Where does quantum field theory come from?
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Friedan, Daniel
This will be an interim report on a long-running project to construct a mechanism that produces spacetime quantum field theory; to indentify possible exotic, non-canonical low- energy phenomena in SU(2) and SU(3) gauge theories produced by this mechanism; and to calculate signals of these phenomena to see if they can be used to check whether the proposed mechanism operates in the real world. The last effort is still ongoing.
 I'm going to be talking about a project that has been going on for ridiculously long time. I should mention it makes use of discoveries by many other people, some of whom are in the room. At the appropriate times I'll try to mention them, but if I forget I think it will be obvious. I'll be sketching a proposal mechanism that produces quantum field theory in space-time, a real quantum field theory, a measure on space-time fields which is equivalent to a Hamiltonian. At the same time produces a quantum string background. I'm reasonably confident that it's the correct idea of a quantum string background, but as things stand that's a purely formal question. Whether this is how actual quantum field theory is produced by which I mean the one that describes the real world, that I don't know, but I would like to. At the moment I'm trying to derive from this proposal a predictions of unexpected low energy phenomena that might actually be seen or not seen in experiments. More explicitly the possibility that this mechanism produces some rather bizarre low energy degrees of freedom whenever there's SU2 gauging variance. These will be associated with well-known non-trivial homotopy group of the space of SU2 gauge fields on Euclidean spacetime. I get this too. Okay, I should put in this caveat. It's extremely speculative fundamental physics. A long shot. I like that word. It comes from when the British were shooting at the French. It goes back as I said a very long time, the beginnings. It goes back to this two-dimensional generalized nonlinear model where doing two-dimensional quantum field theory, the field lies in some manifold M and the action is given by what mathematicians call the energy functional. The parameters in the action are comprised in a Romanian metric on the target manifold M. There are infinitely many coupling constants which you can think of as the Taylor series of the metric at some point, X0 in M, then you get the metric, its first derivative, second ad infinitum, infinitely many couplings, but they are all renormalizable and the renormalization can be done co-variantly with respect to the target manifold and the renormalization group equation has an expansion in sort of the inverse size of the target manifold which starts with the Ricci tensor. This work was inspired by Sasha Polyakov's discovery that the two-dimensional nonlinear signal model is renormalizable, where the target manifold there is the two-sphere with O3 symmetry, the round two-sphere, so the only coupling constant is the inverse size of the two-sphere and Sasha discovered that that was an asymptotically free coupling constant. So this can be thought of as a generalization of that. So what the renormalization, this was, I was a graduate student at the time and Ken Wilson's view of the renormalization group was how I learned renormalization. And one thinks of the renormalization group as a machine that acts on quantum field theories, in this case two-dimensional quantum field theories, and drives them to something new at larger and larger distances. So here the two-dimensional renormalization group, and it's doing this at very short distances, it drives this metric on the manifold M towards a fixed point, which would be then a solution of Ricci tensor equals zero. So it is a two-dimensional renormalization group, it's a mechanism that produces what looks like a classical field theory. Now by mechanism I'm not talking about a mechanism that acts in real time, it's an abstract mechanism that sort of acts on theories. Okay, so at the time that was extremely exciting, though I think only to me. It seemed to me that it could be a clue to where the laws of physics come from. It looked like Einstein's equation. Not exactly, but it looked like it. Okay, so the questions that arose were, first of all, it isn't Einstein's equation. It's a field equation, but not a physical one. So the question was, is there something related that would produce a realistic field theory? I mean it should come from an action principle and it should have a realistic collection of fields, not just a metric on space time. I forgot to mention that for this picture one takes the target manifold M to be space time. So it should have gauge fields, fermion fields, scalar fields. Then there's a question, I was a bit sloppy when I said that the renormalization group drives you to a fixed point. It only does that if that fixed point is an attractor. If that fixed point has unstable directions, then you have to do some tuning to get driven there. And at the time, I didn't see, I mean you can eliminate all but a finite number of directions, but I couldn't see how to get rid of the possibility of a finite number of unstable directions. So you get stability in this sort of scheme. And then the big question, this is what this produces is a classical field theory, where does quantum field theory come from? What makes quantum field theory? So within a few years, the first two were answered when this two-dimensional field theory was interpreted as the string background. So the idea is that the world surface of a string propagating in space time is described by a two-dimensional field theory. And if the space time has a curved metric, the two-dimensional field theory will be this two-dimensional nonlinear model. And the condition that the beta function vanish, that you be at a fixed point, is a consistency condition. You get the virisoral algebra and thereby a unitary S matrix from this two-dimensional construction of the string S matrix. Okay, so that, I can't possibly mention all the names involved in that picture of string theory, but I would like to mention my late colleague Claude Lovelace, who made the really remarkable discovery that string theory to be consistent requires the space time dimension to be a certain number, 26 for the Bosonic string. So that was, I think, the first example of properties of space time being determined by consistency conditions of string theory. Okay, now, so it was realized that this condition, beta equals zero, would be a physical field equation coming from an action principle if you added, in addition to the target metric, additional couplings in the two-dimensional theory. This was Fratkin and Saitlin, who introduced, I think it's called the dilaton coupling. And then, if one goes to the fermionic string, where you add fermionic two-dimensional degrees of freedom to get two-dimensional supersymmetry, you get more renormalizable couplings in this two-dimensional theory. Besides the metric and the dilaton, you can cook things up so you get space time gauge fields, fermion fields, scalars. And of course, two of the main people responsible for the fermionic string are in the room, Andre and John. In the fermionic string, the two-dimensional version of that stability turns out to be basically given by the GSO projection, which gets rid of tachyons. Tachyons correspond to relevant directions under the renormalization group flow. So you're getting rid of the main source of relevant directions. And then, a complication which I didn't point to, the Ricci tensor can have marginal directions. It can have a manifold of solutions Ricci equals zero. The higher-order terms in the beta function, curvature squared and so on, can produce non-trivial flow on that manifold of solutions of Ricci equals zero, and that can produce instabilities. So there was a remarkable discovery by Alvarez, Gourmet and Friedman that at higher-orders, if the two-dimensional theory has an additional supersymmetry, these higher-loop terms cancel. And they discovered that the two-loop term cancels, and then later it was done to all orders. Okay, so now we have additional questions. It doesn't seem right that the string background should be given by a classical field. You would think that the string background, the environment in which strings scatter should be a quantum state of the laboratory where you're doing a string experiment, or a hypothetical string-scattering experiment. The additional question is, what is the quantum string background? And then is there a mechanism that produces a quantum field theory in space-time? And the classical background, the two-D renormalization group flow is producing a solution of beta equals zero. They're producing a string background. And the question is, is there a generalization that would produce a quantum string background? Well, you are getting equations of motion by better functioning, one to zero for space-time, so on. It's classical, but by default. You are getting not a silicon jam in space-time, but rather the classical equation. There's a long way from classical equations to go to Lagrangian. What I mean is that this beta equals zero. Okay. These equations of motion, they're counted to be classical. That's sort of the motivation for these questions. One would like to go beyond that. Okay. I don't know. This is probably personal. I feel an urgent need for a mechanism that produces quantum field theory. And there are too many self-consistent quantum field theories. If we're too much in the position of sort of groping in this huge space of possible quantum field theories, the world is described by one. So a mechanism that produces quantum field theory seems like it might give some explanatory power beyond there being just quantum field theory. After quite a while, I came up with a proposal. So abstract a little bit. Write lambda i for the two-dimensional coupling constants. So lambda i are the modes of whatever space-time fields describe the two-dimensional field theory. By modes, I mean the momentum modes in space-time. And then let, I don't know what's that called, Cal-M in latex, be the manifold of space-time fields. So the lambda i are coordinates on this manifold. Equivalently, we can think of it as, since this is the most general quantum field theory in at least some neighborhood of a certain class of quantum field theories, we can think of M as the manifold of 2D quantum field theories. So to be slightly more concrete, each lambda i couples to some 2D field, which abstractly I'll write phi i, but think of as, say, a vertex operator in, and I'm always taking space-time to be Euclidean to, and leaving wick rotation for some future day. So these lambda i have scaling dimensions, the first term in the beta function, which are of the form space-time momentum squared, minus, slightly negative, where I'm using dimensionless units for space-time distance, and these are all going to be very small in dimensionless units. We're only interested in very long-distance physics in dimensionless units. So in the language of the renormalization group, these coupling constants are slightly irrelevant. They have slightly negative dimension. So under the flow, you're driven, they're driven to zero under the renormalization group flow. That's the idea of stability. Now, let them become sources in two dimensions. So let the coupling constants vary over the surface, and then set them fluctuating. Make them into two-dimensional fields with a natural action. So this is a sort of meta nonlinear model, where the target space is that curly m, the space of space-time fields, where lambda i's are coordinates, and the action is given by the natural metric on the space of space-time fields, roughly the L2 metric, modular gaugian variance, which is in the limit where the target manifold is big, is the natural metric on the space of 2D quantum field theories, as Sascha-Zamologikov defined. And then little g would be the coupling constant in spacetime, the string coupling constant. So here on the right, we have the coupling of the sources to the local two-dimensional fields. And then in the middle there, we have the exponential of the action of this meta nonlinear model. Okay, so in this functional integral over the lambda i sources, I want to do the two-dimensional renormalization in the Wilsonian sense. I want to integrate out fluctuations of the lambda i at small two-dimensional distances, less than capital lambda inverse. So when we integrate out fluctuations, right, we're going to be producing insertions of the phi i in the surface smeared by some functions of z which are, you know, local, over distances less than this distance lambda inverse. Okay, and this action was designed so that these insertions of the phi i would be the same as the insertions produced by tiny handles attached to the surface. What would be produced by string theory quantum corrections, but only the quantum corrections that are local in two dimensions, where the ends of the handle are close together at this two-dimensional distance scale. So this, when I say the lambda model acts, what I mean, I mean the two-dimensional renormalization group in the lambda model, it acts to produce an effective, these insertions of the phi i's produce a new two-dimensional field theory, but with a cutoff at this scale, big lambda inverse. And this is what I claim is the quantum string background. This two-dimensional field theory cutoff at this scale, big lambda inverse. So what the lambda model is doing at smaller distance scales is calculating the effects that a froth, the froth of small handles would have produced. Now, we only understand even in principle how to calculate the effects of those small handles perturbatively. But this two-dimensional field theory is, you know, a sensible two-dimensional field theory. It's non-perturbative. So this is a non-perturbative construction of the quantum corrections to the world surface at short two-dimensional distances. And the crucial design principle here, this limitation to handles that act locally on the surface, is two-dimensional locality. So that at two-dimensional distances larger than this cutoff, we have a local two-dimensional field theory. So string theory, calculations of the string S matrix will work. They depend on integrating correlation functions that are non-singular, except when vertex operators collide. So here, this effective two-dimensional field theory will have that property at distances larger than this cutoff. Okay, now, at the same time, a measure on the lambda i is being produced. So before we've allowed the lambda i's to fluctuate locally, you would have some value for, you'd be in some quantum field theory at very short distance, but it would evolve under the renormalization group along a trajectory. So points in curly M would evolve to different points in curly M. The beta function is, the renormalization group is a flow in the space of quantum field theories. But now, the lambdas are fluctuating. So when you integrate out the short distance fluctuations, you might start with a point, some specific 2D quantum field theory, but you immediately get a measure on 2D quantum field theories as you evolve. And the 2D renormalization group of the lambda model drives that measure to a limiting equilibrium measure. That's the generalization of driving to a fixed point. So the picture is, well, to study what's going on, take some observable at z in this theory. That's a function on the spacetime fields, a function on the 2D coupling constants, a function on this curly M. And under change of scale, well, the coupling constants will evolve in the beta function. But there will also be noise. And you can see this as expanding lambda in the soft modes plus the fluctuations, and you'll get a two-point function of the fluctuations that produces two derivatives of F and some short distance correlation that will go as the inverse of the coupling in the action. So then if you just look at the evolution of the dual measure, it evolves as a random walk in the space of quantum field theories driven by beta, which was the picture I tried to give in the previous slide. So you're being driven to the fixed point, but there's noise. So you wind up, the asymptotic equilibrium is a measure concentrated near the fixed point, but with some distribution around. Now, if the beta function were zero, it would just be random walk, covariant random walk on the target manifold. So you'd be driven to the metric volume element on the space of quantum field, on the space of spacetime fields. If the beta function is the gradient of some function s on spacetime field, then the equilibrium measure will be the metric volume element times e to the minus s. So one gets quantum field theory in spacetime as the equilibrium measure under this flow. Another way to picture this production of a measure is in the radial quantization. So then you have loops in the target manifold are evolving under dilation, two-dimensional dilation. The non-zero modes are strongly suppressed, so the ground state wave function will be concentrated near the constant loops. So the ground state wave function under the radial quantization, its evolution will be just essentially stochastic quantization of the spacetime quantum field theory. It's just driven under the gradient of the spacetime action with noise. Then the distribution around the constant modes would be governed by the target metric. You'll have some Gaussian wave function around the constant modes plus corrections, and that contains all the two-dimensional physics. Now, what's the role of this cutoff in the effective two-dimensional surface? Doing string theory one always just uses coordinate z and takes absolute z minus z1 minus z2, such quantities. I want to think in terms of more physical two-dimensional terms. Can I imagine that our two-dimensional quantum field theory, which has couplings lambda i, is renormalized at some two-dimensional distance, mu inverse. And our cutoff is much, much smaller than the scale at which the two-dimensional quantum field theory is renormalized. So this number, big lambda times mu inverse, is very large. So we have a two-dimensional theory renormalized called a macroscopic scale, distance scale, mu inverse, and we're doing these fluctuations in the coupling constants at a very tiny two-dimensional scale, big lambda inverse. So when you have an irrelevant operator, an irrelevant coupling constant, the effects where we want to use them at scale mu inverse are strongly, well, are suppressed by a power of the difference of scale, where the exponent is the dimension of the coupling constant. So there's the dimension of the coupling constant, there's the ratio of scales, and I write it in this form, e to the minus l squared times the spacetime momentum squared, where l squared is the logarithm of the ratio of two-dimensional scales. So what we see is the wave modes of the spacetime fields at high momentum are cutoff. So we can simply, they are really irrelevant. The ones at small momentum are almost marginal, but the ones with large momentum in spacetime are genuinely irrelevant, and we can disregard them. They have no effect, the corresponding insertions have no effect on the macroscopic two-dimensional physics that we're going to make use of, say, for string S matrix calculations. So this spacetime distance l gives an effective ultraviolet cutoff in spacetime, which renders the target manifold effective of the lambda model effectively finite dimensional. So this really is a sensible two-dimensional theory. So we have a correspondence between the two-dimensional distance scale, at which divides where the lambda model fluctuations take place from the effective world surface, that corresponds to a spacetime distance scale l. Okay, so usually, as I said in the beginning, we just take, you know, we take mu equals one, and then this lambda inverse would be a dimensionless number, which would cut off the integral over moduli in string theory calculations. But I prefer this more physical feeling interpretation. Now, when we're doing string theory calculations, right, and have a two-dimensional, in the effective world surface, we have, this becomes the ultraviolet two-dimensional cutoff, and when we integrate over the moduli, say, for a tube in the world surface, we wind up with a propagator that is not one over p squared for each mode, but has a contribution, a cutoff, an ultraviolet cutoff, in two dimensions. So you notice that for large p squared, this is a lot, the number in parenthesis is very large, so for large p squared, we just get the one over p squared, but for small p squared, the pole is eliminated. So we have, in the string theory in this effective world surface, we have a cutoff in the spacetime infrared. So this effective string S matrix is only described scattering at spacetime distances up to big L, and the quantum field theory, the lambda model has only lambda i's wave modes of the spacetime fields at distances larger than L. So the quantum field theory that produced describes physics at distances larger than L, and the string scattering the S matrix is at distances smaller than L in spacetime. And that choice of big lambda was arbitrary, so we can vary L at will, as long as it's large in dimensionless units, and this construction, I suggest, the fact that both the effective world surface and the spacetime quantum field theory are constructed by this two-dimensional model guarantees that these will be consistent. So if you choose L large and describe some physics by a string S matrix, or choose it smaller and describe the same physics by quantum field theory, they should agree. So I think of this as a realistic version of the string background, where you have a quantum mechanical description of the apparatus in which scattering experiments take place. That's what we ought to have if we're going to do a physical string theory. And on a philosophical level, it seems like a practical theory. The religious S matrix theory, where everything is an S matrix, that's, I think, an idealization that doesn't have anything to do with how we really have done physics. We describe all that we see by quantum mechanics. In particular, we describe CERN by quantum mechanics. On the other hand, the notion that we have quantum field theory all the way down indefinitely, in this sense, that seems also an unnecessary idealization. Okay, finally, in this presentation of the proposal, I point out an intriguing property. The two-dimensional physics, this two-dimensional renormalization group in the lambda model, that's operating from small two-dimensional distances to larger. That's how Wilson taught us to think of renormalization. I guess it goes back to the beginnings of renormalization, that picture. But given this correspondence between two-dimensional distance and space-time distance, the quantum field theory in space-time is being built from the largest distances downward, which is very unlike our picture of effective space-time quantum field theory. I find that intriguing and also might end up having some explanatory power. Okay, so at this point, given a proposal, the proposal offers an endless number of formal internal questions. You know, describe string scouting, you know, some analog of the reduction formula, describe string scattering in a state of the quantum field theory. That's just one of them. But I want to know if this is right or not. I would like it to say something that can be checked against the experiment. And, you know, one could fantasize that this machine, you know, you crank it and pay careful attention to what it does, and you will end up with the standard model. But given the history of that fantasy over the last couple of decades, I think it's not going to happen in my lifetime, and I really want to find this out in my lifetime. And I don't see much possibility of verifying string theory experimentally. So I took another tack. I said, okay, let's suppose that it does produce. The only way it could be right is if it does somehow under some circumstances produce the standard model. So let's say it does produce something roughly like the standard model, something that could include the standard model. Could it be that it produces something else, something exotic, that you would not expect from the canonical quantization of the standard model? Okay, now this lambda model is a two-dimensional field theory. It isn't stochastic quantization of the classical field theory action. It's a two-dimensional field theory. So could there be non-perturbative 2D effects that would show up in this measure it produces on spacetime fields? Okay, so remind you that, okay, so now given this assumption that we've produced a spacetime field theory that includes the standard model, curly n is the manifold of whatever those spacetime fields are, the gauge fields of the standard model, spacetime metric, the fermion fields, some scalars, whatever, on the spacetime where we're going to do experiments, which is R4. Classical does it mean that this over here is the most? Yeah, no, no, I just mean the all connections in an SU2 bundle on R4, right? That's what? But also metrics. Metrics too, sure. I mean, you know, a quantum field theory cutoff quantum field theory of the metric is going to look like classical field theory to us. So that's not a particularly interesting. You can leave the metric out if you want. It's always a waste of motion and it's like a three-dimensional field. No, no, I mean four-dimensional field. Okay, so the, and everything I'm assuming is happening at weak coupling. So, so non-perturbative effects would be semi-classical effects. And in two-dimensional field theory, semi-classical effects, the ones that I know about are given by winding modes, which come from non-trivial loops in the target manifold, and two-dimensional instantons, which come from pi2 of the manifold. So I guess this is the first, there'll be more mentions of instanton, but two of the inventors of that, two of the, the, the discoverers of that, Sasha and Sasha. Paul-Yakov and Balabin are here. Okay, so the mathematicians have long known that, that, that there are non-trivial homotopy groups in the space of gauge fields on R4. And when I say the space of gauge fields, I mean, of course, the physical space of gauge fields, gauge fields, mod gauge transformations. So pi1 for SU2 gauge fields, there's one non-trivial loop in the space of gauge fields. That first came into physics, I believe, with Witten's discussion of global anomalies in SU2 gauge theory. And pi1 of the space of SU3 gauge, oh, all the other pi1s vanish. So the only interesting whining, there's one winding mode that might be interesting, and that's for SU2 gauge fields, which of course we should have. Pi2s are always non-zero. There's one non-trivial two-sphere for SU2 gauge fields, and there's a whole integer worth of such two spheres for SU3. So winding modes in a two-dimensional field theory give new local fields. So those, you would think, would correspond to new degrees of freedom in the space time theory. So the question is, are these low energy degrees of freedom? Of course, they would be not canonical because they have nothing to do with the space-time classical action. And then these two spheres would give 2D instantons, and one might expect they would give non-canonical couplings in the 2D field. But I find the first question more interesting, so I haven't done much with the second. Okay, but I wanted to know what those loops look like, right? Because if you're doing a quantum non-linear model with these spaces as target manifolds, the winding mode would be given by the minimal loop, the shortest loop you can find. And if that loop has finite length, it's going to be a high energy, a very massive excitation, and we can forget about it. So the only possibility is if that minimal, if there's a loop of zero length that's topologically non-trivial. So I investigated this in a very roundabout way, and it turns out after I reported my results to the mathematicians that they basically knew about it. But I had to find it out for myself. And I didn't really look, well, that's a somewhat long story. But what I found was that the non-trivial loop in the space of SU2 gauge fields is a loop of gauge fields which consist of an instanton and an anti-instanton. Now these are four dimensional instantons, the original instanton. And the loop, and then pi2 for SU2 gauge fields, you need two instantons and two anti-instantons glued together. And then you modify the gluing parameters and you find a non-trivial two-sphere. And for pi2 of SU3, it's again an instanton, anti-instanton pair. But the interesting thing is that the minimal loop has zero length, and the minimal two-sphere has zero area. So there's a chance of low energy physics here. And as I said, I find the possibility of non-canonical degrees of freedom, given the mysteries we have in physics, that's tantalizing. So that's the one I've been thinking about. Okay, so I'm going to, how much time do I have? Or am I up? Oh, oh, fine. Okay, so here's the construction of this non-trivial loop. You take an instanton on R4, and then you glue in a very small anti-instanton. So A plus is the instanton, and it has size rho plus, and it has a center, X plus. So X plus is in R4. And there's a possibility of an internal rotation in SU2, but you fix that by global gauge transformation. And then you take an anti-instanton, a very small size rho minus, centered at another point, X minus, and you rotate that by an element of SU2 acting in the adjoint, and then you glue it in to the instanton. Well, the possible ways you can rotate that anti-instanton are given by SU2 acting in the adjoint. So the parameter space is SU2 mod the center, plus or minus one, so SO3. And that has a non-trivial loop in it. Pi 1 of SO3 is Z2. And that loop in this family of gauge fields is a representative of the non-trivial loop in the space of gauge fields. One can show that very explicitly. Well, do you know that it is kind of contracted by modifying these solutions in some other way? Yeah, you can explicitly show that you go around the loop and you get a gauge transformation that is explicitly the gauge transformation that cannot be contracted to the identity. Well, that's about statement about the space of SU2 gauge field. Why does it come from instanton? I mean, the instanton is a subspace and a space. Yeah, so I'm constructing a loop in that subspace and then verifying that that loop is non-trivial in the space of all gauge fields. And you use it as a self-dual gauge field. Where does it come from? It's a combination of those. Oh, that comes only if you take A plus and A minus to be one instant and another one to be instanton. It's essential. The gluing, yeah. You have to glue. This has topological number zero. So what we are doing, we need to have the metric. We want to find out what's the length of this non-trivial loop. So again, this is from the mathematicians. So we parameterize that anti-instanton that we're gluing in. Well, leave the location fixed. We parameterize the size and the internal rotation in SU2 by an element in C2. So we take a reference point in C2. We apply the SU2 rotation to it and we get an element in C2. So that's the standard parameterization of SU2 by C2. And then we multiply. But it's also the two-spheres. I'm sorry, the three-spheres in C2. And then we take the radius to be rho minus. So we get an element in V minus in C2, which parameterizes the size and the SU2 rotation. And then we calculate the metric. Take small perturbations. Take the L2, mod out by gate transformations. Take the L2 inner product. And the metric is smooth in this parameterization at the origin. And the actual metric on the instanton-modulized space has nice properties. But here we only are talking about near the origin where the instanton, the anti-instanton decouples from the instanton. When rho minus is finite, you have to glue them together. You're not in the space of self-dual. You don't have the nice mathematical properties. But anyway, the metric, remember, has a 1 over coupling constant squared. And then it's just the Euclidean metric in this space C2. And the non-trivial loop, that's the loop in G minus. So it's a loop in the three-sphere. Oh, V minus is only defined up to plus or minus 1, because G minus is only defined up to plus or minus 1. So we're really looking at the orbital, C2 mod Z2. So we're looking at the non-trivial loop from the north pole of S3 to the south pole of S3 for any given size, rho minus. If we take rho minus to zero, the length of that arc, that longitudinal line, goes to zero. So the minimal non-trivial loop has zero length. So in the lambda model, the winding mode for such a loop is just the twist field for this orbital. And we did this for fixed values of the location of the instanton and the anti-instanton, and for fixed size parameter for the instanton, so they're collective coordinates. And now, again, given this assumption that we somehow got a field theory that includes the standard model, we'll have fermion fields. And those coupled to the SU2 gauge field, and they'll have zero modes, localized in the instanton and in the anti-instanton. And it's very pretty. You can actually show how, as you move in this non-trivial loop, the zero mode in the anti-instanton, the eigen, you know, changes sign. And that's an explicit realization of this SU2 global anomaly. But when we do an orbifolding, we project out all the two-dimensional degrees of freedom that are odd under the Z2. So we project out the zero modes in the anti-instanton, this tiny anti-ant. And we leave the zero modes in the instanton. They survive. Okay? So it looks like there are possibilities for interesting quantum numbers in this object. But at this point, I want to emphasize that it's a bizarre looking object. It's bilocal. It depends on two points in spacetime, X plus and X minus. And then this additional parameter, rho plus. There's an analogous object, the Cp transform, in which you have an anti-instanton with a tiny instanton embedded. Anyway, it looks very bizarre. And this orbifolding, right, where you eliminate the degrees of freedom that are odd under the Z2, it looks like that ought to make sense of field theories which have global SU2 anomalies. Because this gets rid of the global SU2 anomaly, but leave that. Okay. We need to understand the flow in this neighborhood. And one can actually calculate the Yang-Mills action, of which the flow is the gradient. I'm not going to go through the details, but you find that there is one unstable direction. So there's a trajectory, right? So you have a fixed point. I normalize the Yang-Mills action so the instanton has action one. So we have an instanton and an intense instanton, that gives two, plus something proportional to the sizes. Okay. At least one of which is small. And most of the form of that is determined by conformal invariance in R4. But if you study the behavior near the fixed point, when the little anti-instanton has zero size, that's a fixed point, that's a genuinely self-dual field. You study the behavior near that, you find one unstable direction. So there's a flow that leaves from near this fixed point, this non-trivial fixed point, and flows down to the flat connection. And then interesting things happen when both instantons become small, but the flow becomes sort of like asymptotically free. You don't have actual instability, but just marginal instability. So at this point, my thinking is still quite muddled. The question is how these degrees of freedom will contribute to the space-time physics, if they do. So my best understanding of how to proceed is to introduce new coupling constants that couple to this winding mode, and then study the dynamics. I mean, one can fantasize all sorts of things, a condensation, well, who knows. But in order to study the dynamics of these couplings, one has to know the beta function of these new degrees of freedom. So one has to understand how products of the twist field and ordinary two-dimensional fields evolve under two-dimensional scaler. And in particular, zero length just guarantees that the energy will be zero classically. But we know from ordinary orbe folds that the twist field, the scaling dimension of the twist field gets quantum corrections immediately at one loop. And the question is, they would ruin this idea of low energy excitation. So that has to be checked. And then this, it seems to me the basic object that is this unstable trajectory from the instanton-anti-instanton pair down to the flat connection. But I have yet to work that out. Thank you. So it looks like the lambda model is actually, it's a single model on the product of the original space-time manifold and the space-time manifold. No, no, no. I mean, you can think of it that way. I think it's, yeah, but it seems to me to analyze the structure. It's important to understand that the couplings, lambda i, which in this picture are the fields on space-time, right? They have non-trivial renormalists. They aren't dimensionless, right? They have dimensions that come from quantum corrections in the original nonlinear model. And one, the structure of the lambda model is based on the properties, the quantum properties of that two-dimensional, of the original two-dimensional field theory. So I don't think it's all that useful to study, to quantize them simultaneously. The picture I presented is, first you quantize the original model and then you add fluctuations of its quantum coupling constants. I mean, it might be that there's some fruitful way to do it all at once, but it hasn't seemed like that. Another option would be to do something like a spin glass model, where you would take the first quantized, the way you say the original model, then you take a, you know, connect. That's interesting. Yeah, I have no idea how to, but that's very interesting. So you'd get an effective theory just, yeah, that might be, that might be. Other questions here? Yes, maybe it's related to the previous question, but somehow M is the space of all 2D quantum field theories, and you use it to build another quantum field theory. So somehow this new field, the lambda model, should be a point of M, isn't it? No. It's a, it's not all 2D quantum field theories. It's a special class. Oh, I forgot one thing I really wanted to say. Back when I started talking about string theory, that this is where Demeric Kniznick and I intersected. We were both interested back in the mid-80s in finishing the covariant quantization of the fermionic string, using Sasha Polyakov's super-diffinomorphism 2D ghost fields. Yes, I definitely wanted to mention that. My motivation was this, I need a covariant two-dimensional field theory. But, so it's a special class of 2D quantum field theories, the original 2D field theory. And the lambda model is not a globally defined 2D field theory. It's only operating at short two-dimensional distances. So I don't think that the picture is, as you say. You refer to this, twisted the loops of instantons as a minimum of a positive of I1 over 2B. Does that mean that you have further configurations? Well, I mean, you, you, okay. So you have this parameter space, which is the size of the tiny anti-instanton, and then a three-sphere of orientations. An SU2, or MAZ-E2, but think of it as a three-sphere, okay. And the non-trivial loop goes from the north to the south pole of that three-sphere, okay. So take the size to be non-zero. Really, what you can think of this is describing, you have all the other modes of the gauge field, all the other ways, the irrelevant directions you can go away from this family. And what this is describing is a cycle in, so that, so that any non-trivial, any loop that, that, that surrounds this cycle will be non-trivial. But, but for the two-dimensional physics, I'm interested in the one that shrinks down to zero. I think that's the finished one. And what do you need the lambda model to study those exotics? I don't know. I think of it the other way around. I need those exotic twist fields to study because they're in the lambda model. I mean, you could go, you could go and study them, but I wouldn't have a motivation. Thank you. Thank you. Thank you.
 I'm going to be talking about a project that has been going on for ridiculously long time.  I should mention it makes use of discoveries by many other people, some of whom are in the room.  At the appropriate times I'll try to mention them, but if I forget I think it will be obvious.  I'll be sketching a proposal mechanism that produces quantum field theory in space-time,  a real quantum field theory, a measure on space-time fields which is equivalent to a Hamiltonian.  At the same time produces a quantum string background.  I'm reasonably confident that it's the correct idea of a quantum string background,  but as things stand that's a purely formal question.  Whether this is how actual quantum field theory is produced by which I mean the one that describes the real world,  that I don't know, but I would like to.  At the moment I'm trying to derive from this proposal a predictions of unexpected low energy phenomena  that might actually be seen or not seen in experiments.  More explicitly the possibility that this mechanism produces some rather bizarre low energy degrees of freedom  whenever there's SU2 gauging variance.  These will be associated with well-known non-trivial homotopy group of the space of SU2 gauge fields on Euclidean spacetime.  I get this too.  Okay, I should put in this caveat. It's extremely speculative fundamental physics.  A long shot.  I like that word. It comes from when the British were shooting at the French.  It goes back as I said a very long time, the beginnings.  It goes back to this two-dimensional generalized nonlinear model where doing two-dimensional quantum field theory,  the field lies in some manifold M and the action is given by what mathematicians call the energy functional.  The parameters in the action are comprised in a Romanian metric on the target manifold M.  There are infinitely many coupling constants which you can think of as the Taylor series of the metric at some point,  X0 in M, then you get the metric, its first derivative, second ad infinitum, infinitely many couplings,  but they are all renormalizable and the renormalization can be done co-variantly with respect to the target manifold  and the renormalization group equation has an expansion in sort of the inverse size of the target manifold which starts with the Ricci tensor.  This work was inspired by Sasha Polyakov's discovery that the two-dimensional nonlinear signal model is renormalizable,  where the target manifold there is the two-sphere with O3 symmetry, the round two-sphere,  so the only coupling constant is the inverse size of the two-sphere and Sasha discovered that that was an asymptotically free coupling constant.  So this can be thought of as a generalization of that.  So what the renormalization, this was, I was a graduate student at the time and Ken Wilson's view of the renormalization group was how I learned renormalization.  And one thinks of the renormalization group as a machine that acts on quantum field theories, in this case two-dimensional quantum field theories,  and drives them to something new at larger and larger distances.  So here the two-dimensional renormalization group, and it's doing this at very short distances,  it drives this metric on the manifold M towards a fixed point, which would be then a solution of Ricci tensor equals zero.  So it is a two-dimensional renormalization group, it's a mechanism that produces what looks like a classical field theory.  Now by mechanism I'm not talking about a mechanism that acts in real time, it's an abstract mechanism that sort of acts on theories.  Okay, so at the time that was extremely exciting, though I think only to me.  It seemed to me that it could be a clue to where the laws of physics come from.  It looked like Einstein's equation.  Not exactly, but it looked like it.  Okay, so the questions that arose were, first of all, it isn't Einstein's equation.  It's a field equation, but not a physical one.  So the question was, is there something related that would produce a realistic field theory?  I mean it should come from an action principle and it should have a realistic collection of fields, not just a metric on space time.  I forgot to mention that for this picture one takes the target manifold M to be space time.  So it should have gauge fields, fermion fields, scalar fields.  Then there's a question, I was a bit sloppy when I said that the renormalization group drives you to a fixed point.  It only does that if that fixed point is an attractor.  If that fixed point has unstable directions, then you have to do some tuning to get driven there.  And at the time, I didn't see, I mean you can eliminate all but a finite number of directions,  but I couldn't see how to get rid of the possibility of a finite number of unstable directions.  So you get stability in this sort of scheme.  And then the big question, this is what this produces is a classical field theory,  where does quantum field theory come from?  What makes quantum field theory?  So within a few years, the first two were answered when this two-dimensional field theory was interpreted as the string background.  So the idea is that the world surface of a string propagating in space time is described by a two-dimensional field theory.  And if the space time has a curved metric, the two-dimensional field theory will be this two-dimensional nonlinear model.  And the condition that the beta function vanish, that you be at a fixed point, is a consistency condition.  You get the virisoral algebra and thereby a unitary S matrix from this two-dimensional construction of the string S matrix.  Okay, so that, I can't possibly mention all the names involved in that picture of string theory,  but I would like to mention my late colleague Claude Lovelace, who made the really remarkable discovery that string theory to be consistent  requires the space time dimension to be a certain number, 26 for the Bosonic string.  So that was, I think, the first example of properties of space time being determined by consistency conditions of string theory.  Okay, now, so it was realized that this condition, beta equals zero, would be a physical field equation coming from an action principle  if you added, in addition to the target metric, additional couplings in the two-dimensional theory.  This was Fratkin and Saitlin, who introduced, I think it's called the dilaton coupling.  And then, if one goes to the fermionic string, where you add fermionic two-dimensional degrees of freedom to get two-dimensional supersymmetry,  you get more renormalizable couplings in this two-dimensional theory.  Besides the metric and the dilaton, you can cook things up so you get space time gauge fields, fermion fields, scalars.  And of course, two of the main people responsible for the fermionic string are in the room, Andre and John.  In the fermionic string, the two-dimensional version of that stability turns out to be basically given by the GSO projection,  which gets rid of tachyons. Tachyons correspond to relevant directions under the renormalization group flow.  So you're getting rid of the main source of relevant directions.  And then, a complication which I didn't point to, the Ricci tensor can have marginal directions.  It can have a manifold of solutions Ricci equals zero.  The higher-order terms in the beta function, curvature squared and so on,  can produce non-trivial flow on that manifold of solutions of Ricci equals zero, and that can produce instabilities.  So there was a remarkable discovery by Alvarez, Gourmet and Friedman that at higher-orders,  if the two-dimensional theory has an additional supersymmetry, these higher-loop terms cancel.  And they discovered that the two-loop term cancels, and then later it was done to all orders.  Okay, so now we have additional questions.  It doesn't seem right that the string background should be given by a classical field.  You would think that the string background, the environment in which strings scatter should be a quantum state of the laboratory  where you're doing a string experiment, or a hypothetical string-scattering experiment.  The additional question is, what is the quantum string background?  And then is there a mechanism that produces a quantum field theory in space-time?  And the classical background, the two-D renormalization group flow is producing a solution of beta equals zero.  They're producing a string background.  And the question is, is there a generalization that would produce a quantum string background?  Well, you are getting equations of motion by better functioning, one to zero for space-time, so on.  It's classical, but by default.  You are getting not a silicon jam in space-time, but rather the classical equation.  There's a long way from classical equations to go to Lagrangian.  What I mean is that this beta equals zero.  Okay.  These equations of motion, they're counted to be classical.  That's sort of the motivation for these questions.  One would like to go beyond that.  Okay.  I don't know.  This is probably personal.  I feel an urgent need for a mechanism that produces quantum field theory.  And there are too many self-consistent quantum field theories.  If we're too much in the position of sort of groping in this huge space of possible quantum field theories,  the world is described by one.  So a mechanism that produces quantum field theory seems like it might give some explanatory power  beyond there being just quantum field theory.  After quite a while, I came up with a proposal.  So abstract a little bit.  Write lambda i for the two-dimensional coupling constants.  So lambda i are the modes of whatever space-time fields describe the two-dimensional field theory.  By modes, I mean the momentum modes in space-time.  And then let, I don't know what's that called, Cal-M in latex, be the manifold of space-time fields.  So the lambda i are coordinates on this manifold.  Equivalently, we can think of it as, since this is the most general quantum field theory  in at least some neighborhood of a certain class of quantum field theories,  we can think of M as the manifold of 2D quantum field theories.  So to be slightly more concrete, each lambda i couples to some 2D field,  which abstractly I'll write phi i, but think of as, say, a vertex operator in,  and I'm always taking space-time to be Euclidean to,  and leaving wick rotation for some future day.  So these lambda i have scaling dimensions, the first term in the beta function,  which are of the form space-time momentum squared, minus, slightly negative,  where I'm using dimensionless units for space-time distance,  and these are all going to be very small in dimensionless units.  We're only interested in very long-distance physics in dimensionless units.  So in the language of the renormalization group, these coupling constants are slightly irrelevant.  They have slightly negative dimension.  So under the flow, you're driven, they're driven to zero under the renormalization group flow.  That's the idea of stability.  Now, let them become sources in two dimensions.  So let the coupling constants vary over the surface, and then set them fluctuating.  Make them into two-dimensional fields with a natural action.  So this is a sort of meta nonlinear model,  where the target space is that curly m, the space of space-time fields,  where lambda i's are coordinates,  and the action is given by the natural metric on the space of space-time fields,  roughly the L2 metric, modular gaugian variance,  which is in the limit where the target manifold is big,  is the natural metric on the space of 2D quantum field theories,  as Sascha-Zamologikov defined.  And then little g would be the coupling constant in spacetime, the string coupling constant.  So here on the right, we have the coupling of the sources to the local two-dimensional fields.  And then in the middle there, we have the exponential of the action of this meta nonlinear model.  Okay, so in this functional integral over the lambda i sources,  I want to do the two-dimensional renormalization in the Wilsonian sense.  I want to integrate out fluctuations of the lambda i at small two-dimensional distances,  less than capital lambda inverse.  So when we integrate out fluctuations, right,  we're going to be producing insertions of the phi i in the surface smeared by  some functions of z which are, you know,  local, over distances less than this distance lambda inverse.  Okay, and this action was designed so that these insertions of the phi i  would be the same as the insertions produced by tiny handles attached to the surface.  What would be produced by string theory quantum corrections,  but only the quantum corrections that are local in two dimensions,  where the ends of the handle are close together at this two-dimensional distance scale.  So this, when I say the lambda model acts,  what I mean, I mean the two-dimensional renormalization group in the lambda model,  it acts to produce an effective, these insertions of the phi i's  produce a new two-dimensional field theory,  but with a cutoff at this scale, big lambda inverse.  And this is what I claim is the quantum string background.  This two-dimensional field theory cutoff at this scale, big lambda inverse.  So what the lambda model is doing at smaller distance scales  is calculating the effects that a froth, the froth of small handles would have produced.  Now, we only understand even in principle how to calculate the effects of those small handles perturbatively.  But this two-dimensional field theory is, you know, a sensible two-dimensional field theory.  It's non-perturbative.  So this is a non-perturbative construction of the quantum corrections to the world surface at short two-dimensional distances.  And the crucial design principle here, this limitation to handles that act locally on the surface,  is two-dimensional locality.  So that at two-dimensional distances larger than this cutoff,  we have a local two-dimensional field theory.  So string theory, calculations of the string S matrix will work.  They depend on integrating correlation functions that are non-singular,  except when vertex operators collide.  So here, this effective two-dimensional field theory will have that property at distances larger than this cutoff.  Okay, now, at the same time, a measure on the lambda i is being produced.  So before we've allowed the lambda i's to fluctuate locally,  you would have some value for, you'd be in some quantum field theory at very short distance,  but it would evolve under the renormalization group along a trajectory.  So points in curly M would evolve to different points in curly M.  The beta function is, the renormalization group is a flow in the space of quantum field theories.  But now, the lambdas are fluctuating.  So when you integrate out the short distance fluctuations,  you might start with a point, some specific 2D quantum field theory,  but you immediately get a measure on 2D quantum field theories as you evolve.  And the 2D renormalization group of the lambda model drives that measure to a limiting equilibrium measure.  That's the generalization of driving to a fixed point.  So the picture is, well, to study what's going on, take some observable at z in this theory.  That's a function on the spacetime fields, a function on the 2D coupling constants,  a function on this curly M.  And under change of scale, well, the coupling constants will evolve in the beta function.  But there will also be noise.  And you can see this as expanding lambda in the soft modes plus the fluctuations,  and you'll get a two-point function of the fluctuations that produces two derivatives of F  and some short distance correlation that will go as the inverse of the coupling in the action.  So then if you just look at the evolution of the dual measure,  it evolves as a random walk in the space of quantum field theories driven by beta,  which was the picture I tried to give in the previous slide.  So you're being driven to the fixed point, but there's noise.  So you wind up, the asymptotic equilibrium is a measure concentrated near the fixed point,  but with some distribution around.  Now, if the beta function were zero, it would just be random walk, covariant random walk on the target manifold.  So you'd be driven to the metric volume element on the space of quantum field,  on the space of spacetime fields.  If the beta function is the gradient of some function s on spacetime field,  then the equilibrium measure will be the metric volume element times e to the minus s.  So one gets quantum field theory in spacetime as the equilibrium measure under this flow.  Another way to picture this production of a measure is in the radial quantization.  So then you have loops in the target manifold are evolving under dilation, two-dimensional dilation.  The non-zero modes are strongly suppressed,  so the ground state wave function will be concentrated near the constant loops.  So the ground state wave function under the radial quantization,  its evolution will be just essentially stochastic quantization of the spacetime quantum field theory.  It's just driven under the gradient of the spacetime action with noise.  Then the distribution around the constant modes would be governed by the target metric.  You'll have some Gaussian wave function around the constant modes plus corrections,  and that contains all the two-dimensional physics.  Now, what's the role of this cutoff in the effective two-dimensional surface?  Doing string theory one always just uses coordinate z and takes absolute z minus z1 minus z2, such quantities.  I want to think in terms of more physical two-dimensional terms.  Can I imagine that our two-dimensional quantum field theory, which has couplings lambda i,  is renormalized at some two-dimensional distance, mu inverse.  And our cutoff is much, much smaller than the scale at which the two-dimensional quantum field theory is renormalized.  So this number, big lambda times mu inverse, is very large.  So we have a two-dimensional theory renormalized called a macroscopic scale, distance scale, mu inverse,  and we're doing these fluctuations in the coupling constants at a very tiny two-dimensional scale, big lambda inverse.  So when you have an irrelevant operator, an irrelevant coupling constant,  the effects where we want to use them at scale mu inverse are strongly, well, are suppressed by a power of the difference of scale,  where the exponent is the dimension of the coupling constant.  So there's the dimension of the coupling constant, there's the ratio of scales,  and I write it in this form, e to the minus l squared times the spacetime momentum squared,  where l squared is the logarithm of the ratio of two-dimensional scales.  So what we see is the wave modes of the spacetime fields at high momentum are cutoff.  So we can simply, they are really irrelevant.  The ones at small momentum are almost marginal,  but the ones with large momentum in spacetime are genuinely irrelevant,  and we can disregard them.  They have no effect, the corresponding insertions have no effect on the macroscopic two-dimensional physics  that we're going to make use of, say, for string S matrix calculations.  So this spacetime distance l gives an effective ultraviolet cutoff in spacetime,  which renders the target manifold effective of the lambda model effectively finite dimensional.  So this really is a sensible two-dimensional theory.  So we have a correspondence between the two-dimensional distance scale,  at which divides where the lambda model fluctuations take place from the effective world surface,  that corresponds to a spacetime distance scale l.  Okay, so usually, as I said in the beginning, we just take, you know, we take mu equals one,  and then this lambda inverse would be a dimensionless number,  which would cut off the integral over moduli in string theory calculations.  But I prefer this more physical feeling interpretation.  Now, when we're doing string theory calculations, right,  and have a two-dimensional, in the effective world surface,  we have, this becomes the ultraviolet two-dimensional cutoff,  and when we integrate over the moduli, say, for a tube in the world surface,  we wind up with a propagator that is not one over p squared for each mode,  but has a contribution, a cutoff, an ultraviolet cutoff, in two dimensions.  So you notice that for large p squared, this is a lot, the number in parenthesis is very large,  so for large p squared, we just get the one over p squared,  but for small p squared, the pole is eliminated.  So we have, in the string theory in this effective world surface,  we have a cutoff in the spacetime infrared.  So this effective string S matrix is only described scattering at spacetime distances up to big L,  and the quantum field theory, the lambda model has only lambda i's wave modes of the spacetime fields  at distances larger than L.  So the quantum field theory that produced describes physics at distances larger than L,  and the string scattering the S matrix is at distances smaller than L in spacetime.  And that choice of big lambda was arbitrary, so we can vary L at will,  as long as it's large in dimensionless units,  and this construction, I suggest, the fact that both the effective world surface  and the spacetime quantum field theory are constructed by this two-dimensional model  guarantees that these will be consistent.  So if you choose L large and describe some physics by a string S matrix,  or choose it smaller and describe the same physics by quantum field theory, they should agree.  So I think of this as a realistic version of the string background,  where you have a quantum mechanical description of the apparatus in which scattering experiments take place.  That's what we ought to have if we're going to do a physical string theory.  And on a philosophical level, it seems like a practical theory.  The religious S matrix theory, where everything is an S matrix,  that's, I think, an idealization that doesn't have anything to do with how we really have done physics.  We describe all that we see by quantum mechanics.  In particular, we describe CERN by quantum mechanics.  On the other hand, the notion that we have quantum field theory all the way down indefinitely,  in this sense, that seems also an unnecessary idealization.  Okay, finally, in this presentation of the proposal, I point out an intriguing property.  The two-dimensional physics, this two-dimensional renormalization group in the lambda model,  that's operating from small two-dimensional distances to larger.  That's how Wilson taught us to think of renormalization.  I guess it goes back to the beginnings of renormalization, that picture.  But given this correspondence between two-dimensional distance and space-time distance,  the quantum field theory in space-time is being built from the largest distances downward,  which is very unlike our picture of effective space-time quantum field theory.  I find that intriguing and also might end up having some explanatory power.  Okay, so at this point, given a proposal, the proposal offers an endless number of formal internal questions.  You know, describe string scouting, you know, some analog of the reduction formula,  describe string scattering in a state of the quantum field theory.  That's just one of them.  But I want to know if this is right or not.  I would like it to say something that can be checked against the experiment.  And, you know, one could fantasize that this machine, you know,  you crank it and pay careful attention to what it does, and you will end up with the standard model.  But given the history of that fantasy over the last couple of decades,  I think it's not going to happen in my lifetime, and I really want to find this out in my lifetime.  And I don't see much possibility of verifying string theory experimentally.  So I took another tack.  I said, okay, let's suppose that it does produce.  The only way it could be right is if it does somehow under some circumstances produce the standard model.  So let's say it does produce something roughly like the standard model,  something that could include the standard model.  Could it be that it produces something else, something exotic,  that you would not expect from the canonical quantization of the standard model?  Okay, now this lambda model is a two-dimensional field theory.  It isn't stochastic quantization of the classical field theory action.  It's a two-dimensional field theory.  So could there be non-perturbative 2D effects that would show up in this measure it produces on spacetime fields?  Okay, so remind you that, okay, so now given this assumption that we've produced a spacetime field theory that includes the standard model,  curly n is the manifold of whatever those spacetime fields are,  the gauge fields of the standard model, spacetime metric, the fermion fields, some scalars, whatever,  on the spacetime where we're going to do experiments, which is R4.  Classical does it mean that this over here is the most?  Yeah, no, no, I just mean the all connections in an SU2 bundle on R4, right?  That's what?  But also metrics.  Metrics too, sure.  I mean, you know, a quantum field theory cutoff quantum field theory of the metric is going to look like classical field theory to us.  So that's not a particularly interesting.  You can leave the metric out if you want.  It's always a waste of motion and it's like a three-dimensional field.  No, no, I mean four-dimensional field.  Okay, so the, and everything I'm assuming is happening at weak coupling.  So, so non-perturbative effects would be semi-classical effects.  And in two-dimensional field theory, semi-classical effects, the ones that I know about are given by winding modes,  which come from non-trivial loops in the target manifold,  and two-dimensional instantons, which come from pi2 of the manifold.  So I guess this is the first, there'll be more mentions of instanton,  but two of the inventors of that, two of the, the, the discoverers of that, Sasha and Sasha.  Paul-Yakov and Balabin are here.  Okay, so the mathematicians have long known that, that, that there are non-trivial homotopy groups in the space of gauge fields on R4.  And when I say the space of gauge fields, I mean, of course, the physical space of gauge fields, gauge fields, mod gauge transformations.  So pi1 for SU2 gauge fields, there's one non-trivial loop in the space of gauge fields.  That first came into physics, I believe, with Witten's discussion of global anomalies in SU2 gauge theory.  And pi1 of the space of SU3 gauge, oh, all the other pi1s vanish.  So the only interesting whining, there's one winding mode that might be interesting,  and that's for SU2 gauge fields, which of course we should have.  Pi2s are always non-zero.  There's one non-trivial two-sphere for SU2 gauge fields, and there's a whole integer worth of such two spheres for SU3.  So winding modes in a two-dimensional field theory give new local fields.  So those, you would think, would correspond to new degrees of freedom in the space time theory.  So the question is, are these low energy degrees of freedom?  Of course, they would be not canonical because they have nothing to do with the space-time classical action.  And then these two spheres would give 2D instantons, and one might expect they would give non-canonical couplings in the 2D field.  But I find the first question more interesting, so I haven't done much with the second.  Okay, but I wanted to know what those loops look like, right?  Because if you're doing a quantum non-linear model with these spaces as target manifolds,  the winding mode would be given by the minimal loop, the shortest loop you can find.  And if that loop has finite length, it's going to be a high energy, a very massive excitation, and we can forget about it.  So the only possibility is if that minimal, if there's a loop of zero length that's topologically non-trivial.  So I investigated this in a very roundabout way, and it turns out after I reported my results to the mathematicians that they basically knew about it.  But I had to find it out for myself.  And I didn't really look, well, that's a somewhat long story.  But what I found was that the non-trivial loop in the space of SU2 gauge fields is a loop of gauge fields which consist of an instanton and an anti-instanton.  Now these are four dimensional instantons, the original instanton.  And the loop, and then pi2 for SU2 gauge fields, you need two instantons and two anti-instantons glued together.  And then you modify the gluing parameters and you find a non-trivial two-sphere.  And for pi2 of SU3, it's again an instanton, anti-instanton pair.  But the interesting thing is that the minimal loop has zero length, and the minimal two-sphere has zero area.  So there's a chance of low energy physics here.  And as I said, I find the possibility of non-canonical degrees of freedom, given the mysteries we have in physics, that's tantalizing.  So that's the one I've been thinking about.  Okay, so I'm going to, how much time do I have? Or am I up?  Oh, oh, fine.  Okay, so here's the construction of this non-trivial loop.  You take an instanton on R4, and then you glue in a very small anti-instanton.  So A plus is the instanton, and it has size rho plus, and it has a center, X plus.  So X plus is in R4.  And there's a possibility of an internal rotation in SU2, but you fix that by global gauge transformation.  And then you take an anti-instanton, a very small size rho minus, centered at another point, X minus,  and you rotate that by an element of SU2 acting in the adjoint, and then you glue it in to the instanton.  Well, the possible ways you can rotate that anti-instanton are given by SU2 acting in the adjoint.  So the parameter space is SU2 mod the center, plus or minus one, so SO3.  And that has a non-trivial loop in it.  Pi 1 of SO3 is Z2.  And that loop in this family of gauge fields is a representative of the non-trivial loop in the space of gauge fields.  One can show that very explicitly.  Well, do you know that it is kind of contracted by modifying these solutions in some other way?  Yeah, you can explicitly show that you go around the loop and you get a gauge transformation  that is explicitly the gauge transformation that cannot be contracted to the identity.  Well, that's about statement about the space of SU2 gauge field.  Why does it come from instanton?  I mean, the instanton is a subspace and a space.  Yeah, so I'm constructing a loop in that subspace and then verifying that that loop is non-trivial in the space of all gauge fields.  And you use it as a self-dual gauge field. Where does it come from?  It's a combination of those.  Oh, that comes only if you take A plus and A minus to be one instant and another one to be instanton.  It's essential.  The gluing, yeah.  You have to glue.  This has topological number zero.  So what we are doing, we need to have the metric.  We want to find out what's the length of this non-trivial loop.  So again, this is from the mathematicians.  So we parameterize that anti-instanton that we're gluing in.  Well, leave the location fixed.  We parameterize the size and the internal rotation in SU2 by an element in C2.  So we take a reference point in C2.  We apply the SU2 rotation to it and we get an element in C2.  So that's the standard parameterization of SU2 by C2.  And then we multiply.  But it's also the two-spheres.  I'm sorry, the three-spheres in C2.  And then we take the radius to be rho minus.  So we get an element in V minus in C2, which parameterizes the size and the SU2 rotation.  And then we calculate the metric.  Take small perturbations.  Take the L2, mod out by gate transformations.  Take the L2 inner product.  And the metric is smooth in this parameterization at the origin.  And the actual metric on the instanton-modulized space has nice properties.  But here we only are talking about near the origin where the instanton, the anti-instanton decouples from the instanton.  When rho minus is finite, you have to glue them together.  You're not in the space of self-dual.  You don't have the nice mathematical properties.  But anyway, the metric, remember, has a 1 over coupling constant squared.  And then it's just the Euclidean metric in this space C2.  And the non-trivial loop, that's the loop in G minus.  So it's a loop in the three-sphere.  Oh, V minus is only defined up to plus or minus 1, because G minus is only defined up to plus or minus 1.  So we're really looking at the orbital, C2 mod Z2.  So we're looking at the non-trivial loop from the north pole of S3 to the south pole of S3 for any given size, rho minus.  If we take rho minus to zero, the length of that arc, that longitudinal line, goes to zero.  So the minimal non-trivial loop has zero length.  So in the lambda model, the winding mode for such a loop is just the twist field for this orbital.  And we did this for fixed values of the location of the instanton and the anti-instanton,  and for fixed size parameter for the instanton, so they're collective coordinates.  And now, again, given this assumption that we somehow got a field theory that includes the standard model, we'll have fermion fields.  And those coupled to the SU2 gauge field, and they'll have zero modes, localized in the instanton and in the anti-instanton.  And it's very pretty. You can actually show how, as you move in this non-trivial loop, the zero mode in the anti-instanton, the eigen, you know, changes sign.  And that's an explicit realization of this SU2 global anomaly.  But when we do an orbifolding, we project out all the two-dimensional degrees of freedom that are odd under the Z2.  So we project out the zero modes in the anti-instanton, this tiny anti-ant.  And we leave the zero modes in the instanton. They survive.  Okay? So it looks like there are possibilities for interesting quantum numbers in this object.  But at this point, I want to emphasize that it's a bizarre looking object.  It's bilocal. It depends on two points in spacetime, X plus and X minus.  And then this additional parameter, rho plus. There's an analogous object, the Cp transform, in which you have an anti-instanton with a tiny instanton embedded.  Anyway, it looks very bizarre.  And this orbifolding, right, where you eliminate the degrees of freedom that are odd under the Z2, it looks like that ought to make sense of field theories which have global SU2 anomalies.  Because this gets rid of the global SU2 anomaly, but leave that.  Okay. We need to understand the flow in this neighborhood.  And one can actually calculate the Yang-Mills action, of which the flow is the gradient.  I'm not going to go through the details, but you find that there is one unstable direction.  So there's a trajectory, right? So you have a fixed point.  I normalize the Yang-Mills action so the instanton has action one.  So we have an instanton and an intense instanton, that gives two, plus something proportional to the sizes.  Okay. At least one of which is small.  And most of the form of that is determined by conformal invariance in R4.  But if you study the behavior near the fixed point, when the little anti-instanton has zero size, that's a fixed point, that's a genuinely self-dual field.  You study the behavior near that, you find one unstable direction.  So there's a flow that leaves from near this fixed point, this non-trivial fixed point, and flows down to the flat connection.  And then interesting things happen when both instantons become small, but the flow becomes sort of like asymptotically free.  You don't have actual instability, but just marginal instability.  So at this point, my thinking is still quite muddled.  The question is how these degrees of freedom will contribute to the space-time physics, if they do.  So my best understanding of how to proceed is to introduce new coupling constants that couple to this winding mode, and then study the dynamics.  I mean, one can fantasize all sorts of things, a condensation, well, who knows.  But in order to study the dynamics of these couplings, one has to know the beta function of these new degrees of freedom.  So one has to understand how products of the twist field and ordinary two-dimensional fields evolve under two-dimensional scaler.  And in particular, zero length just guarantees that the energy will be zero classically.  But we know from ordinary orbe folds that the twist field, the scaling dimension of the twist field gets quantum corrections immediately at one loop.  And the question is, they would ruin this idea of low energy excitation.  So that has to be checked.  And then this, it seems to me the basic object that is this unstable trajectory from the instanton-anti-instanton pair down to the flat connection.  But I have yet to work that out.  Thank you.  So it looks like the lambda model is actually, it's a single model on the product of the original space-time manifold and the space-time manifold.  No, no, no.  I mean, you can think of it that way.  I think it's, yeah, but it seems to me to analyze the structure.  It's important to understand that the couplings, lambda i, which in this picture are the fields on space-time, right?  They have non-trivial renormalists.  They aren't dimensionless, right?  They have dimensions that come from quantum corrections in the original nonlinear model.  And one, the structure of the lambda model is based on the properties, the quantum properties of that two-dimensional, of the original two-dimensional field theory.  So I don't think it's all that useful to study, to quantize them simultaneously.  The picture I presented is, first you quantize the original model and then you add fluctuations of its quantum coupling constants.  I mean, it might be that there's some fruitful way to do it all at once, but it hasn't seemed like that.  Another option would be to do something like a spin glass model, where you would take the first quantized, the way you say the original model, then you take a, you know, connect.  That's interesting.  Yeah, I have no idea how to, but that's very interesting.  So you'd get an effective theory just, yeah, that might be, that might be.  Other questions here?  Yes, maybe it's related to the previous question, but somehow M is the space of all 2D quantum field theories, and you use it to build another quantum field theory.  So somehow this new field, the lambda model, should be a point of M, isn't it?  No.  It's a, it's not all 2D quantum field theories.  It's a special class.  Oh, I forgot one thing I really wanted to say.  Back when I started talking about string theory, that this is where Demeric Kniznick and I intersected.  We were both interested back in the mid-80s in finishing the covariant quantization of the fermionic string, using Sasha Polyakov's super-diffinomorphism 2D ghost fields.  Yes, I definitely wanted to mention that.  My motivation was this, I need a covariant two-dimensional field theory.  But, so it's a special class of 2D quantum field theories, the original 2D field theory.  And the lambda model is not a globally defined 2D field theory.  It's only operating at short two-dimensional distances.  So I don't think that the picture is, as you say.  You refer to this, twisted the loops of instantons as a minimum of a positive of I1 over 2B.  Does that mean that you have further configurations?  Well, I mean, you, you, okay.  So you have this parameter space, which is the size of the tiny anti-instanton, and then a three-sphere of orientations.  An SU2, or MAZ-E2, but think of it as a three-sphere, okay.  And the non-trivial loop goes from the north to the south pole of that three-sphere, okay.  So take the size to be non-zero.  Really, what you can think of this is describing, you have all the other modes of the gauge field,  all the other ways, the irrelevant directions you can go away from this family.  And what this is describing is a cycle in, so that, so that any non-trivial, any loop that, that, that surrounds this cycle will be non-trivial.  But, but for the two-dimensional physics, I'm interested in the one that shrinks down to zero.  I think that's the finished one.  And what do you need the lambda model to study those exotics?  I don't know.  I think of it the other way around.  I need those exotic twist fields to study because they're in the lambda model.  I mean, you could go, you could go and study them, but I wouldn't have a motivation.  Thank you.  Thank you.  Thank you.
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