 So a better title, a more informative title for what I'm going to say would have been brains in anti-deciderspace and conformal field theories on the Coulomb branch. And what I plan to do is to present a world volume action for certain P-brains in anti-deciderspace and to interpret the result as an effective action for a super conformal field theory on the Coulomb branch. And the principle that's guiding me in doing this is to follow a principle that I've been espousing to others for a long time, which is to take coincidences seriously. There are several examples in the recent history of theoretical physics where this has been a fruitful thing to do. And there are other examples where there were missed opportunities because people didn't take advantage of this principle, which we now can recognize with the wisdom of hindsight. So in certain cases, I will boldly conjecture that the P-brain action is the exact effective action for the super conformal field theory on the Coulomb branch, capturing the entire theory on the Coulomb branch, and call it the highly effective action. Exactly what is being conjectured should become clearer as we proceed. So I'm going to discuss systems with lots of supersymmetry where I have more mathematical control. These are systems which, from the point of view of the previous talk, would have been regarded as trivial because he was dealing with systems less supersymmetry, where he could ask more difficult questions. Here, by the same token, that means if I'm going to say anything interesting, I have to make a stronger conjecture. So the examples that are going to be discussed are D3 brains in anti-decider 5 times S5, the basic example of ADS-DFT duality. The M2 brain in ADS-4 times S7 modded up by a cyclic group ZK. The D2 brain in ADS-4 times CP3. These two examples are both related to what's known as ABJM theory, which is a super conformal churn simons theory in three dimensions. And let's say a little bit maybe about the M5 brain in ADS-7 times S4, which is related to this mysterious 2-0 super conformal field theory. Now these actions, so what I'm going to do is discuss the world-volume actions of probe brains in these geometries. And these probe brains actions involve various well-known approximations. So there's the probe approximation, which involves neglecting the back reaction of the brain on the geometry and the other background fields. Since the brain is a source for one unit of flux and it's in a background of n units of flux, neglecting its effect is closely related to n being large. Also, these D-brain actions are going to involve born-info-like terms. That's a function of a U1 gauge field strength. And the formula in born-info theory has no derivatives of the field strength, just the field strength itself. So it's customary to say that this requires assuming that the field strength is slowly varying so that its derivatives can be neglected. So these are the well-known approximations associated with probe brain theories. Now despite all that, the formulas for P-brain actions have some beautiful exact properties. They precisely realize the isometry of the background as a world-volume super conformal symmetry, which in the case of the D3 brain would be PSU2 comma 2 slash 4, or in the ABJM examples would be OSP6 slash 4. Now in the formulas I'm going to be showing you, I've only looked at the bosonic degrees of freedom. I haven't put in the fermions. This is technically challenging, but straightforward. And it would have just taken too much time, so I postponed it. So that means I will only see the bosonic subgroups of these supergroups. Brain actions also implement the duality symmetries of the background theories as dualities. So for example, SL2z is an exact symmetry of type 2b super string theory. And so since the D3 brain is living in a background of that and the D3 brain is invariant under SL2z, this group has to show up as a duality group of the brain probe action. And there's a duality that turns out to relate the D2 and M2 examples as well, which I'll discuss. So the brain world volume actions have local symmetries. They have general coordinate invariance. This is familiar with all brain stories, starting with the Nambu Goto formula for the string, where you made a reparameterization invariant formula when you write down the area of the world sheet, which doesn't depend on how you parameterize the world sheet. And that generalizes all these brains. So these brain actions have general coordinate invariance. And they have something called local caposimetry. But that only becomes relevant when you include fermions, and I'm not going to be describing that. For the purposes of making contact with effective actions that I'm interested in, there's a natural gauge choice in discussing these brain actions. And that's called static gauge, and I'll show you that. And this results in formulas, after you make this gauge choice, that have the expected field content for an effective action. However, this modifies the formulas for the global symmetry transformations. So I said the super conformal group is a global symmetry of the world volume theory. And the formulas for how that symmetry is realized is very straightforward in the gauge invariant formulation. But once you go to static gauge, the formulas become much more complicated because you have to include compensating gauge transformations that preserve the gauge choice. And this will be illustrated and interpreted more later. So let's talk now about super conformal field theories on the Coulomb branch. So forget brain actions for a while. And so I'll be very specific to start with. And then we'll have a couple more examples. So let's discuss the example of N equal 4, D equal 4, super Yang-Mills theory, with the U2 gauge group. Now for most purposes, one could say that a free U1 multiple decouples leaving an SU2 theory. However, this decoupled U1 is actually necessary for the theory to have SL2z as its duality group. So it's not strictly speaking true that you can ignore it. That's been clarified by Cyberg and collaborators recently, exactly how that all works. So with this caveat, I'm going to speak of SU2, but I know about this. And on the Coulomb branch, just as Samson was saying, we have of the three supermultiplets that make up SU2, we have one that I'll call the photon supermultiplet that remains massless on the Coulomb branch, and two that I'll call W supermultiplets that acquire a mass when you're on the Coulomb branch. Now in principle, one can integrate out the massive fields exactly, thereby producing a very complicated formula in terms of the massless photon supermultiplet only. The resulting effective action would capture the entire theory and be valid on the Coulomb branch at all energies. It may not be the most useful form at very high energies, but it's all there. So this defines what I'm calling a highly effective action. It's the result of doing this impossible calculation. Obviously, we can't do this calculation. However, in some cases, we know many of the properties that the answer should possess. So what are the general requirements? It should have all the unbroken and spontaneously broken global symmetries of the original Coulomb branch theory. So conformal symmetry, of course, is spontaneously broken just because you're on the Coulomb branch, but it's being spontaneously broken. It's only broken because it has to do with the choice of the vacuum, not of the action or the equations of motion, though it's an exact symmetry at that level. It should have the same duality properties as the Coulomb branch theory with explicit W fields. It should have the same BPS spectrum. So what does that mean? Because you only see the massless multiplets. Well, you know in the formulas with the Ws, you can find monopole solutions, as Sasha showed us. And so what I'm going to argue is that this HEA should have solitons that give all the massive fields. Not only the monopoles and dions, but also the Ws themselves should appear as solitons, giving a complete SL2Z multiplet of solitons. So the claim is that the probe P-brain construction, which is an approximate solution to the problem it purports to address, gives a compelling candidate for the HEA in a few special cases. So these cases include N equal 4D equal 4 Super Yang mill theory with U2 gauge group, and N equal 6D equal 3 ABJM theory with U2 level K times U2 level minus K gauge symmetry, subscripts being the levels of Churn-Simons terms. Note that we're using gravitational theories, super string theory and M theory, as a tool for studying non-gravitational theories. But our goal here is to say something about non-gravitational theories. And an important remark is that these non-gravitational theories are going to have general coordinate invariance. And I think that's fundamental importance. In d-dimensional conformal field theory, every term in the action must have dimension d, since there's no dimension of full scale in the problem. Now, in the Coulomb branch, there is a scale, which is the VEVA of the scalar field. But as I said, it doesn't need to be specified when you write down the action. And so the full symmetry is realized on the action and the equations of motion, and only the choice of vacuum break symmetry. And in fact, all choices are equivalent. So really, you have just two discrete choices. Either you're on the Coulomb branch or you're not on the Coulomb branch. And I'm only discussing the Coulomb branch. So I'm only looking at that half of the problem, if you will. So the only unusual feature of the effect of action is that it contains inverse powers of the scalar field. And so individual terms in the action can be arbitrarily complicated with more and more fields and derivatives and whatnot, but end up with dimension D simply by including an appropriate inverse power of the scalar field. This is well known to most people, but if you haven't thought about effective actions before, then you might not know that. So now, since I want to work in ADS space, I have to tell you a little bit about it. So it's very similar to formulas that Slashe wrote down except for a minus sign, and to be ADS rather than DS. And so I'm going to work in ADS P plus 2, because I want to talk about P brains. And the formula is just given by a hyper surface, just like the one that Slashe wrote down, except Y dot Y has Lorentzian signature. So it has one time and P spatial directions. And UV gives you one time and one space in some funny way. And so that formula has SOP comma 2 symmetry. And the corresponding metric is just this thing up here, up there. And so if you just make the simple change of variables, going to x by dividing out v from y and solving this equation for u and substituting into the metric, you get this formula for the Poincare-A patch metric. And the only thing that's not manifest in this formula is the conformal symmetry. And that's trivial in the hyper surface. There's just some trivial transformation which you can easily trace through to this formula. So now let's talk about the D3 brain in ADS 5 times S5. So the 10-dimensional metric is just induced from what I just showed you. And so the first two terms at the top there are just the ADS. And then you add this metric for a 5-sphere. And so the 10-dimensional metric can be recast by using a standard trick, which is to take the d omega 5 squared and put a v squared in the numerator and a v squared in the denominator. And then you see that you have an R6 written in spherical coordinates. And you just recast it in Euclidean space. So you get this formula. Now I'm going to take the interpretation that x has the dimensions of length and v has the dimensions of mass. And v is going to be identified as the scalar field that gets the vacuum expectation value in the theory. But it won't be normalized very nicely as it stands, so I allow myself an extra constant and rename it phi. So C is just a fudge factor to get the normalization right. One of the standard formulas in ADS CFT is this relation between the radius R, capital R there, and the string coupling constant, the number of units of flux in the string length scale. And then there's a phi form flux that threads the phi sphere, which is proportional to n and the number of units of flux. And the coefficient just depends on your conventions. And the actions that one has for all brain theories is a sum of two terms, which I like to call S1 and S2. Often people call S1 DBI for standing for Dirac Born-Infeld and S2 CS, standing for Churn Simons. But you could attach many other names to each of them as well. So I prefer to call them S1 and S2. S1 does have this Dirac Born-Infeld structure. And it involves the embedding functions that I showed you, which tell you, so the sigmas are the parameters of the world volume of the brain. And x is the spacetime, so it just tells you how the brain is embedded in the spacetime. So that's why they're called the embedding functions. And in addition, what you always have when you're talking about D-brains is a world volume u1 gauge field, which is a vector field in the brain. So it has no spacetime index on it. Before you do gauge fixing, it's completely inert under the entire super conformal group. So it only gets a transformation rule after we do gauge fixing. It doesn't transform at all before a gauge fixing. And it has the standard u1 field strength. So everything I'm doing is abelian, a single brain. Relatively easy. So here's the standard formula for the bosonic part of this action. G is, again, the pullback of the metric to the world volume. But just here, yeah, but OK, right. Alpha prime is the so-called registroth slope, which is the square of the string length scale. Then there, so I wrote that up in the upper corners, the alpha prime related to the string length scale. There's the formula for the tension of the D-3 brain, which is standard stuff. It's BPS, so the formula is completely exact and unambiguous. And combining equations, the things, the combinations that ultimately appear in the equations are r to the fourth times Td3 and 2 pi alpha prime divided by r squared. And those are just pure numbers, so all scales drop out of the problem. So there are no dimensionful numbers in this action. Now we come to the gauge stress. So the static gauge simply says, take the coordinates I called x mu and identify them with the world volume coordinates. That's static gauge, and that tells you exactly where the brain is located. Well, in certain dimensions, the other fields tell you where it's located in the radial direction on the ADS and its position on the 5-sphere. That's embedded in the other six scalar fields. And in this gauge, the six scalar fields and the u1 gauge field become functions of x because you've replaced sigma by x. So we now have this formula. And I've left out all the numerical coefficients in the writing just so it would fit on the slide. But later I'll give you the formula with coefficients. The coefficients are dimensionless and only functions of the string coupling constant and the number of units of flux. Now remarkably, as soon as I haven't shown you the formulas or the coefficients, you don't see it. But when I do show you the coefficients, you'll see that this formula is 1 over the string coupling constant times an expression that only involves this combination, g string times n. This fact, I think, is interesting because it suggests that the loop expansion of the, so I'm going to interpret this thing as what I'm calling an HEA, a highly effective action. And this suggests that the loop expansion corresponds to the topological expansion of the non-Abelian theory. So the classical HEA action would encode the entire planar theory if this is correct. So in tree approximation, this means it should exhibit dual conformal symmetry and Yangian symmetry and all that good stuff. That hasn't been demonstrated yet. It's in progress. Should it follow from the T-valent invariance of the reaction? That this would be true? For the dual conformal symmetry. Yeah. So I should say that everything I'm doing is conjectural, that this has anything to do with this HEA as a conjecture. Because you could imagine that there's another formula that has all these good properties. And this is the wrong one. Right. But as you will see, I don't have a good interpretation of the parameter n. And in making this argument, I'm assuming that n is the usual n we're familiar with. And so this part of this discussion is a secondary conjecture that could be false, even if the first one is correct. I haven't discussed what I'm going to do with n. Because what I'm going to find is that I don't get a unique theory with all these properties. I get a theory labeled by an integer n. And I only want one formula. And so what I'm going to do is put n equal to 1. Because that's the only natural choice, which is kind of remarkable from the pro-brain point of view, because that's where the approximation is worst. So let me just remark on induced transformations. So I'm just looking at the bosonic subgroup. And in static gauge, we've fixed what x is. It's been related to sigma. And so if we have some transformation of x, it's part of our super conformal group under which x changes. I have to add a general coordinate and variance to bring the total change back to 0. So that's the compensating gauge transformation to stay in the static gauge. And that general coordinate and variance then infects the transformation rules for the other fields in the theory. When fermions are included, there's also local kappa, which also has to be gauge fixed. And the compensating local kappa transformations will give rise to complicated supersymmetry transformations. The supersymmetry transformations will be easy before you gauge fix, but they'll be very complicated after you gauge fix. And these extra contributions can be interpreted as quantum corrections due to massive fields that have been integrated out. People who've tried to construct effective actions order by order by what I call brute force, using Nuva method or something, find that they have to keep modifying supersymmetry formulas, transformation formulas, as they do it. So here's kind of an organizing principle for how that's supposed to go. So let me come to the second term in the action, S2, the Chern-Simons term. So for this particular example, there are two pieces. There's a Ramon-Ramon-4 form, C4, that has to be integrated over the world volume. And if you want, you can have a C0, which is a Ramon-Ramon-0 form, which is like an axion. And its expectation value is chi, which is something like a theta angle. So the Ramon-Ramon-4 form has a self-dual field strength. And that can be written down explicitly. It's the volume phi form of the five-sphere plus the volume phi form of the ADS space. And that's manifestly self-dual. Constant chi, as I said, is the expectation value of C or 0 in the background. Of course, that's a topological term. It doesn't contribute to the classical equations of motion, but it can be there anyway. So S1, remember that S1 involved the square root of determinant. And that determinant, the thing inside the determinant, instead of gauge, started out with 8MU nu plus stuff involving fields. So if you expand out that determinant in a series in fields, it starts with 1. And that 1 multiplies phi to the fourth. So the 1 contributes a so-called potential term, phi to the fourth d4x, which, if it were there, would be a disaster, because it means there's a force on the brain. And we know this thing is BPS and the forces have got to cancel. So this term has to be canceled. Wann discusses this in his original famous paper with over 9,000 citations. And he says exactly the right thing, which is that there must be a minus 1 added to that determinant so that the potential is not there. And so the thing is static. The small contribution I've made is to evaluate the contribution of C4 and to show it gives precisely the right answer with the correct coefficient. The complete answer, aside from fermions for this d3 brain then, is given by this. And here I show all the coefficients. So lambda is g string times n. Chi, I've already explained. And this g mu nu, I've only written out in second lines because this wouldn't have fit in here otherwise. So the scalar fields, the kinetic pieces of the scalar fields come from here, and there's the f. And so that's the bosonic part of the formula. So this formula is, if the conjecture is right, this formula should already be of some interest because it's what you will get from the complete answer when you just cross out the fermions. So you can see how the phi of the fourth has been canceled with that minus 1. So let me now turn to the issue of estuality. So we introduced a modular parameter tau, which is tau 1 plus i tau 2. So the real part, tau 1, is what is called chi up there. And tau 2 is 1 over g string, which appears both out front there and inside lambda, because you remember lambda was g string times n. But I'm going to put n equal to 1 eventually, right? So estuality. So in the manuscript that I will post after I get back home, the complete proof of the estuality of the formula is given, but I'm going to give you a simplified version where I omit the scalar fields, which will give you the flavor of the argument, but it's not complete as it stands. So if I take the formula I just showed you and throw away all the scalar fields, more or less, then you get something that looks like this. And where delta is the square root of the determinant. And it's a 4 by 4 determinant, and you can do it, and there it is. That's well-known in many places in literature. So next, to analyze the estuality, we're going to treat f as an independent field in Adelaide-Grantz multiplier term that implements the fact that f is dA. So h is the Lagrange multiplier 2 form, whose equation of motion tells you that f is dA. And the a equation of motion tells you that dh is 0, which is solved by making h the exterior derivative of a dual potential. Then you have the f equation of motion. So varying f in this term gives you h dual. And then varying f in the rest of the formula gives you some complicated expression, which is written down here. So this comes from the topological term, and this comes from the determinant term, just for differentiating them with respect to f mu nu. Now to demonstrate estuality, what you need to do is to solve this equation for f in terms of h. As you can see, that looks like difficult algebra. The amazing thing is you can do it. And the answer is that f, or the dual of f, is given by the same function of h and tau prime, where tau prime is minus 1 over tau. And so that means that there's a dual action written here, which gives all the same set of equations. And so that is the estuality of the theory. And I've worked it out with the scalar fields included. So now let me turn to ABJM theory. So the second example is a candidate for u2 level k times u2 level minus k ABJM theory on the Coulomb branch. So ABJM theories, as I've said, are n equals 6 turns Simons theories in 3D with this orthosymplectic super conformal symmetry. And I'm going to come at this problem from two directions. So the first direction will be to discuss an m2 brain in ADS 4 times s7 mod zk. So this k is the same as that k. This action has the field content and structure of a u1 level k times u1 level minus k ABJM type theory. So that's what you would expect on the Coulomb branch. Everything else has been integrated out. The d2 brain scenario starts from ADS 4 times cp3. And this action, turns out, has the field content and structure of an ambillion three-dimensional n equals 6 super Yang-Mills theory. So the one theory has eight scalar fields. The other theory has seven scalar fields and a u1 gauge field. But the u1 gauge field is related by duality to a scalar field in three dimensions. So they're going to turn out to be equivalent. So all the calculations I'll show you are carried out for n units of flux. But as before, I conjecture that n equal 1 is the relevant choice. So for the m2 brain theory, the radio coordinate and the s7 coordinate combine into four dimensions. So we want dimension and a half scalar fields when we're in three dimensions. And so these phi's have dimension and a half rather than dimension one when we're in three dimensions. And so this is an analog of the formula I showed you before. But now we have to build in this modding out by zk of the sphere. The way that's done is that these are covariant derivatives. So phi a are four complex scalar fields rather than real ones when we want to do this. And we couple them to a u1 gauge field so that they have charge one and minus one in this way. And we want to build in to this the requirement that b is actually the gradient of a scalar field, or the exterior derivative of a scalar field, and that the scalar field is periodic with period 2 pi over k. That's what we want. Now if that's the case, then this will be an equivalence on these scalar fields. And that will represent the modding out by the zk. And so the way we're going to achieve this is by adding a term Simon's term that ensures that there is a signal to that property. And so there are other small changes compared to the standard formulas, but these are formulas you find in the ABJM paper, where the relation between radius and n has this extra factor of k in it. And the formula for the quantization of the flux is you only integrate it out over the modded out space to get in. So those are formulas you find in their paper. So the analysis is similar to before. Again, there's an integral over Ramon Ramon III form that provides the crucial term, provides this crucial minus 1 that cancels the one from the determinant so that there's no force acting on the brain. And so in this case, we can define it as a parameter n over k. That's the standard choice, which you also find in the ABJM paper. And written in that language, the final answer I obtained from the m-theory point of view is the formula shown here, where I've just written the definition of y-mu-nude below. And again, we're getting k or n times the function of lambda. So again, if you take the n dependence of the formula seriously, again, there's this idea that you're going to get the full Yangian symmetry and the tree approximation. Here's the Chern-Simons term I mentioned. This form of A is a Lagrange multiplier here. It tells you that B is the gradient of a scalar. And the coefficients, if you do things properly, tells you that that scalar has period 2 pi over k. This term can be written as the sum of difference of two Chern-Simons terms by taking the sum and difference of A and B. And then you see the u1k times u1 minus k structure when you do that. So for the d2 brain on CP3 is a similar story. You have to use the metric of CP3. Again, you pull a scalar out of the ADS and combine it with the CP3. And then you get some funny seven-dimensional space written this way. This looks like it has eight scalar fields, but there's actually a local u1 symmetry. So it really only encodes seven of them. And the pullback of the metric to the three-dimensional d2 brain world volume action is denoted that way. So that describes the s1 term. And the s2 term in this case has two pieces. One is the integral of CP3, which, as before, cancels the one from the determinant. That's all it does. And there's a second term, because as ABJM already noted, there's also a Ramon-Ramon 1 form in this construction, which contributes an amount that can be written this way. So the k over 4 pi times w is basically the c1, is the background value of c1, which is substituted into the standard formula, which is c1 wedge f. And w is given by this formula in terms of the complex scalar fields. And it has a nice geometric interpretation, because dw is just the CP3 scalar form. So putting these ingredients together, the complete d2 brain action, aside from fermions, is given by this formula. So here the g mu nu, which was on some previous slides, incorporated the seven scalar fields. And here's the u1 gauge field. So by the time we go to static gauge, this will have the field content of Super Yang-Mills theory in three dimensions. It's surprising that you would find that in a theory that has no scale in it. Because you say, what about the kinetic term of f? Integral of f squared has the wrong dimension. Well, you don't get the integral of f squared. You get the integral of phi to the minus 2 times f squared, which has dimension 3. So by manipulation, it's very similar to proof of estuality for the d3 brain. One can demonstrate that this is equivalent to the one I got from the M theory picture. So sometimes people think that the type 2a picture is less powerful than the 11 dimensional picture. But here you reproduce the same formula. But there is one piece of information you don't know if you only do it from the 2a point of view. Because I cheated in writing it in terms of k and lambda. The parameters that you have when you do it is a 2a problem of g string and n. You don't have a k. And it turns out that what you don't know if you only know the 2a stories, you don't know that this is the formula for gs in terms of n and k. See, gs is not a continuous parameter in this problem. It depends on two integers in this way. And if you only knew type 2a string theory, you wouldn't know that. But otherwise, the two formulas are equivalent. So I only have one slide about this M5 brain. So I'll be very quick. The M5 brain, in ADS7 times S4, unfortunately, there's no analog of this modding out by zk. And because of that, the theory one we'll get here doesn't have a small parameter in it, and doesn't have a perturbation expansion. So finding a classical formula for a theory without a small parameter is a questionable value. I think it's not a zero value, but it's not as much value as you might like. In any case, if you do it, here are the steps. You turn the S7, the S4 into an R5 by the standard trick. And after some song and dance, you end up with this formula, just like before. This h-twittle mu nu is a complicated story for how you represent a two-form or the self-dual three-form field strength in this kind of a theory. And that was discussed in papers I coauthored in the 90s, but I don't want to redo here. In any case, it has that information in it. And again, well, there's only one parameter. That's N. And I want to put N equal to 1. I didn't show the N here. I didn't show any coefficients here. They exist. So I should emphasize something that even though this is the classical approximation to a theory that doesn't have a small parameter, it's nonetheless interesting because the corresponding formula with unbroken gauge symmetry doesn't exist. There is no analog of the SU2 gauge theory for the two-zero theory. So even though it's not a terribly useful formula, it at least it exists. So I think that's already progress. And I think it might be useful for studying solitons. So this should have a self-dual string as a soliton. And I think that since that's BPS, maybe this is good enough for doing that. So one of the motivations for my doing this work was to understand the two-zero super conformal field theories in six dimensions on the Coulomb branch. And so the M5 brain action I just showed you is a candidate for the simplest such one. But as I just said, its applications are limited because it doesn't have a small parameter on which to base a perturbation expansion. Now, besides looking for solitons, another thing that might be good for is if you compact it out on a circle or a torus or something, you might introduce additional parameters that allow you to discuss some limits in which is weakly coupled. Of course, when you do that, you're going to be breaking the conformal symmetry explicitly. And I'm a little uneasy about that. And as I already said, there's no known Lagrangian formulation in the unbroken phase. And it's widely believed that one doesn't even exist. So that makes this formula more significant perhaps. So there are lots. So all this is obviously very speculative. The reason that I'm sticking my neck out and saying that the speculation should be taken seriously is because these systems have so much symmetry that I think that's very constraining. So there's this large super conformal group. There are the dualities. There are solitons which haven't yet been constructed. But I'm confident it will be found. If the dual conformal symmetry of the tree approximation is demonstrated, of the classical approximation, that would be very encouraging, I believe. But one wants to do other things. For the first, one of the important things, of course, is to incorporate fermions. And that's technically somewhat challenging, but there's no question it can be done. And I know how to do it. I've done analogous things in a flat space. And it's just a little harder in ADS space. When we have that, we will want to compare expansions of these formulas with the results for Coulomb branch low energy effective actions. By brute force, found several terms in the expansions of most of these theories. And so obviously you want to compare. It's clear they weren't very careful, though, because they didn't discover that there was freedom of N in the formula. So the people who write down these expansions using another method missed the fact that there's a freedom of an integer. And carrying out the comparisons, there are various things that might look wrong at first. They might have terms that involve more derivatives of the fields than I've shown. Because remember, I have no derivatives of F in the formulas and no second derivative of the phi's. But there are all sorts of possibilities for how that might get sorted out, if such discrepancies were to appear. There are the possibility of field redefinitions, which could shift around the number of derivatives back and forth. And also, as Samson pointed out to me yesterday, there's also the freedom of just adding total derivatives to the formula, which is equivalent to partial integrations, which can also shift derivatives around. So when you put all these things together, you would have to take those possibilities into account in making comparisons with existing formulas in the literature. So I think it'll be interesting to explore soliton solutions, so these things, and to explore. So if these things do have this young in symmetry, the formula for tree approximation scattering amplitudes should have some very pretty properties. And so I think it'll be interesting to compute them explicitly. And with all the technical tools that exist out there, there are people who are really well-equipped to do this. I'm not one of them. But I think it may turn out to be a good problem. We want to understand the extent to which symmetry and other general considerations should determine this thing, which is to make the case more compelling. I'm not exactly sure how to get a handle on that. But if all this holds up as people examine it, if people decide this isn't completely crazy, then, of course, one will become bold and make stronger conjectures that you should apply to the systems with less symmetry, and so on and so forth. And there are other obvious questions. You'd like to generalize, even with this much super symmetry, you'd like to generalize the higher-range gauge series. So instead of SU2, you'd like to do SU3, which would have two abelian multiplets rather than one. Now, I found the formulas with these arbitrary integer n in them, so those other formulas should have an application. So I think those formulas will play a role in answering these type of questions. But I doubt that they'll enable you to write down the full answer. So I think there will be terms in the formula that involve these things with various values of n, but that there may be an additional term, which is much more complicated and really hard to find. So I'm a little worried about that. So I might turn out to be a real hard problem. Once one has a good understanding of the classical theory, and it's convinced that it makes sense, one will want to understand its quantization and loop amplitudes. So we've integrated out all the massive stuff, but you still have the integrals over the massless stuff. Then you're going to have to confront all the, and the formula has a general coordinate variance in local caposimetry, so you have to do all this gauge fixing, ghosts, BRST, blah, blah, blah. And so that'll be interesting. And furthermore, you'll have infrared divergences when you start integrating massless stuff. So this is going to be challenging. But I think it's analogous to things that people have done in other settings. So I'm sure there are people who have the requisite tools. So I can conclude by saying we've conjectured that the world-wide action of a probed grain in a maximally supersymmetric or three quarters maximally supersymmetric spacetime with anti-decider can be reinterpreted as what I'm calling a highly effective action of a super conformal field theory in P plus 1 dimensions on the Coulomb branch. And the main evidence is that the actions incorporate all the required symmetries and dualities, which in the case of the D3 brain are the super conformal group when the Grammins are included in the SL2Z duality. And verification of dual conformal symmetry would strengthen the case. So I'll stop there. Thank you. Questions? Maybe a clarification on my trivial misunderstanding, which is maybe I didn't understand how you defined highly effective action, which my problem is it seems that if you define the story, you define it by creating out W bosons, then the answer should be non-local. It should display some momentum space, some poles of those bosons and so on. And you wrote view of the local actions. Well, it's local, but it has arbitrarily many derivatives, not higher derivatives, but many first derivatives to arbitrary powers. So I guess that's local. So you're asking where the Ws are. But I tried to address that when I said they should appear as solitans, just like your monopole. You see, the theory, as I demonstrated, has the SL2Z symmetry. So if one of them is a soliton, the other one's a soliton. But how can it be that you really integrate out the W bosons and you get, is that the procedure you're suggesting just at the time? Well, I said that's impossible. So I'm conjecturing that I found there. You're not going to conjecture it there. But also only in slower fields. I mean, here's an higher derivative. Well, I'm conjecturing this exact answer, regardless. I think this formula, Born and Infeld wrote down, is better than anyone ever dreamed. It has the general coordinate invariance. It has the SL2Z duality. And I think the fact that they didn't put in the higher derivatives of F, if you try and try to maintain the general coordinate invariance, not so easy. And they had SL2Z. So the mass of the W boson is what's the phi square? Is it being by the bandwidth scalar? Yes. That's right. So it's possible that it actually, if the external momentum is of that order, that you will be determinant revenge. So if you have a similarity with that, that you actually have to branch out. OK, let me address that. That thing inside the determinant can, in principle, vanish. What's been shown in flat space, and I haven't investigated here, is that in the limit where it vanishes, the energy goes to infinity. Yes, so that you wouldn't get there. Like in relativity, you can't go bigger than C. This would be analogous. Another thing I would remark is that if you clitianize, which you would need to do if you're going to do loops, then the thing inside the square root is positive definite. That's not obvious. Just looking at it for me, I have to think a bit to see that. But it's true. So that, I think, is encouraging also. But if you have the young males at small calving, you can integrate out the derivative. That's right. And the derivative, if you do it like I guess, you'll get some genome locals, but you have to worry about it. No, the things that people have written down, although I haven't tried to make detailed comparisons, but they have the same general appearances which you would get from the first few terms. So that they would have infinite number of derivatives, please, if you. Well, only if they go to infinite order, at any finite order, they'll find a finite number. And if they do find higher derivatives, there are all sorts of possibilities for getting rid of them, as I was mentioning. The most exciting theory you would see, actually, at the pictures of that new square term with the kind of similarity of the p squared, which is the same as that new square. But here it's castle because of symmetry. That's right. Some of the things you worry about are sort of guaranteed by symmetry. Another thing I worried about for a little while was if this is a 1 over n expansion, I'm putting n equal to 1. That doesn't sound very good for the expansion. But you have to look at the formula more carefully because there's still a coupling constant there. And you know that the theory started out as weakly coupled. So I think that's OK. Any more questions? Yes. So the action is known toward the alpha prime to the fourth. So people have done this. So have you tried to check whether that action has? Oh, you're talking about by doing out the integrals. Yeah, by actually doing out the path integral thing. Yeah, the perturbative expansion. Yes? Yeah, that's right. I said that I haven't tried to compare because some of the more interesting terms they found in file of the fermions, like there's a side of the eighth over phi to the eighth that's known to be present. Because it's interesting that there's a term like that without any derivatives, but only starting at side of the eighth of that. And even though I haven't done the fermions, I looked at it enough to know there will be a term of that general sort, and of course, there are lots of indices to go with it, which I haven't explored. So I haven't tried to do detailed comparisons. The answer is I haven't tried. Yeah, so the theory is just the bosonic term, so for the f field, right? So the derivatives of the field up to field with definition. The people who do these things by new of the method find unique answers. So unless they've made a mistake, I've got to agree with them, at least for some value of n. That might be enabled me to confirm my speculation that n is 1 if it might be realized that they inadvertently put it n equal to 1, right? Not recognized as a possibility it could take other values. If I have more questions, let's stand John again.