 Yeah, it's very, well, pleasurable to be here and indeed pleased to be talking to Samsung because I know you very long time already. I mean, we were together at the Institute. I'll talk a little bit about indeed the things we discussed there. And we also played a lot of tennis there, which was fun. You taught me how to play a nice surf and taught me many things about that, but also about physics. I'm going to talk about that. Indeed, I didn't announce the title yet because I had sort of two topics in mind. But this is the title, the world as an anomaly. I mean, we're talking about anomalies and the idea is basically that I think that eventually the physics laws that we have in nature are possible to derive from something deeper, something more underlying. And then, well, what is controlling the interactions and everything like that? I mean, since Samsung has been working on anomalies, there's actually an idea I want to present in this talk that has to do with the fact that we want to derive basically all of physics from an anomaly in some form. So I know Samsung's work, I mean, from early on when he was already, well, I think he was still in the landing ground where he worked with Anton on this idea of quantizing the fear-zero group using, well, a method that used a geometric action on the orbit of this fear-zero group. It's work that influenced my own, well, papers a lot actually. I thought about it at the time. It was just before, at the time when I had moved to Princeton and within a year from that, I met Samsung also there. And we spent two years at the institute discussing another topic and we almost collaborated on a paper on a background independent open string theory, string field theory. Yeah, it's just learning that, yeah. I put it there. Indeed, the acknowledgment says, I would like to thank Eric for in the four collaboration in the initial stages of the work. So I dropped out. I still feel guilty about this, so I want to make up that a bit. And so I'm going to talk indeed about also open string theory. But since it's an idea that I've been having, like about 20 years ago, I had to, I never published that one as well, by the way. I had to sort of remind myself of what my ideas were. And so this is one reason why I wasn't sure I should be able, it was possible to talk about this one. So I chose two topics, which has to do actually in both of these papers. But one will be basically thinking about the symplectic form of the low energy theory. So we have physics in our world, and we can describe it in terms of quantum mechanics, but also sort of classically. And one of the basic ingredients is the symplectic form. But there may be an underlying microscopic theory, and I'm going to present evidence for the fact that the symplectic form in our sort of physical world is actually derived from a very curvature of the underlying theory. This is actually connected to the work that Anton and Jean-Fan did on this geometric phase, because that's going to be a particular example. Then the part two I'm going to indeed think about open string field theory, and there I'm going to think about it as indeed a sort of arising from an anomaly. That's correct, so the two are kind of related. Although the generalization I'm going to discuss is actually more general. So this is indeed the geometric action that some someone and Anton wrote down on the Viral Zorro group. So it's on the diff S1, the few motions of a circle. And it has the form of sort of like a vessel minoterm, but then expressed in terms of a function that has a reparameterization on the circle. And indeed you can think about this action as a bury phase of a action of the group on a state. If you take a highest weight state, actually this is made very precise in work recently by Oblak, then you can compute the bury phase by looking at the change in the parameter f, and you integrate this change over the orbit, and you find a bury phase, and actually turns out that bury phase has precisely that form. And of course this is a form from which we can derive also a bury curvature, I mean here, and then this would be the same as deriving this emplactic form for this action. So this is an example of a connection between a bury phase and an anomaly, because this is also one way of thinking about this action. So I'm gonna generalize this, well first of all I'm gonna put it in a context of ADS-CFT, and you can indeed talk about this geometric action also in that context by thinking about special geometries in three dimensions in the ADS space, which I depict here as a cylinder with a boundary, which is the circle. If you write down the usual ADS geometry, you can apply a reprametization, or you actually can write down the BTZ geometry anyway, but you can apply a reprametization on the boundary, which is some diff as one. Actually in this case we have two reprametizations that work on the X plus and the X minus coordinates, so X plus minus are the combinations of the time and the angle. And then these functions that appear in here are functions of either X plus or X minus, and they can be written in terms of the Schwarzschild derivative of this function F, and this is a parametrization of a set of geometries in three dimensions. If you insert this into the action in the bulk, you can, because it's a solution to the equation of motion, rewrite this into a boundary action. And then that boundary action exactly has again this form of the geometric action. And this is something that had been worked out also in detail in recently in the paper by Kotler and Jensen. But what I'm claiming is that this is a special case of a much more general story, because I'm gonna be talking about more general boundary, first of all, ADS spaces in arbitrary dimensions. But also, thinking about more general geometries or even classical solutions in the bulk that are, well, obtained from some arbitrary fields. So I'm gonna actually use a notation that was introduced by Bob Wald to talk about just fields in the bulk very generally, where we have a Lagrangian in the bulk where if you vary the Lagrangian with respect to the field, so the fields may include the metric or anything else. And then if you vary it, you get the equations of motion, but then you also get a total derivative which precisely defines for you the symplectic potential. So we're gonna be looking at field configurations in ADS space that satisfy the equations of motion, and they define a phase space. And then you can derive the symplectic form from the symplectic potential by again varying once more. And then you have a two form on the space of variations of these fields. Now, the solutions of the equations of motions are determined by, because they're, well, they can be equations of motion of various type. But we're gonna assume that when we specify the boundary conditions of these fields on the cylinder, that the solution is uniquely fixed. So these parameters actually, the fields on the boundary are parametrized in the classical phase space. And so they will enter also in when we're gonna construct the symplectic form. So this is a generalization of the story for more general fields in ADS. And as I said, I mean, this can be having metric field. So we're gonna assume later on also that this is a repumpetization in variant field theory, and that means also I can construct, conserve charges associated to that. But first let me think about these fields as quite arbitrary. They're actually connected through the ADS-CFT correspondence to operators on the boundary, because the microscopic theory is gonna be living here and it's a CFT. And of course that's the usual rule that when we calculate the classical action of this theory, it will tell us something about the partition function of states computed on the boundary. Sorry Eric, what actually can constitute the symplectic form? Is it 2, 4, 4? What is the concept? Sorry, what is? What does it constitute for? What does it represent exactly? The symplectic form? Yeah. It's something that will allow us to define Poisson brackets or whatever on the phase space. It's a standard, I don't do anything else here. I mean, I'm just thinking about the space of classical solutions of a field theory in the bulk. But it also has an interpretation on the boundary because these objects are actually associated to operators that I can connect to the boundary states. And indeed there's a bulk symplectic form, but I write down here more explicitly. They are closed on shell, that means that I have a equation like this, that if I take the d of this, I actually can show that it's zero if I assume that these fields satisfy the linearized equations of motion. So I have both equations of motion assumed as well as the linearized equations of motion of these variations. Then this form is closed. Now this integral is being done over this slice, which is a time slice in the bulk. But since it's a closed form, I can deform this integration surface because it's gonna be closed everywhere, say below or above. So I can push it to the boundary. And that's actually gonna help me to interpret the same quantity in terms of something that's living on the boundary. Indeed, it's possible to write it like this, where instead of integrating over sigma, I'm integrating over the boundary of sigma, which is the circle or a sphere, times a real line. And it's actually the real line, just so you can go into the past. I have a similar expression going to the future because it's closed in the bulk. And this is an expression that you can evaluate now in terms of the boundary values. So you will see that this expression actually has components that are related to the fields phi. But there's also a symplectic form that basically tells you that these fields, which are the boundary fields, are dual to objects that are, well, basically the derivatives perpendicular to the boundary. Those derivatives are related to the expectation values of operators on the boundary. So there's gonna be an expression I can write down here that is a boundary interpretation of the same form. And as I said, it's a bulk quantity, but it's also a boundary quantity because of this closeness of this omega. In two dimensions, a special thing happened that the equations of motion are holomorphic equations, like d bar of something. Everything gets simplified because of that. Because in two dimensions, of course, of the gravity equations. But I will come to you. If it's C of t, then equation of motion is some kind of holomorphistic equation, like d bar of j equal to zero or d bar of z equal to zero. It seems to get simplified. This is true when it's pure gravity. But here I'm actually talking also about other fields in the bulk. And so there can be any operator on here. So I can switch on couplings for these operators. And so I have a huge class of perturbations I'm looking at. And this is an infinite dimensional space, much larger than just the space of repartmentalization. I meant only the case of gravity. Yes, I understand. So what I'm gonna do now, I'm gonna think about states. In the C of t, that are gonna be associated indeed with these boundary values. So you can think about them as being constructed by a path integral, where I specify these boundary conditions. They also represent, of course, states in the dual theory, because of the correspondence. But I'm interpreting this now as a calculation on the boundary, where I have a family of states. And then if I vary it, I can define the connection. And then in this case, it would be the berry connection. And the berry curvature can be written in this form. I'm gonna make this more precise because of course here I have to specify which fields are we really, the states depending on. Because we are dealing with a phase space and therefore we should choose some polarization. Where the states can only be dependent on say half of the variables in your phase space. There's a natural choice for this. Which is coming from the fact that I have radial quantization. So there's one way that I can think about the states here are being created by operators that act in the past. And I'm gonna think about preparing this state in not in real time, but sort of in Euclidean time. And then I'm gonna define what are called coherent states. Because then if you have a radial quantization, you can think about say the origin as being the point that equals minus infinity. And I draw a circle here in the plane, which is this circle here on the cylinder. And then there's a correspondence between I call now the coupling constants lambda because they're actually complex valued. And they are related to the operators here and have a basis of operators. It's gonna be all operators in the CFT. I can also take their derivatives and this is why I'm indicating some labels in here. And so these functions, these actually are then coefficients of functions that are multiplying these operators. So there's another way of writing the same expression where here there's a sum over I, instead of having these derivatives I choose some function that depends on the time, Euclidean time. It's a time that indeed goes up to t equals zero. And then I act with this exponent. There's some time ordered in here on the state zero, which is the the vacuum state that I started from. But by doing this I've turned on couplings which are associated to the boundary values of those fields. And they indeed parameterize part of the state, actually the state in this case, but also part of the phase space. This is a more explicit expression if you wanna think about what this integral is because I'm also integrating over the angles. So I have a lot of information in these coefficients. Indeed in gravity you might think that this is only a function of a little particular combination, as you say. But here I'm allowing even a more general case where these operators are functions of t and phi. More like in this time, like at the end we didn't try to define boundary. I think it's a well-operated boundary. Well, this is in its boundary, meaning in the field theory case as well. Because there's a similar thing of course in string theory where you have the boundary and the open strings and the closed string one. Yes, so there's some boundary couplings that we are switching on here. But of course these coupling constants have to do with something with bulk fields as well because they specify, as I said, the boundary conditions of the values of the fields in the bulk. And so there's also a classical solution associated to these parameters. Provided I also put boundary conditions on this other part. And this is where a nice thing happens. Actually you get here because of, I'm working in Euclidean. And here I'm gonna think about indeed the antipodal map as sort of complex conjugation. And I'm gonna impose that if I have couplings here that they satisfy some symmetry conditions, then we, this is a complex variable. And here I put the same function except complex conjugate. And I re-invert the time t. And so now I have a complete phase set of phase space variables. But it actually describes a scalar structure on the space of couplings. And so I can calculate, I think I missed the slide here. So first of all, there's an inner product that I can take because the partition function in the bulk will be the inner product of those two states. And that defines also the scalar form in my scalar structure. And so there is a symplectic form that I get from these parameters by evaluating these inner products and varying in this way. And that's a symplectic form that is defined on the boundary in terms of these coherent states. But as I wanted to argue that this symplectic form is actually the same as the symplectic form you would derive from the bulk equations. By, well, the procedure I described earlier. But this, as I said, is a boundary construction in terms of coherent states. Any questions about this? And indeed, when I take this bulk symplectic form, which was defined as an integral here, and I push it to the boundary. Then I did this earlier, then I get this integral on the boundary. If you do this for the parameterization in terms of coherent states, you will get this answer. But there's also an explicit expression you can write down. Namely, one other way of choosing a polarization, and actually I learned this from Samsung. There's always Darbou variables. And the Darbou variables are actually the ones where we take the couplings on one side and take the operator expectation values on the other. And so there is a Darbou way of writing this in a product where indeed you have the association between couplings and operators as being derived from this symplectic form. So in this case, Darbou is considered as local coordinates, right? Well, I have to say that even these operators, if you think about the whole structure of the coupling constant manifold, they may be defined only locally. I mean, if you do, there may be contact terms that are like curvature. I mean, anyway, this is probably a very similar local formula. Of course, I made a choice to write it in this form, but there must be a way in which I just exchange lambda bar and I change some other operators I integrated over that side. But the two expressions must be the same. Now note that here I still have an integral over all of this geometry here. If you think about the Virazoro case, this would be the stress tensor, but this would be the metric. So it's not yet in the form where we used the dual field sort of like the on the orbits. For that, namely, we expect to have an integral only over the boundary here. So this expression is more general. Actually, there's independent work. Actually, we discovered this in Amsterdam, but there was also work by these authors that sort of made the same observations. Although what we also were interested in is indeed how this worked for gravity. Because one of the applications of this formalism is to try and derive equations that are equivalent to the equations in the bulk. To the field equations in the bulk. So this is just a definition of this symplectic form. But I'm gonna now study a more general case. Name me that indeed of Divian morphisms. And here again, I use this ideas of Wald because there's a symplectic current. Actually, there's a, sorry, a nutty current where I used the potential. I mean, I just write it in the usual form. So we have some symmetries associated to reprimandizations. These can be arbitrary vector fields. In particular, I'm gonna use this to construct the VeroZero generators. And so this is some transformation on the fields. And then this is the general form of a noted charge. Actually, you can add a term to it, which is a total derivative. This charge, of course, is conserved. But actually, due to the reprimandization invariance, it turns out this J actually is the D, identically to the D of some other object, which we call Q. Though that means that when we construct the Hamiltonian or the other neutral charges associated to all these vector fields, they are again written as integrals over this surface. But since J is a total derivative, they become integrals on the boundary. And indeed, this is what translates on the boundary to the integrals of the stress tensor. So this is a different expression, as I said again, as what I had on the previous slide, where I had integrals over the entire boundary. But now I have, for the gravity case, I can have integrals over only the slice here. And this has to do with symmetries and actually word identities if you think about it from the boundary perspective. Excuse me, am I right that J is not the current, but the divergence of the current? And this is what I learned as sort of writing down a neutral current. It's the divergence of the neutral current. No, the D of this thing is zero. You want to have a bulk integral. This thing is the D minus one form which I can integrate. And it's the D of it is zero. Maybe I misunderstood the notation, sorry. So the object here is the- It's the dual of the current as a vector. It's a hodge dual. It's a hodge dual. It's not a vector, but it's a form. It's a form, the D minus one form. Actually, this is the most natural way to think about it, because then the conservation law is just closed. The D dimensions, the current is the D minus one form. Actually, it comes from this whole formalism, because even it's related to the symplectic potential. And here, of course, you also see it's a D minus one form. And actually, this term is important later, because one thing we're going to be interested in is also the Hamilton equations. This is what Walt shows is that this term is important. Anyway, what I'm going to- before I get there, let me indeed translate this now to a boundary statement. I'm going to specify what B is in a minute. There's something needed here, namely to make sure that this Hamiltonian that defined by integrating this or actually discharge satisfies the correct properties with the symplectic form that I also defined. And then this term is important. But it doesn't mean anything, because it's basically adding a total derivative to a current which is like kind of an improvement term. There's a nice paper now by Daniel Harlow, which explains the origin of this thing more precisely by making- by discussing actually that the action you write down in the bulk cannot be fully defining a theory if you have a boundary because you also have to write down boundary terms. And if you include those boundary terms, you actually understand also where these terms come from. So the water identities, what I'm going to think about actually indeed is this equation, namely if I take this charge, which is an integral over this, I can write this again as an integral over the current. And now if I insert this expression, this psi I'm going to be assuming is going to be transferred along the boundary. And then these objects don't have any components along the boundary when you integrate. You only have this object that's being integrated. So this is the symplectic potential, which also on the boundary is the symplectic potential for this form that I also wrote down. So I had delta lambda delta. Oh, and this is actually a way of writing then the symplectic form. If you interpret this form this way, the other terms are also very natural. This was the integral of the stress tensor. This is the D of it. We normally call a water identity because it basically tells you that the divergence of the current actually tells you how all the fields are transforming with the appropriate parameter that I have in here. So the water identity is basically naturally following from these manipulations by writing these currents as boundary terms. So that is actually derivation almost of this fact that I can write J as the D of a Q. You can also look at the variations, and this is where indeed this term actually is important. If I didn't have this term in here, this equation would not fully work. This is an important equation because if you take the variation of this current, you get the delta of the first term, but you also get the equations of motion in here because I take the variation of Lagrangian, but then also there's a term that gives you the potential again, symplectic potential, but there has to be a variation of this term that also combines together in such a way that you precisely end up with this combination. And this, of course, is again the symplectic form, but now evaluated the two variations, one being the gauge transformation associated to psi and the other one, an arbitrary variation. But now when you insert this into the definition of the Hamiltonian, which is the integral of this object, you find that we integrate this side, which is the symplectic form, and this is the Hamiltonian, and then this actually is nothing but the Hamilton equations. So you derive the Hamilton equations from this identity, but it's important, and actually, of course, it's not a surprise that we use the equations of motion in the bulk because this is another way of writing the equations of motion. So this is one way I actually wanted to, why I was interested in this. These equations also now have boundary interpretations of a more microscopic nature where we think about this as a symplectic form, but in the bulk it's the equation of motion. So there's some way in which the equation of motion must be derivable from the boundary by interpreting this equation. Because phase space is the space of solutions of the equation of motion. That's right, but we have the phase space in our disposal. You're right, in a certain sense, we already projected on the solution space, but you want to know, of course, which equations are being obeyed in the bulk. But you're right, I mean, it's true that there's a... kind of a little bit of a tautology in the sense that the boundary only knows about the boundary states, and therefore it should be... there's some equations that need to be imposed. But there's a whole industry of people that have been trying to derive the Einstein's equations from these kind of manipulations, and that actually works in a slightly different setting than the one I described here. Namely, this is a picture I should have... Yeah, this is the one I wanted. So, first of all, I can look for banyanos geometries. And this is a little bit of a complicated picture. I mean, here I've done something slightly different than the setup that I had before. Here I had time going upwards, you think about just... but these vector fields can be chosen to be quite arbitrary. There can be boost generators in the bulk as well. So, if I think about boost, they actually create horizons. And this is the picture I have here. So, here I have a generator which is a boost generator which has a horizon in the bulk, which is these planes. And this is a picture of a winder space, actually, in... I think I'm missing a slide. So, this is a winder space where there's a time flow this way, and there's a generator of this which is this operator, which again can be written as an integral of the charge. So, this is called the model Hamiltonian. It's the integral of the stress tensor times this vector of psi again. This equation actually is an equation for the variation of what's called the relative entropy. I'm actually a little confused by my slides, sorry, not... Oh, here. You mean Poznan entropy, right? Poznan entropy? It's classical, right? Classical Poznan. No, it's... Actually, I think I... This slide should have been without these equations first. Just look at the top equation here. There's an object that's called... which is the stress tensor integrated against psi, which is called the model Hamiltonian. If you take the difference with that and the entanglement entropy, you get a combination that we call relative entropy. And the entropy is the entanglement entropy, the quantum entanglement entropy across this surface. So there are various contributions, one that is an integral over this boundary and this is an integral over this boundary. And this is the symplectic form again, but now only integrated over this part of the space. And so there's an identity that tells you that the difference between the entanglement entropy across this surface and the charge there is equal to this symplectic. So you have here a situation where we split the space into two parts and so there's an entanglement across this region. I'm not gonna spend a lot of time on this because I mean it was sort of a side remark. There's an expression which is due to Cardi and Calabresa of what kind of entanglement entropy I have here. But why is it defined? Usually it is ill-defined the entanglement entropy if you have two ranges adjacent to one another. So that's a very good point. I also made that point when Van Ramson was talking about it. There's some way of regulating this at these points. You can even split these points a little bit by having another interval in between and then you can have a finite factor in the middle. Actually this has been discussed quite recently by Faulkner in quite a beautiful way that you indeed have the finite. You can regulate this in a nice way. And actually these expressions also come from regulating these entanglement. Anyway it's just a generalization of these neutral charges which Walt introduced where instead of integrating this quantity Q over this region you integrate it here. But the identities are the same because they can use the fact that the variation of the current is the symplectic form and that's what's here in the book. Anyway, what I wrote down these equations is actually the application if you would do this for Bagnata geometries. So think about this as ADS-3 and I have here a boundary state that's parameterized by a function F, the same function that Samsung considered namely in Diff as 1. Then there is actually an entanglement entropy expression that is given by the difference between the locations of these endpoints. So there are two points X1 and X2 and this is the reprimatization that you would do and this is the expression of the killing vector field that actually you have to integrate on the boundary. It's zero on these endpoints so that's the boost generator and it's normalized in a particular way and then it's multiplying the Swachin derivative because that's the expression for the stress tensor on the boundary. And then this variation minus that variation must be some expression that we can integrate in the bulk that would should be the symplectic form in the bulk and it's some way of sort of deriving these equations of motion. And as I said, this is a way of thinking about this that this implies the linearized equations. I want to close this part. Anyway, I just want to summarize that I discussed the relationship between the barrier curvature in the boundary and the symplectic form in the bulk. It's much more general than just for gravity. It also has other fields associated to that. You can try to derive the bulk Hamilton equations from the boundary and I think this sort of hints towards derivation of the bulk equations from a more microscopic perspective. And there are some ideas of how to do this and I describe that if you do this for banyardous geometries you actually connect to this work of Samson and Anton. So this is the first part. Now I want to get to the other discussion which is sort of related to the things that we were discussing at the time which had to do with background independent open string theory. I also indeed want to connect to the idea that indeed that my thing is that there is some way in which we can derive our field equations from an underlying theory. What I talked about here was sort of like where we go from open strings to closed strings which is one way in sort of thinking about an emerging step. But now I want to even go one layer deeper where I want to think about where the open strings may come from. So this was the idea that the open string field theory Lagrangian can be written down as a Turing-Simons action sort of suggests this relationship. So this was an idea I worked on about 20 years ago. I have tried to remember what I all thought about. I didn't write it down because indeed in the end there are some things that I'm not totally sure how they work but I had some nice observations that I want to tell you about. So this is Ed Witten's original paper about non-commutative geometry and string field theory. Here he writes an action that looks indeed like the Turing-Simons action but it involves all kinds of quantities that they had to define like star products and he did this by gluing together the open strings in some way like this where he chose a midpoint and there are two sides of the string that have to be matched together but since this looks like a Turing-Simons action the natural question is can we sort of think about this as a coming from an anomaly? And there were many other reasons why I thought there might be another formulation that is dealing with well this interaction slightly differently. The idea would indeed to write down an action that is not defined on the open string but sort of on half of it. I want to think about this actually there's of course an issue actually the same issue that you mentioned before actually Witten had the problem of defining the half string because he wants to split the Hilbert space of the open string in the two parts which is not a very easy thing to do in this case as well because actually even the open string Hilbert space doesn't factorize in the left and the right and there were all kinds of other issues but anyway what I wanted to indeed think about is if I think about this as an algebra what is the space that it's acting on and it's motivated also by Consway of thinking of course Witten was already inspired by that but if he would have followed this idea of having a spectral triple I'm actually denoting the gauge field now with the same notation as the algebra because I'm thinking about A as being the algebra of open strings which indeed this is star algebra but then there's a Hilbert space and then you may ask what is this Hilbert space actually consisting of and looking in the picture here the Hilbert space must be associated to these half strings so the idea would be to write down this action where these are half string fields and then integrate out these well fermion fields do a computation of this sort and get back the other action so what does it mean? Ah, so what I'm thinking about is there is a Dirac operator and I want to actually have some way in which I'm going to have a family of Dirac operators parameterized by my open so it is not connection here I'm going to think about it as the connection and this is an element of algebra A well I mean there's some let's put this in quotation marks in the sense that I want to think about indeed the open strings actually is the thing that are defined in algebra just like here so these are connections these are connections they are not connections, they are open string fields it's just a question of notation A is an algebra I know, I know, I'm sloppy here but the idea would be like this I mean if I have these open strings and sorry for using this notation I'm multiplying because I mean any states open string states should be possible to act on a state which is sort of half of a string state and then the multiplication is like gluing those together and then I obtain a new half string state which is the product of those two and so there's a midpoint that I have to put here which is where the two sort of have to meet and so my half strings have one midpoint there and the other point which is the open string on the other side and the proposal I thought about was thinking about these half strings that sort of strings with mixed Dirichlet and Neumann boundary conditions because then I have a point which I fix and the other point is sort of free to move and there's another reason why I like this because it's, I'll explain in a minute actually it's sort of natural form T-duality or things like this Dirichlet and Neumann boundary conditions are exchanged and this string doesn't know so much about which space it's moving on because it doesn't know whether it's, this is, it's one free side or that one and as I'll explain here indeed you can think about these end points sort of as being D-1 brains because they're kind of objects that are floating around in space and as I said T-duality exchanges Dirichlet and Neumann boundary conditions and there are many cases where the end strings play an important role and often appear to be more fundamental one example I like is if you think about a D-brain system on D-4s and D-0s you can write down the ADA HM equations namely because if I have different kind of D-brains then there are strings that are connected to one brain say 4-4 but there's also the 0-4 strings and they are like the end strings and then you have some equations of this sort where you can indeed transform well construct the dual gauge fields sort of in the sense of T-duality by again computing baryfaces I mean I'm not going to go into the details here the ADM equations have data in them which are actually having two numbers namely K being the number of instantons N being the rank of the gauge group but then these objects are actually can be thought of as D-N strings that's actually the way that these ADHM equations appear in string theory and similar in NAMD constructions Neumann, just one question can I, for the sudden dynamics of the scattering objects of the D-1 branch can I form open strings or connect them together to open source ones? so one way I'm going to think about this and actually this is what this equation suggests is that if I glued together two of them then they would again become an open string but there's another way in NAMD you can also glue them on that side and then actually they construct an object that lives in the T-dual space sign because T-duality actually makes from listing a Norman boundary condition and so there's some way in which T-duality can be thought of as cutting an open string in the middle and gluing it back together on the other side and that actually is the way that the NAMD transformation actually works on the torus is basically it changes the rank of the group with the gauge, with the instanton number and the other way around so the boundary conditions are being exchanged so these objects actually they live kind of on the space and it's T-dual in some way anyway this was a... A point in space time or a line? Yeah, a good point I'm now thinking as a point in space time because it's D-1 but there are examples like if you do D0, D4 but then I have an additional time in them you can also think about them as Euclidean D3 and D-1 I have not, as I said I mean these are ideas I was developing and there are various variants that you can go into but the most natural one would be take D-1 brains which are points and then you also have to do a T-duality in all directions including time What do you say of this ADHM data on living on torus? I don't understand If you take the... there is a construction of the ADHM equations in string theory using open strings where we have two kinds of brains one describing the instantons and one associated to the rank of the case group and the data are basically objects that have two indices namely k times n, k being the instanton number and being the rank of the case group those objects are strings like this sort where you have a D-slab boundary condition and an Ormian boundary condition and so there is some way that the ADHM equation has this structure in it So Eric, the idea is basically that the ADHM equations 86 paper in a modern way which are degrades rather than thinking about strings containing the middle or something That's right Think about degrades and give a notion of the A as an object which acts Yes changes the boundary condition Yes, but anyway if you are integrating them out you should get the action back Anyway, this was motivation not a full description So what I want to do is actually construct sort of like what looked like a Dirac operator and the idea would be integrating them out to get this action and there are ideas that namely you can mimic what we do in two dimensions with the chiral anomaly or there is a parity anomaly in three dimensions that also produces transimons actions and I want to do a calculation, so you can compute a determinant expand it out use things like collapsed propagators or whatever, in some way they usually calculate these anomalies The other idea is namely there is a three-dimensional space hidden here even in witness action that space is actually the mobius group that acts on the sphere because the reason why there are three terms in here had to do with ghost number counting these objects should be thought of as being one form in some way or an integral that wedges them There is I'm going to actually define that now more precisely because that was actually a problem I would have said in Witten's construction Witten defined his states first of all as states in Hilbert space this open string had ghost number minus a half and in order to absorb all the right ghost numbers he had to define something called the midpoint operator which was kind of a little bit of a funny thing because it's some exponent of a bosonized ghost field with a power of three halves if you would calculate the conformal dimension of this thing it's some funny number like minus nine over eight another problem is that the splitting of the string in two halves is not very well defined actually the Hilbert space doesn't factorize while if you think about these things as operators you would like them to be sort of in a tensor product of the dual vector space with itself I'm going to try and solve both problems I'm going to define a new midpoint operator that's going to turn the usual open string field into an operator of this kind and it worked quite beautifully and precisely in a way that sort of uses the critical dimension the idea is actually the following if you think about the open string modes they have mode expansions are using just the usual string theory modes and the ghosts and I'm going to split these modes into the even modes and the odd modes when you have the odd modes when you restrict them to the half string they already obey the mixed Dirichlet and Neumann boundary conditions so one way I think about actually the open string of course the open string is a line but you can think about the modes going to the left and to the right as if they're going around the circle and I'm thinking about that circle now as being doubly winded where there's a crossing point that's going to be the midpoint so I'm going to split the string by making this crossing disappear so they're going to be two untwisted ones so what you then work out actually that the operator that I have to insert there actually is a twist operator which also twists the X coordinates as well as one of the ghost fields so I have two modes which has two fields namely I'm going to actually split all the fields in the even and odd one so I have fields which are called C- which are the odd ones and are plus fields and then you twist the even ones that's this operator this is going to be the twist operator for the even ghost fields and then I have this the operator as well together this has ghost number 3H which is the same number that Witten required and so this new twist operator actually precisely splits it and actually makes of my string state something that is in this product and actually is an operator that multiplies acts on half strings and what's the dimension of it it's conformal dimension is minus 1 this is 26 over 16 which is 13 over 8 minus 5 over 8 is 0 so this is an operator that you can put down in 26 dimensions that has conformal dimension 0 and it does exactly what you want namely it splits the strings into in a way where it becomes a product so it really becomes an algebra now because now I can start multiplying these things so my star product is basically just really multiplying in this way can that be repeated for Neville-Schwarn a first super string that Witten was not able to remember he had a problem using feature-changing because there was no other way of defining super strings with him I didn't think about it anyway that was a problem I had this construction 20 years ago and the fact that I have a twin-border can also be proven by the fact that Herman actually invented this same construction recently when he was doing the TT bar definitions he needed the same thing and I could tell him about this fact he hadn't noticed the C-mon that I could multiply it so that gets dimension 0 and an important thing that they need furthermore is with the BOST charge which is also a very important thing indeed you can show this that Q this is the BOST charge with this new midpoint operator actually it's 0 and there's a very funny thing that I have to admit I don't fully understand what is happening there so this is a C-1 but this object therefore has a spin 1 this has the form of a C times a spin 1 if you have this operator this is the spin field of 26 bosons so I say the bosonic stream this is the so this is the twist field I should say this is the spin field for the ghost so this object is a current that I can integrate and defines for me a new charge which has ghost number half if you work out its square you can really work it out using the operator product expansion that gives you the BOST charge something I don't fully know why that is possible but if you take this operator and it's operator product expansion with itself actually the sigma with sigma gives you the identity but then you have to expand because of the 0's here and you pick up the stress tensor and the same way you pick up these other terms so you get actually exactly this combination from the operator product here this relation I kind of don't understand exactly the meaning of it how am I doing with time by the way you running short of time okay I'm going to finish oh it's a little over it's 15 minutes you're a little over okay alright I'm going to stop I'm going to actually stop where I don't know the answer I mean I try to construct a good Dirac operator the most natural thing is to write down the BOST charge on the half stream where you integrate say to the midpoint the only thing that I had chosen is namely that also the ghost satisfy dearly boundary conditions maybe I should modify this because actually then in that case this operator is not very well defined because these ghosts then have half integer mode expansion well the stress tensor likes to have an integer mode expansion so I don't know how to do this integral the other thing as I said I actually have ghost number minus one which means you have to absorb something like two additional ghost zero modes so I was not really able to solve these problems actually this is where I got stuck a bit anyway I'm still there with what I wanted to achieve but I have to find the correct definition of this operator and it's just an idea that I had at the time to see if there's some other formulation of open strings where we can think about this action as being induced from an action of that sort anyhow that was what I wanted to live with thank you any questions? when you had this square root of q yes so that current is that a conserved current which you integrate or does it depend on the contour of integration you know for the BRSD current usually you integrate the BRSD current which is conserved so it doesn't matter it's a conformal state good point so there's a spin one which you can write down a 1,0 component and a 0,1 component I mean as a total current probably it is conserved anyway it's something that needs to be worked out so when you start splitting this string into two halves so this sounds like you work on some kind of manifolds these corners right you have a bulk you have your string you have those corners where you split it so what would live in the corner right you put in there some operators but what kind of is it some kind of algebra or what's in general would live in in the corner is there some kind of idea of what's the structure you mean on the world street you're talking about I mean there are certain boundary conditions that can be modified I mean this is what usually we have no I don't think I don't have a lot to add to what I said here to be honest maybe I just kind of a second question because from the very beginning I think it's something strange because it's 120 degrees maybe you should define the location depending on the angle it's not just an algebra the angle was the thing that was part of the story of witness construction I think my construction actually goes away with it because I merely get to an algebra so I don't even need to work with the same pictures of course I have an arbitrary midpoint sort of still I mean that issue is still there there's some way you have to break the re-parameterization invariance to do this but once you calculate only B or C invariant quantity that dependence must disappear which is the same actually as in witness construction no and that's I think it's almost similar he had to go through some consistency conditions to make sure that everything worked and but this maybe even make it make it easier I did some conjecture how to relate boundaries in field theory with witness Kubik so remember in the paper you mentioned I derived the tension for boundaries in field theory partition function and things like that so I will explain the literature there are two terms in the action I derived for boundaries in field theory I'm impressed that you remember it but that's good so there are two terms and one of the terms was partition function boundaries I wanted to interpret as taking the disk to my fingers maximum is 120 degrees and that's upside, upside, upside I got 80 the first term was derivative partition function along the wet function so that one is AQA where Q and beta are identified in terms of the space of deformations so I wanted to interpret that AQA comes from beta to Z and the AQ comes from Z A is 120 degree divided by three parts on one side you have boundary condition as you say arbitrary and on the other side no problem always goes in the middle because you need A of X and here is X and here is the boundary of X okay anyway we will discuss that so let's think over again