 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 the super conformal index which is elliptic genus which has this form, and is independent of Q bar, is a 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 lead to the idea of specialized super conformal indices that is to 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 is changing 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 new trigger in realization group flow, but these indices will remain constant along the renormalization group flow. Then 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 vacuums, which have 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 j is just the number of BPS states, I mean here really BPS multiplets in that sector. The sign here depends on subtle things, it is the corresponding phenomena in two-dimension that in four-dimension lead to the Gaiotto-Mursnowski 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 in which combine all sector by taking the product of these matrices in an order which is anti-clockwise ordered with respect to the phase of the central charge, and the symbol here means just anti-clockwise order. This matrix M is what we call the 2-2 quantum monodrome is just an n by n matrix in two-dimension. Now in two-dimension we have also a chiral algebra associated to this problem, because the 2-2 super conformal field theories contains a topological sector, I just twist topological theory, and I get the 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 theory is passive 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 charator. We have three objects, which are the three objects in the title, and they are related essentially F 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 charator 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, which is purely algebraic. But for a given super conformal theory in the ultraviolet, we get many different BPS Petra, which depends on the particular deformation we can 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 M to the N should be continuous across a wall of marginal stability because on the two sides we have the same with a very 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 of in two-dimensional. But everything will be essentially the same except that the structure will be much more complicated because everything will be infinite-dimensional essentially. So we start from the first ingredient, the super conformal index for a 2D four-dimensional theory, 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, will not depend on beta as always. This is just for conversion in 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 other 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 my off-surface. It is essentially this relation with Q and P being these two parameters. But we have a further fugacity, T, so we have to insert a fugacity, T, explicit in this game. So we have the party integral on this geometry with an explicit insertion on 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 super conformal invariance. Let me start with an intermediate specialization, which is for technical reason. I set T equal QPM plus 1, where N is still an integer, so I get this. It still depends on two, a complex fugacity and an integral variable N. And here we have various possibilities. If N is zero, this is just a 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, the Rastelli, Ratham, and Jan, and it reduced to a character of a two-dimensional chiral algebra, which was defined by the same P for Rastelli et alp. And since it is a character of this algebra, two-dimensional algebra, it's independent of P. It is just a fun, an onomorphic function of Q. For general N, you should think of I, N, P, Q as analogous to the function we have in 2D with P and Q, but variables give you a non-trivial dependence. And in order to get something which is invariant, we have to specialize even further the state. This is, again, this is the analogy between D equal 2 and 4. You can do this argument with D, anything. So let's 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 T R for convenience. I set already N here, so this is just the same, but written in the same form. So this is the index, again, which is now a function of Q and N. And as before, this synthesis has an interpretation as a part integral, or a partition 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. So let's take the off-surface before we 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. And you get this geometry here, together 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 play this N equal to specialized index, which plays the same role as a Yang in 2D, and is an index, which you see very easy by the standard manipulation, which is preserved under all deformations, even the ones which break conformal invariance. That is an infinite family now of indices with the pent on Q, where Q is in the unit disk. And, indeed, just as in two-dimension, the fact that Yang was well-defined was the world-crossing formula. In four-dimension, you just write a condition that these indices are 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 man 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 in a paper with Vafa and Nayski. And so we, 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, IJI 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 multiplet. 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 UFS before the BPS phase, which is the argument, the phase of the central charge. To each sector, 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 for a product, well, a product here. I need to take the product over all particles or BPS particles I have in that sector with the charge. It is an 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 dirac pairing, electromagnetic dirac pairing. So I have this, and just as before, I define the quantum monodromy taking the product of these sector operators on all sector and ordering the product in the anticlockwise way with respect to the phase just before, and I get this. And this is the 4D quantum monodromy. And again, calling it insurmountable 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 this, and F being plus minus 1, the first component, which is the fortino. 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 it's 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, and these are the formations and wall crossing invariant of the theory. And so should be indices of the ultra-value super conformability theories and state 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 spatial indices, which are just the limit of the standard indices. So we can compute the specialized super conformal indices 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 F and path integral representation as a partition function on a geometry, this is defined by this geometry. This is defined by the degeneration of the off-surface, which is this, but these two spaces are the same. So that is, the sphugacity is inserted in both cases in the same way along the same cycle. So it is just taking the degeneration of the off-surface and say that there's geometry we had before when we defined these things. But a special case of this identity was checkered by Kordover-Schau before 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 charator of the 2D 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 integrally. Let me see. Let me consider the case of the simple class of n equal to 2 theory, at least conceptually, that is the Lagrangian n equal to theories which have a weakly couple Lagrangian formulation, which are super conformal. The index as this form, for J is the gauge group, I 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 on this statistical distribution. So for Lagrangian theory, it is enough to specialize the integrant because this where P and Q appear, you specialize it for free hypermultiplets with given fugacity. You get this expression here, where this is the standard theta function defined like this, and notice that this is just the n power, where n is this n here, of the partition function of a complex ping-wan-alf 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-garma zero moles. You do just the same things, so defined, and after that you get a finite quantity which is like this, where this is the root lattice of the gauge group. So for free or abelian gauge theories, this equality is just autological because the same expressions just on the nose are just both the special 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 you have just this integral, and so everything is autological. What is not autological 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, and so on and so forth. 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, that the full contribution of all this infinite tower of states is just the operator one, and so you can neglect and you get the same. Again, this was checked before for SU2 super-QCD by Cordoba and Chao for the special case N equal 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 one example, the free hyper-multiplet, which for N positive was just a partition function of N free spin one-half chiral complex fermions, which is a charter 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 just interchanging bosons with fermions. 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 general Lagrangian case, and we will see that is true, which is due to Rastrelli et al. If I start with a four-dimensional super conformal field query, which has central charges C and A, then the special eye index is a charter of it to do chiral rational conformal field theory, which has a Viral-Zorov 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 unitary for N negative. For N positive, it's 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, and in fact the full chiral algebra, which was introduced by Rastrelli and collaborated. 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 eye index here is in particular a Q series, this time power series in Q with some coefficients i, n, which are integers. And given a Q series, I can define a concept, which is this operative central charge by this formula. Essentially, I suppose this to be a charter of a conformal field theory. I use the card, the formula backwards, and from the behavior of the coefficient for large N, I deduce 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 for unitary and also for large N 0, in this case it is just the central charge, and so it is given by this formula with N positive. And for N negative is given by this formula here, we are seeing these numbers here, the four dimension, 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 characters here, a Katsnudi charator 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 Katsnudi characters of one kind or the other, if I have a flower group or a branch 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 of a bona fide 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 the assumption that the theory was super conformal and our expressions mixed only under that case. Let me speak about the related setup, which is different, but related, that has a compatification of the fourth dimensional equal to theory on a geometry which is T2 times U2. Now this procedure gives me in two dimension, in this T2, always a theory which is unitary if I start with the unity theory, and this is a full quantum field theory in two dimension, not just an index computation, it is always unitary, it satisfies all the good properties. So the way you do, you start from N equal to 2, Euclidean super conformal in S2, T2, S2, 2, 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, not the US2 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 a 2,2 theory in R2, which is in some sense the obvious competition, if N is equal to minus 2, you get a chiral theory which is 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 a Lagrangian theory, I mean a theory which has a Lagrangian, we do this, you can do field by field and you get like this, the 4D multiplies becomes 2D02 multiplies. So for N bigger than zero, the vector multiple becomes a vector multiple plus N plus one chiral multiplies of 0,2 in the adjoint for the hypermultiple became N fermi multiplies in the corresponding representations. If you have N negative, the vector multiple gives still a vector multiple, but now you get that essentially fermi and chiral multiplies 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 coupled with fermi and chiral multiplies of 0,2 in the suitable way and you have also the J superpotential of 0,2 which comes just by. 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 Virazoro central charge of the 2D02 theory. Left-moving is the part which is not supersymmetric, so it is the central charge you see in the elliptic genome. So this means just the effective central charge of the elliptic genus, which is the central charge of the left moving, and it is given by this formula. You have a formula for n positive, one formula for n-nevity, nH is the number of multiplies n in the vector multiplies. This is true even if the theory is known as grand genre with the all-suitable definition of 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 spatial index or the trace of the monogamy for the same n 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 the grand genre model 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 essentially 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 Kear one for example. So the two things are certainly different for instance for n negative this is unitary and the other is not, so it cannot be the same, but have the same C effective because they are part integral representation with the same integral and different prescription. 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 complex point up to using identities from Kirilov-Nam about the hologram. You just see how many vector multiples you have. You just write the product. You have, before I brought this explicitly, this, and this you just do the localization computation this computation give you the same integral, but the prescription from the contour is different. 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 I can multiply vector and multiply UF in which representation and they are just the same rules for the same n with the 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 in two-dimension which is there, but in two-dimension from the side of the special eye indexes, we have just this charater, we don't know anything on the right movers and things like that instead the compatification give you a quantum field theory which have all the ingredients and everything. 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 all towers of dions that is hard, but once you know that everything else is easy, 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 prefer conformal in theories which are labelled by two ADE dinking diagrams, G.N.G. Primer to simply lased dinking diagrams and the particular case in which the second dinking diagram is just A1, this is just a g-redaglass model of type G or G is in ADA. We know the central charges, the four dimensional super conformal central charges of these models which I write here under the condition that G.N.G. Primes have co-prime coseter numbers for simplicity. I know the formula in general but we'll take more lines to write because it depends 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 the non-trivial flavor symmetry of at least of G.R. Douglas, the ones which are not written here have no flavor. This U1 should be A1, the same sprint and this is the flavor group. This is the central charge of the flavor group in four dimensions, the conformal central charge of the flavor group. In particular we know from Rastelli's artwork that the n equal minus one case, the sure case, contains the index, a 2D chiral algebra. The 2D chiral algebra contains the current algebra of the flavor with a level which is minus one-half K or K is the four-dimensional charge. So this is the model. So computing this from the BPS spectrum, we know the spectrum, it was computed by Buff and Iski and myself. It is given by the minimal chamber where we have the lesser number of states. It is given 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 tensor product of the two root lattice and this is like this, where one is simple and so this gives me and this is positive root. Using this BPS pattern, we can describe explicitly MQ and y and write 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 the sum of the sum of C to D, the theory is unitary on the two-dimensional rational conformal p-tree is unitary, so it is equal to CFT. It is given by this, where two R is the correction in the case H and H of G and H E prime is not co-prime. So this expression here which is rather complicated but it is always equal to 12 and C where C is equal to the central charging for the mention for co-prime, you just check with the previous one otherwise they find C like 12 divided by 12 n. For n negative, it was not computed in general, it was computed for this class of model, you can compute but this seems to be 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 four-dimension and it is equal to the effective central charge of the minimal models PQ with 2 to L plus 3 for n equal 1 minus 1 and this is S to the power of S predicted by Rastrali et al. In particular you check from C and see effective the value of the minimal conformal weight and this is just, I suspect, from the generality. I stress that these computations here are then 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 explicit chargers from BPS spectrum. So the simple case is a to n a1 and n equal minus 1 which is the Schur index and we know what should be because it was computed with different techniques. I mean Rastrali et al computed us in the ultraviolet from the super conformal theory. We compute here from the BPS spectrum and we need to get what you 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 so very complicated but using various techniques from the theory of Q hypergeometric function and identities and Andrews identities you rewrite it as a finite product which is like this. It is just n the product of two n Pozheimer symbols here and this expression here is exactly the Feggis-Nexiani-Auguri way of writing the vacuum chargers of that BPS automata. So the fact that this is equal to that proves 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 n plus 1 trace where we have no prediction because no 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-Stoyanovski for any Carton matrix here of any simplest Lie algebra and what you get is that this is the vacuum chargers 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 decomposition, then you go to the loop algebra and you make the central extension and you take only things which came from here. All other currents are set to 0. This will return in a moment and so in this case we get just the vacuum chargers 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 instead of SU2nS2n plus 1, so in terms of the rank it is just the same. Instead for trace m to the minus 1 often even are very different. Let me go to this case, probably I have here, considering instead the chargers of the Argirae Douglas of type DR which are much more, here we have a phenomenon that I call C saturation. If you recall on 20 we had talked about n equal 1 extension of supersymmetry of 2n equal to 2 cases in which n equal to 2 enhance to n equal 1 and n equal to 2 and the story was like this. You start in the ultraviolet with n equal to 2 theory. You give a perturbation, a very special perturbation with preserved only n1 and you get here a thing that a priori should be n equal 1 but instead you discover that this is an answer to n equal to 2. So the infrared fixed point is magically n equal to 2. This, the example for a certain class of n equal to 2 super conformal theories in the ultraviolet you start here I mean talking exactly on the same class, same characterization and we will see what are this. In this language you start from one super conformal theory in the ultraviolet which has some chargers. You end up in the infrared with another n equal to 2 which has also some chargers and you can ask is there a simple relation between the two chargers. So what I call C saturation 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 greater or equal here because you can have other degrees of freedom beside the current algebra but if you have equality then the current algebra is the full rational conformal field theory and the chargers of rational conformal theory are just good chargers. Now since we have a 4D to D dictionary due to rastrali et al between the two dimensional central charts and the four dimensional central charts which is like this this became like this if I have a flavor group GF because the group is the flavor so if this equality holds for an n equal to super conformal field theory I say that this is G saturated. Why I call G 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 and nothing 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 an inequality so this is really a saturation inequality. If this is true you can see that for n positive you have a general formula like this that the trace of the monodromy to the n with n positive is a theta function of some lattice which depends on n and so on. These are the flavor fugacity divided by Q to 12 and C so this you have essentially eta to the effective central charge which is the central charge in this case so you have just a bunch if you wish of free fields if you bosonize the theory as many fields as you have central charge. 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. So this is very magical because in general it is a very complicated story here. 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 over infinite many things. So reshuffling this in this form is not obvious and in particular for this class of models as we see they are C saturated so you expect an identity like this that the Q sum from the spectrum of this class of agiradaglaces of this form and this is very similar to an old conjecture in Q hypergeometric function theory that you should have an identity between a sum of the four similar to this with some decoration and C the carton of the end and a theta function of this form. But this conjecture was very old because it was motivated by the geometric or combinatorial interpretation of this Q function but was obviously wrong because you just expand the two sides of the proposed 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 deformation in the formula, a very small change but no one was able to find what that variation was. So what you see here is that the formula says is well one side is what we want to go, this is on general ground of that form essentially from non-work 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 monodromy operator we just get to Varna, we have the conjecture said this is the precise conjecture and in two weeks was proved the general case because once you have the precise conjecture it is easier to prove than say there is a very conjecture that something should be true. So now what I say is really new theorems and proof in full. So let me start with Algiridaglas with AutoRank. AutoRank has SU2 flavor, has this central charge for SU2 flavor in formation, C is just n over 2, these are classical you put here the number and this inequality so it is saturated. The Schur index is just the vacuum charator 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 charator of SU2 of whatever level you get. You get a negative level so it is some less not standard but it is in mathematical literature you have and you check and everything is correct. 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 ERC2N is just the Carton matrix of A to N so this is just a specialization of a theta function for the SU2M-1 root lattice. The most beautiful form you can have, the chiral algebra here is just SU2M-1 at level 1 plus some free scalars in order to get this A-task input. So this in this case and it is even better you see for N positive you get unitary theory for N negative you get non-unitary theory in two dimension. Strangely enough people prefer negative Ns but N positive are better because they are positive, they are really two dimensional rational conformal free theory. If we consider even his trickier and was not correctly studied in the literature because it is flabberg SU2 times U1 with this central charge for SU2 and this central charge for the conformal anomaly and it was thought not to be saturated but it is saturated because U1 gives me C1. U1 current algebra is just a free scalar so if you put here the one corresponding to the U1 this is just saturated and you get the same formula where now this depend on two fugacity just because the rank of the flabberg group is two so you can put two flabberg fugacity you have a set and 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 data 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 was this dr a1 model here and you get here a r minus 1. The current algebra here for any plus one is just affine SU well specializing something in the odd case for odd odd is just this otherwise times you want to some power will not consider free. I did couple free fields also here and here is SUR 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 light. I don't know how this is related to realization group and so on and so on and so on and so on and so on 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 group and instead of having all currents you set to zero all currents but the upper triangular one and the one in Np and that is it. Both level one so nothing else change. It is just that you take subalgebra instead of taking the maximum nilpotent subalgebra instead of taking the full algebra. That is it.