 Okay, thank you very much for waking up early in the morning and be on time and I'd like to thank the organizers for this fabulous workshop, which I learned a lot and also for the very impressive and exciting environment that you created here in this workshop. So today I want to talk about two somewhat different topics, just mix them together, I mean to put them together. One is about broadly called the Newton's constant and the other one is about the symmetries all associated with localization and disclaimer, I will not talk about anything about the Wilson loop though. It's just a, you know, straightforward corollary from what I'm going to talk about. So this is all about my ongoing project. So this is not published yet. And so I think it's very ideal to discuss with the people and exchange the the idea and also get the comments from the expertise audience. The first issue is about the very old subject, which is large and scaling behavior of free energies and entropy and the known super conform theory theories and its avatar. Second one is issue which I think many people here expressed an interest in in this workshop, which is about the conserved and anomalous symmetries, all related in the context of a supersymmetric localization computation of some d-dimensional super conformal field theory. Schematically, that object I'm discussing here is this partition function and of course, it also means that they are all the co-relators, which is derived from this generating functional. And the first issue is about the dependence on n and also some coupling constant, g-square, which is generic expansion primary in the theory. And the second issue is associated with this functional integral measure. So let me start with the first topic, this large scaling of the free energy. If you are tabulated all known super conformal field theories and it's free energy scaling behavior respect to a large n, then you will notice that they have this regularity, namely it scales with non-universal dimension dependent order of one constant times n to the d over 2. For d called 3, 4, 5, 6 that we all know, it falls into this category. It's a very simple form and this clearly cries out for the explanations why it has a very simple form like this one. And of course, there's some caveat for computation of this scaling behavior and that we have to keep in mind. And we want to anyway, but we want to understand what this relation, how this relation comes about and what is the implication of this scaling behavior. By the way, I will not cite any, sorry, so I will not cite any papers here. It's just my habit and please arrest, be assured that your important original works are all well received and Okay, so yeah, now a question. Massive theory. No, I'm not, yeah, so that's the one important caveat. I'm not going to talk about here. The simple one. Yeah, that's different behavior. That's right. Yeah, okay. Yeah, I will not touch upon it. Yeah, that's because it's some of the normals from what I'm going to talk about. Thank you. So, you know, so let's just recall what ADS-CFD is after so many years. So in nutshell, the ADS-CFD correspondence states duality relation between some physical observables you compute in some ADS-D plus one dimensional gravity and the conformal field theory in D dimension, which shares the same kinematics in the spacetime symmetries. Each theory contains two coupling constants. That's a very important caveat for ADS-CFD to work. At least we can compare the two side by computing two of the observables in the two ADS and the CFT side. Otherwise, there's only half, therefore, it only remains as a conjecture. And ADS-D plus one, we have a Newton's constant, of course. And the other one is ADS-Radius scale, which sets the characteristic momentum or angular momentum. In D dimension and conformal field theory, we have rank of the gauge group or symmetry group, which is N. The other one is just a fixed coupling constant, G square. And weak coupling, we expand this one up for fixed N. And that's our conventional textbook quantum field theory perturbation expansions. Sorry, I couldn't hear you. Oh, that's right. So that's not, okay. That's just M theory, but I'm going to mention. So the way we understand is in terms of string theory reduction to type IIa, right, the five dimensional theories. Otherwise, we cannot compute anything from the CFT side. That's all I just meant. Okay. So physical observables on each side are, they're therefore computed in a double series expansion of these two coupling constants, respectively in ADS and CFT. And we reduce, that's exactly my point here. So reduce an M theory configuration, which has only one coupling constant, to type IIa configuration so that we now have two coupling constants to play with. One is coupling gate, a string coupling constant. The other one is roughly alpha prime, or maybe some other coupling constant. And if you have a more, more expansion parameter, that's better. That's always the corner we look for and do systematic and analytic computations of the given theory. So in ADS D plus one, a schematically, a physical observable called curly A is computed in weak gravity and weak curvature double series expansion. It's just double series form like this one. So you just compute in ADS gravity. And CFT also, I compute some observable, which has the same quantum number and expand in one of, in the, in the double, of the top expansion, namely one of N square plan expansion, as well as top coupling, coupling constant expansion in this way. So each of them are double series expansion. And we compare these two. Now, important thing is that the M theory, again, as I said, they're emphasized because we have only one coupling constant in M theory, namely Newton's constant. We cannot really reliably compute the physical quantity and either in CFT side or in gravity side. So what we always do is either in, we go into a type two way configurations where you do have a two coupling expansion parameters. Like in ABJM, we have the obi-fold parameter K and the other one is string coupling constant or in a five-dimensional reduction of the 2,0 theories where you again have coupling constant and the large M. So ADS correspondence implies that the A and A tilde are two physical observables must be equal. So based on this, we test all the, the consequences and whether, see if they are mutually compatible with each other. This means that these two series expansion must be somehow related. And here we have Newton's constant, ADS radius, the other one is one of N square and top coupling lambda. So that means these two parameters, coupling parameters should be related to these two. Namely, there should be some functional relation of, for example, the gravity side coupling parameters with conformity theory parameters. So this is a very generic relation we anticipate. And what furthermore, the reason why this would work is that in, in, in conformity field theory, in the way that we understand the large N expansion, when we make this double C expansion like this one, so this is one of the N-scaled expansion for fixed lambda and that's of course asymptotic expansion. However, for given N in the display expansion, for given the order in AD, one of N-scaled, namely the handle whole number H, the top coupling expansion, the this expansion is a convergent with the finite radius convergence. Therefore, we can read some perturbation theory in lambda and then analytic continue from the weak coupling regime to the strong coupling regime. So that's why we got all the mileage is in ADS CFT. So that allows this cranking up this whatever expressions we have this observables in CFT side to the large lambda region and then we can relate these two. And so this is a very generic relations that we anticipate just a priori. Now, the prototypical example of ADS CFT is any couple super young mills and there the relation we know is this one. So this relation between ADS and CFT is given by this function called G and R, that's function relation, but very mysteriously this functional dependence is such that this Newton's constant is related to only N, one of N squared and it's independent lambda and the other one ADS radius is related only to the lambda but not N. In general, we would expect some mixed functions of these two variables, but this is very special. However, this we should really view as an exceptional one rather than being generic. So we'd like to ask what our generic behavior of this interpolating function between CFT and ADS or vice versa. Of course, this function has to be invertible, otherwise this ADS CFT equality would not work. Now, this has to the fact that this has very special exceptional form is tied to the fact that among all CFTs we know of, this the any couple super young mill is derived from type 2b string theory, all the others are coming from type 2a or some variant of it. So in general situation we would anticipate somewhat deviating from this one and we'd like to understand how. Let's look at ABJM which is the another one, another the ADS CFT relation where we do have microscopic ABJM theory dual to ADS 4, the supergravity. There it is interpolating function is indeed when lambda this top coupling which is set by John Simon level and the rank N ratio of the N over K when that one is very large, it has this behavior. Indeed, it is function, it is a leading order behavior, it's a function of all of N square and lambda. And the radius of ADS somewhat again analogous to the any couple super young mill, so scared of lambda. But anyway, so this is a better, this is more generic than any couple super young mill theory. And because of this behavior and lambda being N over K, for fix the K in particular and K equal 1, which we go to M-theorem gene, this one gives you a scaling of this interpolating function which goes like N to the minus 3 half and this is a Newton's constant therefore the entropy which is 1 over G Newton's constant scales like N to the 3 half. So this mixing between two coupling constant in CFT is quite crucial to get this anticipated behavior as derived from ADSI supergravity computation. Okay, so again in M-theory case, as I said, this K equal 1 is set to 1, K K set to 1 therefore there was only one coupling parameter N and we artificially divided into Newton's constant ADS radius. They are just interrelated and that's M-theory regime. Now let's talk a little bit more about that, the expansion and see what that the implication is. So here everyone knows by heart that what this ABJM localization result is, this in terms of eigenvalues to UN times UN, I just take only parity invariant one and so this again is double expansion in 1 over N square and here top coupling is set by N divided by K where K is the level of John Simons of each UN gauge group. And we know from this integral the saddle point arise as a master value configuration when these, sorry, this I just mixed the connotations. Namda is the same as phi. Phi, this what I would call this eigenvalue, eigenvalues are scaling like square root of N and this also the other one is sort of square root of N. And in fact this scaling behavior is not so strange even though this just comes out of just the grand computation of the saddle point of this only large N. Remember that in the free matrix Gaussian matrix model, the Wigner distribution mobility edge ranges for this eigenvalue between square root of top coupling constant and its positive value. So the edge of that is precisely like a square root of N when you fix G-square to some constant value. So there must be some generic behavior, generic reason why the eigenvalue scales like square root of N even though this is not the general rule. There are some always exceptions. For example as this massive case is that already alluded. Now the interpolating function, the free energy has been computed from this localization and this function f of lambda which summarizes this interpolating function. This is the free energy Gaussian n-square times f of lambda is given by this one extreme weak coupling or strong coupling regime. And let's look at the strong coupling regime where top coupling is very large. Then we have this indeed the proclaimed one of a scale to lambda behavior which eventually gives you in the M theory limit n to the 3 half behavior. So this is it. So this we identify from this one and from the thin air that the ADS4 action, your own share action for this configuration is pi divided by 2 Newton's constant. We infer Newton's constant is given by one of n-square times f of lambda not one of n-square as in ADS5 case. So this f of lambda it is interpolating function here is plays a very crucial role in understanding the M theory behavior of this one. Now we have to do a little more test. I mean this is very thin the leading behavior in semi-classical approximation with a classical ADS gravity the action and we'd like to do more. And in particular as I said ADS CFT is a comparison between observable to observable. We just compare the identity operator. There are infinitely many other gauging very physical observable so we'd like to check the equality between the two. Now gravity is universal meaning that it just attract every energy source of mass source in the universal manner. In other words there's a coupling constant g Newton for every energy source. Therefore in gravity side correlation function is set by uniquely by this Newton's constant whatever it is. All others are kinematical factors like the polarization the prefactors. So that's not really setting the coupling constant. So we'd like to check this Newton's constant being exactly this form and see whether other physical observables in particular their correlators obeys same relation. For example suppose I compute M point correlation functions on the ADS side or equally in CFT side gauging variable operator. Here I suppress all the tensorial indices and I simply separate the positions and then I normalize all this operator by the two point functions. Good so if I do compute this one being the gravity observable the endpoint function should behave universally as g Newton to the n minus two over two power times order of one the position dependent factors as well as a tensorial index dependent factor. So we'd like to check whether this comes out right not just comparing only identity operator. So to do this we just follow textbook of quantum field theory we just introduce source and the value of the specter source and then find the response of the system. Okay this we can do one more for example when these operators are some tensorial press like energy momentum tensor we can squash the the the base manifold which is s three from round s three to squashed s three that's not the only one but that's one available already you know many people compute this one nicely so we can just take advantage of the result and we then extract the response of the CFT three around the round three sphere and see whether we get indeed this form. So here it's just a caricature of the squash three sphere here just I just you know the emphasize that a is a parameter which characterized the squashing a zero is round three sphere and a is one is extreme squashed one and this is the metric and also as you see the when I squash this three sphere I have to also turn on background for the the R current. So if I do this one then many people here compute this as an original beautiful work and the large and limit the localization of the free energy shows a very interesting factorized behavior namely it is given by the squash of free energy is given by round the three sphere free energy times some factor which which captures all the information about squashing. There is no dependence or mixture of the squashing into the function expression of this free energy on round three sphere and this is quite important even though this looks very trivial and seemingly the very innocuous actually the fact is precisely what we need to get the desired expression of universal behavioral gravitational couplings. So let's do a little bit of computation I will just sketch very briefly this very elementary computation so like any quantum field theory if I just introduce some source like in this case a squashing of three sphere then I can expand that is small squashing limit as a set of local operator insertions around round three sphere. I have just expanded here a a square a cube and associated with it there is an operator 010203 so on. In fact you can easily compute it by brute force that leading order leading order in A the operator precisely some linear combination of stress tensor as well as R current and this combination is quite reminiscent of some topological twist here. A second order you can expand this one now it's given by this O1 square and O2 however one point function in CF is always 0 therefore this two point function of O1 integer quantity is related to second derivative free energy respect to squashing parameter and lo and behold if I just look at this expression of this one and expanding powers of A I find that's just a number order of one times free energy of round three sphere. I can do the third order it's again given by the round three sphere free energy times order of one some coefficient and fourth order again is some linear combination of order of one coefficient and F0. What are these alphas they're coming in here that has to be the fact that here I have only one minimum coupling operator which is energy momentum tensor and R current and all other operator O2, O3 are different moment expansion of this operator with respect to angle among three sphere. So more generally sorry I missed the integral notation so here integer quantity by the way these are integer quantities because I'm taking here A to be the rigid value the constant number independent of the position on round three sphere. So the integrated two point function is given by the round three sphere free energy times order of one coefficient which depends on moment of moment expansion around three sphere which is the order of one integral and so this does not depend on n or lambda at all and then this one and we already know scales like n square F of lambda and likewise endpoint function is again order of one coefficient times n square F0 therefore n square F of lambda. Now the reason why all these different the moment the correlation functions scale like F of not and not in very differently is precisely because the innocuous of form that free energy is a factorized into round three sphere free energy times the squashing dependent factor otherwise it's all mixed up and different correlators will have different dependence and n and lambda and it's it will have a very complicated form. Now because it's a universal form if I compute endpoint correlation function normalized in two point function it will have this very universal form which we then read this as a Newton's constant expansion. What okay I'm okay so if that's the case I'm just re-deriving this from the localization. All right so all these operators as I said here the minimum coupling is only in term stress tensor and r column and all o n's are simply this energy momentum tensor then the weighted with furia mode of the three sphere or the spherical harmonics of three sphere and integrate over that's basically the the o n and therefore all these correlators are correlators t and j expanded in different moment. So these are in gravity side are nothing but the correlation functions of the graviton and gauge free or associated r current. So therefore this shows that the correlators of stress tensor are current universally controlled by Newton's constant and this given by this one scales as n to the minus 30 half. So this really completes not just comparison of the free energy but in terms of correlators that we have a Newton's constant coming out this way and this is the expansion parameter in m theory which goes like n to the minus 30 half. Okay so now we just illustrate this one in a specific example of ABJM. Now let's look at free energy of the dimensional CFT. Here I want to give some sort of heuristic maybe intuitive picture how we understand this the proclaimed behavior n to the d over two. Now all known CFTs that we define in the that we know of are defining some certain limit of d-brain world volume theory. In case of m theory m brain case we reduce in type 2a therefore you see certain suitable limit of the d-brain dynamics. So therefore the dynamical variables are the matrix value trivially. This implies that free energy of the mid-plan expansion one of n-scale expansion and there is an additional coupling constant which is top coupling parameter. So therefore that should play a certain role and here if I look at only fixed the top coupling coupling parameter and look at the large end behavior it goes like n to the d over two which I can decompose into this manner n square times the rest. Now n-scale the reason I pulled out n-scale is precisely like good old matrix model any matrix value the quantities should give free energy which scales like n-scale leading order. This is spherical topology of the warship. Now the other part is something we should explain. As I said this is for the I'm looking for the subset of those CFTs which has sort of maximum supersymmetries and then we can derive from the d-brain or m-brain dynamics. So the question is whether we can understand this extra this you know red colored extra factor which I will call anomalous factor and is there any simpler way of understanding the scaling. Now remember in ABJM example as well as you know Gaussian matrix model the mobility edge tells you that the the eigenvalue reaches up to order of square root of m and that I will use as the hint for the whole argument. Yes at least from localization we can explain does not mean that I understand the microscopic the dynamics or the nature of microstate in detail but just a number wise I can understand the scaling. Here I'll give only intuitive and heuristic argument rather than giving the details and in fact that the older the the core of the physics has nothing to do with special functions or the exact expressions of localization computation it's just only the physics. Recall that localization organizes computational partition function into classical part and one loop part and the rest does nothing maybe there's an instant on here I will press or the instant on assume I will I will assume instant on does not contribute in the large and limit in all those dimensions. Now if the integration variable eigenvalues have order of one distribution then the free and scarce like as n square exactly like a usual Gaussian matrix model and reflecting precise the fact that this is a matrix variable has a matrix variable origin. So this is precise comes out when tough coupling is very small but that's not the regime that we compare with the ADS ADS part and there we have to really take the tough coupling very large now the anomalous part which might scale like n to the alpha which we heard many times in this school and workshop scaling of integration variable that should arise in the large tough coupling regime from one loop part of this localization computation nowhere else in the classical part there's now nothing like this okay so how does that come about now remember what is a one-loop integral like any any quantum field theory one loop is around critical point in this case a q-core model you call a critical point you just compute the one loop determinant that's trace law that's this computation now here the m is set by the eigenvalue typical average value eigenvalue which is nothing but the Coulomb branch the values of the the scalar field and so one is computing this quantity well of course we are talking about supersymmetric system so you have to take a super trace now by supersymmetry leading divergence but not diverging leading power in m simply cancels out super trace m to the d whatever power that or cancels out and the in the in a maximal supersymmetric case the the first non zero non trivial contribution after taking supersymmetric trace it goes like precisely m square to d minus four over two that's from the sum null of the super trace m to d uh 2d on the Coulomb branch this master set by as i said this eigenvalue or scalar field value and this you know we always in the free pre potential we have the dimension of phi set to be the engineering dimension one so therefore we have this kind of relation in any dimension irrespective of the cft dimension now there from this one if phi is distributed by square root of n characteristic average value then we simply find that this one is precisely n to a d minus four over two i'm sorry which is precisely this value okay so this one is a leading power just coming from the trace and the top to the scaling and that comes out and the rest is precisely computed by the fact that phi is not order of one but scales like some order of square root of n the distribution in the eigenvalue distribution that's why this n to the d minus four over two is coming out okay well there is this is not always correct and there's a lower supersymmetric case i still don't have this universal behavior or something else i cannot make any universal i can extract any universal pattern but at least in maximum supersymmetric case this is this seems to precise the case and again again as i say that you know special functions we only need the asymptotics of the special functions the one of determinant and here all i allude to is a simple physics that is a boson fermion cancellation leading behavior which is not set to zero by supersymmetry has precise this behavior of this one okay now warning again localization is a way of computing number it does not tell you the microscopic dynamic there's no reason why eigenvalue should be weakly interacting effective degrees of freedom describing whatever n to the d over two entropies so i have nothing to say about it in fact i even suspect that using localization and trying to guess what the microstates are would be totally misleading so like i said so this is only for the maximum supersymmetric case extension to lower supersymmetries is you know still open and well okay good so i suppressed many steps here but this order of scale to the end in it's a winner distribution that comes out any couple super young male as you know very well in the wilson novel so the case and also the a b j m and the five-dimensional case that's always universal pattern okay so if i know if you ask me did i prove that this one is following from director of supersymmetry no probably okay i shouldn't say that i just inferred from the pattern of maximum supersymmetric case that this is uh the universal behavior independent of space time dimension so maybe i should ask if anyone has question at this i don't as i said i know that that's a very good question but i don't have any direct way of comparing this sort of reasoning to the m2m5 picture because i you know i don't know what the there's no direct relation so okay so what i'm alluding to is that the fact that we know that the five-dimensional super young male in the u v flows into the two comma zero that conjecture which i'm i'm tested assuming that okay so all what i'm based on is basically five-dimensional super young male theory argument actually i think that's because the cumulon is here i think it's uh you know uh just tangentially a question i always had is that when you have a two comma zero theory okay not the leader string theory and rank n then when you are in a cool tensor branch they have all these tenson i mean tension full m2 brain right like n or w bosons now if i bring them into the the center of the the tensor branch you know the critical i mean the conformal point then all these tension full string becomes tensionless there are infinitely many degrees of freedom at that point okay even even though we don't know how to count and how to you know explain or the describe the dynamics but it is the fact or at least we anticipate infinitely many degrees of freedom coming down compared to a coincident d-brains on the other hand d3 brain well there okay um but here is even more right six again haggled on like behavior that that's right so in that case you know d3 brain case is a composite one here it you think the higher massive string stays composite okay that's one way but otherwise i just want to bring up the puzzle that if you have infinitely string degrees of freedom coming down massless then you would say that there are infinitely many degrees field degrees of freedom at the conformal point on the other hand if i go to ads7 compute the free energy then it goes like n to the q so there's a no trace whatsoever of the tensionless string degrees of freedom there okay so maybe as as cumulon say one should view that those string states as composite unlike a fundamental string and it would be very interesting to check that carefully the next symmetries let's go back again to quantum field theory 101 unlike classical field theory quantum field theory requires not just writing down Lagrangian that's classical physics quantum fields require specification not only up operators in canonical of quantization formalism but also the choice of the vacuum without choice the vacuum it's like solving differential equation without boundary condition and or in path integral terminology quantum field theory requires specification of not only the action but also the integration measure integration measure is very important once you specify this one then you are completely spaced by the problem at quantum level and you can proceed to compute all the physical observables so here for example the generating functional in the with the source j you just compute the vacuum to vacuum transition amplitude in the presence of that source in the Lagrangian path integral okay so that clearly tells you that you have to specify what the vacuum is the second in pathogram terminology you are computing this one where phi is now classical field and now how on the other hand you have to specify what the measure is now in all the supersymmetric localization which is the local way of computing partition function and physical observables in supersymmetric conform field theory imagine that we do this one it's a path integral or maybe operator formulations where did we specify functional integral measure is there any option it's this unique and can it relate one if there are many prescription you can do can it relate one prescription path integral measure to another one and if so how so these are the questions we never as far as I remember we will never encounter this kind of discussions in the literature and here I'm bringing up the question I would illustrate the importance this point in operator formulation in a very simple example just to illustrate what the physics is and relegating path integral formulation and supersymmetric localization you know that's all you know sense just the details and the physics is not the more different and in my forthcoming paper the theory for illustration is a supersymmetric 2d free field theory and free field theory is the limit extreme limit of localization computation in other words you're doing how the computation and set the coupling constant to zero but you can just view it as just you know the standalone theory and supersymmetric 2d quantum field theory which is a free and this is of course you can view this as a string worksheet of the super strings and also it has a direct implication in in entanglement super some entanglement entropy computations that's how I come about this problem I will then show that two different choices of the vacuum of this free theory leads to two or equivalent two different choices of path integral measure leads to totally different theory totally different spectrum operator spectrum if I interpret this two-dimensional free field theory super some free field theory as the world should of nsr super string in string interpretation one choice of the world should vacuum leads to usual string theory of infinity tower started from massless and to all massive infinity tower the other one if I choose the other vacuum then it leads to the type two super gravity only there's no massive tower just super gravity okay this is bizarre but I just want to illustrate this one not because there's much to do with the localization because so trivial in localization technology but just to emphasize the vacuum choice is so dramatic let's illustrate first the simple physics that you know very well it consider two-dimensional fermions likewise any even dimension it's the same and it in two-dimension the fermions are the two separate conserved columns one is holomorphic j and the other entanglement j bar equivalently I guess recombine them into a diagonal vector and the axial orthogonal axial current now I can I decide to gauge this global symmetries and one way of the gauging it is just coupling minimally coupling this vector current to the vector potential a and that's the conventional QED two-dimensional baby version or four-dimensional QED and then in that case we know from textbook that the the axial j which was classical symmetry becomes anomalous in the background of this uh the the vector gauge field on the other end there's another one another choice you can do you can gauge axial u1 by coupling to the vector potential a and a bar then this one has to be conserved because you're coupling to the gauge potential on the other hand in that case a vector current total fermion number becomes anomalous and this is the familiar the baby version two-dimensional baby version of our wall this is something on the totally unfamiliar and of course standard model which is a chiral gauge theory is some linear combination in which we gauge this one and the the entanglement one in a different gauge symmetries the classical symmetry of two theories are exactly the same in these two conventional QED and axial QED you wouldn't distinguish them by just simply staring at classical Lagrangian classical occasional motion yet if you quantize the system in this manner then they will be completely different because they preserve different symmetries and make the the orthogonal other the alternate symmetries are anomalous so they are very very different okay now let's do the gravity now let's think about the you know as i as i alluded the two-dimensional free field theory as the warship theory of some super stream in that case we just use uh usual polar-capacitive integral formulation of the string theory and then we have it basically 2d super conformal matter and the coupled to uh 2d gravity or super gravity now the matter sector has two separate conserved column one is diffeuomorphism the other one is a biosymmetry these are classical column that are conserved at the quantum level what we can casually do is we preserve diffeuomorphism sensible and we uh sacrifice biosymmetry of course once we quantize coupled to 2d gravity and then there will be gravity compensating precise this uh vial anomalies but at the level of the meta-cft this vial column becomes anomalous okay i can choose alternative quantization which i will allude now which is that i gauge this vial column in other words i take this vial column as non-anomalous and conserved while sacrificing diffeuomorphism well it's not completely sacrificing it because i'm trading one degrees of freedom from one to the another it turns out that even though diffeuomorphism becomes anomalous it's not all the diffeuomorphism but only the area changing diffeuomorphism which becomes anomalous and there's a still on the area preserving on diffeuomorphism as the intact symmetry in the system again in this system like this q ed example classical symmetries of two systems exactly the same yet quantum symmetries are very distinct what are the consequences for the spectrum of the theory okay so i will just mention uh i'll just go through this one okay let's just recall very again very elementary uh chapter two in polchinski and i just consider this the bosonic part first of the supersymmetric free field theory in two dimension and i just just mode expand and then this is just usual the mode expansion of the holomorphic and holomorphic part of the the the free field and in canonical quantization okay these are operators now as i said we have to choose the vacuum to completely specify the theory i will choose the vacuum in such a way which is very standard namely it has zero momentum that's trivial central mass momentum is zero and an old annihilation operator in this heisenberg algebra analysis that vacuum okay in other words the vacuum is nothing but an element of a corner of this uh central mass momentum operator as well as a and a bar annihilation operators in this case the dual vacuum the sitting in the dual fox space is in one-to-one correspondence with the vacuum in the fox space however there's one caveat this is only if and only if the vacuum state is the zero is normalizable if it is not then there's an issue now let's choose different quantization now i will this is the conventional wheel everybody knows i will now pick up different one so in the same operator relation now i choose the the vacuum in such a way that for this right moving sector is exactly same as conventional one left moving one i choose the vacuum condition not to the ket state but the dual foc space brass state so these it has zero momentum with respect to brass state as well as the annihilation operators act on brass a vacuum is zero now you would say well this i can always relate to the uh originally the fox space relation by taking uh conjugate however if the vacuum is not normalizable then this is distinct from just the it's a the ket state statement and this uh non-normalizable state arises in many contexts simply the canonical example is Bose-Einstein condensate or in the quantum mechanics we encounter this kind of thing in rigged Hilbert space it's all non-normalizable mode it's background if i do this one then operator relation is exactly same with a with a simple composition one can show that actually the the two-point correlation function free boson actually takes a different form it's actually this one instead of z z uh z times z bar is a z or divide by z bar and this one as you can see preserve different symmetry from the conventional vacuum choice why the load and suppose the corresponds to z and z bar uh the uh scaling as gamma or gamma inverse this you can see its boundaries it's a gamma factor numerator gamma inverse in denominator so it goes like gamma square with respect to conformal transformation z and z bar scales the same way and here it cancel with each other remember in the usual case this is log z times log z bar at the plus log z bar in that case the with respect to conformal transformation z and z bar scales same way therefore becomes uh the anomalous whereas rodent's boost cancels off with each other so clearly even from this two-point function alone you can see that this choice of vacuum will preserve completely opposite symmetry complementary symmetry compared to standard quantization so here biosymmetry is respected whereas the different morphisms now you can just go uh the the counter quantization by looking at the uh the constraint of the verisoral algebra so here's the l0 and l0 bar and then here operator and then this side is as usual conventional vacuum this alternative vacuum where I choose the bra vacuum condition then this n bar is just you know passive integer that's the because this one is acting on the bra rather than the k now at the zero mode of this verisoral the constraint we have level matching condition as well as a mass share condition and it turns out that the the usual the uh the conventional vacuum choice of the the supersymmetric string the mass share condition acquires like the uh some anomalous uh the the cashmere energies and that sets the the uh the the intercept of the string excitation whereas the level matching condition simply says n and n bar equal the total of the oscillator number however here is the opposite here you have left moving the conventional vacuum and the right moving the alternative vacuum have opposite zero intercept number such that in mass share condition there is no anomalous factor here and only level matching condition that it appears and of course there's other higher verisoral moment uh to be satisfied so now we have basically n plus n bar is as number two and n minus n bar equal zero okay now if you look at the spectrum which satisfy that condition you find that in the let's say I here I also include the bosonic case but the first line is for the supersymmetric string with the GS projection if I have oscillate number one left moving an alternate vacuum choice number one one plus one equal two and that gives precisely massless super variety multiplied in nsns sector in the bosonic case for the curiosity if I just look at the further state then there are possible states namely n equal to n by equal zero or vice versa and they are nothing but just tachyon and it's a counterpart in the in this alternate vacuum so it just has the you know the cut two pairs of the tachyons so in supersymmetric case the only one which is allowed by this level matching condition is the the massless mode obviously which is the super gravity so in caricature the spectrum conventional string I have a left to moving a world oscillator state including zero mode and right moving state like that and level matching condition is left to moving minus right moving oscillator total number has to be zero so level by level there is always pair so we have infinitely many possibilities these are string states in this case alternate vacuum I have vacuum starting from zero and oscillate number and in the alternate vacuum it goes negative direction because of this brass state condition and when I demand the level matching condition the left to moving and right moving oscillator is the same then only zero mode remains okay so I intentionally did it in the operator formulation or one can do this all in the path inter languages now in a quantum filter we know that by adding local counter terms one symmetry one or normalist part of the symmetry can be moved into alternative part now there are many varieties of this one many infinite possibilities and complete classification alter quantization in the case supersymmetric localization is it's still to be studied because it's a cute homology which constrained possible the local counter terms you can add so the question is does the localization method work for all possible alternate definition of quantum field theory because if someone gives me supersymmetric quantum field theory and I didn't know any localization I would proceed with the definition of the vacuum or equivalently the function integral measure now somebody asked me I repeat the same computation with localization maybe there's constraint and in that case there is a tension why only a particular subset of possible theories automate the localization method computation so that's all and thank you very much for attention questions