 Okay, good afternoon. We can now start with the afternoon session. Okay, so we will now have the third lecture of Laura Donné on Celestial Amplitudes. Thank you, Francesco. Thanks everybody for coming. Good afternoon. I hope you can hear me in the back, yes. Okay, so today we will finally come to talking about Celestial Amplitudes. After these two first lectures where we had to make a detour to understand what was the detailed structure of the flat space in the infrared and we saw these nice symmetries and that they had charges and that these charges were constraining the scattering problem for masses particle in flat spacetimes. And today we will see, so we will go to now the definition of what Celestial Amplitudes and we see that this essentially amounts to have a very nice organizing, a very nice way of organizing scattering elements in flat spacetime which will make the SL2C transformation of the Lorentz group very manifest. And these eventually will help us to understand what is the potential holographic structure of quantum gravity in flat space in terms of the so-called Celestial conformal field theory. So let me recall briefly what was the last message of last lecture. So yesterday I tried to advocate that there was a nice interpretation of soft photon and we mostly discussed actually some graviton insertions in the quantum gravity S matrix as a word identity for some two-dimensional currents and these two currents manifest the rich asymptotic structure at the boundary of flat space. So for gravity we had these currents were given by important translation currents which I will come back to it when we will talk about Celestial CFT. We will come with a nice definition for this current. We also had for gauge theory this U1 catch movie current which is whose insertion is responsible for the leading soft photon theorem. This guy gives you the leading soft graviton theorem and I mentioned and we will come to a concrete construction for this object but that there is an object which plays a role of the stress tensor in very much like we are used to it's in usual 2D conformal field theories. So that was the main message of yesterday and today we will now go to the core of these lectures which is to talk about Celestial amplitudes. So let's in order to present you this Celestial map that maps the particle states to objects in the to this sphere. Let me just briefly introduce some aspects about Lorentz transformations and the Celestial sphere which I recall is the two-dimensional Euclidean sphere at future null infinity which is identifiably I antipodally identify with the personal infinity. So we have let me write down the Lorentz algebra is to start slowly where I use the notation j i so i is one two or three j's are my three rotations j's one two three are three boosts and these are the commutation relations between these objects and we're also very high when we know that there is in theomorphism between the Lorentz algebra and the SL2C algebra which is also something you have seen in other lectures here so the Lorentz generator are here denoted by the usual lm's mode and this is their commutation relation where m and n are also minus one zero one and okay there is a precise way how to obtain this l in front in function of the rotation in boost generator let me not write all these down because you can find this in most of textbooks but let me just write one of these relations just have an id okay now we will need I and actually I have already kind of introduced that in the in the previous lecture but we'll make a choice for an embedding of how the celestial sphere is embedded into a minkowski of the celestial sphere into actually the the null the null code so there is no unique way to to do that but there is a nice form of the embedding which I can take and I've already actually introduced that when we were talking about scattering a massless particle and this is the choice for you remember the particle has this momentum p mu which I written omega the energy time this null vector yesterday so we have seen this expression already before and it gives me an embedding of the celestial sphere and of course there is not you could make another choice for this embedding but this one turns out to be very convenient convenient for this all this celestial amplitude purposes so now a very I mean the the key observation which is behind behind all this story is very simple it's just the fact that the Lorentz group acts on the celestial sphere via an abuse or sl2c transformations on these sphere coordinates w and w bar is the usual sl2c transformations similarly on w bar so this is just a symmetry observation ed minus bc what one oh yes yes yes so now by using this just a simple observation and I will come to precisely how one can perform this map explicitly if I I write the s matrix element into a basis which manifests which express in a manifest way the sl2c invariance or covariance that acts on the celestial sphere then by construction these objects written in this sort of basis will transform covariantly under the action of sl2c and this is precisely what we are aiming at we are aiming at eventually interpreting these s matrix element in this basis and again I will I will just now define how you can perform this change of basis so by construction you will have this sl2c manifest transformation laws and celestial holography or celestial amplitude is the statement that we will want to interpret these elements as a correlation function so it's something totally different of operators which represent the massless particle in the theory which is this so-called celestial conformity theory so that's the main goal of what's the main objective of this transformation that I will I will present now in more detail talking about conformal primary wave function Laura sorry subject question so n is the number of particles involved into the scattering yes thank you yeah sorry so basically the the usual someone I guess online so the usual drawing that people do is that if you have a n a scattering process involving n particle here n equal to 5 you will see for re-interpret this thing as some correlation function as insertion of operator so each operator here operator which represents massless particle which here sees the celestial sphere at a point the given point w w bar and the delta I will come to it but it would be the dual variable of the energy I will precisely come to that so you you will see this scattering process as some insertion of operators so if you have two incoming and three outgoing they will all live on the same celestial sphere so this follows from the fact that remember we had some antipodal matching between the past and the future in principle you might think there is one celestial sphere at the past and one of the future but this antipodal matching is precisely made in such a way that these these two regions are identified to each other in such a way that a free massless particle will enter and exit the celestial sphere at the same point so this all comes from this antipodal matching condition I've written the main message is that there is only one celestial sphere and we will have insertion of operators which represents creation well the massless particle coming from the past and first is that's a given point the sphere w w bar and we have n of them so this correlation can it be interpreted as coming from some specific null coordinate because presumably the scaling dimensions referred to the Fourier transform in the retarded or advanced null coordinates right so in some sense they are smeared all over scri minus or scri plus so this dimension is not quite the Fourier transform of the of the neither the energy nor the there is a malintransform associated to that and I will come to this precise mapping I see but I mean does it correspond to strictly local insertions on some no no no it as you say it it involves at the annual integral over all all retarded so it's some there's some smearing yes there is some smearing all right thank you yeah so okay let's go to this this map I mean what is this delta how is it defined so conformal primary wave functions abbreviated by cpw so let's um so all the magic will be done by this integral transform called um malintransform so let's consider again a non-channel plane wave can be incoming or outgoing uh plus or minus i omega q mu uh x mu where x are the Cartesian coordinates so here plus or minus is in versus out so as we have seen we can parameterize a null momenta by three quantity the energy and the point w w bar which the particle passes the sphere and this malintransform that will make the magic for us namely trading the omega for this delta and this is how the celestial map is implemented so malintransform the definition of of this transform is as follows an integral transform over a variable that is here will will be the energy on the particle so this is the malintransform of a function f which depends on omega it returns a new function which now depends on this variable here that appears in the integral transform that's the malintransform there are some assumptions on on this function f for the malintransform to be well defined if you're interested you can go in you can ask me reference for for this but i don't want to enter into too much detail so delta in principle can be uh is generally complex in this expression and take so this form and i will come back to to that so in principle delta is an arbitrary complex there is um a well defined um inverse transform so you can invert this malint integra modulo some some some assumptions but um but there is a way to invert uh inverse this map so notice that there is an another notice that there is an inverse malin under some assumption it's about the analyticity of this f tilde on some complex strip we will not need all these details but just to tell you that you can go back and forth under some assumption so what i will call a conformal primary wave function is simply a malintransform of a plane wave packet so let me start with uh so now i'm starting with a scalar conformal primary wave function so this is a definition of what i mean so we'll have a bunch of labels so we have to be careful about all these labels because you don't want to neither confuse you nor be too sketchy so a conformal primary wave function which can be incoming or outgoing depending on whether the plane wave incoming outgoing is defined as a malintransform of the wave packet and it will carry now it will no longer be labeled by energy omega but rather now by this quantity this complex delta so i'm just writing this malintransform for a plane wave and this epsilon here is some regulator which is there to ensure that the enter in integral converges which is positive so that's the definition i will let me introduce some stupid extra label that is not necessary here but when we'll be talking about spinning spinning wave functions we will have to introduce another label which is spin and here since i'm talking about scalar conformal primaries this level will just be done and just be equal to zero but in in more in general we'll have we can define more general spinning primaries so this integral can be computed actually you can the mathematical just gives you what's the answer and the result is given by this where you have disappearance of this regulator here in the dominator so it's plus or minus i to a delta gamma function of delta divided by q dot x to the power delta so why are we interested into this well because now the nice thing about these functions as their name suggests is that phi transforms now as a bulk as a scalar under the bulk law and transformations so you see that this object depends both on the bulk coordinate x and also on on the point w w bar on the sphere there's just a scalar under the lawrence but under the action which of the lawrence which is induced by by this embedding so on this sl2c action on acting on w w bar it transforms as we will see a 2d conformal or quasi conformal primary because now i'm just looking at the global part of of the conformal group with a given weights like the weights that we are used to write down into dcfc h h bar under sl2c so i will write down the transformation rule so let this sentence take some flesh so if i do a lawrence transformation on the coordinate which is accompanied by a mobius transformation on the angles w w bar and then i could have this written down w prime w bar prime the transformation rule that you can check that this is true and there are two ways of express expressing the weights either in terms of h and h bar either in terms of the conformal dimension in the spin and they are related via this equation the conformal dimension as usual is the sum of h and h bar while the spin the 2d spin is the is the difference so here it's actually zero but when we will generalize this to spinning particles it will no longer be zero so equivalently this this is actually well too if you're more familiar with this way of writing the conformal transformations here is an unequivalent way of writing that so you should recognize this as the transformation rule of a primary of which weights h and h bar it's clear so what we are just doing is we are mailing transforming this integral transform on the plane waves and this the reason why we are doing that is because now we have made the sl2c action manifest so in summary usually we write in momentum basis where particles are labeled by their energy here in this boost let's say let's call it celestial basis particle will be represented by this complex number delta and we go from one to the other via the smelling transform and if we have a spinning particle in the bulk it will carry a that's the energy it will carry a helicity l and this will be simply identified with the 2d spin in this this will be equated so this is 2d spin conformal dimension okay so now there is a state a comment i would like to make which turned out to be whether important before before i write that maybe i just mentioned that this phi delta another way to state this is just that they are a complex now highest weights with respect to the loins group so this is just another way just to write down what the weights are so since this is a scalar zero they satisfy this relationship where this l are the loins generator i've written before so basically the now the action of the boost is that analyzed while in the usual momentum basis we have a nice the plane weight transform manifestly nicely under translations but the sl2c transformation is obscured now is the other way around we will have in this boost basis they said the transformational analysis to see which will be very nice but now the price to pay is that the transformation's under translations will be more obscured and i will get back to that later on there any question on the on that yes i have two questions first of all this action of sl2c on w can be just derived from the definition of the complex coordinate on the celestial sphere up to while there is no unique way that this would induce a unique way to write this down so this this will depend on on on this embedding so it's actually equivalent to choose a little group choice for a little group but if you give if basically if i give you if you give me a given embedding then yes you can derive this you can derive this in a unique way okay so okay but okay this two coordinate then w is essentially that does not contain more information than x no it's just a way to parameterize the limit for for for the distance going to infinity yes so the the the law and group will the transformation on the bulk coordinates will will induce a transformation on the on the angles at the sphere at infinity so they go hand in hand okay so but i i'm not understanding why when you write the fields you you write both x and w like if they are independent coordinates well so in principle it's it's so it's like you know a plane wave in principle depends on the on the on the on the bulk point and which is defined but also has a momenta which is pointing towards a point in the sphere so if you want here also this plane wave is also an object which depends both on x and on this q but now this q i'm i'm choosing an embedding which where this q is parameterized by w w bar okay so it's just it's the same stuff but now in this in this basis somehow it's just we are not used to that because we are not used to write the momenta i guess like so but it's nothing okay so the w is the large distance limit of the dual variable of of x yeah exactly okay so exactly so this this object is defined at any bulk point x but now i will want so it's a bit what we did yesterday when we pushed this wave this this function to the boundary by taking an enlarge r limit and then i mentioned this saddle pronged approximation computation that i did in a show but basically when when you will push this to the boundary it will uh it will make you know the angles let's say z here which are included you know in bond decor you have urs as the bar it will make that and the bar to coincide with w and w bar when you take the enlarge r yes i know it's a bit might be confusing this double thing but it's nothing but again if you look at the plane wave just inherited from from this and this relation now let me make a comment so the statement that plane waves form a delta function normalizable basis which is statement where these things you know the Klein Gordon in your product i can write down quickly the the definition i'm using for the Klein Gordon or two to pass the stuff so we are familiar with the fact that the plane waves from a delta function normalizable basis which press like so so this statement translates into the celestial language in the following way if this statement will impose some constraint on the value of delta more precisely what you can do is just you take this expression here you do two malin transform one for the first plane wave one for the second and you will find the Klein Gordon in our product of two primary wave functions let me just write w and not every time w and w bar but there is one evaluated that on that depends on w one on w prime so it can be in and out so if you do a malin transform a two malin here you can compute this so the left hand side is equal to that by definition of what is informal primary but at some point you will hit some integral and you will see that this integral doesn't converge unless you restrict delta and delta prime and see here sky depends a priority can depend on you can carry another conformal dimension delta prime so it will constrain delta and delta prime to take this form where lambda is and lambda prime are real and you've encountered similar stuff in Mathias lecture before so these are the continuous here it's called the principal series of irreducible representation of s o 1.3 so this is imposed by the convergence of of the malin integral and this is something that was worked out by Sabrina Pastorsky and Shuhang Xiao maybe I can give you some reference by the way for these conformal primary wave functions um so I guess Dirac Dirac already looked at this but more recently they were investigated by the Boo and Solodukin and also Pastorsky Xiao and Strominger 2017 here's the reference for that and you can find the reference for the De Boer Solodukin paper inside this paper so basically this this this statement about the delta function normalized normality of the plane waves is translated into the malin basis as some restriction on the conformal dimension to lie on the principal series so um I will come back to that because there are a lot of subtleties about this spectrum if you want but I think this is something that is worth mentioning because actually these modes these celestial primary wave functions which which have these values of delta correspond actually to normalizable states which have the usual radiative fall off when they go to the boundary of of spacetime but in general you might wonder what happens if I go off the principal series so if I take the real part of delta to be two for instance and these are also very interesting thing to to to look at as we will see the stress tensor obviously lies outside this this principal series and we will come back to that when I will talk about tomorrow about 2d currents any question or comments this I guess this is just not a question but just a microphone left open otherwise I didn't understand the question so let me say you think two comments about now spinning conformal primary wave functions because at the end of the day we'll want to discuss catering of of photons gluon and gravitons so spinning so this was I I've written here the the scalar case formal primary wave functions in one first they equal plus or minus one so now the confusion is arising because there will be some plus or minus coming and mean plus or minus elasticity and there are some plus or minus which meant outgoing incoming I will I will actually now I will suppress the in and out labels because otherwise we will have too many indices to carry on and so a spin one primary wave function will carry now well it would be we'll have some tensor indices and let's see let me write it here so j equal plus one so I'm looking at electricity plus here so it has been j equal plus one so it is defined roughly speaking as a scalar conformal primary times these tetrad m u and there is another I will I will explain that but let me just write the expression for the negative electricity just to be complete so now j is equal to minus one we'll have m bar now times the conform the scalar conformal primary so this carries the electricity piece and you might think that this I could have obtained a spinning primary just by multiplying the scalar by the polarization tensor but this is not quite true this m is not the polarization tensor it is the polarization tensor of a plus elasticity this is a notation that I used yesterday corrected or shifted by a quantity that depends on the bulk coordinate x and this q mu times q dot x and similarly m bar gives you the object for the opposite electricity so again here plus a minus they don't denote incoming outgoing but this so these things are required in order to have a nice transformation under sl2c if you just take this polarization tensor times the scalar primary you will see that this guy doesn't transform nicely under sl2c you have actually to to take this tetrad thing and so now with this spinning one primary so transforms under Lorentz and the Mobius now is it will be also transformed as a 2d primary of weights h and h bar but now it's it's also transforming as a vector under the bulk transformation so this is the same transformation as for the scalar except that now is transformed as it should under the Lorentz action notice that there is some intrinsic gauge fixing into this this definition I don't want to mention too much on that but basically these guys are naturally in Lorentz gauge but just because of the equation satisfied for the scalar primaries and similarly and then I will stop bombing you with definitions but we have a spin to spin to primary which is plus or minus two and this will define for us later on the currents that play a role in in celestial cfc so for plus electricity this is nothing but now you need to take two of these amps here and similarly for j1 minus two where you take the bars and now these guys will transform as a tensor under the Lorentz group and it satisfies the so this thing satisfies the linearized equation of motions so yeah basically if it's in the so-called under gauge this will just be a box of h minus equal to zero yeah a bit more complicated if you relax the trace and the Lorentz condition on it but yeah so just to finish so it's an important it's an important thing that the fact that to have this nice transformation you have to add this piece will make this object quite not equivalent to just the polarization tensor time the scalar one but they will actually be gauge equivalent to that but we we have heard in the first lecture that the gauge all that has to be with gauge we have to be very careful because there are some gauge transformation or diffeomorphisms that do not die at the boundary and we have to keep track of all that and actually it will be crucial to to keep these gauge transformations because some of them can be non-zero at the boundary and so just to tell you that the relationship between simply the polarization tensor time the scalar primary these guys is gauge equal or diffeomorphic equivalent to to it so i'm not writing the explicit expression for xi ximu you can find it in the in the in the pastor's key paper this the form the exact form is not very important but you can already see that there's a weird thing happening when delta equal to one so for this value of the conformal dimension you can see that the naive spin to conformal primary is no longer well defined it just reduces to a pure diffeomorphisms and this is actually a titan in one to one correspondence with the presence of super translation and of this super translation current that i have presented yesterday and i will go back to it so here i just want to say watch out when delta equals one and also when delta is equal to minus one but this i don't want quite to discuss this when delta equal to one this guy is a pure diffeomorphism or sometimes called a world stone mode associated to the breaking of super translation symmetry so far it's not clear why this has to do with super translations but i wanted to mention this already at this stage and similarly for the emu so the emu is gauge equivalent to to epsilon mu times this yellow guy right and the factor is delta minus one over delta so you can see again that delta equals zero and delta equal one seems to be important value of the conformal dimension and indeed this guy will define for us some world stone mode associated to the breaking of large gauge transformation at the boundary so now this alpha gauge can be large namely it doesn't die at the boundary gauge transformation again watch out when delta equals one and this will actually define for us what people have been calling a conformalist operator in celestial safety for these specific values of delta any question for em i now we have all the all the relevant quantities and transformations to to talk about celestial amplitude to define a celestial amplitude the last 10 minutes i didn't quite understand are you saying that all emus are gauge equivalent to that particular form with delta minus one over so this is an exact statement this this is just so what i mean is that emu so this is just an equality that follows if you take remember you the expression for phi delta was roughly speaking one over to the text to the delta times some i and some gamma of delta so if you take this and multiply it by that you will be able to write it in this form so the first term will just give you this piece and the other guy i'm sorry that i'm not sure i have the the expression for for alpha here oh yes i have it so that you can take if you want that you agree with this but the alpha the expression of alpha is just epsilon times x divided by the delta times minus q dot x to the delta so this is just uh an equation you can check easily what i mean is that this emu is not exactly equal to epsilon times the scalar one but it's gauge equivalently this is just the first bar gauge transformations but usually we don't care about like gauge transformation we just say yeah i mean we drop this but as we have seen in the first lecture there are some gauge transformations that are non-zero at the boundary and they act non-trivial in the states so so we better keep them and indeed we'll see that this will define for us when delta equals one these two d currents that i call gz and the point is you need that extra term for you to transform nicely in the sl2c yes exactly so it's because of this guy doesn't uh doesn't transform in a nice way but this object does okay do you also get like a funny zero at delta equals to two is some other thing that you have the super translations not to so yeah so the special value for us will be one zero and two so delta equal to two as you might guess we have to do with the stress tensor for gravity for the graviton conformal primary and the delta equal did you ask about delta equal zero or delta equal two i think it is the super rotation always this called a bit of sort of thing yeah delta equal two and then i will mention some this this shadow transform which exchanges okay with the two minus delta so the delta equal two and delta equal zero are related via shadow transform okay okay that game okay thanks i will i will uh define what is a shadow transform tomorrow good so in the last seven minutes we can finally find us um what is um celestial amplitude so basically uh we're we're over over we have already encountered that but oh a celestial amplitude 3.3 so if i write a usual s matrix element these are the usual momentum basis we talk about state somehow and now we'll talk about wave functions we'll talk about um amplitudes of n particles now we have a precise way before i was a bit sketchy but now so we have a precise way to define that so as i've said over and over massless particle is parametrized by an energy and points at the bar because it versus the sphere and this is the radiosity so this is the usual way we express amplitude in this momentum basis so for massless actually yeah i'm focusing on on massless case maybe i would say a word on massive case the celestial amplitude is the so i will call it uh curly c now involving n particle would be labeled now by two other quantum numbers this is conformal dimension and the spin and it's just defined as taking n times the malin integral one for each leg of the usual momentum basis amplitude where here the identification between the l and the j is just really needed in the identification between the elicity and the spin as i've mentioned before and we trade the energy for this delta and this is definition of a of a celestial amplitude celestial amplitude for the massless case this map is simply defined with the malin transform but for and for a massless case it's a bit more tricky but there is also a way to define to make an integral transform which is such that the scattering of massive particle will always transform nicely under sl2c so far by this in words so this is just to go a little bit beyond what um what i have presented so far we were focused only on on massless particles is so involves now no longer the malin transform another transform which is built from the tool from a cft embedding formalism and involves the bulk to boundary propagator for medias a gator so this is just a common if you're interested in into the massive case which i have to say is much less understood from a celestial holography perspective than the than the massless case so i'm just writing down the reference for that if you're interested in looking at this how you can obtain a celestial amplitude for massive particles but so the important point and the upshot of all this is that for both cases here uh so this is another not a malin but i mean it's not a malin transform that's something else but for both cases by constructions by construction celestial amplitudes transform covertly under sl2c this meaning again that if i act with the sl2c by by design i will have an n for endpoint function will end time this these sort of factors where the h is all the weights of each associated to each uh scatter okay so hi is again it's just delta one half delta i delta i minus okay so what you can do now that you have this recipe to build celestial amplitudes you can take any formula for your favorite amplitude and the first one that were computed in the celestial basis were the three point maximally elicity violating glue on amplitude and i will not write down this because i'm running out of time but just so that you have an idea of what it looked like you can really explicitly compute check that the amplitude you obtain uh so example three point m hg m hg blue one amplitude you can do the exercise the reference for the paper you do three malin transform and then you get so you start from the amplitude you know from the formula and then you obtain the celestial amplitude for let's say two minus elicity and one plus glue on so there are some that you have to go in the split signature because and take z and z bar real and independent but basically you will find sort of expressions that you recognize as a three point function in a 2dcft there are some subtleties that i don't have time to discuss but you can ask me there is a there are some delta function in the z bar coordinates this is inherent to the slow point functions and you can ask me more on that but you recognize this as the usual form of a three point function with the given weights because this standard expression with definite weights where this lambdas so remember that delta i so here they are also lying on the principle series and delta i is one plus i lambda i and this is just i'm just and then i'm done writing this the specific form it takes so you can actually check for a bunch of amplitude that performing this integral makes all this cft looking like transformation to be very manifest so this has also been checked for massive particle and for other sorts of particle but i will just stop here so thanks very much for listening thank you very much Laura and we also discussed a little bit about the topological string which we propose will be holographic if you do so today we're going to study the back reaction in the topological string this is the first thing we started so last time we also we wrote down the field sourced by a brain in the topological spring this was a certain beltrami differential but we solved that equation the geometric role of the beltrami differential is that it changes the Cauchy Riemann equations in our case in the deformed geometry the original coordinates for w1 w2 and z and here the new Cauchy Riemann equations are these equations here which tell us what does it mean to be a homomorphic function in the deformed geometry so let's solve these equations and then we'll find that the deformed geometry is ads3 times s3 let's go up a little bit if we look at the equation no more in the equation is there a d by dw1 so w1 and w2 these are still solutions to these deformed Cauchy Riemann equations so these still define homomorphic functions however if we look in the equation we see that there's a d by dz so the function z will no longer be homomorphic you can instead see that there are two new homomorphic functions which i've called u1 and u2 here given by these expressions u1 is zw1 plus n over normal w2 w bar 2 u2 is something similar and it will take you half a minute to stick u1 and u2 into these equations and check that these equations are solved so w1 w2 u1 and u2 are the homomorphic functions in the new geometry however they're not independent as you can see u1 w2 minus u2 w1 is equal to n which is again entirely elementary so we conclude that the deformed geometry is that defined by this equation so we can think of this as the space of two by two matrices like this the column vectors u1 u2 w1 w2 whose determinant is n now i can absorb the factor of n into a rescaling some of the coordinates so you might as well think of this as matrices determinant is one so that the geometry is s l2 c okay so this this is our the topological string derivation of the back reaction um for the experts i want to point out that this is a little simpler than what normally happens in the physical spring because we don't have to pass to any near horizon limit once we back react the geometry always already has a homogeneity it means passing to the near horizon limit does nothing are there any questions about this computation okay so let's so our goal is of course to compare the chiral algebra as we've been discussing to quantities built from the topological spring in this deformed geometry so if we go back to think about the more familiar ads 5 times s5 polygraphic dictionary the very easiest thing you might see is that in ads 5 times s5 the symmetries of the cft are the same as the isometries of the geometry this is so this is a very basic and elementary check the ads cft correspondence so well the isometries of ads 5 times s5 are s o 6 times s o 4 2 and here s o 6 that's that rotates the six scalars many plus four young melds and s o 4 comma 2 that's the conformal symmetries of the force sphere and the symmetries of the force sphere which preserve the metric up to available scale so because it's a cft these are symmetries of n equals four young melds in the force sphere so what we're going to do in the twisted holography setting is we're going to check that the statement holds but what we'll find is that it's a much richer statement for us because the group of symmetries will become infinite dimensional and therefore will give us a very very strong constraint before we move on let me explain briefly how to think about the symmetries in the case of the cft in a more abstract language your cft will have a certain number of conserved charges especially in a chiral cft any operator gives you a conserved charge I integrated around a circle that's my conserved conserved charge however we're interested in those conserved charges which preserve the vacuum at zero and at infinity like that so if I take my charge integrate around the three sphere and I hit it with the vacuum at zero I get zero similarly with the vacuum at infinity so think of these as the initial and final states just want to add another peg why so why is this a good thing to do so charges like this preserve all correlation functions so get a picture like this where I have local operators and if I sum over integrating a charge around the local operators that's going to have the same effect by change of contour argument as integrating the charge around all of them that's going to be zero because there's the same as integrating my charge against the vacuum against that infinity okay so we'll come in a minute we'll come back to calculating these chart these symmetries in the cft side so back to the topological string what will be the analog of the isometries well we know that you know general relativity is covariant under diffeomorphisms of course diffeomorphisms are the gauge symmetries of general relativity and the isometries are those gauge symmetries which preserve the field configuration you start with like they preserve the vacuum field configuration which you know the metric on ads 5 times s 5 so the kind of thing we will study that will replace isometries will be gauge symmetries preserving the vacuum field configuration so let's study what that is let's look in the type one case so the fields well the solutions to the equation's motion more precisely in the closed string sector there's a complex structure and a homomorphic volume and in the open string sector we have an SOA bundle like a homomorphic SOA so what are the gauge transformations of this data well for the complex structure that's really a part of the metric so the gauge transformations are are diffeomorphisms those which preserve the vacuum field configuration though because we don't have all of the components of the metric we're not just looking at isometries we're looking at those diffeomorphisms which preserve the complex structure and they also have to preserve the volume so at an infinitesimal level diffeomorphisms are given by vector fields so these are homomorphic and divergence free vector fields on SLTC so if you're familiar with complex geometry you might notice that this is a very very very large the algebra and we're going to see it that it also appears in the CFT side something that's a little easier to study is what happens for the open string fields in the type one spring there we have you know our gauge field describes a homomorphic bundle on on the manifold so those gauge transformations which preserve the homomorphic structure of the bundle they're not constant but why would they satisfy this equation so the gauge transformations are homomorphic maps from SLTC to SOA so again this is a very big space because there are lots of homomorphic functions on SLTC so these Lie algebras are infinite dimension so our goal today is to find these Lie algebras in the chiral uh inside the chiral algebra as symmetries which preserve the vacuum at zero and infinity I have a question yeah so since SLTC is non-compact is there some condition on the homomorphic vector fields at infinity that one has to impose no no you you look at all of them okay so that's why it's so big so we try to write a little more explicitly what the open string algebra is we're looking at homomorphic functions from SLTC to the Lie algebra SOA so what kind of thing is that an element of it can be looked it's going to look like product of an element of SOA a is an adjoint index for SOA and a homomorphic function in SLTC which is a function of the variables ui wi modulo the equation u1 w2 minus u2 w1 is n so let's look at the quantum numbers of these symmetries how do they transform under the natural kind of evidence symmetries one has well of course under the global SOA these expressions live in the adjoint representation what's more interesting is to think about how these expressions live under the left and right action of SO2 on itself so the group SO2 acts on the left and the right by matrix multiplication so on the CFT side we also have those two SO2 actions the left one is the global conformal symmetry to rotate CP1 so this is kind of the analog of SO42 and the right one it rotates you know in our theory we have two matrices and they have a symmetric role this right action rotates those matrices so this is the analog of SO6 now how does a function on the group behave under the action of left and right multiplication unfortunately there is an old theorem from the 1920s or something which tells us this is really just harmonic analysis of the group or the compact group would be like but it says that every polymorphic function can be expanded to modes and the modes living the tensor product of the spin j representation of the left with the spin j representation on the right so our symmetries are then in that joint of SO8 tensor spin j on the left spin j in the right so we should keep this in mind because we want to reproduce this from the CFT side well what we want to identify the algebras we see on the topological string of the CFT we also want to identify them not just as vector spaces but also the structure constants of those algebras so let's write down some structure constants as well and then we'll see if we can reproduce them on the CFT side oops I don't know what that comes later sorry take that back I will discuss the structure constants later so what we'll do now is find symmetries with exactly these quantum numbers in the CFT so how does that work so if you remember in the CFT the elementary fields are eight vectors and two matrices these matrices have certain symmetry properties the gauge invariant operator is going to be interested in the single trace ones are vector bunch of matrices vector because of the symmetry properties of x1 and x2 this expression is anti-symmetric in the ORS indices so that it's in the adjoint of SO8 and under the conformal symmetry for the CFT this is an expression of spin m plus n over 2 plus 1 because the eyes eyes and the x's are all of spin half now it also lives in some representation of the SO2 which rotates x1 and x2 and it's in the representation of spin m plus n over 2 of that SO2 so we've you know that is we've listed the open string operator and which representations it lives in but to reproduce the quantum numbers we saw on the holographically joule side we should recall that we're not interested in all operators or all modes of operators rather we're interested in those modes that preserve the vacuum at zero and infinity so this picks out a sub algebra of all of the modes um some people call this the wedge algebra so how is this defined i take the contour integral of some operator i take some mode you ask okay when does this preserve the vacuum at zero for that you need k to be positive like non negative because if k was negative then i hit it with the vacuum at zero i'm going to get some derivative of my operator then you ask does it preserve the vacuum at infinity will we change coordinates so that goes to one over z and i pick up some powers of z from the way the operator transforms just dictated by its spin and you find it preserves the vacuum at infinity if k is in this range or j is the spin of my operator so for example uh if i have a spin one operator then only that is in the algebra sorry i think i need a page to explain this so spin one like an ordinary current j a so the globe no if i have a spin one operator maybe it satisfies it cast new d o p e then the global symmetries will be a finite dimensional dimensionally algebra i have a spin two like the stress energy tensor then this constraint tells me that you know we have not less than equal to k less than equal to two j minus one and if j is two we have okay is not one or two so for spin two operator like the stress energy tensor the global symmetries are the sl2 global conformal transformations as you increase the spin of the operator getting more and more global symmetries so let's go back to a little bit and see what are the global symmetries of associated to our basic currents our basic currents are i x1 to the n x2 to the n i well if you look at the spins you see k is allowed to be in the range from zero to m plus n so these expressions they're in the adjointive s away of course and they're in a representation of spin m plus n over two with the respect to both the sl2 conformal symmetry and the sl2 symmetry which rotates x1 and x2 so this is the same quantum numbers as we saw before it's like sum s o a tensor e j l that's e j or if there are no other are there any questions there so what have we seen so far on the topological spring side we looked at the analog of isometries we found this big algebra maps from s o to c to s o eight and on the cft side we looked at those modes which preserve the vacuum at zero infinity and we found something which looks the same at least something that has the same quantum numbers and i haven't explained how to calculate the quantum numbers that closed string operators but it's not hard to check that they are so much so what we haven't done so far is match the structure constants easily outwards so the way it's going to work is that here t is some element of s o eight i'm writing this for or an s our vector indices for s o eight these are anti-symmetric and or an s so homomorphic functions from s o to c to s o eight they look a little like this at the lowest level i just have those constant functions here which are these then i have those functions which are linear in the coordinates t or s ui and t or s w y these are going to get sent to i x i are the distinction between the ui and the wi is whether or not there's z dz or just a dz the same happens as you go down so next so we need to be able to match the non-trivial commutators so let's examine one of these commutators for example if i take t or s times u one t b q times w two and i can mute them all i'm doing is i'm taking the s o eight commutator of these matrices these are anti-symmetric matrices i'm just commuting them and then i'm multiplying the functions in particular if i specialize these indices to particular values you find t one two u one t two three w two is u one w two u one w two t one three similarly t one two u two t two three w one is u two w one t one three but now we should we should recall that s of two c is the as a defining relation for u one w two minus u two w one is equal to n so we get such a relation in the commutator like this so this is the relation we're going to want to match so let's translate this into the cft language u one is going to be a mode of x i one x one i two like this and w two will be like this an x two and a z and the other commutator there's going to be an x two with no z and an x one with a z and we expect that the modes commute according to this relation and we're going to do this computation in a minute we find it works well what i find kind of cool about it is that it's a way here we're just doing some computations in the mode algebra of the cft and we're seeing the holomorphic functions on the dual geometry appear like the defining relations okay so let's let's go and do this computation so before we do so i think i should remind people of something pretty elementary of how how one does how do you commute elements of the mode algebra of some chiral algebra so in general five two operators in a 2dcft o one and o two i can take their modes with some powers z to the k one z to the k two and i can try to commute them so the commutator is just the difference between two contour integrals for here on this contour on this contour here o one is on the inside no two is on the outside and this counter here o two is on the inside and o one is on the outside so one i take the difference between those two contour integrals you can write it as a contour integral where here o one moves around in a circle and o two orbits around o one so to write that in coordinates we're going to do contour integral of z just go around in a circle and a contour integral of w also go around in a circle but o two is placed at z plus w and then i also have this power z plus w to the k two now when we do the w contour integral what kind of non-zero terms can we find well of course we're only going to find terms which have the power of one over w because you want to get a residue expanding this expression will give us positive powers of w and the ope's here will give us negative powers of w so we see that the terms that are relevant are the terms in the ope which have a first error pole second order pole do a pole of order k two plus one okay in our case we're doing a commutator of something here where there's no powers of z and something here where there's one power of z so by this argument the commutator will have terms from where there's a first order pole in the ope between the two operators or a second order pole now now in our CFD each the ope's involving a pair of elementary fields always has a first order pole so a wick contraction contributes a first order pole since we want a first or second we find that the relevant terms involve two wick contractions so let's write out diagrammatically everything that involves two wick contractions here i can contract x1 with x2 but this is bad because this is a non-planar diagram so since we're looking to CFT in the planar limit this will not contribute i can contract i1 and i2 that's good and that contributes this expression to the commutator of the modes or i can contract x1 and x2 and i1 so and i2 and i2 this will contribute something with an n and i1 this will contribute n i1 i3 and the reason for the n is this the usual large n combinatorics when i have a loop like this when i have a loop like this here it contributes a factor of n from like the trace in that loop so i think so now we've pretty much gone through the proof if i commute these guys i get i1 x1 x2 i3 z dz and a factor of n for the other guy for the other one you find the same thing in minus n i1 i3 so you find pretty much the same thing except for an extra sign so we see in the algebra the analog of the commutation relation of the defining relation of sl2c by commuting the modes each other okay so this is really easy but fun so any questions how does it work more generally so the theorem is that the the global symmetry algebras for the planar CFT at the top logical string are isomorphically algebras even when you include the closed spring sector so let me say a little bit about this so the general case maybe i can ask you a question actually sure yeah so on the carol algebra side if i understand correctly we don't have to take large n right we can understand this carol algebra finite n is more than where do we see the need for a large n to match with the topological string i mean what would i mean here we had a match but we've dropped the the one of our n corrections we don't play the diagrams so so where would we see those those corrections right so this could seem to things but you're a little tricky but possible to do with involving you can try to compute by various methods so the diagram might be dropped looks like this these are contracted here um let's go off like that so it's like a two to two scattering process i'm sorry i haven't drawn it very well that's what that's what one can compute so there are only corrections to the ob in the topological string and these capture it it is possible to see it right it's a little tricky so i think in general you know i like to be optimistic and think that loop corrections on the topological string side should be much more accessible than they are in the physical strain and i think they are but they're still not easy probably the best way to approach them is to know that they're very tightly constrained by crossing symmetry thanks um of course the other way and large n appears is that they're trace relations i don't know how to see those in the topological string at all so in general statement days that if i take some kind of open string thing like this the planar ones i have to contract a bunch of adjacent stuff some other ones go off this is a two to one kind of open string process in cft that if we're looking at the global symmetry algebra that this matches the algebra polymorphic maps s l to z to s o 8 similarly on the closed string side you take modes of this guy the global symmetry algebra vector fields the holomorphic conversion is free vector fields on s l to c so how would we prove this so for the open string side because the algebra is generated by ta ta ui ta w i so really once we've checked this commutation relationship relation algebra basically the jacobian entity tells us the two algebras must be the same what about the closed string set that's going to be much more challenging on the cft it's more challenging to do the cft computations we're going to use the following trick vector fields on s l to z derivations of functions on s l to z and so i'm going to use this symbol for functions and so there's symmetries of this reality and on the cft side you can kind of see explicitly why the closed string fields have to be symmetries if i take a closed string field like this the commutator of this with an open string field looks like into these go past and you can check this kind of well this is compatible with the jacobian entity so i've got a very briefly sketched why the argument we just gave well very simple it is enough to allow us to constrain just using algebra all of the symmetries and to see that for the closed string side too would also get vector fields when s l to z okay so the topic i haven't talked about so far let's see which i'm going to have to go spend more time on tomorrow is states from the point of view of holography of the top logical string so in general ad s cft our fields have an asymptotic boundary condition on ad s so suppose we write ad s in utility and signature as like the upper house plane then roughly there's the fields someone wants to consider have some prescribed pole in the radial parameter or a local operator is modification of this asymptotic boundary condition at a point on the boundary which means away from this point we're going to have usual asymptotic behavior and at this point we're going to have modified asymptotic behavior so i don't know if this is a familiar picture to people but it relates to the more standard one by saying that so equivalently if we modify the boundary conditions we can find a solution to the equations of motion with the modified so this solution to the equations of motion is the state on ad s corresponding uh cft operator so i want to explain how to implement this in the case or considering of sl2c so to understand this we need to to think about what does sl2c look like near its boundary sl2c it's like u1 w2 minus w2 u1 was n that's up so to understand the boundary let's make these into homogeneous coordinates and then the equation will be u1 w2 minus w2 u1 equals n v squared so now i'm lifting everything into homogeneous coordinates and then writing sl2c as an open subset of a quadrant in cp4 as i'm imposing this quadratic equation so you know just like it started life as a solution of a quadratic equation in c4 and i'm adding on some points at infinity to c4 if you get cp4 then i want to see what the boundary of sl2c is in this compactification the boundary divisor is a locus where v is zero in which case the equation looks like u1 w2 minus w2 u1 is zero this is a quadric in cp3 as you might have seen in anus lectures it's kind of standard in the study of scattering quadrics in cp3 are parameterized by cp1 times cp1 so the boundary looks like s2 times s2 now should we think of this ads3 times s3 is something see the boundary so one s2 is the boundary ads3 the other is well almost the s3 it's s3 modulo u1 so that we're related by a half vibration i didn't get very far with the discussion of states so i told you what what it means to get a state we're going to modify the boundary conditions now we've studied the boundary of sl2c and we find it's a product of cp1 so the next thing to do is to look at the boundary conditions and to see that modifications of the boundary conditions do indeed give you the states of the duality of t after that we can study we can try to study scattering opportunities matching the correlation functions okay so i'll start there thank you very much