 Okay, let's get started for the last session of this workshop. We have Cyril who will tell us about line operators and Cypher. Thank you. Thank you to the organizers for inviting me to this wonderful conference. So yes, I would like to discuss some work that appeared on Monday. We see on Kim and Brian Willett and Brian already talked about some of the main ideas on Monday. So it is a story about three-dimensional gauge theories. And of course 3D happens to be between 2 and 4, and gauge theories in two dimensions are a bit too easy. In four dimensions they tend to be a bit hard, so 3D is just a sweet spot. For instance, while all 3D gauge theories are UV-free, but they can still have very interesting infrared physics. So just even for Abelian gauge fields, for instance, if I have an Abelian gauge field, so-called U1-1-1-2 gauge field coupled to a fermion of charge 1, then what is infrared physics? And it turns out that it's a non-trivial fixed point, which is the O2 with centrifugal fixed point. So that's something that is well-known, but has been readdressed in the FTH language by Céber and collaborators in recent years. And this talk I will consider these kind of gauge theories, but with four supercharges. So that's 3D and equal to supersymmetry. Of course that close cousins to 4D and equal 1, and two-dimensional and equal 2, two supersymmetric theories. And from the get-go we assume that we have an arc symmetry, and as we know that we have laws and we have many exact results in all the localization results and so on, there is always at least four supercharges, or sometimes two, but we always need this kind of arc symmetry. Then we can consider very arbitrary gauge theories. For technical reasons, in what we did, we considered pretty much arbitrary gauge group, except that it will be a product of some simply connected, simple lig groups, compact groups. And you can also have unitary factors, including U1 theories. And for this regression purpose in this talk, I will often just take the simplest such a gauge theory, which is the inequality version of this system with a vector and a fermion. So in the inequality version, there is a vector multiplet and a carol multiplet of charge 1 under the vector multiplet. And again, you can ask what is the arc physics. In that case, it is well known from the late 90s, at least, that this simple system flows to a free carol multiplet in the infrared, which has identified the UV gauge theories with a disorder operator or monopoly operator T. Now our current understanding of this kind of gauge theories has been exponentially stronger in the last 10 years or so, and this is mainly because of all these exact localization results that we've heard about. And the most interesting, the most important certainly is such result, and the first one historically in three dimensions was the three sphere partial functions computed by Kapov-Simlet and Yakov and Dijant Lehmann in the n equal 2 case. And it takes a simple form where you try to do a pass integral over a three sphere for a CFT, but more generally for this n equal 2 theories with an arc symmetry. And you have this, you reduce the pass integral to the simplest matrix model when there is an integral over sigma, some scalar in the vector multiplet, so you have integral over constant mode of sigma, and the integrand is just a classical piece from terms and terms, and some one loop piece, one determinants, which is easy to compute. And a few properties of this object is that first of all it is Argin variant, that's the main reason why we care about it, because we can compute it in UV free theories, and then it tells us something very deep about the infrared physics, about the fixed point in particular. So when we have interesting fixed point, SCFTs, when we turn off this mass parameter m, then literally the minus the log of the partial function, the real part of it is this so-called F quantity, which is a C function in three dimension, and even the imaginary part is some interesting physics. And in general it's also be like a neuromorphic function of various mass parameters, this m, which are coupling to, you should really think of it as coupling to all the global symmetries of your theory by having background vector multiplet. So for the same price we can also study various line operators on the three sphere, the same formula. What you can do, for instance, you consider Wilson loops, and to preserve half of the supersymmetry, they will need to be wrapped on some S1 in S3, which is the OPF fiber, so you see the S3 as the OPF vibration over the two sphere, and then you can wrap any OPF fiber, and you need two Wilson loops to be off-linked, and then you can compute expectation values of these Wilson loops by just plugging in some character of the representation for the Wilson loop into the integral formula and that. And of course I can start studying more complicated line operators in particular, various defects operator, vortex loops and so on, and that's the kind of thing we want to do today. So on general ground there should be many exact location results, not only on the three sphere but on many other geometries, while in fact if we want the same kind of half VPS geometries that can be preserved on the three sphere, so two supercharges of a particular kind, then the answer is known and it's just any Sefert manifold. So I will discuss Sefert manifold in more detail later in the talk, but really the Sefert manifold is just a three manifold that just admits a natural S1 action, and then we want to compute the supersymmetric portion functions on this guy, like we did for the three sphere, and in general it will again depend more morphically on various mass parameters new, which is just to complexify real mass in 3D. Here that's a dimensionless parameter, where beta here is just a radius of some S1 fiber here, and A0 is some polyethylene loop on the S1 fiber for background gauge fields. And in general these guys might also depend on the various background gauge fluxes, bundles for the flavor symmetries. So until now the exact result for this kind of object is known for relatively short list, just essentially spherical topologies, there is some squashing that we will discuss. There are also land spaces and there's squashing, there is then S2 times S1, that compute a super conformal index, and then there is another S2 times S1, which is a different way of preserving supersymmetry, which we've heard a lot about in this week, which is a two-set index that was discussed by Benin and Zaffaroni. And I should commend that then we want to generalize this thing to M3. Mathematician might not be that impressed because for them Sefert manifold are kind of boring, in particular they are never hyperbolic, but they're still a much more general class of topologies than just spheres. So to give you an example, a nice example of a Sefert manifold is the Poincare homology sphere, so that, of course, it's the only homology sphere with a finite phononatal group, you can obtain it from the decahedron by gluing the 12-phase pairwise, or you can also see it as a quotient of the three-sphere by this binary Acozyada group of order 122, 20. Now you could try to naively localize on this, this three manifold, and you would get some general formula as the first pass, where you would have a similar formula as Estruy, there is an integral of a constant mode of sigma, there is a classical piece that you can compute, there is a well-known piece that with a bit more effort you can compute as well along the trivial connec- with this background, with trivial connection. And that's indeed the result for an Abelian theory, essentially because it's homology sphere, so H1 of this space is 0. But on the other hand, there is a non-trial phononatal group, so for non-Abelian gauge groups, you will have to worry about the fact that you will have some of the political sectors that includes many more flat connections or flat bundles on these things, and well, if you want to do that explicitly, it becomes more complicated, obviously. Even for Tian-Samen theory, this kind of problem is non-trivial. So that should generalize that this kind of brute force localization that we've learned a lot about becomes increasingly hard when the topology of the manifold increases, the topological complexities of the peak increases because of all the sum of our topological sectors for the gauge fields and so on. So in this talk, in our paper, we bypassed this difficulty entirely that the main idea was already explained on Monday. So our approach will be to consider an auxiliary two-dimensional topological field theory. So a two-dimensional field theory, which n equals 2 to supersymmetry, which is just the reduction of the 3D n equals 2 theory on a circle of fields, then, which are all the KK modes. And you can deal with this 2D gauge theory by standard methods. And the point is that in 2D, it's very simple, it's very natural to do then a further topological twist along the remaining two directions, and then you can define the theory, let's say, on a Riemann surface. So the basic setup will be discriminative for Riemann surface times S1. That's, again, those things that compute those three set indexes. With the topological a twist. And that kind of setup we call the 3D model. Then we will see that there exists particular line operators that wrap the S1, and then it's just a point-like operator in the two-dimensional setup, which we will call the geometry-changing line operator. And inserting such line operator at a point on the Riemann surface is equivalent up to Q exact terms that we don't care about to introducing a non-trivial vibration of the S1, so that we obtain a generic Sefer manifold. And here, so we can, for instance, start from S2 times S1, and then just insert a line on S2 along the S1 at a point on S2 where S2 times S1 is a two-set index. So all these partial functions, including the twist sphere and so on, are just insertion of lines in the topologically twisted index from that point of view. So that's the plan of my talk. First I will discuss in 3D and equal to theories in more detail. Then I will have an intermezzo about just Sefer geometry to review the necessary background about geometry. And then we will see that this basic picture of Sefer 3 manifold we can mimic it in the supersymmetric context by introducing this line operator. And with this tool we will derive this formula for the partial function that I will explain. And then with that we will revisit as an example, various previous computation on land spaces and just as an appetizer I will at the end discuss briefly the Poincaré homogenous sphere. Okay. So let's consider first the theory, as we said the 3D theory, but compact if I am a circle. So then the supersymmetry algebra N equal to is the usual one, but here it's viewed as the 2D N equal to 2 superalgebra in the remaining two dimensions where of course I have the momentum along the circle P3 here, Z is the real central charge in three dimensions and in 2D these guys here becomes the usual complex central charge in the N equal to algebra in 2D. And we will be interested in preserving the supercharge Q minus and Q bar plus here in this talk. So that's all these RVPS geometries and then we can consider also lines in these setups which should preserve the same supercharges so any operator that preserves Q minus and Q bar plus this is just a twisted carrying condition in two dimensions, but this thing of course is covalent in two dimensions it breaks SO3 covalence SO2 in 3D so what you have here are lines that indeed are wrapping the S1. So of course the basic example is precisely those super symmetric Wilson loops that we are familiar with are the Wilson loops where you also integrate some scalar across the same one cycle. Now given any such twisted carry operator or RVPS Wilson line I can fuse them, I keep them parallel they are still mutually BPS, I fuse them and as usual for carry rings it is the fusion is non-singular in Q-comology so I have a ring structure like that for Wilson loops in pure transformant theory with an equal to super symmetry that's equivalent to the usual Verlin algebra but in general is some modification of the Verlin algebra that depends on various flavors, parameters and so on that depends on the matter fields so this kind of algebra I've been studying in a number of papers and then you can like in any such setup in 2D and here in 3D you can study the ring structure by just computing three-point functions on the adjunas 0s on the two sphere here so I have, I can wrap line on S1 at 3 points on S2 and I compute NIGK like this and of course here I never put insertion points of any of my lines because the theory is topological along the rim and surface so I can put them anywhere I want and then in particular the 3D twisted carry meaning the algebra Wilson lines or any line operator is governed by the twisted super potential in two dimensions okay so that's the main player then in this game we've seen also in Itamar's talk before lunch the twisted super potential in an equal to theory like any of those twisted super potential in 2D in equal to two theories and so that's in fact the classical piece which comes from 3D transformant terms so it's KU squared it's a master essentially because the transformant term lift the classical Coulomb branch in three dimension and then there is a one loop piece which is just a dialogue for each kind of multiplet so in our example we call that I wanted to study this example of just a single vector with a kind of charge one then I have this explicit formula super potential I have one dialogue so here U U or the exponential 2 which is X will always be gauge parameters that I will integrate over at the end of the day while the news or Y will be fugacity or flavor parameters okay and here note that for as a side remark I also introduced some new R which is also fugacity for the r symmetry that's just the fact that in fact even R2 times S1 there is a choice of spin structure that we can have on S1 and we can keep track of that as well here I would like to make a comment about the parity anomaly just to get on the same page about what we mean because this of course in something is well known but we actually had to look at that a bit more carefully so this is a well known fact just in 3D in flat space that we have a parity anomaly for any 3D so parity anomaly so parity is just an inversion of one coordinate in 3D and the parity anomaly is just a mixed gauge parity anomaly or you can also have gravity parity anomaly so it's a mixed anomaly meaning that we cannot preserve one in the quantum theory we cannot preserve both parity and gauge invariance so of course because at the end of the day I want to integrate over gauge field and turn I want to preserve gauge invariance but then I will necessarily break parity for the effective action let's say of a single a free direct fermions so that's the usual regularization of a direct fermion the absolute value is an ambiguous as a staff function regularization let's say of your direct operator but then you also need to regularize the signs of the deline values and that's one way to do it that was done by the gentleman in the 80s is to just use the 18 variant which for many purpose behaves like a so called level one of transformant terms but it's important for finer checks to realize that this is not a transformant term it's just a gauge invariant but non-local functional of the gauge fields so but this regularization is slightly ambiguous in the sense that I can still add all local terms in 3D that would affect the face of the partial function but the only local terms I can add in three dimensions are transformant terms and as we know transformant terms background gauge field have a level the coupling k which is quantized so I can always add a transformant term here and that will just shift the face of the partial function by a particular phase but which is integer so it's all captured by this so called effective transformant level sometimes called effective transformant level SCAPA which has a piece which is let's say intrinsic to our quantization minus one half or minus q squared over two for a kind of charge q sorry plus an integer transformant level and when we integrate out then if we integrate out the field by giving some real mass positive or negative we can shift this effective transformant level by half integers so this is all well known but I wanted to review that because in a lot of the locational literature it turns out that the regularization and volume determinants that people used were actually breaking gauge invariance at question and point while preserving parity where you should really do the other way around so going back to my example I have this dialogue and the claim is that this dialogue that's just contribution to the two super potential from a single carol is precisely in this what we will call u1 minus one half quantization meaning that we have effective transformant level minus one half and then for instance if I just integrate out this guy by giving a real mass which is large in positive or large in negative if it's positive then I end up with zero transformant levels and if it's negative I end up with a minus one transformant level but the point is in the IR the transformant level are properly quantized okay so now given the two super potential it would be interesting later on to define this explanation function where I just take e to the 2 pi i dw du where u is any parameters and here for the gauge parameter that we will call the gauge flux operators and then the vacuum equation are just setting this thing to one as we've seen in a talk earlier so the vacuum equation in the 2D theory so the 3D model is just setting this guy to one and there are some extra conditions for Nabil and gauge groups that the solution to this equation should be acted on freely by the VIG group so the solution to these equations are called the Bethe Vagua because of the work of Nekrash Shashvili this kind of system are in correspondence with so-called integrable systems but for us it's just a name those are two-dimensional Vagua and the two-dimensional Vagua we call Bethe Vagua to distinguish them from let's say 3D vacuum concept and we also assume that the Vagua are isolated which would be the case if we have enough flavor symmetries let's say well here it's precisely so the 2D super potential is only defined up to some shift by linear by linear piece in u times an integer but that's why we take the exponential here so that this is well defined in fact this would become rational those are rational functions of x where x is e to the 2 pi iu so those are like polynomial functions really and so you just solve a bunch of polynomial equations so those are literally some roots those Bethe Vagua so again in our example I have a single equation because I have single gauge field in general there are as many equations as the rank of the gauge group so that's the explicit equation so there is a single solution and the building blocks are these particular flux operator and here I just write it down but the point is of course in any Lagrangian theory you can build this thing by adding the contribution from matter fields, from transformant terms, from f i terms and so on so now how do we compute the observables in this a to the theory so we want to compute the expression values of line let's say on a 2 times s1 or more generally on the Riemann surface well it's a 2D topological field so the expression values of operators on Riemann surface in any TFT we can just bring the problem to having operators on the torus and then on the torus it's just a trace so what we do is that let's say I have a 2 Riemann surface here I can always shrink a handle all the way down to a point and because it's a TFT that doesn't do anything except that in this extreme limit the handle becomes a local operator on the torus and then I can just do the local operator on the torus and in the case of in this gauge series it was computed this H explicitly for those gauge series by Necrophyshatechvili a few years ago it's given in term of the so-called effective dilaton which I won't get into but for example it's just this simple function for a single-carol multiplet and then the hn of the twisted superpotential the determinant of the hn and then in general in the general TFT we can compute with this formula for expression values where we have a trace over the Hilbert space a genus one where insertion of this angle operators to go to any genus some h to the power g minus one so in our case this trace so the the space of state are in one-to-one correspondence with the state of with the two-dimensional vacuum but the usual state operator math and we can diagonalize all these operators and so we just have a sum like in a brand stock so for the fact that we have a Riemann surface we just insert this angle operator to power g minus one and then we just insert our line and so our various lines will be various functions of this so-called 2D Coulomb branch parameter this use but now we evaluate all these operators on the vacua on the solution to the beta equations so that's it that's my formula for line operators in the in 3D so in the absence of line of course this is a two-set index we've seen that formula before so in particular a genus one that's just a width and index and the partial function on t3 which is regularized for matter fields by turning on these various fugacity is new but then it's just a number that's just a number of beta vacua and of course in general we can write this guy like this where the 2-set index is just the insertion of h to the power g on s2 times s1 so that's the first example of this geometry changing line operator that was discussed in 2014 okay so now what I want to do is start fibering my s1 over the Riemann surface to get much more general topology so that's the that's what happens in safer geometry safer geometry is really topologically just a safer manifold with a safer vibration and a safer vibration is a particular vibration of a two-dimensional Riemann surface with orbital points I can have n orbital points and so this is a smooth Riemann surface topologically but near each orbital points it's locally I mean the geometry is really conical so there is some zq orbital singularity now we can consider line bundles of our Riemann surface allomorphic line bundles as usual they're determined by their version class or degree but now because we have an orbital we also need to say what happens over the orbital points and that's another bunch of integers which are valued in zq so each orbital points has some parameter q some zq orbital and so there is some zq valued integers what we will call the fractional fluxes okay so I can describe line bundles like this in particular I can consider the circular line bundle of degree d and those parameter n equal p but when I ask that all the qnp qi and pi at each orbital point are mutually prime then I can consider the associated circle bundles which is three-dimensional so the full line bundle is four-dimensional the circle bundle is three-dimensional and that's precisely my safer manifold and the condition that qnp are mutually prime is so that this is a smooth three manifold are presented in that way that's the standard sefered invariant in terms of the degree the genus of the base and the exceptional fibers qp, qp fibers and of them so that's the trivial example of a safer vibration the product space then I can start obtaining those various exceptional fibers by surgery all we're doing is cutting a disk then we have a disk times s1 so a solid torus that we glue back to itself with some sl2z twist so if I do that in this way with this s2z matrix I could just get a land space or a tree sphere for p equal 1 and more generally some land space and if we add more and more of those we get general safer manifolds so a nice example again are things like just variation on the tree sphere I can question the tree sphere by some SO4 if it's just a zp cushion it's just a bunch of land space and all the land spaces are safer manifold with only two exceptional fibers then if I start having let's say three exceptional fibers that's one of the example is this Poincare homology sphere that we saw before one comment maybe more for the expert about all these supersymmetric backgrounds so what we have is that on any safer manifold like this we have supersymmetric backgrounds and in fact most of the time we have essentially unique backgrounds the exception is topologically if you take the topology of a tree sphere or more generally a land space those are the only safer manifold that admits an infinity of different safer vibrations and in that case that's related to the notion of squashing that we are familiar with on the tree sphere in particular so in general those supersymmetric backgrounds have this squashing parameter b and in our formalism we will naturally consider only the case where b squared is a rational number q1 and q2 which you can take positive so as an example let's take the tree sphere that's the usual supersymmetry algebra of the squash tree sphere where there is a keying vector here that is where you see the tree sphere as a t2 fiber over an interval and then you have a keying vector in an arbitrary dimension arbitrary direction in a torus so in general the orbits do not close but when b squared is rational then the orbit close and we really have a safer vibration so we really have a vibration along the keying vector that appears in the supersymmetry algebra so that's why we consider this kind of setup where the rational squashing then the vibration over a two sphere with two orbital points so-called spindle okay so coming back to gate theory no in general it's just yeah it's not allowed by supersymmetry but also there is no most safer manifold are unique there is only a so in this case on the lens space what happens is that there is an infinite number of safer vibrations of squashing in general what happens is that indeed you know if you start from s2 times s1 let's say and you can start having a rotation along the s2 because there is an isometry but if and that's the case also for if I have only two orbital points I see an isometry but if I have more orbital points or if I have higher genus there is no isometry so there is really nothing to deform supersymmetry algebra so there is no squashing yes so that's the only one we see to action in fact with yeah okay so given any background well any u1 gauge field I can try to turn on some supersymmetric background flux or flux on the Riemann surface times s1 here and supersymmetrically it doesn't matter where I put the field again because it's a tft so I can the flux sorry so I can concentrate the flux to a point and that's my notion of flux operator that we saw before and it's just given in terms of the exponentiated first derivative of the two super potential with respect to flavor parameters that the flavor flux and with respect to gauge parameter that's a gauge flux and by definition the gauge flux operators are trivial because they're trivial on every two dimensional vacuum so similarly I can view safer geometries as coming entirely from a few defect line operators on s2 like we said and all we want to do is understand these building blocks all to introduce this degree and more generally those exceptional fibers so how to do this kind of surgery prescription at the level of the quantum field theory so what we will describe is these safer fibering operators where you see there is this so called ordinary fibering operator that was discussed by Brian that we did a year ago a year and a half ago and more generally there is this GQP operators that insert these QP fibers and I should say that in general those QP fibering operator depends also on various fluxes which are localized at the herbifold points in the 2D language which are those fractional fluxes yes so it's the same well it's the same for the vibration for instance I will explain I guess yes but only the integer value not on the profile I'm just saying I have one unit of flux let's say it doesn't matter if I put this profile or this profile is the same thing but I have quantized fluxes so a few comments let me skip some of it but I'm just saying here that everything is properly gauging variants and so on as it should be so there are various large gates transformation and generic topology that you can do and then there are various concept checks on general ground that those operators should satisfy so then to take the first example of a fibering operator is this ordinary fibering operator that changes the degree of the vibration this is just given in terms of potential literally the exponential twist is exponential minus some things so that this whole thing is some derivative so that this whole thing is well defined and this is just a flux operator again for KK momentum because fibering DS1 is just introducing some fortune class for the KK momentum along the third direction and of course that's explicit formula here again in our examples for a single carol multiplet that's a very explicit formula in term of exponential dialogue but this thing if you write down it's not obvious because it's a dialogue, a log and so on so there are branch cuts but in fact it's all such in a way that this whole thing is meromorphic so those are meromorphic function of U and U and whatever parameters okay so similarly we can write down the QP fibering operator I will not explain the derivation it's in the paper it's a bit technical but it goes as follows so for any QP let me choose an integer T which is the modular inverse of P so TP equal 1 mod Q and then you can write down building blocks again for all the elements of my UV free get theory so for a free carol multiplet let's say for zero charge there is this particular function which generalize this guy so it's again a particular dialogue and so on but it's all meromorphic at the end of the day and you see that there is a sum of this parameter L here all the way to Q so for Q equal 1 we recover this guy there is the classical action contribution for each QP fibering operator it's just usual E to the U squared thing with the chance amount level K here and of course it depends on the Q and P and some fractional flux and some particular sign that I won't get into but it's all very explicit and similarly there is a contribution from vector multiplets in particular for W boson so you can build up this thing and here all these things depended so far on N and M let's say here which are gauge flux and fractional flux on my orbifold but now when I want to do the pass integral I should integrate some of our old gauge fluxes so that's what we're going to do in the last step so there is this sum of our old so those are essentially Z Q value so for a U1 theory that just Z Q value flux it's generally some Z Q reduction of the magnetic charge lattice of the gauge group so I sum over all these fractional flux at every orbifold point on my 2d base and that's my full severed fiber operator for my the QP fiber and so I just multiply all the QP fibering operator like we did like we did here I multiply all the QP fibering operator and I get the full well defined severed fibering operator that introduce my complicated severed vibration so that's the formula then for the super symmetric partial functions again it's just an insertion of a line on S2 times S1 the twisted index so I just insert my line which is this guy with all the bending blocks so that I just insert this thing which is again combative explicit and I add this EH the G-1 also for the genus of the base and then that's our main formula that I wrote here and the point is that it's again a sum of orbite vacua I should mention so there are many checks of this formula I will mention just a few one thing that is partly work in progress but you can check that this whole thing also when you specialize to chance amount theory so unequal to chance amount theory that's what produces well known surgery formulas from the work of Witten and so on 30 years ago so in a way this is just a generalization of the usual surgery prescription for interpreting partial function of chance amount theory but now with matter fields and so on and supersymmetry so let me make one more check of this formula again to infer duality so we say in our example we have a U1 theory with a single carol it's dual to free theory so I can compute this various operators in one theory and they should match on shell so on the vacua to the dual theory so here for this theory there is a single beta vacuum denoted by UAT we saw the XPTX equation earlier in this gate theory here that's the simple answer so there is that's the particular function that we saw here for a carol multiplet there is a contribution because of our choice of quantization there is contribution from a chance amount level this is contribution from the FI term and this is a pure phase that is important but that has to do with the quantization of the gegeno and then there is a sum of our fractional fluxes from 0 to Q minus 1 and in the dual theory just a free carol so it's just a single function so with the shift has to do with the R charge here it's a single function that this this JFI QP and so you have to plug in U equal UAT here and see that it's equal to that for any parameter zeta and any background flux and so on and that's not easy to prove I mean we didn't prove that except in some special cases but you can plug it in Mathematica take in any arbitrary value and that's certainly old so that's very interesting there should be an antique proof more generally for any gate theories which have infra-dual for example sabre-lar dualities you can then check that all the partial functions of dual pairs match exactly you just need to show that the endanguin operator and the sephirth-fibring operator match on shell in the dual theory so there is a vacuum there is a dual beta vacuum in the dual theory and they should match for each vacuum and then by construction all the partial functions and all correlators and so on match so we check that again numerically those identities, this one we proved actually in a paper by Weizian two years ago this one we checked it numerically and well it works very nicely there should be some proof maybe using number theoretic methods the point is that for instance for sabre-lar dualities all the beta vacuum are really roots of polynomials and those are miraculous identities which tells you that when you plug in roots of certain polynomials in those crazy functions they all can sell and get zero okay so now we'll use this formula in a few examples and just compare to more familiar integrals so for the squash-fibre so that's the usual formula for the squash-fibre there is again classical piece vector-multipede piece matter piece I just write it explicitly that's all well known there are a few details about the phases and so on that we discussed in our paper let's do with the quantition of the genomes for the experts who might be interested in the rational squashing limit then we can compare to our results so again the squashed s3 s3b for b squared equal q1 equal over q2 sorry is just this severed vibration with two with two exceptional fiber q1p1 q2p2 such that q1p2 q2p1 equal 1 and then there is a match of the parameters here the sigma at here in my notation to my parameter u and u like this and so you can actually rewrite even the integrant here in term of our gqp at zero flux so that's one way to write it and then you can essentially do the integral or close the contour and show that this is equivalent to the sum of our beta-vaqua like this we are now the full severed fiber operator that I insert here in the suset index formula and we just this guys with the flux for the gauge field and summed over all the fractional fluxes at the two or before points on the sphere so that's an explicit evaluation formula for the three sphere partial function when b squared is rational so interestingly that in a very special case of a u1 theory with charge one matter fields that was derived in a different language by this gentleman a few years ago in the context of they were looking at complex gentleman theory where these things appear as state integrals so called state integrals so they found this thing and of course here we generalized we need a gauge group this kind of explicit evaluation formula similarly you can consider the twist set index and in particular the refined twist set index so as the same twist set index with this epsilon parameter which is some kind of spinning of the two sphere so that's again the explicit formula in term of gq residue we don't need to look at the details but yeah we don't need to look at the details in particular for epsilon equals zero that usual twist set index so that's the one of the terminal for instance for a carol multiplet let me just point out that this is the form maybe you've seen this formula with square roots and that's the correct form where you treat the parity anomaly correctly so again when epsilon becomes rational this is a safer vibration so it's s2 with two orbital points with the same zq anisotropy at the north pole and south pole and then again we have the explicit evaluation formula where the safer operator here for the rotating twist set index is this guy g epsilon similarly we can consider any land space lpq with squashing so the easiest way to discuss those is in term of the so-called holomorphic blocks of demoft and pasquetti they remarked from a bunch of example and then you can generalize to any lpq that you can factorize this lpq partial function in term of this so-called holomorphic blocks where you have a sum over alpha which is precisely a sum over the betae vacua so the same betae vacua that appear and the blocks are morally speaking they're just a partial function on a solid torus or on a disk times s1 and of course any land space has some eager splitting of genus one precisely where you glue two solid torus with sl2z and you get a lpq land space so morally speaking then those holomorphic blocks should be somewhat equivalent to our fibering operators more precisely those holomorphic blocks depend on these tor parameters and they are singular in the limit that we want where tor becomes rational which is the case that would correspond to our fibering operators but you can remove this singularity with the whole argument in our paper but essentially there is a limit where you take the holomorphic block with some sl2z transformation of the parameter divided by the original holomorphic block and this has a smooth limit which is just our fibering operator so that's another derivation so to speak of our fibering operators so to conclude let me just give these examples I promised so all the known examples and all the examples are reviewed but more importantly where I have only two orbital points so I have a sphere with north pole, south pole, something happens and that's it but now we can have any orbital point as you said and get any separate topology any separate manifold with very complicated topology in general so one with relatively simple topology but this very interesting case is this Poincare homology sphere which is again a separate manifold with three separate vibrations type 2, minus 1, 3, 1 and 5, 1 in my notation here so it has anisotropy Z2, Z3 and Z5 by the way I denoted S3 of E8 because of course the group that we quotient by here is the corresponding to E8 in the eddy classification of subgroup of SU2 and in general of course I can have any eddy quotient here so here I have again this formula for this Poincare homology sphere that's just my separate fibrinic operator adjusting the product of the 2, 1, 3, 1 and 5, 1 QB fibrinic operators so that's my answer so let me give you a small consider checks of that, maybe you don't believe me so I mean that's a almost trivial check but it's still cute so what if I consider a free carol multiplet so one reason why I like this Poincare homology sphere is that it's actually the most of the separate manifold I can define super conformal field theories on it meaning that there is no quantization direct quantization on the arc charges so I can take the super conformal arc charge in the UV and then it will be like we do on the three sphere and that should compute some kind of observables in any super conformal field theory so the simplest super conformal field theory is a free carol multiplet so that's my formula then for the free carol multiplet let's me consider the real part minus the real part of the log of the Poincare function for S3 so as a generalization of F of S3 here it's F of S3 Poincare homology sphere so that's all my ugly function GQP and you can plug that that's the particular thing that's the number, this number and you want to compare to something but it's a free theory so you can just go and compute that one loop so I didn't do it but thankfully those gentlemen from Argentina did that credit computation they have some typos but when you check their typos they actually have the correct zeta functions regularization of a fermion a real scalar and a fermion that they compute it on this space on any spherical manifold really so a carol multiplet is just two times a real scalar plus a direct fermion so I just add those and that's precisely this and of course you can generalize this kind of check for free theories, it works very nicely so to conclude we computed partial functions on any safer manifold it always takes a form of a sum of a two-dimensional VACWA, the so-called BETA VACWA where I insert this geometry changing line operators which is this safer vibration times the undone operator so let me just conclude with some more speculative comments for this audience, maybe a challenge to you so I did everything by inserting some lines on s2 times s1 because from the point of view of the A model it's more natural but from the point of view of both mathematicians studying three manifolds that would typically take s3 and then do some surgery on the nuts on s3 and also from the point of view of ADS-CFT where s3 is really the vacuum in the ADS-4 vacuum then it might be more natural to just start from s3 and insert line but of course this f operator are invertibles to speak so I can just write the same formula by being on s3 and saying I insert this anti-fabric line operator to get back to s2 times s1 so what this f1 this f-1 is doing is really creating some magnetic charge ADS-4 black hole, the kind we've seen and similarly if we insert this G epsilon we get the black hole to rotate like in the last slide of Keras talk yesterday so of course that just morally speaking that should be true but it would be very interesting to make this point of view more precise in particular well really even from real theory this line operator we described them as local operator in a 2D theory that's for simplicity but we know they are really line so even in the 3D get theory there should be some more UV definition of it as boundary conditions and so on for all the fields and similarly this line should extend to some kind of shrink defect in ADS-CFT that is concentrating the geometry so that you can go from a DS-4 to black hole and so on so that's the challenge to you guys because that's it for this video thank you so much for watching see you next time