the charge and when counting the number of remnants will be fine any question about this this is a heuristic argument hello can I ask a question so what you're saying here if I'm if I understand correctly is that so you calculate any process in an EFT couple to gravity and because you have an infinite amount of states you have sort of an infinite amount of states and loops that you need to yeah sum and you get infinity for anything that you try to calculate but is there a possible caveat that there could be a form factor for for these black hole states and the loops because it's an extended object or you know some some kind of form factor that allows you to regularize that sum has this been yeah so four factors are actually very important I mean when you and you can wonder also why a black hole already a single black hole no it's not a spoiling the loop computations that we do in the standard model and that's because of these four factors right and also because the yeah if you try to compute it you will see that you get a suppression in terms of the entropy like e to the minus s so that's important for that now if you have an infinite many then at least the naive computation if you try to do it actually is that this e to the minus s will precisely cancel by the number of remnants like you have infinite many so that they should not be suppressed now this is like the I don't know like I don't know it's to say naive like I don't know if you do it the vanilla way I don't know if there is any way to complicate things such that you have some form factor that could suppress it and I think I mean nobody has proposed anyway like there is no quantitative way to do that but I mean I for these type of reasons I just I mean I like to emphasize this is just like a heuristic argument I mean it's not a proof because there can always be caveats if we don't know what is the quantum gravity completion right like how do we do this computation I don't know maybe we are missing know what that is form factors and how and with quantum gravity and we change it in a way but you always have to take these arguments as a motivation of that this is telling you okay here there is something funny like if you try to do it since you have infinite many states it's true that there is a sharpness and like gravity seems to distinguish whether it's a global or a asymmetry and black holes are not sensitive to that are blind so it seems that this gives rise to an infinite entropy so this something here that is weird let's now study this more quantitatively in string theory or ADCFT and see if we have a proof for that that's the way you should take it so yeah I mean it could be currents okay see there's another question can you say something about sl2c in type 2b say how why is it gaited yeah so all the duality symmetries that we have in string theory are gaited okay the dualities in general right they are like this kind of gaited transformations um and sometimes we can even understand how they arise from from a diffio or a gaiting variance in higher dimensions so for example the sl2c and you can see that this gaits you can or you can see it in a worksheet like you can describe sl2c in type 2b perturbative string theory and see that it corresponds to something global in the worksheet that becomes gaiting spacetime and something that you can do sharply in a worksheet but also in a spacetime like without having a worksheet description from ftod perspective I mean since it becomes part of the diffios of the torus uh the form of the torus of the elliptic vibration uh you can kind of see you know that it's gonna be a gaited symmetry because it's associated with gravity to the geometry okay so let's now define this more precisely so the point is okay so this is the situation that we had a few years back okay uh now in parallel in quantum theory there is like a mini revolution regarding generalizing the notion of symmetry okay so there are many more symmetries in addition to this axionic symmetry as I told you about we have two form of our symmetries non-invertible symmetries many things and this also has an effect here because all these global symmetries they are I mean these new notions of symmetries are still global symmetries so this should still be absent in quantum gravity and we can learn a lot by understanding how they are absent because it's not so obvious okay and many familiar phenomena in string theory or quantum theory can actually be already arrived just from requiring the absence of all these symmetries so we'll see but I don't know like for example if you start with the supergravity I mean if you work in string theory maybe you know a bit about this but if you work with the supergravity description of type two in ten dimensions like just the closed string sector uh you can start playing the game of how many things can I say about the rest of the theory just from requiring the absence of global symmetries and it's very cool that the existence of brains also the chair simons terms and so on and all these processes of brains intersecting brains and dissolving in brains and so on can be already arrived just from requiring that the theory doesn't have global symmetries so it's actually more constrained in that word we think and also other things that can be derived from we see is that gauge groups are compact or that the spectrum of charges is complete okay so I will give you examples of this and just so you know now the completeness of a spectrum it was another form plan conjecture I mean that was written also it's also quite old that the spectrum of charges have to be complete and now we are relating this to the absence of global symmetries precisely making use of all this generalized notion of symmetries okay so the goal here on the sum plan program is also try to unify these conjectures to try to identify you know what is the minimal set of statements not that reproduces everything to identify what is the quantum gravity underlying principles behind sorry so all the symmetries have to be gauge symmetries and now you're saying that all the gauge symmetries have to come from compact gauge groups also yeah yeah like exactly and here I was thinking of continuous and we put continuous right okay otherwise fine because if you talk about these dualities no no this is about continuous gauge groups they have to be compact and discrete they don't have to I mean you can see that if you have a continuous gauge group that is non-compact then you can have like a global symmetry in the sense that you have like a particle with charge one and a particle with charge square root of two and they can never be k one into the other so you have like selection like different sectors but that's the output thanks okay so for the rest of maybe less five minutes I can take um okay and then we should go to the discussion arena okay so I will be I will try them to be less than five minutes okay so just I want to give you just to finish um what is the sharper definition for these global symmetries okay and tomorrow we will start with generalized notions of symmetries okay but at least for the zero-formular symmetries we are talking about let's try to make it sharper so uh recently the the formal is to talk about global symmetries is in terms of topological operators okay so this is something that is going to be useful for you regardless of the swan plan program because there's a lot of activity here so in order to describe I mean a global symmetry is going to be characterized by a unitary local operator okay that lives in d minus one dimensions we will see why and is labeled by an element of the group such that this operator is topological okay so the I mean it's a symmetry so it has to commute it's like saying that commutes with the Hamiltonian right so in point of view it has to commute with the stress energy tensor so it has to be invariant and their small diffomorphism which means that this operator has to be topological okay this is the main point and also needs to have something that is charged under it okay otherwise there is no symmetry to talk about so it must act non-trivial on some gauge invariant charge local operator okay so that if I apply this unitary symmetry operator this is not invariant okay and this charge operator creates particles okay creates a particle that is charged under the global symmetry and the third condition which can see actually can be relaxed is that it satisfies a group law but for the ordinary notion of symmetry I mean we require that the the satisfy a group law so in the example I mean with this I finished example we studied right was just some scalar field the unitary operator that we had is just the the exponentiation of the charge of the global charge so in this case since we have a current right is the exponentiation of the integral of this current which if you remember we said it was the integral of star d phi and now you can see that this right star d phi so d phi is a one-fold current so when you compute the hot dual you get something that is a d minus one a form okay so this is the number of dimensions okay this is dimensions so this has to be integrated on a d minus one for manifold okay so that's why these symmetry operators okay for zero form global symmetries like this one live in d minus one and the charge operator is just a exponentiation of the action okay so what happens now we are saying and with this I finish we say that the symmetries have to be broken or gauge so if the symmetry is broken right then this is no longer true the operator is not going to be topological is not going to be conserved okay and if it's gauge what happens is that the second condition is not satisfied because then I don't have gauge invariant charge operators because this operator that I'm writing here is no longer gauge invariant when coupled to the gauge field right because the transformation of the action has to be reabsorbed which are gauge transformation of the gauge field so this charge operator is no longer no longer gauge invariant okay and this is the the start of the definition I was talking about you just have to check what are the topological operators that you have in your theory so that tells you what are the possible global symmetries and then you have to see if there is any gauge invariant charge operator under it if there is then it's a global symmetry and that's to not be compatible with having a quantum gravity description at high okay so let me finish here thank you for listening and tomorrow we will continue with the analysis notion of symmetries and jump the guild so you're ready to go Alex yes I think so everyone can hear me okay okay thank you yeah great so it's a great pleasure to be speaking today you know I wish I was there in person I haven't been to the ICTP in almost 10 years great coffee if I remember correctly so I wish I could be there in person but we'll have to make do so I should begin really with an apology because I see the names of many people who who I know you know attending today unfortunately if I know you probably you know I'm going to bore you today because what I'm really going to try and do at least for this lecture today is keep things really quite elementary so my goal for this first lecture is to give a very general kind of schematic overview really emphasizing some basic stuff some background so that we really can all be on the same page when I get into some of the more detailed aspects of this of these lectures starting tomorrow so I'll begin by just describing a bit of a plan for what I'm going to do today so I'm going to be presenting this sort of new view of holography and the ADS CFT correspondence that's emerged over the last two to three years I would say although there were some precursors going back maybe more than a decade but really to begin before I present this new view of holography what I want to do is start by spending perhaps half of the first lecture today reminding everyone of the standard view of holography you know the one that you would learn if you picked up any one of a number of excellent lectures on the ADS CFT correspondence and then after spending a little bit of time reviewing the standard picture of holography and ADS CFT I'll move on to talk about some recent indications some hints that this might not be the whole story in particular I'll talk very schematically about the Euclidean path integral and how when we try and construct precisely theories of gravity based on path integrals we discover things that are known as Euclidean wormholes and these inevitably require us to change a little bit our point of view of holography and in particular they introduce this notion of ensemble averaging which I'll be talking about in great detail as these lectures go on and finally and the main focus of my lectures this week and something that I'll really only introduce today very schematically if I have time I'll talk about the two main examples of theories of holography that incorporate this notion of ensemble average so the first of these is at this point famous theory of gravity in two dimensions known as JT gravity which can be formulated in terms of a matrix integral so that is to say as a matrix model and then I'll be talking about a theory that was described more recently in a paper that I wrote with Edward Whitten as well as in another lovely paper by Afkami Jetty, Tejdini, Hartman and Cohn where this describes a different theory of gravity known as U1 gravity which is related to an average over a space of CFTs and then I'll end this isn't something I'll even get to today but probably later on in these lectures by talking about some speculations about what this means more generally for our understanding of quantum gravity of general relativity and where all of this stuff is going okay so that's my plan really for these lectures today what I'll really just be focusing on are these first two points which I'll introduce rather schematically but as the other lectures have mentioned and before I get started I want to encourage everyone to stop me interrupt and ask questions if anything is unclear or on the other hand you know if you feel like the things that I'm doing are too simple too elementary then feel free to interrupt me as well and encourage me to go faster or bug me about the details because I love nothing more than getting distracted to talk about you know you know other sorts of details so um that's all just to say please feel entitled to interrupt me and ask questions at any point either just unmute yourself and ask a question or you could go ahead and type something into the chat um good so let's begin by just reminding ourselves of what exactly we mean by holography and in particular holography is an equivalence or at least the standard version of holography that you might read in a textbook is an equivalence between a theory of gravity in anti-decider space and a conformal field theory living on the boundary of anti-decider space so before I get into any details about how we're going to modify this picture of holography to incorporate ensemble averaging I'd just like to talk for a few minutes about where this comes from and why it's interesting so to begin and just to set some sort of common notation so that we all are on the same page let me just think about the simplest setting for ADS CFT which is an example that I have come back to many times in my work on the subject which is three-dimensional anti-decider space so what is anti-decider space? Anti-decider space is the maximally symmetric solution of general relativity with a negative cosmological constant so typically we would write the cosmological constant as 1 over l squared where l is a parameter with dimensions of length known as the ADS radius and then one way to describe anti-decider space is by embedding it in one higher dimension so for example if you were to talk about ADS 3 then you would embed it in a four-dimensional space with coordinates that I have called here TSX and Y this is a Lorentzian space and I'm embedding it in a higher dimensional space time with two Lorentzian directions so that when you look at the induced metric in three dimensions you'll get a Lorentzian signature spacetime with one time like direction and if you were to try and write down metric on this three-dimensional anti-decider space then there are of course many different metrics that we can choose I'll just introduce a couple of them that will be helpful to us as we move forward in our discussion of ADS CFT so here I'm introducing a radial coordinate rho a time coordinate t and an angular coordinate phi so that if you were to compute the induced metric in terms of these t, rho, and phi coordinates that's a straightforward exercise which you can almost read off from the embedding coordinates that I've written down here basically I've written down these coordinates just so that we can all have a clear picture of anti-decider space in our heads today as we move forward with these lecturers so you can see here that we have a time coordinate an angular coordinate and a radial coordinate the picture that I want you to keep in your head is of ADS as a sort of solid cylinder so here in this diagram that I'm drawing of ADS t is a coordinate that runs vertically that's the time coordinate phi is an angular coordinate that's periodic you can see that it's periodic because the coefficient of d phi squared in the metric shrinks to zero okay so that means that it has to be periodically identified so near the origin the metric looks like d rho squared plus rho squared d phi squared that means that the metric is periodically identified phi with phi plus two pi and then finally you have a radial coordinate that fills in the solid cylinder and in particular at rho goes to infinity we have a boundary and you should think although this boundary sits at infinite distance from points in the interior of ADS I really want you to try and think so I've drawn a picture here where everything has been rescaled and I've pulled that boundary into finite distance and this boundary is a cylinder which in this picture is a time coordinate times a circle so these coordinates that I've written down are the global coordinates of three-dimensional anti-decenter space for the purposes of comparing to other metrics that I'm going to be writing down later I would just like to rewrite these coordinates a little bit to make them maybe a little more familiar to you so in particular instead of using rho as a radial coordinate instead I want to use a radial coordinate that is the coefficient of d phi squared in the metric okay so if I define a new coordinate r which is going to be l times sinh rho then it's easy enough to go ahead and write down what the metric looks like in those coordinates so not much happens to the metric this is just a straightforward exercise and these are the conventional coordinates known as the global coordinates of ADS they cover all of the Lorenzian ADS geometry okay good now we began today by saying that holography is a relationship between a theory of gravity in ADS and a conformal field theory but there's a very important point that I want to make which is that when we study gravity in ADS we are not just studying anti-decenter space instead what we are studying is not just ADS but we are studying all space times that look like anti-decenter space out at infinity that is to say that look like ADS asymptotically so asymptotic infinity so not just ADS so we're not just studying ADS or even just perturbation theory describing small fluctuations around ADS and so what that means is that when we define theories of gravity in ADS we need to be very careful about describing the boundary conditions of our theory of gravity under consideration so gravity in ADS is not just you know a theory of metrics related by diffeomorphisms but it's also a theory that includes a very careful and precise definition of boundary conditions and in fact it's those boundary conditions that really will lead us to think about holography and the ADS CFD correspondence okay so for example there's another core an easy way to describe this is to actually use a different coordinate system that will also be useful to us as we move forward so these are not going to be a global coordinate system but what are known as planar or Poincare coordinates so this is a different set of coordinates on ADS so where I'm going to have a radial coordinate R and now I'm going to introduce coordinate systems a coordinate system that is adapted to the planar symmetries of ADS so here we have a metric on a two-dimensional plane that I'm describing using a complex coordinate z along with a radial coordinate okay and these are supposed to be lectures so because these are lectures I feel entitled to assign homework so your first homework exercise is going to be to find the coordinate transformation that relates these global coordinates to these planar coordinates that I have written down here and so now given these planar coordinates it's easy for me to define exactly what I mean by geometries that look like ADS at infinity so what I mean is that I want to consider the class of metrics that approach the metric that I wrote down above in planar coordinates plus terms that are subleading so what do I mean by subleading well in terms of these planar coordinates on ADS that I have written down the boundary of ADS is often R goes to infinity and so all I want to do here is I want to consider the class of geometries that look like this boundary metric this ADS metric plus terms that are subleading as R goes to infinity okay so the reason why I'm being very persnickety about defining my theory of gravity in terms of some set of boundary conditions is that once I've defined these boundary conditions I can talk about the symmetries of the theory and in particular the symmetries of ADS and I should be a little bit more specific and say the symmetries of gravity in ADS are the diffeomorphisms that is to say changes of coordinates that preserve this form of the metric and so given this form of the metric it's a straightforward exercise okay again I'm using the word exercise because I'm too lazy to work it out feel exactly so instead I'll assign it to you as a homework problem it's a straightforward exercise to find the class of diffeomorphisms that is to say the class of changes of coordinates that preserve this asymptotic form of the metric so in particular what are they well z here is a complex coordinate and so it turns out that for any holomorphic function epsilon and anti-holomorphic function epsilon bar you can find the diffeomorphism of this three-dimensional geometry that takes z to z plus epsilon and z bar to z bar plus epsilon that preserves these boundary conditions with a little bit of a wrinkle which is that if you work it out there are corrections that you need to include in order to get everything to work out so that you end up preserving these boundary conditions and here the radial coordinate will have to change as well so the reason why I've gone through the trouble of writing all of this definition of exactly what I mean by gravity and ads and exactly what I mean by the symmetries of a gravity of gravity and ads very carefully is the following okay so usually when we think about a theory of gravity it what we do is we talk about the diffeomorphisms of our theory of gravity as gauge symmetries okay but that's not actually the right way of thinking about gravity in ads when we think about gravity and ads as a theory defined with a set of boundary conditions in particular it's only those diffeomorphisms where this function epsilon equals to zero that we regard as pure gauge and what do I mean by that well what I mean is that metrics related by these diffeomorphisms are regarded as being identical so usually when you think about a theory of gravity like when you learn general relativity you learn that general relativity is a theory of geometry not a theory of metrics and what do we mean by that what we mean by that is that any two metrics that are related by a diffeomorphism a change of coordinates describe the same geometry so for example if you were trying to construct a path integral they would give you the same they would give you one contribution to a path integral not two different contributions to a path integral or if you were talking about you know classical solutions of general relativity any two classical solutions that are related by a change of coordinates you know they're not different classical solutions they just describe one classical solution and the rule of the game when we're studying gravity in ads is that the diffeomorphisms that are regarded as gauge symmetries in this sense are only those where the function epsilon is equal to zero and diffeomorphisms with epsilon not equal to zero are not pure gauge instead they generate symmetries of the theory so in particular what do we mean by that what we mean is that two metrics in anti-decider space here let me scroll up so you can see the metric up above two metrics in anti-decider space that are related by a change of coordinates are not regarded as being identical if that function epsilon that i have written here is non-zero so what does that mean that means that these diffeomorphisms of general relativity in ads are actual symmetries of the theory in the sense that they take you from one state to another okay right what is the symmetry a symmetry is something that takes you from one solution of the equations of motion say to another solution of the equations of motion different solutions of the equations of motion or it's something that in the quantum mechanical picture would be a unitary operator acting on the Hilbert space that would take you from one state of the theory to a different state of the theory the important ingredient that I'm trying to emphasize is that when we talk about gravity in ads the diffeomorphisms that act non-trivially out at infinity in the sense that epsilon is not equal to zero are actual symmetries of the theory excuse me not gauge symmetries yes please uh you mean pure gauge is epsilon equals to zero does that mean you only have the sub leading corrections or that's correct so a diffeomorphism that is purely sub leading in the sense that it only lives in those dots that dots that I wrote down above is regarded as a gauge symmetry okay thank you we have genuine we have other things where epsilon is not equal to zero these are not symmetric these are not gauge symmetries these are actual physical global symmetries if you like of the theory and what are they in order to understand what they are let's ask how they act on the boundary of the theory how do they act on the boundary of ads well they take z to epsilon plus z plus epsilon of z which where epsilon is some holomorphic function of z so out at infinity these symmetries of the theory not gauge symmetries real symmetries of the theory take z to some holomorphic function of z and z bar to some anti-holomorphic function of z okay how do these act on the boundary well out at infinity the boundary looked like some radial factor times dz dz bar and if you were to write this in terms of the w coordinate that I wrote down over here on the left side of this slide well you'll see that this change of coordinates that is an actual symmetry of the theory not a gauge symmetry an actual symmetry of the theory ends up just multiplying our metric by an overall factor now we have a name for a transformation of a metric that multiplies the metric by an overall factor it's a core it's a conformal transformation so this is a conformal transformation that takes the metric that takes the metric to some function times the metric so in particular coordinate transformations or transformations of the metric that just multiply the metric by an overall function preserve angles but not lengths that's why these are known as conformal transformations okay so just to summarize what we have seen is that diffeomorphisms that act non-trivially on the boundary of anti-decider space are two things the first is that they're not gauge symmetries right because of our definition of what we mean by gravity and ads and the second is that they are conformal transformations of the boundary and these are the two ingredients that really underlie the ads cft correspondence which is the idea that a theory of gravity and anti-decider space is a conformal field theory so this is the basic kinematic statement of ad scft now what do we mean exactly by this statement well when i mean when i say equals here i really mean equals okay and so there are really a couple of different ways of thinking about this duality uh one way of thinking about this is to think about it in lorenzi and signature so in lorenzi and signature you define a quantum theory by a hamiltonian and a hilbert space okay and so when i say that these theories are equal i mean equal in the sense sorry my pen is acting up in the sense that the hilbert space of quantum gravity and ads is supposed to be the hilbert space of the cft and that the hamiltonian that describes time evolution of our theory of gravity and ads is the same as the hamiltonian of the cft okay so when i say equal the standard view of the ads cft correspondence is that these are really two quantum theories that are equal same hamiltonian and same Hilbert space or if we were to talk about the theories in euclidean signature what i mean is that they have euclidean path integrals that are equal so here on the left hand side of this equation i am thinking about the gravitational path integral of a theory of gravity and ads regarded as a function of boundary data and that is supposed to be equal to some cft path integral so this is the traditional point of view of gravity and anti-decider space and if this uh fixed perspective is true then this is great okay okay because if this is true it answers some of our most confusing questions about quantum gravity and in particular it answers what i think of as the big question of quantum gravity so in particular i think that the big question of quantum gravity is is quantum gravity a normal theory that is to say a normal quantum theory with a hamiltonian and a Hilbert space and with time evolution described by you know a standard unitary evolution with this hamiltonian on the Hilbert space or uh to really state that a little bit more differently it also answers the big question uh is black hole entropy counting the number of states in some standard quantum system in the usual way where an entropy would be the log of the number of microstates so why does this ads cft of correspondence answer these questions well in particular if you can identify the uh Hilbert space and hamiltonian of ads gravity with that of a cft and it says yes quantum gravity is a normal quantum theory uh but it also answers the second question because gravity in ads is a rich theory of gravity that includes black holes and uh if those black holes are living in some nice quantum theory uh then black hole entropy is just doing the usual thing that we expect in any sort of theory of statistical mechanics where an entropy is counting the logarithm of some number of microstates and indeed uh if you um read any standard review of ads cft uh they'll present a huge amount of evidence that this works and that the ads cft correspondence is correct for certain what i will call fancy theories of gravity so the standard example the stand which is probably the defining example of a fancy theory of gravity would be something like type two b string theory on ads five cross s five which is via ads cft supposed to be dual to a conformal field theory that is super symmetric n equals four su n yang mills theory okay so uh of course there's you know probably 10 000 papers probably more than 10 000 papers written uh collecting evidence uh and understanding this duality unpacking it in various ways um every calculation that we have done so far seems to indicate that it is correct but the problem is that since this is a strong weak coupling duality really as any duality worth its name is it is difficult to study both sides of this at the same time so in particular uh if you were to try and study the classical limit of gravity on the left hand side then that corresponds to uh strongly coupled yang mills theory on the right hand side so in particular uh super gravity on ads five cross s five corresponds to the large n limit of yang mills theory but it's strongly tuft coupling rather than a weak tuft coupling and uh even worse if you wanted to try and understand genuine quantum effects in our theory of gravity then that would mean not just studying su and yang mills theory it's strongly tuft coupling but also understanding it at finite n okay which is a very difficult thing to do it's a very complicated problem in strongly coupled gauge theory and so in fact pretty much every nearly every check of the ads cft correspondence involves working perturbatively on one side or the other and what we would really like to do is to understand this more precisely in a setting where both sides of this correspondence can be studied explicitly in particular I think most people would would agree that we don't have a non-perturbative definition of what we mean by type two b string theory on ads five cross s five we have a lot of uh you know incredible evidence that you know such a theory exists uh that it can be studied in various different limits uh using various perturbation theory techniques but we don't really have a rigorous non-perturbative definition of the right hand of the left hand side of this correspondence independent of the right hand side and in fact I think the modern point of view is probably that we should define type two b string theory on ads five cross ads five via this duality and say that there's a sense in which uh it is equivalent almost by definition to n equals four s un super yang mills but we would really like to try and understand you know whether there are theories of gravity where we can compute both the left hand and the right hand side of this correspondence uh exactly and independently for example what we'd like a gravitational theory where we can compute something like a path integral over a space of metrics so we don't have a definition of of string theory which would allow us to compute this you know independent of some ads cft correspondence so instead what I'm going to do is just try and find some sort of a simpler thing okay where we can hope to compute this thing exactly but even before we do so what I want to emphasize is that uh in many cases there is a strong tension between the existence of such a path integral and the ads cft correspondence as I have defined it above okay so what is that tension so in order to understand that tension let's just begin by asking if we can't compute this path integral over a space of metrics exactly how would we go about approximating it well the way that we would approximate this path integral is the way that we always you know approximate path integrals which is through a saddle point approximation so for example you know I left out my h bar here I should really put a one over h bar here and then in the h bar going to zero limit you would approximate this path integral as a sum over classical saddle points so these are saddle points that I have called g naught which are going to be solutions to the equations of motion and here I've just written down the classical contribution coming from each saddle point uh if we're thinking in Euclidean signature we would call these saddle points instantons and so I'm thinking about the classical instanton contribution to a gravitational path integral now in order to understand what this would tell us about ADS CFT ADS CFT tells us that this gravitational path integral so let me try and define it a little bit more precisely so this is going to be something like a sum over a space of metrics weighted by some kind of Euclidean action because I'm now imagining a Euclidean path integral and I have a boundary that I'm calling sigma here and I want to sum over all geometries whose boundary is equal to sigma so you should think about this as a sum over manifolds m whose boundary is given by some surface sigma and the claim is that this should be equal to some CFT partition function on sigma now what CFT partition function it is depends on well what CFT sits on the right hand side of the ADS CFT duality but it might be something like for example a theory of a bunch of bosons okay that would be the simplest possible example and then what do I mean by this CFT partition function well I literally mean the path integral over the CFT degrees of freedom and here the boundary is literally just the surface that these fields live on so for example if we're thinking of a bunch of bosons then these bosons would be mapped from that surface into whatever the target space of my CFT is so that's just the sort of example of something that you might expect to live on the right hand side of the ADS CFT correspondence okay it's an equivalence between two functions one of which is supposed to be a gravitational path integral of sum over all metrics with some specified boundary and the other is supposed to be a CFT partition function where you compute the partition function of a bunch of degrees of freedom that live on that surface out of the boundary okay but a basic fact Alex yes can I ask you a question please um regarding the semi-classical approximation should we be worried about the action being bounded below you should be worried about all kinds of things okay in order for me to make this story precise I'm going to need to talk exactly about how I do this Euclidean path integral so for example um you know when I talk about an action of Euclidean of some geometry then I need to regularize it in an appropriate way so when we talk about gravitational theories in ADS we have a machinery holographic where we introduce boundary counter terms that allow us to regulate this in the appropriate way so when I talk about a classical action I really need to talk about a regulated classical action in that sense um there also some dot dot dots in that expression appear that I very carefully left out so for example and I think this might be one of the things that you're alluding to which is that when we talk about Euclidean path integrals of gravitational theories in ADS it actually turns out that some one of the degrees of freedom appearing in this path integral appears to be a Gaussian with the wrong sign in the effective action so you get a so you actually would appear to get something unbounded below when you look at linearized perturbations around Euclidean saddle points in the study of perturbative gravity there are ways of dealing with this that involve very careful contour manipulations that allow us to turn this from a wrong sign Gaussian to a right sign Gaussian and so when we compute the subleading loop corrections of a theory of gravity in ADS so here in those dot dot dots that would be like a one-loop effective action or two-loop contributions and so forth these are all things that we're going to need to try and deal with precisely okay I'm trying to be a little schematic right now so I'm kind of brushing that under the rug but those are all things that we're going to need to try and deal with precisely but in that case with the wrong sign right sign Gaussian nonsense that's a standard field theory thing but all of this can be summarized by saying that I agree with the spirit of your question which is the following which is that I am trying to study a theory of gravity and Lorentzian signature but at the end of the day only a dummy cares about sorry let me restate let me restate that sentence carefully what I'm studying here is a path integral in Euclidean signature right I'm doing a Euclidean gravity path integral but really at the end of the day we don't you know only a dummy would care about Euclidean signature we care about Lorentzian signature physics Hamiltonians and Elbert spaces and to go from Lorentzian to Euclidean signature what you need to do is understand very carefully what kind of contour deformation you're doing to go from Lorentzian to Euclidean signature and so when I wrote down this equation up above I was really making a subtle assumption which is that there is a contour when I go from a Lorentzian to Euclidean signature there is a contour through the space of metrics which is you know the stationary phase contour and that the saddle points that appear as contributions to this integral in the stationary phase approximation are just the usual solutions to the Euclidean equations of motion now nobody really understands exactly how to define precisely the contour that would define a Euclidean gravity path integral and all we really know how to do is follow our nose so to speak and try and see what makes sense and so what I'm going to do today is kind of write down some things some contributions that seem to make sense but what you're saying is absolutely the right question and I think one you know really what we should try and do is define a theory of gravity completely precisely by defining exactly what the contour of integration through the space of metrics we are studying and then that would allow us to determine exactly which solutions to the equations of motion would need to appear in that top equation on this slide but for now let me just you know I don't have a ton of time left so let me now just try and get to the punchline of what I was trying to get to today thanks okay so a basic fact about CFT partition functions is that if our surface that our CFT lives on is the disconnected union of two other surfaces then the CFT partition function factorizes so it'll just be the product of the partition function on those two surfaces in many gravitational theories this does not appear to be true in particular if I try and study these CFT partition functions and compute them by doing a gravity path integral what we find is that they don't factorize okay so let me just present a sort of simplified version of the argument that these gravity partition functions don't factorize okay so let's just talk about a silly okay two-dimensional example so this would be ads2 CFT1 so here let's imagine that our boundary is a circle and it's going to be a circle of some length beta I've called it beta because a circle of length beta is what you would study if you wanted to compute the finite temperature partition function so here if you I'm imagining that we have a CFT in one dimensions this is just a sort of dumb toy model the Hamiltonian is the thing that generates time translations around the circle and the partition function of the theory on the circle is trace of e to the minus beta h that's the usual statement in thermal field theory you know e to the minus beta h is the Euclidean time evolution operator that moves you a distance beta in Euclidean time okay so how would you go about computing this via some sort of ads CFT correspondence well in the bulk what you would need is a geometry m whose boundary is the circle that is to say we would need a disk and indeed we're imagining a theory of gravity in ads so you're looking for ads metrics on the disk or in Euclidean signature we would call them hyperbolic metrics on the disk and there is a simple hyperbolic metric on the disk it's just the two-dimensional version of a metric that I wrote down for you above here I'm calling the angular coordinate Euclidean time uh because I'm thinking about this in terms of the calculation of a finite temperature partition function and this here is the metric on what we would call the Poincaré disk otherwise known as Euclidean ads 2 or two-dimensional hyperbolic space okay all of this is fine it means that if you were to compute the CFT partition function at finite temperature it should be something like the gravitational action of the disk good okay so now it's the end of the lecture and I can stop and we can all go home declaring victory okay except I'm going to steal a few more minutes of your time and show you that life gets more complicated once we consider more interesting observables so what about our boundary being the union of two of these circles okay so what sort of geometries could live in the gravitational path integral that computes the CFT partition function on a pair of circles well one option is the disk union the disk okay there's a picture our boundaries are disconnected and our bulks are disconnected but if you think about it there's another option which is the cylinder so in particular there's another locally ads metric that is to say hyperbolic metric whose boundary is two circles and I chose to work in two dimensions because it's very easy to write down the metric there it is this now has two boundaries one at t u at r goes to minus infinity and one at r goes to plus infinity okay it's locally ads so if ads is a solution then this cylinder is going to be a solution as well and this is what we would call a euclidean wormhole it is a euclidean wormhole and that is a euclidean solution with two boundaries two disconnected boundaries one question so what does this mean yes um this I mean here in the in the first example and we could also have for example a torus without a circle and in that case we all we would also have like a two-dimensional manifold whose boundary is is a circle so why are not we taking it into into account in the full because I am being simple here and I am just writing down two representative solutions in particular I wrote down the cylinder because it's the simplest one that's going to lead to a puzzle but of course there are going to be other options as well okay so for example um you know there are things that look like this yeah and in a full theory of gravity I would need to sum over all of them and indeed that's exactly what happens in jt gravity okay now in a theory of gravity you would weight these guys by their euclidean action in the present case the euclidean action is just you know includes a factor of the order character so there's going to be a genus expansion in our theory of gravity so all that higher genus stuff is going to be suppressed okay but it will be there good great question actually okay thanks good thank you okay but this is a problem so it appears that the gravitational calculation of this thing does not factorize so here I've given you a super dumb example in a two-dimensional theory of gravity but these euclidean wormholes it turns out are ubiquitous in all sorts of theories of gravity this was originally pointed out uh in the ads cft context in a lovely paper by Muldesino and mouse that's probably you know written 15 years ago at this point and I think of this Muldesino mouse paper as the starting point for this whole story of uh gravity and ensemble averaging so what I've done here is present the puzzle and what I now want to do in the last negative five minutes that I have remaining in my lecture today is present one potential resolution of this puzzle so what is the possible resolution of this puzzle so the possible resolution is that we should interpret our gravitational partition function by which I mean an integral over a space of metrics with a specified boundary not as a cft partition function but as a sum of many cft partition functions so here what I'm going to do is I'm going to label my cfts by a capital letter c and I'm going to imagine that my gravitational partition function is going to be an average of many cft partition functions averaged with some probability density over the space of conformal field theories so this is a kind of radical reimagining of the ads cft correspondence right ads cft is supposed to be a relationship between a theory of gravity and the cft okay you might think in that statement in that sentence the words theory of gravity and conformal field theory might be the most confusing words in that sentence but instead the most confusing word in that sentence is a because in this interpretation ads cft is not a relationship between a theory of gravity and a cft but a relationship between a theory of gravity and an ensemble of cfts described by some sort of probability distribution so let's now return to the calculation I did a moment ago and understand how we would interpret this cylinder amplitude in terms of this probability distribution so in particular let me think now about ensemble averaging so if I have any cft observable let me denote the average of that quantity using angular brackets so for example uh the gravitational path integral I wrote down above is going to be the average in this sense of the cft path integral then let me repeat the calculation that I did above so the gravitational path integral on a circle is going to be basically so that's the thing that at leading order was the gravitational action of the disc this I interpret as the average thermal partition function of my cft but now what is the interpretation of the gravitational partition function on the union of two circles well one of the bulk saddle points was two disconnected discs and another was the gravitational path integral on the cylinder at leading order the action of the cylinder and this is the thing that I interpret as the expectation value of the partition function squared but what is this term here this term is just the square of the expectation value that's just this first line up here okay so what does that mean that means that this cylinder amplitude of my gravitational theory using very sophisticated mathematics where I move that trace of e to the minus beta h expectation value squared onto the other side of this equation is the variance of the partition function so the lesson that we draw is that the existence of euclidean wormholes implies or at the very least strongly indicates that ads cft is describing an ensemble of cfts with a variance that is not equal to zero and for example the classical action of the cylinder solution of our gravitational theory is the variance and if it's non-zero then our variance is non-zero and we have some sort of probability distribution so let me just end well first I'll apologize for going over time I think I was only supposed to take 50 minutes so I apologize for that and let me just end with an advertisement for the next few lectures so so far I've been extraordinarily schematic what I'm going to be trying to do doing over the next three lectures is to describe to you two different models the first is jt gravity the other is a theory of gravity that I'll refer to as u1 gravity where this ensemble averaging can be made precise and along the way what I'll try and do is sort of you know be honest to you about the things that we do and the things that we do not understand in this emerging picture of holography and averaging but why don't I stop there and take some questions let's thank Alex for this opening lecture