 So, as a motivation, let us recall the situation to two dimensions, 2,2 supersymmetry. The situation in four-dimension will be exactly the same, conceptually is the same story without any difference, and even the proof, the strategy of the proof is the same, actually even the proof is conceptually the same for all results. But of course, in four-dimension, you get more complicated structure, richer structure, more information, more nice things, so it will be interesting to go to the four-dimensional theory. So, in 2D, start with a 2,2 super conformal field theory, and you write a super conformal index, which is elliptic genus, which has this form, and is independent of q bar, is the salomorphic function of q. J naught is the left, r-charge, and so on, and notation, I think, it is explained here, it is clear. So, this is an index, it's a super conformal index, which means that it is a quantity which is invariant, and the continuous deformations of the theory provided this deformation preserves super conformal values. You can ask if you can do something better than that. You can see this as a family of quantities parametrized by these two complex numbers, and you can ask the question if, for some value of these parameters, the index is actually more stronger than just a super conformal index, that this is invariant under a more general class of deformations of the theory. In particular, deformation which breaks conformal index. And this leads to the idea of specialized super conformal indices, that is, you specialize the index to some value of the fugacity such that that particular index is invariant under the formation of the theories which break conformal symmetry, although they should preserve supersymmetry, otherwise you have no index theory at all. So, n is an integer, and you consider this where you put this fugacity z equal to e to pi n, which is one, but in fact this is not an entire function of z, so this depends on n in a trivial way in general. This is independent of q now, you can prove just by standard techniques of written index theory, and so it's a set of numbers labelled by n, and it is, in fact, an integer just by modularity of the starting theory. You just write the index as a trace, exchanging space and time. Now, these indices are more robust than the general index here, because they survive massive deformation of the super conformal theory. I perturb the theory also by adding a relevant deformation, I go to a massive theory, I get a non-trivial organization group flow, but these indices will remain constant along the randomization group flow. At the end, we get a massive theory in the infrared, and the massive theory in the infrared is characterized by a BPS spectrum. We have many sectors, the BPS spectrum splits in many sectors labelled by i and j, where i and j go from one to m, where m is the number of duty vacuum, which has a central charge, which is anti-symmetric under the exchange of i and j, it is a complex number. And for each sector, we form a BPS operator, which is just an n by n matrix in this case, which has this property under the exchange of i and j, and which is written like this, where n i j is simply the number, the absolute value of n i j is just the number of BPS states, I mean really BPS multiplets in that sector. The sign here depends on several things, it is the corresponding phenomenon two-dimension, that in four-dimension lead to the Gaiotto-Mürnowski quadratic refinement. You have the same story also in two-dimension. Then, when you have this operator, which are defined sector by sector, you get something which combines all sectors by taking the product of these matrices in an order which is anticlockwise ordered with respect to the phase of the central charge, and the symbol here means just anticlockwise ordered. This matrix M is what we call the 2-2 quantum monodrome, it is just an n by n matrix in two-dimension. Now, in two-dimension we have also a chiral algebra associated with this problem because the 2-2 conformal field theories contains a topological sector, I just twist topologically the theory, and I get a chiral ring, and I can add the gradient operator in the conformal limit, and I can consider just the character of this chiral algebra over a module, which is the chiral algebra itself. So, I have a character here. Again, the character depends on the complex parameter, but I can specialize it to this, and just as before, while these characters are defined only at the super conformal point, the specialization f and meaning can be computed or well-defined even if the theories are massive and are independent of the massive deformation. So, in two-dimension we have a relation between the super conformal indices, specialized, the BPS Petra, and the chiral character. We have three objects, which are the three objects in the title, and they are related essentially equivalent information, and indeed the specialized indices are equal to the trace of this BPS matrix m to the n, n is an integer, n is equal to the specialization of the character of the chiral algebra, which is here in two-dimension a stupid algebra, but it will be interesting for them. So, in particular, the specialized super conformal indices can be computed in the infrared just from the BPS Petra or from the way the ring behaves, the spiral algebra. But for a given super conformal theory in the ultraviolet, we get many different BPS Petra, which depends on the particular deformation we do. We can deform in many ways. We get different infrared theories with different BPS Petra, and these different BPS Petra are usually called BPS chambers, but all the BPS Petra we can get are restricted by the condition that this trace should give me the super conformal invariant, which are independent of the deformation. In particular, trace n to the n should be continuous across a wall of marginal stability because on the two sides we have the same super conformal theory, and indeed the wall crossing formula is just a statement that the specialized super conformal index exists. They are well defined. This gives a condition for m, which is just the full information you have from the wall crossing formula. The problem of this talk is to find exactly the same relation, super conformal indices BPS Petra and chiral characters, but in four dimensional and equal to two quantum field theory instead then in two dimension. But everything will be essentially the same except that the structure will be much more complicated because everything will be infinite dimensional extension. So we start from the first ingredient, the super conformal index for a 2D for dimensional theory. I guess we are... OK, this is it, which is an expression which contains three fugacity, P, Q and T. These are the generators of the symmetry. Here we have written the dictionary. It will not depend on beta as always. This is just for converging regions, but everything is independent of beta. And what is interesting for us is that this index here can be given a geometrical interpretation as the partition function on an off-surface. The off-surface is a complex manifold which is topological S1 times S3. So compute the index with some fugacity, the partition function on S3 with some fugacity, and P and Q are just the complex structures, parameters of the surface. The partition function for a supersymmetric theory will depend only on the complex structure of the surface, not on the details. And for off-surfaces of this class, the complex structures are parameterized by two numbers, P and Q, which are in the unit disk. Well, the puncture unit disk. And this is the definition of the... of surface. It is essentially this relation with Q and P being these two parameters. But we have a further fugacity, so we have to insert a fugacity, the explicit in this game. So we have the party integral on this geometry with an explicit insertion of the operator to this fugacity here. As in two-dimension, what we want to do is to specialize this index to get something which is robust against breaking superconformal invariance. Let me start with an intermediate specialization, which is for technical reason. I set T equal QP n plus 1, where n is still an integer. So I get this. We still depend on two, a complex fugacity and an integral variable, n. And here we have various possibilities. If n is zero, this is just the partition function on the off-surface of the theory with the original... with the anthropological twist, the one which gives me the Jones polynomials. And so this is purely topological, depends on the topology, but not on the complex structure, so it is independent of Q and P. If I take n to be minus 1, I get the Schur index, which was defined by God, Rastelli, Razzamatt and Jan, it reduced to a character of a two-dimensional chiral algebra which was defined by the same people, Rastelli et alp, and since it is a character of this algebra, a two-dimensional algebra, it is independent of P. It is just a fun... an homomorphic function of Q. For general n, you should think of i and PQ as analogous to the function we have in 2D with P and Q, but variables give you general dependence, and in order to get something which is invariant, we have to specialize even further this thing. This is again... well, this is an analogy between D equal 2 and 4, you can do this argument with D, anything. So let specialize further, so I specialize just the same way I did in two-dimension, where I had the first variable set equal e to pi i n by here, for convenience, I set already n here, so this is just the same, but written in the same form, so this is the intersection, which is now a function of Q and n, and as before, this synthesis has an interpretation of this function, and it's just the partition function of the n equal to 2 in four-dimension computed in a degenerate limit of the off-surface. Just take the off-surface before and take this limit of that geometry. You get a singular geometry, in fact, a non-compact one was compact, but in this limit, they're compactified with the fugacity corresponding to this factor here, and this is what is called, well, it is the Melvin cigar, which is this geometry times S1. Now, these indices is the same index which plays this, is this n equal to specialized index, which plays the same role as Ia in 2D, and is an index which you see very easy by the standard manipulation, preserved under all deformations, even the ones which break conformal invariance. And it's an infinite family now of indices with the penton Q where Q is in the unit disk. And indeed, just as in two-dimension, the fact that Ia was well-defined was the well-crossing formula. In four-dimension, you just write a condition which is well-defined, or you wish the partition function on this geometry is well-defined, and it is just on the nose the same as the concept which saw the well-crossing formula in four-dimension. So everything remains conceptually the same, but this geometry here, the partition function of this geometry here was already started before with Wafa and Nasky, and so in the massive case, and so we can borrow from that. So let me take now after the deformation of the theory, I get a massive theory and go to the Coulomb branch of this theory, and here I have BPS states as before, as in two-dimension, which are characterized now by the charge, we take the place of the sector, Ij as as before, and also they can have a spin here, charge and spin means, in fact, charge and spin of the lowest component of the BPS multiplier. It has a central charge which is an additive function on the charge lattice, the lattice in which the gamma takes value, the mass, and then you have as before the BPS phase which is the argument, the phase of the central charge. The factor that is to each charge here, you associate a quantum operator which is analog to the finite matrix MhA of two-dimension which is defined by this, it's essentially a quantum dialogue or a product, a product here, I need to take the product over all particles, all BPS particles I have in that sector with the charge. This is the element of the quantum torus algebra where the quantum torus algebra is generated by operators which satisfy this algebra here where gamma, gamma prime is an anti-symmetric pairing on the charge lattice which is given by the direct pairing, electromagnetic direct pairing. So I have this and just as before I define the quantum monodromy taking the product from all sector and ordering the product in the anti-clockwise way with respect to the phase just before and I get this. And this is the 4D quantum monodromy and again, calling it in sovereign member crossing in this formula is obvious, it's just the fact that the conjugation class of MQ is in inverse. However, in 4D if we are along the Coulomb branch we have in addition to massive BPS particle also massless BPS particle which are just the BPS photon the infrared BPS photon which F charge 0 from these and F's being plus minus 1 the first component which is the 14 and so I have to add here a factor to take care of these things which are however centrally in the quantum algebra here so it is just a C number factor which depends on Q but is otherwise totally trivial. And this is standard. R is the complex dimension of Coulomb. Now this script quantum monodromy which is just the same essentially up to normalization I take the traces to a power n these are the formations and we are crossing invariant of the theory and so should be indices of the ultraviolet super conformal theories and take the same value for all massive deformation and the analogy with 2D suggests that the traces to the hem should be the same as the specialized indices which are just this limit of the standard indices. So we can compute the specialized of the ultraviolet theory just by infrared information that is by PPS spectrum and the proof of equality is essentially obvious because both sides of the identity of an integral representation as a partition function on a geometry this is defined by this geometry this is defined by the the generation of the off surface not this but these two spaces are the same the sphugacity is inserted in both cases in the same way along the same cycle so it is just taking the generation of the off surface and see that this geometry as before when we define these things indeed a special case of this identity was checkered by Kordov and Schaub for the case n equal minus 1 in that case this is the Schur index which has a stronger property in particular you don't need to put p equal e to the 2 pi i because this will be already independent of p so it is an even stronger indices and then in that case it is just the vacuum character of the two to D chiral algebra defined by the rest of the entire so this equality was shown in this paper before and what is done here is to extend this to all integral let me see so let me consider the case of the simple class of n equal to 2 t conceptually that is the Lagrangian n equal to t which have a weekly couple Lagrangian formulation which are super conformal the indices as this form produce the gauge group have the flavor group and you write just this where IV and IH are just the three super conformal indices and the integral over the R measure of the gauge group just project on the singlets distribution so for Lagrangian it is enough to specialize the integrand because this where p and q appear you specialize it for free hyper multiplets with given fugacity you get this expression here where this is the standard data function defined like this and notice that this is just the n power where n is this n here the partition function of a complex pin 1 half fermions into the dimension theta divided by eta up to a trivial factor q to the minus 1 so this is already something we suggest relation with two-dimensional conformal field theories for free vector multiplets the story is slightly more complicated because we have zero moles which are both of fermions dependent if n is positive or negative and you need to cancel them by a procedure which is essentially the same you have in super string theory to cancel beta gamma you do just the same things so they find and after that you get the finite quantity which is like this where this is the root lattice of the gauge group for free or abelian gauge theories this equality is just autological because the same expressions just on the nose are just both the specialized indices and you just write this in this form and this is just the product I had before defining the quantum monotony in the abelian gauge theory integral and so everything is tautological what is not tautological is this equality for non abelian gauge theories think of the simple non abelian gauge theory as you do with four levels the bps spectrum consists of w bosons quark states but then you have infinite towers of dions of unbounded mass unbounded electric charges so in the quantum monotony you have infinite many factors besides the obvious one which give you a very complicated operator and you need to prove that the factor of all the bunch of dions is just one the full contribution of all this infinite tower state is just the operator one and so you can neglect the same again this was checked before for SU2 super Q2D by Kord of Ainshaw for the special case n equal to minus 1 so the story is not surprising but it is technically difficult to prove the quote the third ingredient is chiral algebra we saw in one example the free hyper multiplet which for n positive is the partition function of n free spin one half chiral complex fermions which is a character of the chiral algebra of a chiral algebra with central charge equal to n if n is negative we get the absolute value n of free spin one chiral bosons changing the partition function in one of the partition functions for a free theory changing bosons with fermions so for n positive we got a good unitary two dimensional theory with this central charge for n negative we get a non unitary theory which has the opposite central charge which is negative this result here although this is an example but we will see the general case this is at a standard general result which is true even in the nola-grantzian case and we will see that is true which is due to rastrelli et al so if I start with a four dimensional supra conformal field query which has central charges c and a then the special i index is a character of it to do chiral rational conformal field theory which has a Vrazaro central charge which is 12 n c where c is the four dimensional central charge if n is less than zero then c to d is negative and the rational conformal field theory is never is never unitary for n negative for n positive is usually unitary but we have no result in general the most important case was n equal minus one which is the sure case which is of course negative and what we get here it is minus 12 c which is exactly the central charge in fact the full chiral algebra which was introduced by rastrelli and collaborate for n equal minus one it is just the story of rastrelli but we have an infinite story for all integer n positive and negative now the special i index here is in particular a q series which is some power series in q with some coefficients i n which are integers and given a q series i can define a concept which is attractive central charge by this formula essentially I suppose this to be a character of a conformal field theory I use the card formula backwards and from the behavior of the coefficient for large n it would be central charge which in fact is not really the central charge it is central charge minus 24 the smallest dimension of a primary field with full unitary theory and also for n large n 0 in this case is just the central charge and this just and so it is given by this formula with n positive and for negative is given by this formula here we are seeing a this numbers here and this is always positive if n is negative in addition if my fourth dimensional theory is a flower group I get in this two dimensional character here a cat smoothie character all level which is minus one half k is the central charge of the fourth dimensional current algebra and so the special eye indices became cat smoothie characters of one kind or the other if I have a flavor group or related to that and we can ask since this is very similar to a character of a rational conformal field theory in two dimension if it enjoys modular properties as a character in a field that two dimensional rational field theory should have and you can show at least for Lagrangian models that modularity of the integrals I had written before are equivalent to asking that all gauge beta function vanish that the theory is super conformal in the infrared and this is of course correct because we started with assumption that the theory was super conformal and our expressions are the same only under that case let me speak about the related setup which is different but related that has a compatification of the four dimensional equal to theory on a geometry which is t2 times u2 now this procedure give me in two dimension in this t2 always a theory which is unitary if I start with a unity theory and this is a full quantum field theory in two dimension not just an index computation it is always unitary to satisfy all the good properties so the way you do you start from n equal to 2 Euclidean super conformal this makes sense only if it is conformal because we do a partial topological twist but we twist with the u1 part of the r symmetry group two r like in Witten case so Witten twisting makes sense in general but this makes sense only if this is non-anomalous which is just the same as asking the theory to be conformal so you split this group like this and you twist by this combination of operators where n is an integer and so what you get if n is zero you just get 2,2 theory in R2 which is in some sense the obvious complication if n is equal to minus 2 you get a chiral theory which has symmetry 0,4 on this t2 or 2 and in all other case it is a chiral theory with supersymmetry 0,2 really this is correct for n even because you see if n is odd I get one quarter spins which make no sense so since it is important to extend this also to odd n's we need to have an extra global u1 in order to make a twist on that to compensate this mismatch by one quarter but in the example you get so if we start in four dimension with a free theory or wiki couple Lagrangian theory I mean a theory called Lagrangian we do this you can do field by field and you get like this the 4d multiplet becomes 2d02 multiplet so for n bigger than zero the vector multiplet becomes a vector multiplet plus n plus one chiral multiplet of 0,2 in the adjoint for the hyper multiplet became n fermi multiplet in the corresponding representations if you have n negative the vector multiplet still a vector multiplet but now you get that essentially fermi and chiral multiplets get shifted one into the other the gauge grouping 2d will be the same as in 4d and the flavor group remains the same and you have just this gauge theory with the same group as in 4d which is couple with fermi and chiral multiplets of 0,2 in the suitable way and you have also the j super potential of 0,2 which comes just this is a chiral theory in general well it is always a chiral theory so it can be anomaly anomalous and you check that the condition of being anomaly free in two dimension sense is just equivalent to the condition of being conformal in four dimensional sense as before so this is the same now we can consider the left moving of the 2d theory left moving is the part which is not super symmetric so it is the central charge you see in the elliptic genus so this means just the effective central charge of the elliptic genus and it is given by this formula you have a formula for positive one formula for and h is the number of multiplets vector multiplets this is true even if the theory is known with the all suitable definition what you mean by the number of nh and v in such a way that you get c and a correct and you check that this is the same as the effective central charge of the special index of the trace of the monodromy and for all n just identically this is always positive consistent with the fact that this theory is unitary this result is not accidental and for Lagrangian models it is obvious to understand when you have because the traces is just given by the integral this partition function which is given by the integral yet before where this is the product of all three things and the elliptic genus for this theory is given by this in the partition function of this geometry which is just an integral with the same integral but a different contour which is in this case the prescription given by Geoffrey here one for a contour so the two things are certainly different for a negative this is unitary and the other is not so it cannot be the same but at the same c effective because they are but the integral representation with the same integrant and different prescriptions and the effective central charge depends only on the integrant on general grounds not on the contour just because it is the value of the leading compressor point up to using identities from Kirilov's about the hologram you just see how many multiplicity we have you just write the product you have before I brought this explicitly and this you just do the localization computation this computation give you the same integral but the prescription from the contour is different so this is computed by localization and this is computed by saying since it is an index the theory is a marginal coupling that I can send to zero so I get the free theory essentially and it just remains the trace which is the projection implementing the Gauss law that the states are neutral here instead is really a localization computation and for Lagrangian theory it just depends on how many Ip multiplies and vector multiplies you have in which representation and they are just the same rules for the same with that dictionary and you know that this is very easy to show that the effective central charges are the same for two expressions which have the same integrant even if the contour are different and in this case since this theory is unitary the effective central charge is the central charge so the central charge of this theory is equal to the effective central charge of the other for N positive in fact you have a theory into the equation which is but in two dimension from the side of the special eye indexes we have just this character we don't know anything on the right movers and things like that the compatification give you a quantum field theory which have all the ingredients so this is different but related and so this was for Lagrangian models which are not very interesting I mean everything is almost autological you have the same integrants and you just need to well you need to prove that in Lagrangian theories you can forget of the ions that is hard but once you know that everything else is trivial so let me go to the new Lagrangian examples well we understand something particular new theorems about hypergeometric functions so as an example I can consider G.G. Primes for conformal in theories which are labelled by A.D. dinking diagrams G.G. Primes are to simply lay dinking diagrams and the particular case in which the second dinking diagram is just A.1 this is just a G. Redaglas model of type G or G is in A.D.A we know the central charges the four dimensional super conformal central charges of these models which I write here and the condition that G. and G. Primes have co-prime coseter numbers for simplicity I know that the formula in general that will take more lines to write because depend on the common divisors of the two coseter numbers in a complicated way and R is the rank and H and this is C this is A and let me list of G. Redaglas the ones which are not written here have no flavour this U1 should be A1 and this is the flavour group this is the central charge of the flavour group in four dimensions the conformal central charge of the flavour group in particular we know from Rastelli's artwork that the n equal minus the index a 2D chiral algebra the 2D chiral algebra contains the current algebra of the flavour with a level which is minus one half K where K is the 4D central charge so this is the models so computing this from the BPS spectrum we know the spectrum was computed by itself it is given by the minimal chamber where we have the lesser number of states it is given that by the product of the two ranks of the two groups G and G prime times the cosector of the second divided by two and they have these charges here I can identify the charge lattice with the type of the two root lattice and this is like this where one is simple and this is positive using this BPS pattern we can describe explicitly MQ and we are in the trace of the monogram to some N as an explicit although very cumbersome multiple Q hypergeometric sum we know techniques to compute the effective central charge of any hypergeometric sum however complicated and this is the result for N positive I get C to D the theory is unitary on the two-dimensional rational conformal theory it is given by this where two R is the correction in the case H of G and HG prime is not co-prime so this expression here which is rather complicated but it is always equal to 12 NC where C is equal to the central charging for dimension for co-prime you just check with the previous one otherwise they find C like 12 this divided by 12 for N negative it was not computed in general it was computed for this class of model you can compute but this thing is boring we did some example not the most general one and you got this which is this expression as predicted in terms of the central charge of further dimension and this equal to the effective central charge of the minimal models PQ with L plus 3 for N equal 1 minus 1 and this is as predicted by Rastrali et al in particular you check from C and see effective the value of the minimal conformal weight and this just I suspect from the generality I stress that these computations here are done from the infrared theory I know the BPS spectrum and I make computation and I get quantities that I compare to the computation done in the super conformal theory and they will develop and we get always correct matching which just means that this identity is correct which it is strange not to be since it is essentially tautological let me do some an explicit character from BPS spectrum the simple case is A to N A1 and N equal minus 1 which is the shoe index and we know what should be because it was computed with different techniques Rastrali et al computed this in the ultraviolet from the super conformal field theory we compute here from the BPS spectrum we don't need to get what we got so this is the expression for trace N minus 1 in terms of BPS states in the minimal chamber it is a very complicated Q hypergeometric sum in two way integral variables very complicated but using various techniques from the theory of Q and N identities and Andrew's identities you rewrite it as a finite product which is like this it is just N the product of two N polygymers symbols here and this expression here is exactly the Feggis-Nexchiani-Auguri way of writing the vacuum character of that BPS automotor so this is just the fact that this is equal to that proved that what we got is equal to what Rastrali got and everything is correct and everything is analytic here you can do the sum explicitly for even the situation well it is the same but now I am computing M plus 1 trace where we have no prediction one computed from the other side you get this expression here where C is the Carton matrix of A to N minus 1 you go down by 1 in rank and this sum was computed by Feggis-Stoyanovsky for any Carton matrix here of any simplest Lie algebra and what you get is that this is the vacuum character of the loop algebra of the maximum important sub-algebra of G1 at level 1 in other words you take your algebra G you write like the standard Carton, the composition then you go to the loop algebra and you make the central extension and you take only things which came from here except to say this will return in a moment and so in this case we get just the the vacuum character of this algebra where G is the affine SU2N if I go to the opting case which is more complicated to compute but you get the same story with just in terms of the rank it is just the same for instead for trace m to the minus one often even are very different let me go to this case probably I have here considering instead the characters of the Archirae Douglas of type DR which are much more here we have a phenomenon of saturation if you recall on trans we had a talk about any equal one extension of supersymmetry of 2N equal to 2 cases in which N equal to 2 and the story was like this you start in the with N equal to 2 theory you give a perturbation a very special perturbation with preserve only N1 and you get here a thing that a priori should be an equal one but instead you discover that is an answer to N equal to 2 so the infrared fixed point is magically N equal to 2 this the example to offer a certain class of N equal to 2 super conformal theories in the ultraviolet that you start here I am talking exactly on the same class same characterization and we will see what are this so in this language you start from one super conformal theory in the ultraviolet which has some character you end up in the infrared with another N equal to 2 which has also some characters and you can ask is there a simple relation between the two characters so what I call cheese factorization is just the definition of what are the good ultraviolet fixed point we start and they are good for my courses too so in 2D we have the Sugabara formula like this and for a unitary theory if this is in equality in general for unitary you have here because you can have other degrees of freedom besides the current algebra but if you have equality then the current algebra is the full rational conformal field theory and the characters of the rational conformal theory are just characters now since we have a 4D to D dictionary due to rasterality between the two dimensional central charge and the four dimensional central charge which is like this if I have a flavor group GF we call that the group is the flavor group so if this equality holds for an N equal to super conformal field theory I say that this is cheese saturated why I call cheese saturated because you have this experimental fact that if this is true you can always find a finite BPS chamber that is a chamber with finitely many hyper multiplies something else as BPS stays such that the hypers are 12 times C that is if you compute C just from the BPS spectrum thinking of them as free fields you get the current C although A will be not correct but for C the BPS spectrum saturate about and in general you have inequality so this is the equation of inequality if this is true you can see that for N positive you have in general formula like this the trace of the monodromy to the N with N positive is a theta function of some lattice with a penton N and so on these are the flavor fugacity divided by Q to 12 and C so this you have essentially eta an effective central chart which is the central chart in this case so you have just a bunch if you wish free fields if you bosonize the theory as many fields as you have central chart so for this class of theories this is which are that you have this formula here and the only thing you need to find is the precise lattice you have here because in general it is a very complicated story this is what you expect from this story of C saturation on the other hand what you have from the actual spectrum here is a very complicated Q sum of the infinite many things well many things so reshuffling this in this form is very obvious and in particular for this class of models as we will see they are C saturated so you have you expect an identity like this that the Q sum from the spectrum of this class of agiridaglases of this form and this is very similar to an old conjecture in this function theory that you should have an identity between a sum of the four similar to this with some decoration and see the carton of the A and a theta function of this form but this conjecture was very old because it was motivated by the geometric interpretation of this Q function but was obviously wrong because you just expand the two sides of the proposal conjecture and just the first term is not correct so the problem was what is the correct conjecture the precise conjecture the conjecture was essentially correct but there was a small information in the formula a very small change but no one was able to find what that variation was so what this formula says is well one side is what we want to go this is on general ground of that form essentially from Namward with appropriate definitions and they should be equal so this is the correct conjecture we replace the not precise conjecture people had and indeed when we come to this through the monotomy operator we just give to Wagner we have the conjecture so this is the precise conjecture in two weeks was pro of the general case conjecture it is easier to prove than say there is a very conjecture something should be true so now what I say is really new theorems and proof in full ok so let me start with Algiridaglas with Autrank Autrank FSU2 flavor have this central charge for the SU2 flavor in two dimensions C is just N over 2 this are classical you put here the number and this inequality so it is saturated the SU index is just the vacuum charge of SU2 at that level because if it is saturated the two dimensional is just pure current algebra and then it is just the vacuum charge of SU2 of whatever level you get you get a negative level so it is some less it is in the mathematical literature you have and you check and everything is correcting trace MQ is more interesting since you expect to be of this form and indeed after we proved in the physical sense and then Wagner proved in mathematical sense that you have this equality where this is very complicated to write explicitly in the original form I will not write and here C2N is just the carton matrix of A2N so this is just a specialization of the theta function for the SU2M-1 root lattice the most beautiful form you can have the carton algebra here is just SU2 2M-1 at level 1 plus some free scalars in order to get this A-tas in front so this in this case and this even better you see for M positive you get unitary theory for M negative you get non-unitary theory in two dimensions strangely enough people prefer negative ends but M positive are better because they are positive they are really two-dimensional rational conformal theory if we consider even even is trickier and was not correctly studying the literature because it's Flavre SU2 times Q1 with this central charge for SU2 and this central charge for the anomaly conformal anomaly and it was thought not to be saturated but it is saturated you want carton algebra is just a free scalars so if you put here the one corresponding to the one this is just saturated and you get the same formula well now this depends on two fugacity just because the rank of the Flavre group is two so you can put two Flavre fugacity you have a certain explicit lattice that I will not write that give you this and this is really a theorem because Wagner proved it in functionality to give you a flavor of this function if you take the coefficient of v to the zero here this is just the standard function of the SU2 and root lattice if you want other example in the paper we computed a very large example so let me return to this one particular case this was the R A1 model here and you get here AR-1 the carton algebra here for any plus one is just affine SU well, specializing something in the odd case for R odd is just this otherwise times you want to not consider free the couple free field also here and here is SU are restricted to the nilpotent part so the relation is just that you take instead of the full the algebra just its nilpotent part and take the character of that and you get the infrared story from the ultraviolet one I don't know how this is related to realization group and so on and so forth but the map between the starting data and the final data is easy, it is just you have the same the group and instead of having all currents you set to zero all currents but the upper triangular one and P and that is it both level one nothing else change, it is just that you take a subalgebra instead of taking the maximum nilpotent subalgebra instead of taking the full algebra that is it okay