 Marcos Krushigno and Darmesh chain and most of these people are here at this workshop so we can talk more with them and So to give some very general background I'm going to be studying partition functions for supersymmetric quantum field theories in various dimensions And these have very interesting physical and mathematical properties. I Think the most striking of these are dualities. Many of them exhibit non-trivial dualities such as 3D mirror symmetry Cyberglick dualities s-duality for 4D n equals 2 theories and then of course there are many interesting Examples of holographic or ADS CFT dualities involving these theories So many of them and and these these dualities can be understood in terms of decoupling limits of string or m theory on Some system of brains or some singular geometries And also from this perspective we can understand many relationships between Quantum field theories in different dimensions. So even if you're only interested in say four dimensions It's often very useful to consider theories in higher or lower dimensions as well however, I think the most interesting behavior occurs in theories, which are strongly coupled and These are very interesting, but they're also very difficult to study directly So we need some tools using super and supersymmetry gives us many tools So one of these which I'll be talking about today are supersymmetric partition functions So to give some Again very general background These are computed by supersymmetric localization. We localize to take a particular example of the S3 partition function We localize this by This infinite-dimensional path or go to some finite-dimensional matrix model with a classical and one loop contribution and this is Various interesting applications. For example, it computes the free energy which is monotonically decreasing under our G flow It admits a continuous family of squashing deformations with a metric parameterized by variable B So called squashed fewer S3 be and this will also play a role in the second part of this dog and finally it leads to because we have these exact results in terms of a More easily computable matrix model. We can often solve these at large and and perform precision tests of holography So they've been in addition to S3 many other examples that have been studied in the literature So I've listed a few here. So there've been spheres in various dimensions from one to five There have been supersymmetric indices, which is a sphere cross S1 There have been land spaces and land space indices and more recently topological indices Which Jim talked about in his talk on Sigma G times the Taurus of various dimensions and many others So there've been many nice results found on all of these manifolds But I think there's some shortcoming in this in this story, which is that essentially all of these computations are independent and in addition all There are many more backgrounds in principle that can be computed many more supersymmetric observables than have so far been computed and And these two points are related because each computation is sort of we have to start from scratch But there there should be an easier way There should be a way to make our life easier using the locality of quantum field theory namely we should be able to Take some complicated manifold and chop it up into simpler pieces associate some observable to each of these pieces and glue them together in arbitrarily complicated ways to build complicated Topologies of complicated geometries from some simple set of building blocks So the idea will be now to construct all of these many of these spaces and new spaces In terms of some simple building blocks and really we should think of these building blocks as the basic Observables and these partition functions is some derived object from those So the approach I'm going to take in this talk is is something that I'll call the higher-dimensional a model And the idea is basically to take these partition functions, which I should stress are not topological For example, the S3B partition function depends on this geometric squashing parameter But nevertheless they often can be understood in terms of some underlying two-dimensional topological quantum field theory So the idea is we take some higher-dimensional theory compactify D minus two dimensions on some compact compact manifold And this can be equivalently described by some two-dimensional theory with infinitely many fields coming from the various kk modes And then if we can take this 2d theory and perform a full topological twist the so-called a twist and obtain a partition function on a compact manifold sigma g a Riemann surface and then this Applying this to this theory here will will effectively give us the sigma g times md minus 2 partition function for the higher-dimensional theory So this gives us a way to build partition functions at higher dimensions and more over it's it's often possible with in this perspective to insert Geometry changing operators. So these are operators in the tqft which have the effect of changing the way Changing this from a trivial product to some kind of non-trivial vibration And in this way we can often obtain more more complicated and interesting D-dimensional manifolds That are all described in terms of some underlying 2d tqft So the outline for the rest of the talk. So I'll start by giving some general background on the 2d on the higher-dimensional a model I'll start by describing some 2d background that will then uplift the higher dimensions I'll briefly review the 3d and 4d version of this of this story and give a few applications and Then in the second part of the talk, I'll describe a computation in 5d. So I'll be slowly moving up in dimension on s3 times sigma g I'll give derive this partition function I'll talk a little bit about the large and limit and some holographic interpretation and then I'll talk about applications to six-dimensional theories So let's start in two dimensions So if we have a 2d theory with n equals 2 comma 2 supersymmetry so 2 left moving and 2 right moving supercharges This has some special short multiplets the chiral and twisted chiral superfields whose bottom components obey these shorting conditions And these are very useful for various purposes. For example, we can Consider the OPE of these operators and these are closed up to some q-exact terms And finite and and defined for us a finite-dimensional ring called the chiral or twisted chiral ring Moreover for for Appropriate 2d theories with a discrete set of vacua We can we can consider these vacua in a one-to-one correspondence with these chiral operators by considering the partition function On a disk which prepare us inserting this operator on this disk prepares a vacuum state So we get a state operator correspondence between the the supersymmetric vacua and these chiral or twisted chiral operators another Observable we can associate to these theories are is the is through the topological twist So this is a very old idea where we define a modified action of the of the in this case u1 rotation symmetry by mixing with the u1 r symmetry and The stress energy tensor becomes q-exact. So this effectively has Is Turning on a non-zero flux or magnetic flux for the r symmetry on these on these spaces and using this this way of coupling the theories to curvature We can define Partition functions on general compact reman surfaces sigma g So I'm going to focus on the the a model which is a particular choice of of this twist This is going to be what's naturally going to uplift the higher dimensions And in the a model the natural operators are the twisted chiral operators as opposed to the chiral operators and so the observables in this a model are In general correlation functions of twisted chiral operators on a reman surface on a higher genus reman surface and But this this correlation function just using the topological invariance We can argue that it has to have a very simple form namely We can first of all deform this higher genus reman surface to a torus with any additional handles Essentially glued at a point and so implemented by some local operator, which I'll call h the handle gluing operator Now that we have a partition function on torus we can interpret it as a trace over the space of acura of the theory by the state operator correspondence And we and we can now Consider the this trace because these operators all commute with each other because we can just move them past each other on the torus We can pass to a basis where they are all simultaneously diagonalized and then we just have a sum of some contribution from these various operators including the handle gluing operator over some set of Some basis of the set of acura So this part correlation function has this very simple form in general which we'll see throughout this talk And for example, if we have a land out Ginsberg theory Then this this handle gluing operator is just given by the determinant of the Hessian of the super potential Okay, so that was the 2d story, but now we want to lift this up to higher dimensions And the idea is that as I described in the introduction We're going to compactify Some of the extra dimensions on some compact manifold and leave two non-compact directions on which we have an effective n equals to come to Supersymmetric theory So some examples we could take a 3d n equals 2 theory on r2 times s1 4d n equals 1 theory and r2 times the torus or 5d n equals 1 theory on r2 times an appropriate 3-matter fold and I'll give an example in the second part of the talk and Necrosalvin-Shatashvili Studied some of these examples and what they showed is that for many interesting gauge theories in higher dimensions when we place them on these on Geometries and consider the effective 2d theory the vacuum equations for this 2d theory coincide with the beta equations for certain integral systems and so in this way we get a nice correspondence between higher-dimensional gauge theories and Intercable systems called the gauge beta correspondence In addition to the vacuum the the vacuum of the theory we can also study the twisted chiral ring and Now while in two dimensions the twisted chiral operators were local operators in these higher-dimensional setups It turns out that to preserve supersymmetry these operators have to wrap the extra dimensions So for example if we consider the 3d theory on r2 times s1 These are line operators for example Wilson-Lipp operators Which sit at a point on the r2 and wind the extra circle and so this twisted chiral ring That we compute in 2d is is really telling us about the fusion algebra of these line operators or in higher-dimensions surface operators and Finally the other natural thing we can up there from two dimensions is this a twisted partition function So if we take these effective 2d theories compute their a twisted partition function on sigma g This gives us the partition function on md minus 2 times sigma g of the higher-dimensional theory Okay So in more detail the way that this computation is done is to use the effect of an effective low energy description of these 2d theories And you can think of this as being very analogous to the low energy solution of 40 n equals 2 theories by cyberg and witton and In particular this law energy this law energy theory is controlled by a single function called the twisted super potential Which is a function of the twisted chiral field strength So this is a twisted chiral multiplet constructed out of the vector field and its bottom component is the complex scalar in the vector multiply and So the action can be written up to some q exact terms in terms of entirely in terms of this twisted super potential So once we know this twisted super potential We can now compute for example the vacuum of the theory are given by the critical points of this modulo some branch cut ambiguities and This equation this vacuum equation as I said is what coincides with the beta equation for many integral systems The other thing we can compute from this w is the twisted chiral ring Which is Generated by polynomials and sigma modulo the relations generated by these vacuum equations And as I said before this is tells us about the algebra of surface operators So explicitly this twisted super potential is computed as follows So we can start with the 2d theory and A single massive chiral multiplet at a generic point on the Coulomb branch a charge chiral multiplet becomes massive and can be integrated out And generates this twisted super potential, which is exact at one loop So this tells us how to build the super potential in 2d to get the higher dimensional versions All we need to do is sum over the contributions from the kk modes So for example in 3d we have this sum over kk modes, which have an effective mass and over r and Performing this infinite sum we find a dialog rhythm and Similarly in a 40 theory compactified on the tourists. We have now a double sum. Yes Yeah, so this sum is is the virgin so it needs some regularization And there's some ambiguity in the regularization which amounts to a choice of churn simons term that you That you need to add to regularize the current multiple So this here I've used a specific choice of churn simons term, which is Called the you on one half quantization because there's a single fermion in the current multiple it so the churn simons term has As quantizing as a half integer But there's other choices you can make here and in 4d There's an essentially unique regularization Which gives the elliptic dialog of them which depends now on the complex structure tau of the tourists And it's given by an integral of the log of a Jacobi theta function So from these basic building blocks for the churn multiplets in the theory We can build up the gender twisted super potential for a general 3d or 4d gauge theory So the expression is given here. So here I've decomposed this gauge field sigma into Components in the for the dynamical gauge field and for the background flavor symmetry gauge fields Which effectively act as masses in the theory? and So as a function as we sum over the representations of all the churn multiplets and of these functions above and in addition We can sometimes have some classical contribution. So for example in 3d the churn simons term Gives us this quadratic classical contribution This gives us the twisted super potential w So as I said before we can use this to compute to find the vacu of the theory or to construct a twisted chiral ring But let me talk about the partition functions these higher-dimensional a twist partition functions so we can consider so these were considered by these people and they We they're given by this expression So this is the same form that I argued earlier just general and general grounds that you'd based on this being a 2d tqft So it's given by a sum over a set of vacua. So this is the vacuum equation I wrote earlier and At each of these vacua we evaluate these two functions. So this is an a function here called the flux operator So this inserts one unit of magnetic flux for a background flavor symmetry and This is the handle going operator and you can see it's again given by the determinant of the Hessian of w With an additional piece called the effective dilaton Which could be computed explicitly and This this formula can be derived just by studying this this low-energy effective theory with this This action this gives us the partition function on the space of sigma g times a circle or a torus And we can go a little further So we can consider Introducing flux for the u1 kk symmetry. So by this u1 kk symmetry, I mean the following So if we have a 3d or 4d theory on r2 times a torus Thinking of this as a 2d theory from it from the 2d point of view This has some privileged global symmetries which are translations along the various s1 factors So these are just global symmetries from the 2d point of view And so we can insert fluxes for them just as we did for any global symmetry by this formula And in this case the effect of these fluxes is to is that the s1 or the torus is now non-trivialy fibered over the Riemann surface. So in this way we get non-trivial S1 bundles over a Riemann surface and so more general topologies and it's implemented by this operator called the Fibering operator Which can be written explicitly in terms of the twisted super potential and this for this Fibering operator gives us a new perspective on many known partition functions. For example s3 is a Is a s1 fiber bundle over s2? So it's a degree one of s1 bundle over the genus zero Riemann surface So from this formula here I have a factor of h to the minus one and then I have one insertion of this Fibering operator F And so this gives us the formula for the s3 partition function as a sum over as an observable in some 2d tqft. So this gives a new formula for this s3 partition function But of course we can now consider arbitrary Fibrations and so this gives us many new partition functions and we can actually generalize this much further to arbitrary cipher manifolds Which are s1 bundles over an orbifold Riemann surface, which is a much larger class of manifolds than the ones I talked about here, and I will refer you to Cyril's talk later in the week For more about that. So let me just mention briefly a few applications so one natural thing to To use these partition functions for is checking or studying supersymmetric dualities And the way these dualities manifest themselves at the level of this higher-dimensional a model is That we should be able to find we can construct a set of vacua for these two theories two theories Which are believed to be dual and these vacuers should be in one-to-one correspondence So we can find an isomorphism between them which I'll call the duality map And they should have the property that if we evaluate the twisted superpotential at dual vacua the results should agree And if we check that then that automatically implies from the formulas on this page that all of these partition functions that we compute Here will automatically agree because they're all built out of this the w's for these two theories So rather than check all these these partition functions independently We just need to check this this finite set of relations and and this gives The same information Let me mention a one example in four dimensions So we can take 40 qcd n equals one qcd with gauge group s un And from the formulas I wrote earlier we can write down the vacuum equation for this theory So it's given by this ratio of a lip of jacopee theta functions as a function of the gauge variables You a and the flavor parameters for the s you and f symmetry new J We also have to impose that the trace vanishes And so these equations are Can be are difficult to solve but we can at least count the solutions and we find that the solutions The number of solutions is given by this formula and f minus two choose nc minus one And this is a few nice properties so we can think of this as the wooden index for this theory and First of all you can see immediately that it's invariant under cyber duality Where s u and c is mapped to s u and f minus nc just by a symmetry of the binomial coefficients And it also exhibits supersymmetry breaking so this theory has no supersymmetric vacua one nf is less than nc plus one Which is as we expect so this formula seems to make sense and A much more powerful check of this duality is to construct the duality map between these sets of vacua and We can construct that map and then this identity here Corresponds to this identity I have written in terms of the elliptic dialog rhythms. So this is some non-trivial identity which As far as I know was not known and we check this numerically and it seems to be true And it implies the matching of an infinite set of 40 partition functions for these cyber dual theories Including the supersymmetric index Okay, so another application which I'll briefly mention is to take the large and limit of this partition function So this was first studied by Bernini Hirstov and Zaffaroni They looked at the s2 times s1 or more generally sigma g times s1 partition function in the large end limit and What they argued essentially was that so for finite n we expect a contribution from all of these beta vacua But in the large end limit, there's essentially one vacuum which is dominant And so this sum simplifies to just this simple factorization into a contribution from the fluxes in the hand of gluing operator And so for the AB jam theory on sigma g times s1 they found this explicit result which for the log for the the Entropy or the log of the partition function, which is just linear in these fluxes and They then match this to a class of magnetically charged black hole solutions and n equals 2 gauge supergravity and Found that to leading order and n this precisely agrees with the area of the horizon of these of these dual solutions This is a nice check of ads CFD and of the microscopic Counting of the states in a black hole So this can be extended to more general theories is as Jim talked about in his talk earlier including ads 5 CFT 4 and black holes and black strings and Also, we can consider in three dimensions the holographic dual of a more general set of three manifolds where the s1 is non-trivial II fibered over sigma g and This was work with Chiara todo and she'll talk about this later in the week Okay, so that was the first part of the talk so now I'll move on to up to five dimensions and Talk about a so work in progress on the s3b times sigma g partition function So there are any questions about three and four dimensions. Yeah that what is true? This statement is certainly true for any n This is what was this is just a partition function on sigma g times s1 and was this formula was derived by localization This formula is is not true for finite time So this this the this is only true in the strict large and limited it Even at low orders and then I would expect that there's contributions from more vacuab and so you'll have to Understand the contributions from other terms in this sum So this this is general, but this this factorization is only in the large in-limit. Okay, so You give some brief background on 5d so in 5d we can construct n equals 1 gauge through the grunge ins in terms of a Cubic pre-potential so the classical Lagrangian can be written as this quadratic contribution Which gives rise to a Yang-Mills term and a cubic contribution Which gives rise to a five-dimensional turn simons term And here this parameter gamma is a combination of the theta parameter and the 5d gauge coupling and in addition We have a choice of hyper multiplets and some representation of g So these gauge theories are IR trivial so in that sense they're not interesting but the interesting Use of them is that they can be obtained by a relevant deformation from some non-trivial uvcfts So we can try to use these gauge theories to understand these these uvcfts And in many in some examples, it's believed that the natural uv completion is not in terms of a 5dcft But in terms of a 60cft with an emergent s1 direction And I'll mention some examples later on And so to study these theories we're going to compute an observable The s3b times sigma g partition function where this is the usual Squash sphere background that I mentioned earlier And the strategy as earlier in the talk is going to be to reduce to an effective 2d theory and find this tqft that controls this partition function Okay, so let's start by placing some 5d n equals 1 theory on s3b times r2 and The net this background in s3b preserves four supercharges in 2d and gives us an effective 2 comma 2 theory in on the r2 So for example, we can consider a 5d n equals 1 hyper multiplet and the modes of this hyper multiplet Contribute some long multiplets and short multiplets We expect the long multiplets to drop out so we focus on the short multiplets and these have some twisted masses which we can read off in terms of the Kk momenta on s3 We're given by this formula in terms of the squashing parameter be So as before we just we sum up over all these kk modes and we get some function after regularizing this this sum which all gb which is written most Easily and by considering its exponentiated derivative, which is related to the double sine function Which is given by this infinite product and is The partition function of the of an s3 of a three-dimensional chiro multiplet on s3b So for example, we can explicitly write at b equals one We can write this function in terms of trilog rhythm and dialog rhythm and some cubic piece So this is the the twisted superpotential of a single hyper multiplet Okay, so we can also derive the contribution from an n equals one vector multiplet So this now has a non-trivial contribution in lower dimensions the contribution was trivial and is again given in terms of this function gb by a sum over the roots of the le algebra and Finally, we can have a classical contribution which comes from the Yang-Mills term in the 5v turn assignments term and It's given by this cubic polynomial again in the gauge parameter u So just as before this leads us to conjecture that the twisted superpotential which should control the The low energy theory of this of this 5d theory compactified on s3 is given by this expression However, I should stress that everything we've done so far as perturbative. We've done a one-loop calculation the classical contribution So just to be careful. Let me call this the perturbative twisted superpotential Okay, so let's go forward and ignore the possibility of instantons for a moment And so we everything we did earlier. We just repeat here. So we write the vacuum equations Derived from this twisted superpotential and these give us a product of double sine of double sine functions depending on the representation of the hyper multiplets and this equation which is not so easy to solve has an infinite set of solutions and The partition function should then be given by a sum over these solutions of this flux operators for the flux We insert on sigma g and handle gluing operators as before given by exactly the same formulas in terms of this new twisted superpotential Now it turns out we can equivalently write this partition function in terms of an integral formula Where we have the same ingredients That appear up here, but we're now we're integrating over a contour in this u-plane The Jeffrey Kerr one contour and we also have a sum over fluxes on sigma g for the gauge symmetry and this formula can be argued Just algebraically to be equivalent to this formula. So these two are the same But this also can be derived by an alternative procedure Which is we can start with the 5d theory and instead first reduced on sigma g When we reduce on sigma g we find some effective modes on s3b For each flux sector for each choice of gauge fluxes in sigma g and so this can be interpreted as the partition function of a Direct sum of 3d theories that you obtain by reduction on sigma g and indeed this these contributions here All involve double sine functions and can be recognized as the partition function of a 3d theory So this gives an equivalent perspective perspective on this partition function Okay, but we should be more careful About the possibility that we miss some instant on contributions and indeed One way to attack this is to instead of reducing the theory to two dimensions to first reduce the theory to four dimensions to an effective 40 n equals two theory and If we focus on the case of genus zero So now this is this s3b times s2 can be thought of as a hop fiber vibration over s2 times s2 Where again we insert flux for this kk symmetry on one of the s2 factors and This s2 times s2 partition function was computed by one a at all By equivariant localization and Their result which has contribution what does in fact have contributions from instant times which sit at the fixed points of The u1 times u1 symmetry acting on this s2 times s2 So we find by this method uplifting their results to 5d we find A partition function that looks very similar where this perturbative piece is essentially what we had here but with an additional contribution from instantons and We can again write this in terms of a twisted super potential In this kind of a way, but now there's an additional contribution from instantons So we can write W as this perturbative piece we wrote earlier, but with an additional instant on contribution So somehow this naive kk reduction that we did earlier is missing these instant on contributions So these instant on contributions are difficult to study So for the rest of the talk what I'm going to do is focus on limits some certain simplifying limits where the we expect these Instantons to drop out and where we can get by just using this perturbative twisted super potential that I wrote earlier And we'll find the results seem to be consistent and justify that this this assumption is is correct Okay, so let me first say a little bit about the large end limit So indeed in the large end limit you typically expect the instantons to be suppressed So we should get away with using the perturbative Expression I wrote earlier And so let's consider for concreteness of 5d n equals 1 theory with usp 2n gauge group and nf less than 8 hyper multiplets And an anti-symmetric hyper multiply So we're going to take a similar approach as Benini first up in Zafferoni as I mentioned earlier They argued that in the large end limit There's a single vacuum that is has a dominant contribution to this beta sum And so we try to find this vacuum by taking an ansatz for the the dominant eigenvalue u in terms of some arbitrary Scaling of with n and to the alpha and if we plug this this ansatz into the twisted super potential We find that it is given by a functional of this form and For these two terms to compete with each other We should take alpha to be one half and that tells us that this scales as n to the five halves and in fact this this eigen this functional Is essentially up to some rescaling of various parameters identical to the partition function that appears for s5 is derived by jafferis and pufu And the result we find for the extremal eigenvalue density plugging that in the extremal twisted super potential Is given by this expression and is just proportional to the s5 free energy And this this is an interesting relation and an analogous relation was observed in three dimensions by hossanian zaffroni They found that the s3 free energy at large n is also proportional to the extremal value of the twisted super potential of the 3d theory We find a similar statement in 5d. So now we can take this this extremal eigenvalue configuration And plug it in To the partition function Uh, we're just given by this functional here and the result we find is again can be simply related to the s5 free energy And it's given by the by this expression. So with some proportionality factor Let's see. What is the holographic interpretation of this? So this theory has a holographic description as massive type 2a gravity on a warped product of ads6 with s4 and uh Obev and krashigno considered Gauge gravity Uh minimal gauge to per gravity solutions in the background of a two brain Which interpolate between asymptotically ads6 and the near horizon geometry of ads4 times sigma g So there's some solutions that interpolate between these two These two behaviors and These are the holographic dual of the geometry we're considering on s3 times sigma g And so what what they found was that the asymptotic Ads4 the near horizon ads4 radius can be related using these solutions and by interpolating this these solutions to The asymptotic radius of the ads6 with some proportionality factor and reading off the radii of these Ads4 and ads6 this implies a universal relation Between the s3 partition function of the theory compactified on sigma g and the s5 partition function With exactly the same relation that we found earlier So we've found a field theory Derivation of this result that they found holographically and this was for a particular theory But this result can be generalized to more general quiver gauge theories and we always find the same relation holds Okay, so let me now briefly talk about Some 60 interpretation of some of these computations So one interesting feature of five the gauge theories is that Many of them are expected to have a uv completion in terms of a 60 theory with an emergent s1 direction So for example, the maximal n equals to super Yang-Mills theory Is secretly the 0 comma 2 theory compactified on the circle and similarly this theory that I described earlier in the case of nf equal to 8 Is related to the n equals 0 comma 1 e string theory in six dimensions And in this setup the radius beta of this emergent s1 direction is related to the five the gauge coupling in this way And specifically you can think of the kk modes as being Related to the instanton particles whose whose action is proportional to one over beta So for these theories this partition function that we're computing Is secretly a partition function on s3b times s1 beta times sigma g Um for this 60 theory And so one signature that that this is the correct interpretation is that if we go back to this large end solution Let's consider the case of The nf equals 8 theory you can see that now this term vanishes And there's another term which I haven't written here But which now the competition is between this term and this other term And in that case you find that the the scaling of the twisted superpotential at large end is actually is n cubed Instead of n to the five halves, which is the signature of the 60 behavior So we can now further compactify these 60 theories on sigma g And this Procedure of taking a 61 comma 0 theory on sigma g to obtain to some 40 n equals 1 theories Is by now very well studied in many examples So the setup is that we have some correspondence That this s3b times sigma g partition function Can be circularly thought of as a 60 partition function on s3 times sigma g times s1 And which in turn is equivalent to the partition function of a certain 40 n equals 1 theory on s3b times s1 Which is the computing the supersymmetric index of this 40 theory And the specifically the parameters of the 40 index are related to the parameters of The s3b times sigma g partition function in this way. So the the rotation parameters p and q Are expressed in terms of the radius beta and the squashing parameter b And similarly for the mass the fugacity flavor symmetry fugacities Uh in the index are related to the masses in 5d So you can take some examples the Is very well studied the maximal theory in 5d which the uplifts to the 0 comma 2 theory within some 80 e k Troop g Gives rise in 4d upon compactification to the n equals 1 and n equals 2 class s theories in 4d And the 5d e string theory When compactified on the Riemann surface was recently studied by this group and they found a class of interesting 40 n equals 1 theories That can be derived from this theory Now this tells us that in principle these computations in s3 times sigma g are telling us about the The supersymmetric indices of these theories, but in many cases these theories are non lagrangian And I should emphasize that these 5d theories that I've talked about do have a lagrangian in terms of this gauge theory And so this in principle gives us a way to compute the 40 index of these non lagrangian theories There's an important caveat, which is that In order to do the full computation. We need to understand the instanton corrections Still work in progress So before below i'll consider A couple of special limits where we expect the instanton contributions to To drop out and we can relate this we can interpret this in terms of some 40 index computation Okay, so the first limit i'll consider Is the what i'll call the casimir limit So here we have uh, we take the limit where the radius beta Is uh, which is remember related to the 5d gauge coupling And we take the limit where these go to infinity And in this limit we expect that the 40 index is a dominant contribution only from the vacuum state because we We've taken a very long Euclidean evolution along this s1 And so the it should be expressed in terms of the the casimir energy associated to the vacuum state, which i'll call e4 Now it was argued by these people a few years a few years ago that the This casimir energy can be expressed in terms of an equivariant integration of the anomaly polynomial of a 4d theory So schematically we have this relation that this casimir energy is an integral an equivariant integral of this anomaly polynomial But since these theories have a 60 origin This anomaly polynomial can be derived by integrating a 60 anomaly polynomial Over the reman surface with the appropriate flux is inserted And so we expect this relation to to hold for the 4d casimir energy So we can try to compute the s3b times part times sigma g partition function in this limit And what we find in several examples is that the beta equations drastically simplify. They become essentially linear And uh, we find some some leading behavior, which indeed looks like a casimir energy behavior Proportional to the radius beta. So for example for the n equals 2 theory. This is given by this explicit expression where this is the Dual coxster element and this is the dimension and depends on the various uh flavor symmetry fugacities and fluxes And so we get this this behavior But this factor that we find here is not quite e4, but it's something i'll call e pert And we can write e4 in terms of e pert and an additional contribution, which Is natural to attribute to the instantons that we've so far neglected But actually this incidental contribution has a very simple Uh expression Namely we can attribute this missing piece to uh by integrating this the anomaly polynomial in 60 of some free fields So this this extra piece can be interpreted Can be uh described explicitly in terms of some free fields some free vectors and tensors In in in six dimensions and it precisely the same interpretation was found Uh by these people when studying the 60 theory Or the 60 kasmere energy In the in the limit of the s5 partition function By studying the perturbative limit of the s5 partition function They found that they were missing some pieces which were attributed to exactly an analogous formula for the 60 fields okay, so the last Application that i'll mention is uh that there's this there's a certain limit if we now focus on the n equals 2 The maximal theory with some gauge group g We can find a simplifying limit where we can again understand these instanton contributions or or where they actually drop out So to start let me write the full answer which is again given by this formula Which i can write explicitly in terms of some double sine functions Here i'll mention that this flux for the reman surf that we've inserted on the reman surface here for the flavor symmetry Um Is so this is the flavor symmetry that we have because this has extended uh our symmetry Compared to a generic n equals 1 5d theory So inserting this flux breaks the symmetry to the n equals 1 star theory And in 40 this this gives us the n equals 1 class s theories of of these people Where at at zero flux this gives us the usual n equals 2 class s theories So the so this partition function should compute the 40 index of general class s theories Now we don't know how to do this for general fugacities, but if we take this special limit here Uh for new Then it turns out that the instanton contribution essentially is trivial It at least is completely independent of the the gauge coupling. Oh, sorry of the gauge parameter u And in this limit also these these double sine functions simplify to just a linear piece or an exponential piece And so now the solutions of the beta equations Can be expressed as in terms of the co weight lattice of the gauge group So they're just proportional to the weights to the co weights Which can in turn be identified with the irreducible representations of g So we find that in this limit the s 3 times sigma g partition function is written as a sum over the representations of g Of some factors which are obtained by simplifying this expression in this limit Which are just polynomials in p and q Which are called the quantum dimension of the representation r And this formula which we find in this limit precisely agrees With the corresponding limits of the 40 index of these n equals 2 and n equals 1 theories as derived by these people So this gives us a hint that this partition function that we're computing is indeed computing the 40 index of these theories Okay, so let me conclude So to summarize, uh, I've tried to convince you that it's very useful to to understand these higher dimensional partition functions by reducing to 2d t q f t's Um, these are often controlled by a very small amount of data specifically the twisted super potential Uh, which can be computed explicitly And this gives many interesting new probes of of dualities and this in particular holographic dualities And in the second part of the talk, I told you about a computation in 5d Using this perspective which tell which can in principle teach us a lot about 5d and even 40 and 60 q f t's So as far as some Uh future directions to explore So I think there are many other interesting dualities we can apply this to in particular one example would be the 3d 3d correspondence Um, I didn't say much about these surface operators, but in principle, we know how to we know their algebra Using this this uh a model perspective and so it'd be interesting to apply this to study these operators in more detail An obvious direction that we're working on in the 5d computation is to understand the full instanton corrected partition function Which can help us study these 5d theories and in particular these 4d non Lagrangian theories And finally it would be interesting to study To understand the interpretation of these results in the context of integrable systems And there are many known relations between the 40 index of of certain Of certain n equals 1 theories and integrable systems and perhaps we can understand that in this context And finally to mention a few applications to holography So we can try to find some new holographic rg flows to understand the relation between these partition functions in different dimensions like the example I showed earlier between 5d and 3d to understand the sub leading corrections in 1 over n and finally to Understand holograph holographically in the bulk these What is the way to think about these these local operators which change the geometry? In the field theory, it's it's uh We understand why the partition functions take the simple form that they do But I think in in the bulk this is still mysterious and it would be nice to understand So i'll stop there