Yeah, it's very, well, pleasurable to be here and indeed pleased to be talking to Samson,

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 serve and taught me many things about that. But also about physics. I'm going to talk about that.

Indeed, I didn't announce the talk yet, 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 Samson 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 Samson's work, I mean, from early on when he was already, well, I think he was still in Leningrad, where he worked with Anton on this idea of quantizing the Viorzoro group using, well, a method that used a geometric action on the orbit of the Viorzoro group.

It's work that influenced my own 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 Samson also there.

And we spent two years at the institute discussing another topic. And we almost collaborated on a paper on background independent open string theory, string field theory. Yeah, it's just well in there for you. I put it there, indeed, the acknowledgement says,

I would like to thank Eric Flinder for 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 didn't, wasn't sure I would, should be able, should, was possible to talk about this one. So, I chose two topics, which has to do actually with both of these papers.

Part 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 bary curvature of the underlying theory. This is actually connected to the work that Anton and Sjampon 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 as 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 Sjampon and Anton wrote down on the Fiero-Zorro group. So, it's on the diff, S1, diffeomorphism of a circle.

And it has the form of sort of like a Wesu-Mino term, but then expressed in terms of a function that has a reparametization on the circle. And indeed, you can think about this action as a bary phase of a action of the group on a state.

If you take a highest weight state, actually this is made very precise and worked recently by Oblak, then you can compute the bary phase by looking at the change in the parameter f. And you integrate this change over the orbit and you find a bary phase, and actually it turns out that bary phase has precisely that form.

And of course, this is a form from which we can derive also a bary curvature, I mean here, and then this would be the same as deriving the symplectic form for this action. So, this is an example of a connection between a bary phase and an anomaly,

because this is also one way of thinking about this action. So, I'm going to generalize this. Well, first of all, I'm going to 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 reparametization, or you actually can write down the BTZ geometry anyway,

but you can apply a reparametization on the boundary, which is some diff as one. Actually, in this case, we have two reparametizations 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 that boundary action exactly has, again, this form of the geometric action. And this is something that had been worked out also in detail 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 going to 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 going to 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 going to 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 2-form on the space of variations of these fields. Now, the solutions of the equations of motion are determined because there can be equations of motion of various types, but we're going to assume that when we specify the boundary conditions

of these fields on the cylinder, that the solution is uniquely fixed. So, the fields on the boundary are parametrized in the classical phase space, and so they will enter also when we're going to construct the symplectic form.

So, this is a generalization of the story for more general fields in ADS, and, as I said, this can be having metric fields, so we're going to assume later on also that this is a reparameterization invariant field theory,

and that means also I can construct conserved 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 going to 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 two-form, or what is the content? Sorry, what is it? 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 sign that 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 is a bulk symplectic form, which I write down here more explicitly. They are closed on shell, that means that I have an equation like this,

that if I take the d of this, I actually can show that it's zero, if I assume that these equations, 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 going to be closed everywhere, say below or above, so I can push it to the boundary,

and that's actually going to 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 sphere,

times a real line, and it's actually the real line, just say, going to 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 going to be an expression I can write down here, that is a boundary interpretation of this same form, and, as I said, it's a bulk quality, but it's also a boundary quality, because of this closeness of this omega.

Because, in two dimensions, of course, of the gravity equations, but I will come to... The equation of motion is some kind of homophistic equation,

like, p-bar of j equals zero, or p-bar of r-10 equals zero, and things 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 repurpose. I meant only the case of gravity. Yes, I understand. So, what I'm going to do now, I'm going to think about states in the CFT, that are going to 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 in this case, it would be the Berry connection, and the Berry curvature can be written in this form. I'm going to make this more precise, because, of course, here I have to specify, which fields are we really, these 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 going to think about preparing this state, not in real time, but sort of in Euclidean time. And then I'm going to 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 at t 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 going to 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 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 parametrize 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 want to 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 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 title, like Edwardian tract to define boundary. I think it is a well-operated boundary. Well, this is in its boundary, I mean 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 ones. 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 I'm working in Euclidean, and here I'm going to think about indeed the antipodal map as sort of a complex conjugation, and I'm going to impose that if I have couplings here,

that they satisfy some symmetry conditions, and then we... This is a complex variable, and here I put the same function except complex conjugate, and I invert the time t. And so now I have a complete 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 a 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 the 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 parametrization 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 d'Arboo variables, and the d'Arboo variables are actually the ones where we take the couplings on one side

and take the operator expectation values on the others. And so there is a d'Arboo way of writing this inner product, where indeed you have the association between couplings and operators as being derived from the symplectic form.

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 reparametrization, if you do, yeah, 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 integrate it 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 Virozoro case, this would be the stress tensor, but this would be the metric. So it's not yet in the form where we use the dual field,

sort of like 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 book, to the field equations in the book.

So this is just a definition of this symplectic form, but I'm going to now study a more general case, namely that indeed of Diviomorphisms, and here again I use this idea of Wald, because there is a symplectic current, actually there's a sorry, a neutral current,

where I use the potential, I mean I just write it in the usual form, so we have some symmetries associated to repurmatizations, these can be arbitrary vector fields, in particular I'm going to use this to construct the Virasoro generators, and so this is some transformation on the fields, and then this is the general form of a neutral 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 repurmatization 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 water 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? This is what I learnt as sort of writing down a neutral current. It's the divergence of the neutral current? No, the D of this thing is zero.

This thing is the D minus one form which I can integrate, and the D of it is zero. Maybe I misunderstood the notation, sorry. So the object here is...

It's the dual of the current as a vector. You mean hodge dual? It's a hodge dual. It's not a vector, but it's a form. It's a form, a D minus one form. Actually this is the most natural way to think about it, because then the conservation law is just closed.

D dimensions, the current is a D minus one form. One form. And 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, 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 we find by integrating this or actually this charge 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, 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 O, 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, and this 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 over 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 to two variations,

one being the gauge transformation associated to xi, 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 book, 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 equations 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 Banyananda's 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 this, but these vector fields can be chosen to be quite arbitrary, they 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 Rindler space actually in...

So this is a Rindler 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 psi again. This equation actually is an equation for the variation of what's called the relative entropy. I'm actually a little confused why my slides are not out here.

You mean Boltzmann entropy, right? Boltzmann entropy? It's classical, right? Classical Boltzmann. 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 going to 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 Carty 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 stretches adjacent to one another. So that's a very good point. I also made that point when Van Ramsunck 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 need to have the finite... you can regulate this in a nice way. And actually these expressions also come from regulating these entanglements.

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, there's actually... what I wrote down these equations is actually the application, if you would do this for Banyarda's geometries. So think about this as AdS3. And I have here a boundary state that's parameterized by a function f, the same functions that Samsung considered, namely in DivS1.

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 rate permittization that you would do. And this is the expression of the Keeling 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 Schwarzian 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 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, I mean, 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 the derivation of the bulk equations

from a more microscopic perspective. And there are some ideas of how to do this, and I described that if you do this for Banyarda's 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 emergent step. But now I want to even go one layer deeper, where I want to think about where the open strings may come from. And just the idea that the open string field theory Lagrangian

can be written down as a Chern-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 Chern-Simons action,

but it involves all kinds of qualities 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 Chern-Simons action, the natural question is, can we sort of think about this as coming from an anomaly? And there were many other reasons why I thought there might be another formulation that is dealing with 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 is of course an issue, actually the same issue that you mentioned before, actually Witten had a problem of defining the half string, because he wants to split the Hilbert space of the open string into 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 a left and a right. And there were all kinds of other issues. But anyway, what I wanted to indeed think about is what is the space of, so if I think about this as an algebra, what is the space that it is acting on?

And it is motivated also by Kahn's way of thinking, of course Witten was already inspired by that, but if he would have followed this idea of having a spectral triple, I am actually denoting the gauge field now with the same notation as the algebra, because I am thinking about A as being the algebra of open strings,

which indeed is a star algebra, but then there is 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? It is T plus A actually. Ah, so what I am thinking about is there is a Dirac operator, and I want to actually have some way in which I am going to have a family of Dirac operators,

parameterized by my open... So A is not connection here? I am going to think about it as the connection. And this is an element of algebra A? Well, I mean, there are some, let's put this in quotation marks, in the sense that I want to think about the, indeed the open strings actually as the thing that are

defining in algebra, just like here. So these are connections, but these are connections? They are not connections, they are open string fields. It is just a question of notation, A is an algebra... I know, I know, I am sloppy here.

But the idea would be like this, I mean, if I have these open strings, and sorry for using this notation, I am 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 is 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 as 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 is another reason why I like this,

because it is, I will explain in a minute, actually it is sort of natural form T-duality or things like this, because Dirichlet and Neumann boundary conditions are exchanged, and this string doesn't know so much about which space it is moving on, because it doesn't know whether this is its one free state or that one, and as I will explain here indeed,

you can think about these endpoints sort of as being D minus one brains, because they are 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 D4s and D0s, you can write down the ADHM equations, namely because if I have different kind of D-branes, then there are strings that are connected to one brain, say 4-4,

but there is also the 0-4 strings, and they are like DN 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 baryphases.

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 DN strings,

and that's actually the way that these ADHM equations appear in string theory, and similar in NAMM's constructions. Anyway. Just one question, can I tighten the dynamics of these half-string objects, so these D-1-branes, can I put them open strings, or connect them together to control forms?

So, one way I'm going to think about this, and actually this is what this equation suggests, is that if I glue together two of them, that then they would again become an open string. But there's another way, and you can also glue them on that side. And then actually they construct an object that lives in the T-dual space sign,

because a T-duality actually makes from this thing a Neumann 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 NAMM transformation actually works on the torus,

it's 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 an... A point in space-time, or a line?

Yeah, a good point. I'm now thinking of them as a point in space-time, because it's d minus one. 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 minus one. 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 to take d minus one range, 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 Taurus? I didn'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 gaze group. And the data are basically objects that have two indices,

the k times n, k being the instanton number, n being the rank of the gaze group. Those objects are strings like this sort, where you have a Duschle boundary condition and a Norman boundary condition. And so there is some way that the ADHM equation has this structure in it. So, Eric, the idea is basically that

Interbright written 86 papers in a modern language of Debray, rather than thinking about things containing the middle or something like that. Think about Debray and give a notion of the a as an object which changes the boundary condition.

Yes, but anyway, these objects should be more fundamental in that sense that by 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 looks 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 John Simon's actions.

And I want to do sort of a similar calculation. So you can compute a determinant, expand it out, use things like collapsed propagators or whatever, some way that usually you 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 Möbius 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-forms in some way where there is an integral that wedges them. There is... So you're sort of trying to write on this stuff all the properties on this field?

I'm going to actually define that now more precisely. Because there 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, and they 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 Boussinesq ghost field with a power of three-halves. Of course, 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. Well, 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

and that's going to turn the usual open string field into an operator of this kind. And it works 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 on 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-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 a 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, two fields namely, I'm going to actually split all the fields in the even and odd ones. So I have fields which I call 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 operator as well. Together this has ghost number three halves, which is the same number that Witton 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 acts on half strings. And what's the dimension of it? It's conformal dimension is minus one.

This is 26 over 16, which is 13 over 8. Minus five over eight is zero. So this is an operator that you can put down in 26 dimensions that has conformal dimension zero. And it does exactly what you want, namely it splits the strings

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 this be repeated for Negutron, for superstring that Witton was not able to remember?

He had a problem using picture-changing because there was no other way of defining superstring in the first time? I didn't think about that. Anyway, that was the problem. Anyway, I had this in 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 deformation. He needed the same thing. And I could tell him about this fact. He hadn't noticed the CMON that I could multiply it so that it gets dimension zero. An important thing that they need furthermore, this operator commutes with the BSD charge,

which is also a very important thing. Indeed you can show this, that Q, this is the BSD charge, actually with this new midpoint operator, actually it's zero. And there's a very funny thing that I have to admit I don't fully understand what I'm doing there. So this is C minus one,

but this object therefore has spin one. This has the form of a C times spin one. Indeed if you have this operator, this is a spin field of 26 bosons. So I say the bosonic string. This is the, sorry 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, it gives you the BSD charge. Something I don't fully know why that is possible,

but if you take this operator and its operator product expansion with itself, actually the sigma with sigma gets you the identity, but then you have to expand because of the zeros here and you pick up the stress tensor and the same way you pick up these other terms. So you get exactly this combination

from the operator product here and that gives you this relation. I kind of don't understand precisely the meaning of it. How am I doing with time by the way? You're running short on time. Okay, I'm going to finish. It's a little over.

It's 50 minutes, you're a little over. Okay, all right, I'm going to stop soon. I'm going to actually stop where I don't know the answer precisely. I mean, I tried to construct indeed a good Dirac operator. The most natural thing is to write down the BSD charge on the half string where you integrate say to the midpoint. The only thing that I had chosen is namely

that also the goals satisfy Dirichlet boundary conditions. Maybe I should modify this because actually then in that case this operator is not very well defined because these goals then have half integer mode expansion while 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,

is the half string modes actually have goals number minus one which means you have to absorb something like two additional goals, 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.

Anyway, 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 leave with. Thank you.

Any questions? When you have this square root of Q, so that current, is that a conserved current which you integrate? Or does it depend on the contour of integration? You know for the BSD current, I usually integrate the BSD current which is conserved, so it doesn't matter if it's a component space.

Good point. So there's a spin one, which you can write down a one comma zero component and a zero comma one component.

I mean, as a total current, probably it is conserved. Anyway, it's something that needs to be worked out here, right? So when you start splitting this string into two halves, so this sounds like you now work on some kind of manifolds with corners, right?

You have a bulk, you have your string, and also you have those corners where you split it. So what would live in the corner, right? You're putting there some operators, but what kind of physics, some kind of algebra, or what in general would live in the corner? Is there some kind of idea of what's the structure? You mean on the world feed 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 have a lot to add to what I said here, to be honest.

Maybe I should ask a second question, because from the very beginning, I feel it's something strange, because it's 120 degrees, yeah? Maybe we should find an application depending on the angle. The angle was the thing that was part of the story of Witten's construction.

My construction actually does away with it, because I immediately 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 repurposition 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 Witten's construction. I think it's almost similar.

He had to go through some consistency conditions to make sure that everything worked, but this may even make it easier. I had some conjecture how to relate the boundary string field theory with Witten's cubic theory.

So, remember in the paper you mentioned I derived the tension for boundary string field theory in terms of disc partition function and things like that. So, I'll explain that later. There are two terms in the action I derived for boundary string field theory. I'm impressed that you remember it, but that's good. So, there were two terms,

and one of the terms was partition function, boundary function. I wanted to interpret it as taking the disc, dividing as Maxim said 120 degrees, and then subside, subside, subside, or A cubed. The first term was the derivative of partition function along the wedge 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 dz, and the A cubed comes from z,

where A is 120 degrees divided by three parts. On one side you have a boundary condition, as you say, arbitrary. And on the other side, no, and problem always was in the middle. You need A of x, and here is x,

and here is the boundary over here. Okay, anyway, we'll discuss that. Okay, so let's thank Eric again.