 Let's start lecture four, and I'm going to do something dangerous again, which is that I'm going to declare That the lecture has two parts. So let's do part a that means that I'll have to make it to part b hopefully So let's quickly review and hopefully you're going to see why I've divided it into two parts so We have been discussing a formulation for the complete three-level S-metrics of three theories of Einstein gravity Jagmills and a scalar field in a very strange representation in this by a joint representation Okay, so all the formulas that we have found have this form So taking the advice from the comment that Ashok made yesterday Perhaps even in gravity we should separate the two fuffians and always think about them as being two independent objects So instead of writing a single integrand, I'm going to write an integrand that is going to be an integrand on the left And an integrand on the right now the measure is a measure over the modular space of the n-puncher sphere Connected to the kinematic space by a set of equations That we call the scattering equations So that was a measure of integration and it's all well defined now These integrants I left or I right They belong it to a set Which so far has only two objects one of them is This part Taylor like object of the product of the differences of sigmas going all around a chain in a particular order Okay, of course we can consider different orderings, but this is the canonical one and the other possible choice Was the fuffian or the reduced fuffian the prime means reduce of a matrix That as juji mentioned took me almost an hour to write down. So I won't try to do it again So we can choose left and right to be one of these any one of these and then we get the different theories If we choose the two fuffians we get gravity if we choose a part Taylor like object we get young meals Choosing the fuffian on the other side if we choose two part Taylor like objects. We get this by a joint scalar So in order to save some time I'm going to denote this as part Taylor with the ordering one two Up to it. Okay. I think I did that yesterday, too Good, but I started the whole lectures giving you some motivation about Finding representations that may manifest properties like kk BCJ and we succeeded we found a representation for young meals. These are properties that young meals has That makes manifests these Two relations, but how about gravity? Why is there a formula for gravity? Why is there a formula for a scalars? Okay, so there is another mysterious property that came to be known as the kawaii level and Type relation Discovered in 1986 or For short KLT and this is a relation that connects To different very different objects, so it connects a gravity amplitude to Partial amplitudes in young meals with a particular ordering. So by now you should know what this denotes This denotes a set of labels in some particular permutation Of course here. I mean the set of labels that are not one two three One two three will always appear in this order there is also another piece here and KLT realized that if you sum Over all possible permutations I guess how many there are a minus three factorial and you multiply by a something that depends on alpha and beta It depends on the two permutations but it's only a Function of Mandelstam variables this relation holds true okay No matter what the helicity of the gravitons are so if you have a gravity ton a particular gravity ton gravity ton one has positive helicity Say in four dimensions. He has positive helicity Then you have to choose the two gluons that appear here Bluons one and one on both sides to be also positive helicity So if you do that This formula is true this formula is indeed true in any number of dimensions and the way KLT found it was as I mentioned yesterday by taking the closed-stream formula written as an integral over the sphere and Very cleverly deforming the contours so as to produce two real contours rather than an Integration over the complex over over the complex sphere with DC and DC bar They ended up producing two real contours and therefore producing two Jamiel's amplitudes and then this object came out By moving Passing different poles and I'm producing this object here now this object as I told you only depends on the Mandelstam variables but it's so complicated that I like to explain how complicated it is by Using an analogy that or a story that Sydney Coleman used to say which is that if you are if somebody wakes you up in the middle Of the night and puts a head in your again in your head and ask you to write down this object I think I would say well just kill me. I don't think I'm not even gonna try So I'll try to convince you today that there is a way of thinking about this so that you would survive the test So pay attention because you know Okay, the way we're going to discover how to survive Coleman's test is going to be again using the technology we have developed Okay, so any amplitude the amplitude that I wrote over there Is an integral with this measure, but these delta functions completely localize They integrals over the sigmas and they localize localize you to m-3 factorial Genetic points in the model a space Okay, which points they are well they depend on the particular kinematic invariance that you have Okay, but they are genetic So when you evaluate that integral is not really an integral all you're doing is solving the equations so you have to Sum over all solutions m-3 factorial of them Evaluate the integral on the left on the I solution the integral the integrant on the right In the I solution and divide by the Jacobian yesterday, we also discussed the Jacobian and the Jacobian was the reduced determinant of the matrix Phi and the matrix Phi was the Jacobian matrix of the equations given by S a b Sigma a minus Sigma b if a is different from b and minus Phi a c Sum over c from one to n if a is equal to b We said that these metrics had coran three. So it wasn't as good as having something of coran one. I Stick to my state, but then you see if you choose to Remove or use ESL to see invariance To fix particles I a J and K and you choose to remove equations P Q and R It's the same as removing from the metrics The rows I J K because the rows correspond to the variables that you are differentiating and it corresponds to removing The columns P Q and R because P Q and R are the equations that you're using right or their Columns are the equations that you're using and if you're removing P Q R It means that you're removing those columns. So these are the rows you remove and these are the columns You remove Okay, and the statement we made yesterday Perhaps not very clearly, but I want to amend that is that the determinant of this object divided by the Vandermonde of I J K P Q R is completely independent of the choice and There therefore it deserves a name that doesn't depend on the choice and we call it the reduced determinant of phi Okay, so now I hope you're convinced That this is the formula that if you would like to put in your symbolic manipulation program could be anything maple sage Mathematica This is a formula that you have to put in you solve the equations you put this formula in you evaluate on all the solutions One by one you added it up and you get the answer Okay, now I want to write this in a slightly different form So what I want to do is to write this as follows. I want to sum from I Want to introduce another sum Okay, and I'm going to introduce the identity metrics This is all evaluated on the solution I here. I'm choosing the solution I too and this is going to be evaluated on the solution J Okay, you haven't done anything here Now I'm going to define this as the entries of the diagonal metrics that I'm going to call D and therefore the amplitude is nothing but a vector That I've made out of the integral on the left evaluated on all these solutions So it's a humongous vector, right? It's m minus three factorial in dimension times the matrix D times The vector on the right. So it's just an inner product computing amplitudes is basically an inner product yesterday We discussed in a little in some length They buy a joint scalar Pieces right we said that if we expand all these double traces that we have there traces with respect to UN and the traces We respect to UN tilde. We will get objects that look like this So this is back to the buy a join a scalar We have some part Taylor with some ordering Which I'm just randomly going to choose to be one two three alpha no connection whatsoever with what you have over here times Again no connection So you have this And we said well if you compute this object You're going to find the sum over all possible Feynman diagrams in a lambda Phi cube theory that can be Drone on a plane consistent with both choices of ordering not the same time of course but meaning that if you take the diagram and you paste it on the plane You can choose it in such a way that this ordering is respected You take the same diagram you twist it you do crazy stuff to it and you paste it again And you're going to find that it's now consistent with this order Okay, if the diagram satisfies us that you add it to the sum So this is a fairly simple object to write down. It's a sum of very very simple Feynman diagrams And that's why it deserves a name So I'm going to call it M alpha beta Again, this depends on the permutations of the labels that you choose to put here and here Just as we did here This is an integrand on the left and this is an integrand on the right So I can do exactly the same thing that I did there and write this formula as the sum over solutions evaluated on the I solution evaluated on the I Solution again, but here I want to do something a slightly different Okay, I want to put this part Taylor factor with his own Determinant prime and this one too Well, but in order to satisfy the formula in order to make this an equality I have to multiply by one determinant prime of five. Is that clear? Nothing fancy. I'm just you can just cancel this with this and get exactly this formula over here Where the eyes were chosen to be part Taylor like factors So you haven't done anything special What's going to be special is a following I'm going to define this object. I'm going to call it you alpha I and This object over here. I'm going to call it V beta I Now these objects both of them are M minus three factorial by M minus three factorial matrices How nice They are a square matrices But their meanings the meaning of the raw space and the meaning of the column space is completely different one of them has to do with the solutions of the scattering equations and the other one has to do with Ordering's so somehow it has to do with color orderings So these matrices are somehow transforming from one space to the other now. Let me write again my metric my Object M alpha beta, but now in metrics notation This is a matrix So you take the transpose of this matrix and you multiply it with The inverse of the D matrix Remember the D matrix was defined to have one over the determinant in the diagonal So I can put here the inverse and here we have the V matrix Okay, and I can remove these indices and think about M also As an M minus three factorial by M minus three factorial matrix But M is a matrix where both the row space and the column space are both defined in the Ordering space in the space of possible orderings excellent question the difference between you and V is that There is no difference. They are exactly the same object and the reason is that I chose the same Particles here and the same particles here in exactly the same ordering, but I Forgot to mention that the reason I'm calling them differently is that you are allowed to change that There is nothing that forces me to choose the same order in here and the same order in here Okay, so I encourage you to try the same procedure, but now with this a completely genetic Object where the particles that you fix could even be different They don't have to be one two and three or you can fix one two and three and change the ordering Okay, so you can have in fact To make a connection to the to the standard KLT formula when should use one three and two but Allow me to keep it as one two three. Otherwise, I'll get confused along the way Okay, excellent question. Yes, so you and V in this particular example. They are exactly the same object Well, now you see what I want to do right? So We have this formula Where D enters and we have this formula where we have D as well. What if I get from here D So let me try to Get from here To obtain D from this formula in terms of the matrix M and in terms of these matrices Okay, so what I have to do is to multiply by the inverse Of these metrics and the inverse of the V matrix and I get the inverse so let me do it here and The final step the complicated step is to take the inverse of both sides So taking the inverse of both sides I get the You transpose and M inverse here Okay, now I'm going to use this definition of V or this representation of D in this formula So I'm going to learn that my amplitude or my original amplitude this one can be written as The sum over I or let me write it in vector notation first the vector I left times the matrix V Times M inverse you transpose times the vector R Okay But these are vectors in the solution space and they are contracted with this that has a solution index So this object by itself Doesn't have any solution space indices. It only has a color space or an ordering space that is contracted with the Color space color indices of the matrix M inverse and the same thing with this object so In equations what I mean is that this thing in parenthesis is nothing but the solution over I from 1 to M minus 3 factorial And I can write that explicitly here I times V But V is part Taylor Evaluated on the solution 1 1 2 3 and I call it what beta divided by The determinant prime of I Okay, that's the first parenthesis. This is multiplied by M inverse beta alpha Times this object here and that object there is the sum again over I from 1 to M minus 3 factorial. I Have part Taylor 1 2 3 alpha Evaluated on the solution I Times the integrand on the left on the right Evaluated on the solution I divided by the determinant prime of Phi I Have you seen what we have done? Now each of these pieces Can you recognize what it is? Well, it's again an amplitude where you choose one of the intake one of the half Integrants one of the two to be part Taylor and the other one something okay So let me choose just for fun Choose I left to be the faffion of the mysterious metrics that we found yesterday And I write to be also the faffion of the mysterious metrics So what do we get? Well, we get that when we make this choice This original function becomes the gravity amplitude by definition So that's a gravity amplitude and what do we get? Well, we get a sum Here the beta index is contracted and the alpha index is also contracted meaning there is a sum over alpha and beta This object over here a part Taylor width Combined with a faffion Is nothing but the corresponding Young Mills amplitude? With the ordering one two and beta chosen Exactly by the part Taylor we chose there, okay times M inverse Beta and alpha and on the other side we find Young Mills and what did we get? Well, we have just redidive KLT and now if somebody wakes you up in the middle of the night and and puts a gun in your head and tells you Tell me what the KLT Metrics is the one that you have to multiply by the partial amplitudes to get gravity You say oh no problem. It's just I it's just the inverse of a matrix that I know how to construct construct It's the inverse of a matrix whose components are the partial the double partial amplitudes of a by-join a scalar field Well, hopefully the person would be happy with that If the person is a physicist they will be happy And by definition the person is a physicist because otherwise he wouldn't be asking about that. Yeah, I think I think that if you see From the beginning we derive everything from the BCJ item and I think BCJ can be really derived from KLT So so the whole the whole thing comes to a circle speaking of circles Let's do a final application of this Which is number two and Say choose to be a faffion As before Well by at this point we only know two objects right the faffion and the partailers you guess what I'm going to choose now here, right? Choose this to be the partailer, but which partailer I Don't have any preference and probably neither do you right? So let's choose the most generic one Pick a random permutation Any one of the n factorial possible permutations of labels. Let's call it w. So this is a completely crazy random permutation Okay Good Now let's pass it through the machine and see what we get again on this side What do we have these are the integrands? So my original amplitude here in this formula is nothing but a partial amplitude with the crazy ordering one w2 Wn completely generic so this is The object I'm using here. Okay Now I want to see what this machine gives as an output. So on this side What do I get? Well, I have I left is the faffion times these orderings. So this is a partial amplitude So again, I get a sum over alpha and beta with young meals again. This is young meals With the ordering one two three and beta Times the inverse of these metrics And on this side, what do I get? I get a partial a part Taylor factor and here I also chose a part Taylor factor so here I get this integral D mu n with part Taylor with the order in one two alpha and part Taylor times part Taylor with the order W1 up to W n once again This is an object that only depends on Mandelstam variables, right? Because this is a double partial amplitude in the bio joining a scalar theory the scalar theory can only depend on Mandelstam variables. Now you see what we have done this whole thing here Let's put the sum over alpha in there or let me write it like this sum over beta a one two three beta sum over alpha of M inverse beta alpha times these Bio joined Object this whole thing is only a function of Mandelstam variables So note what we have done. We have written an Amplitude a partial amplitude with a completely genetic ordering as a linear combination of what? As a linear combination of partial amplitudes that have one two and three consecutive and The other ones completely scramble multiplied by a function. There is only a Function of Mandelstam variables. What does it sound like? Sounds like BCJ, but it's even better than BCJ Why is even better because I didn't have to start with something that had one and two already adjacent So this one does KK and BCJ all in one shot. So this is KK plus BCJ All in one shot and again a formula That you can actually remember Okay, very good. So that concludes part a of the lecture Any questions before we start power B? So let me tell you the reason I decided to split the lecture in two parts The reason is that up to here Modulo of signs and probably misprints Everything has been very rigorous. I've derived almost everything for you. Okay now in part B Almost everything that I'm going to say is gonna be conjecture. So Almost anything that you see that you will see in part B is something that still needs to be proven There is evidence, but it has to be proven So there must be something serious because I'm erasing the whole blackboard it sides I'd really want to think that I'm giving a different talk. I hope not in fact I've decided to write the So I'm going to write them the metrics Yeah, so I've decided to So I won't attempt to write these metrics again. Yes the metrics Muscle as fermions. Oh, I still haven't talked about muscle as fermions. Yeah, so I wish The the problem we have with fermions is that this formulation is dimension independent and fermions are very dimension dependent Well, that's that's the reason I've been giving myself all this time But no in principle There's been some work by Nakulik Introducing to fermions So if you introduce a pair of fermions the formalism can be done nicely But more than two fermions. He hasn't been done Well in ten dimensions, there is a formulation using a bit twisters Strings that allows you to to have all the matter content that you would normally have in type 2a type to be Or even heterotic strengths Yes But a formulation that is valid in any number of dimensions is not it's not it's not available Okay, so instead of writing the the whole monster That I wrote yesterday. I decided to use the Technique of divide and conquer so The metrics was a matrix of side 2n by 2n That's the size. So I'm going to call the first block a the second block B C and minus C transpose a And B are both anti-symmetric C doesn't have any symmetry properties But overall the metrics psi is anti-symmetric So a is the metric is the metrics with components given by this is a B Has components Let me write this in terms of case in order to keep the analogy more precise between a and B and once again is zero on the diagonal Now the matrix C is More interesting So that's a little faster than the exercise. I tried yesterday But I hope you agree with me that doing this makes a little bit unclear How gauge invariance works out and all the things all the exercises with it So I hope you agree that the exercise yesterday was worth it Okay, so what do we want to do? I? said we found formulas for gravity young meals and the by John a scalar and they all fit together KLT mixes them up So you would think that this is a completely closed story and end of the story move on and do something else right Well, I also thought so But then you start thinking well maybe I should do something to this and try to explore the Most generally space of quantum field theories that admit a representation in this form I see that that's also the spirit that you guys have because you keep asking about fermions So you also want to know was there most generally space of field theories that admit a representation like this one So one way to start is by Compactifying what I mean by this is a following a start with gravity in D plus one dimensions Little D is the dimension that you want Okay, but let's start in D plus one dimensions now take the momentum vectors of all your gravitons to be in the dimensions So I'm using the standard notation that capital index means the uncompact if I space and the Greek indices are the compact sorry the uncompact I shouldn't have said that The capital indices are the total space and the mu index the Greek indices are the uncompact if I space Okay, so the momentum of all particles has this structure and it has zero in the last component Now gravity by now We are all very happy With the formula that has these two integrants So I'm going to choose the polarization vectors on the right all of them. I'm going to choose them to be external Okay, they are all going to be External polarization vectors, but on the left I'm going to allow myself to make a choice between having polarization vectors that are external or Completely internal so from the point of view of the small D space-time Anything that I choose with both polarization vectors to be external looks like a Graviton, right? It has two polarizations and Anything that has one of the polarizations chosen to be internal will end up having only one external polarization And therefore it has to be a vector boson So what's the possibility for this theory to be well? Whatever it is. It has to be a theory that has photons and has gravitons. So let me call it Einstein Maxwell So can we write down a formula that computes all possible scattering amplitudes? They've complete three-levels metrics in Einstein Maxwell theory. Well, here is a proposal the proposal is that the integrand is So you just take the integrand on the right For Einstein Maxwell And you keep it as the Fafian of the original metrics, but now evaluated on the vectors in the uncompactified space With all Depolarization vectors and sigma no problem now the left side It's going to be much more interesting because now it depends on which particles you are scattering so This one is going to be the Fafian of a matrix that is going to look like this So let me put all the gravitons here The first and labels are going to be gravitons and here. I'm going to have photons in the next and labels This matrix is 2m by 2n so I can do the subtly the same thing here. I can put gravitons here and photons here Photon labels here. Okay, and the same thing in this direction So we're going to have gravitons photons gravitons and photons Okay, so what's special about this matrix well something very interesting happens So if you have an internal polarization vector dotted with any of the momenta What do you get? You get zero right so Everything here These big blocks they are all zero Remember the definitions right and now this block here Let me let me actually use a color chalk This is tension here is a distinction that appears here Yes, that's exactly what I'm doing exactly that yes So what I get here is a contraction of polarization vectors of only photons Right, and they are all boring the epsilon the epsilon for photons if both are photons it gives me one So these new metrics. Let me call it X is a matrix that has components One over sigma a minus sigma b if a is different from b and zero if a is equal to b The rest here This is still my matrix a nothing has changed and these are some reduced Metrices that I get by doing the proper contractions and here I get the same b matrix but reduced Okay, just restricted to the graviton labels So note something interesting the faffion of these metrics now breaks apart into two faffions Right, so we discover that the integrand on the left happens to be the faffion of a reduced Psi matrix, which is this one Times the faffion of the matrix X So this is exciting because it's saying that well there might be more building blocks than what meets the eye so in the list that I started with it's not only part Taylor factors and big faffions It seems that we also have Other kinds of objects. Okay, so of particular interest for what we're going to do today is Yes, it could be this could be EMS Yes, in fact, I should have called it EMS, but So let's consider the case the particular case when there is a pure photon So all external particles are photon none of them are gravitons or the scales only photons What happens in that case? Well in that case these metrics X grows and Swallows all this block We get a big zero here and a big zero here and this matrix a stays by itself so what we get For the integrand on the left for the pure photon is the faffion of The biggest possible matrix X that I'm going to call X n Times the faffion prime of the matrix a Okay This matrix a that we have here nothing has happened to that matrix a Now you get even more excited. You said, huh, this looks like a canonical object And then you go back and remember that you have done many things with KLT and KLT is an operation that merges two theories with color indices to produce a new theory But you can also think about it In the inverse way you can think about KLT as a machine that takes one theory That has no color ordering and breaks it apart as a sum of things that have color orderings Right, so these pure photon theory Has no color ordering Well, why don't we use KLT inverse and see what we get so using KLT inverse We learn that we're going to get a theory that has a part Taylor with some particular Ordering and I can take I left and multiply it here and this integrant Should be meaningful or hopefully should be meaningful Well, remember I told you I told you that the KLT proof I gave you has lots of flexibility I'll let you find out what the flexibility is Now this is a theory Well, that only has a scalar fixing it Or this is an amplitude that only has a scalar fixing it But it really came if you wish by applying this compactification procedure To something that had a Fafian prime upside here that had polarization vectors But we chose all the polarization vectors to be internal So if this had been our original theory this we have been young mills in one higher dimensions And we are compactifying down in one dimensions and computing the amplitude with only scalar fields So this is an amplitude in young mills scalar theory and This is the pure a scalar Amplitude very good now after doing all this you get very excited and you say well then it seems really That these little pieces that I've been getting the new pieces are meaningful They seem to have meaning by themselves So what are the new pieces that we have found so the new pieces are the Fafian prime of a And the Fafian of X Now in order to play the game of putting together things that are meaningful in the end We need to respect the SL to C invariance Remember everything was built on the assumption that we have SL to C invariance So what is the SL to C transformation of this object? So From the point of view of SL to C the Fafian prime of a counts the same as the Fafian of X and and both of them count the same as a Part Taylor to the one half or a Full Fafian of psi To the one half and those are the rules just start to mix and mash Everything so that you get the same SL to C weight as two Fafians Then you say well, let's take this seriously and The next procedure that I'm going to introduce is guessing Well, what's the first thing that you can try? Well, you say maybe let's try something simple Yes, yes Because remember that we factor out a piece that has a van der Monde the van der Monde compensates for the reduction in the size of the metrics so Let's start our game of mix and match so One of the simplest objects that you can put in is the part Taylor like object, right Well, if you want to get a new theory or you want to get something completely new We have to put here something that we didn't try before if we try the Fafian. We're going to get Young meals if we try Fafian prime of a with Fafian of X we get things that we tried before Something new we need a new combination Well Something that has a correct SL to C weight is Fafian prime of a square Okay Once again These theory has no polarization vectors whatsoever. So it must be a theory of a scalar field scalar field with a un Flavor group and the reason is that it has some ordering Which we can dress with the traces of the corresponding Flavor group Well, just as it is there It's very hard to guess What S metrics this thing computes then you do I guess what people eating a string theory long long time ago, which is You sit down and compute the three particle up the four the five. Well, in fact all even numbers Sorry all uneven numbers of scalar fields give you zero. Okay, so this is This amplitude immediately Implies is that if you have an odd number of particles you get zero. So that's another clue So you start computing all of them and you start building up The Lagrangian or a Lagrangian that we produce this thing so after you do maybe Six or seven terms that are non-vanishing you start to recognize the pattern and you find the following Lagrangian So let me introduce a coupling constant So you find terms that seem to resum if you keep the pattern they resum into something like this and Then of course the next natural step is that you take your Lagrangian and you show it to everybody on the streets And you can you ask them have you ever seen something like this? Is this something that is known or is this a new theory? Well, the answer is that Nobody actually knew But they said well it looks like so you have a scalar theory a scalar field valued in UN And people in the 60s. They were looking a lot of things like that In particular There was a chiral Lagrangian That describes the interaction or the effective interaction of pions Okay, you have a pion field and these are the generators of UN and you make a unitary matrix And you describe the effective interaction of pions by writing a Lagrangian that looks like this This is unitary and this is the pion Lagrangian or the nonlinear is also known as a nonlinear sigma model You say well, okay fine, so people did that but that's I mean it would be too good to be true That that thing happens to be equal to this But then you keep searching and Cali Found a representation of unitary matrices that is of this form So if I is her mission this matrix is unitary and guess what you put this into this Lagrangian and you get exactly this so this Computes the s-metrics for the nonlinear sigma model or the UN nonlinear sigma model So it ended up having a meaning so maybe we just got lucky Yeah, so you can now take you can now take soft limits double soft limits and show that this formula Precisely reproduces the double soft limits that tells you the nonlinearly realized symmetries of the theory Yeah, the single soft limit vanishes We tells you that there is a nonlinearly realized symmetry But the structure you have to learn it or you have to study it by taking double soft limits So even though the single soft limit vanishes when you take two particles to be soft Simultaneously you get something that is non-zero, but it depends on the order in which you do it and the way The order in which you do it reveals the structure of the Symmetry that is nonlinearly realized Okay, so now you have that and you say well, but I've discovered a new theory. What do I do with any new theory? Well, I start to KLT that with anything that I know to produce other things Right This one has some partial ordering so I can pass it through the machine of KLT with things that I know to produce new stuff so let's use a KLT machine with a nonlinear sigma model and Say the young mill scalar theory that I wrote down there So what do you get? Well, you get an amplitude Now doing KLT is trivial, right? We know that all we have to do is to delete this use this as part of the integrand delete this and Get this as the other part of the integrand. So we get this Fafian of x we don't have to go to the pure scalar case we can use the reduced metrics So it's not completely Fafian prime of a times Fafian prime of a square So this new theory has no ordering it has a scalar field. It also has a photon field What could it be? I could give you some hints and could try to derive it and and do things But even that we're running out of time. So let me tell you what it is So here is the Lagrangian the Lagrangian turns out to be sorry It's a standard curvature for a photon field and This object here is what is known as the Dirac theory Okay So we got all the way to dbi and We found a formula for the complete s-metrics of this theory what is spectacular is that you start You don't know what this thing is and then you start to compute the Lagrangian that would produce this and you start to see these Funny coefficients appearing said hmm these coefficients What do they look like? Well then you start to recognize that they look like the expansion of a square root and Of course, you know that any Lagrangian that has a square root at least that is known has to be of this form of dbi Okay, so the last theory that I want to mention is the last one actually that You could imagine doing with what I told you which is again using KLT this time on the non-linear sigma model and And what do you think we should put here? Well, why not let's put another another non-linear sigma model there and see what we get So whatever the amplitude is of this theory is something that is extremely simple Perhaps the simplest of all the possible objects that I've written down because it's a Fafian prime of a Square coming from here and another a square coming from here. So it's just this It's a scalar theory again with no color ordering and it has lots of derivatives Then once again you play the game of computing the Lagrangian and You find that the Lagrangian looks like this You find a Lagrangian that has lots of derivatives and then again You start to talk to people and ask well, have you ever seen something like this a scalar field with lots of derivative interactions and so on this time The answer was yes, we have seen things like this, but maybe they are not related. There are theories called Galilean theories and they have a funny Lagrangian that looks like this where this LM is five Determinant is a well, okay fine. Yes, so that looks pretty good But what are the chances? So this thing has many coupling constants This thing has none at least at most he has one I don't have room to have different coupling constants now. It turns out that this theory is a special Class of Galilean theories where the couplings have been tuned to exhibit an extra symmetry that the original Galilean theory doesn't have so this is a special Galilean theory that was discovered Or it was written down in January Okay, so let me see if I can show you something We're over time, but let me just even that is the last lecture So this is so far what we know. These are the integrants two fafians. You get young meals. You compactify You can get all these change of theories. You can start with young meals compactify You can do different operations to them more nymphs give you dbi There is a new there is a theory that we still we have done the same exercise We have been asking everybody if they have seen this theory and nobody has and we call it extended dbi It's something that interpolates between dbi and the nonlinear sigma model So if you have seen something like that, please let me know But so far nobody has okay, so is this the end of the story or is this just the beginning well You are the graduate students, so you should figure it out. Thank you