 Privolj ste bila, da so poslice, da je vseh modne obišlje v što je, kaj je je zašelite očnevčenje na to droge. V moj priče svoje naredaj, da ustavimo obočkori in kaj je, da je tko čudovati v mojih različna teori. Na mnohem, nekaj ga se zelo izgleda na povrednih nekaj spetite, ki se bolje posledal v kolaboracijenju s Biloh in Frao, tudi z Torino, in sljedeveni Sljeda Sujaja. To je studenča iz Chennai, India. In je bilo na veliko vrste literacije, če bo se prikratilo. To je však in vsih plan, lega je, da imamo prikrati in in zvrša, Zdaj sem tukaj občutnji operatori v n-2-gage teori v 4-dimenčnih, in djeljenje relacije. Zdaj sem predstavljala moje konkluzije. Tukaj, tukaj, nekaj tukaj teori responduje na prezence delovosti, nekaj je izgleda delovosti, delovosti potrebno je vedno, den Profesionalne in in zaop unemployed, separated on submanifold, of the space time over the theory, is defined, and they can be used as useful probes to detect some of the properties of the theory. Perhaps the simplest and almost trivial example of a defect is a point like operator. A zero dimensional defect which is localized in a point. ta vse deljeva v spasih. Zelo pogledajmo vse informacije o tezriji kaj je zelo naprejveni korelacije. Zelo deljeva vse skupaj zelo je objevnjenja operejdu Kaj vse objevnjenje, s katera, kaj je vse objevnjenje vse objevnjenje, zelo je objevnjenje. Surface operators are the two dimensional generalizations of these. And we can even think about a three dimensional generalizations, domain walls, and if we are discussing a theory in four space-time dimensions, this list is exhaustive. In my talk, I will concentrate on these surface operators. V sej sem zelo, da je zelo uživljeno odličen. Jedna je zelo, da modifizirajimo vse prijelje v kvandom pilnosti, zelo vse neselo vse prijeljeno odličenje v kvandom pilnosti. In počak ima je se poživaj, da je da se puneš tudi zvomir, da je to su vse teori. Če, da je to je tudi tudi tudi tudi zo završen, je to vse lahko, nače to, nače to, nače to, nače to, nače to, nače to, nače to, nače to, nače to, nače to, nače to, nače to, nače to, nače to nače to, začušna teori, nešto je nače, da bo obžahan besvetno teori, nače to, nače to. I tako, in tudi se več de секci z vršen, z njegenjem, z vrštih, z zeljah, nekišelj, z njegenjem, z zeljah, do reprezentovati z vrštih. Znamenje mi od tajvaj mod, mod, na kaj je vrštih tega. Nareč sem priklavno koncentrirati na primeri, prikušnjenje, kaj je delovina operatorj. So, surface operators in four dimensional theory are non-local operators, which are supported on a two-dimensional sub-manifold of spacetime. For simplicity, I take the spacetime to be R4, and the two-dimensional sub-manifold, where the defect is localized as a two-dimensional plane. So, this is a picture of R4, this is the R2, where the defect is extended, and this is the transverse plane. These surface operators, as I said before, can be used as probes to detect some of the non-trivial properties of the theories. They sort of thermometers for the quantum field theory. They can be used to define order parameters in the sense that when they are introduced in the path integral, they can provide valuable information on the quantum field theory, like the structure of phases, and even non-pertorbative features that may be not accessible with Wilson lines, or Toft lines, or other kinds of operators. So, they are very, very useful. And as I mentioned before, there are two main ways of describing these defects, in particular the surface operators, namely, as singularities for the gauge field, or in general the fields that you have in the theory, in particular for the gauge field along the surface where the operator is defined. And I will refer to this picture as the picture of monotromid defect, or as a non-trivial coupled 2D4D system, namely a theory in two dimensions, which is coupled in a non-trivial way with a gauge theory in four dimensions. And I will refer to this as a coupled 2D4D system. And the purpose of this talk is to try to clarify the relation between these two descriptions and establish a precise correspondence. So, let me start with the first approach in which the defects are seen as monotromid defects. Take a gauge theory with a gauge group SUN, and we can introduce a surface defect in this four-dimensional theory by assigning a non-trivial behavior of the gauge field when we go around the defect. So, this is the plane that transfers to the defect. Therefore, the surface operator here is a point. And if we move around that point and encircle it and theta is the phase in this plane, then if we want to preserve supersymmetries, we can assign to the gauge field a non-trivial behavior like this, where these gamas are parameters. They have to add up to zero because we are in SUN, but they can be different and they can be grouped into different sets with entries which are n1 here, n2, and so on, so that the total sum of these is n. So, our SUN gauge field, and SUN gauge group actually is broken into a product of several u1 factors with ranks corresponding to these numbers, n1, n2, and so on. This is sometimes called the levi decomposition of SUN. The presence of these unitary factors and the fact that we have a two-dimensional plane, different from the rest of four dimensions, allows us to introduce new terms in the path integral because we can have a non-trivial turn class in two dimensions. So, we can have magnetic flashes for these un gauge fields, which are quantized, and because of the original SUN condition, sum of these numbers has to be zero. So, in the path integral, besides the usual instanton-like factors that we can insert with the second-gen class with the gauge coupling constant to weight in a different way, the different topological sectors of the theory, in this case we can also insert other types of factors related to these numbers in parameters, which I called eta. Actually, the fact that there is this non-trivial behavior for the gauge field here, and the fact that there is a disquantization condition can be used to show that the usual second-gen class is not just an integer k, but actually as a shift, which has this form, where these m are the magnetic fluxes and these gammas are the parameters I introduced before. That means that when we introduce these factors in the path integral, we have sort of an instanton-like weight, which is modified, as I said before, with these extra pieces and using this result, we see that this weight factor actually has the usual form, 2 pi i tau, the gauge coupling times an integer number k, plus a new type of contribution in which you see there is this combination of the gauge coupling and the other parameters, which are specific to the surface operators. The weight in which we can insert in the path integral has this form, so we don't have any more single gauge coupling and a single topological number k, but we have other parameters and other integers and i's. That means that the instanton partition function for this theory becomes like this, with this weight, and with a suitable and trivial change of variables it can be repackaged in a different form by introducing m, m was the number of factors in which the original group is broken by the presence of the defect, by introducing m parameters, which weight the different individual instantons. These are called ramified instantons and this is the ramified instanton partition function. Notice that I have m of these weights and the product of them is the usual instanton weight. It is as if the ordinary instanton that we know in four dimension is broken into little pieces. Just like the ordinary instanton partition function also this ramified instanton partition function can be explicitly computed using equivariant localization method a la nekrasov, and actually that was done a few years ago by Kano and Tachi Kava using an orbifold method. In the usual, as you will know, nekrasov approach deforms the spacetime by introducing the omega background and then that allows to resolve the singularities to compute the determinant and the result of integration over the modulate of the instanton becomes a simple matrix model. Similarly for this ramified instanton partition function Kano and Tachi Kava told us that we can do a similar construction by taking an orbifold, an orbifold zm. Again the m is the number of pieces in which the gauge group is broken. Usually when you take orbifold in the nekrasov approach you mod out the space which is a transverse to the spacetime where the theory is defined here. It's an orbifold in which the orbifold parities act partly on the spacetime. So this c times c is the spacetime which is the form of the omega background. This is the two-dimensional plane where the defect is and the transverse direction actually is non-trivially acted by the zm orbifold. If we introduce this vacuum expectation values for the four-dimensional gauge fields then this becomes an explicitly calculable quantity and you get the so-called ramified instanton partition function. From this instanton partition function by taking the log you obtain information about the nonperturbative quantities in the effective theory. In the limiting which the omega background goes to zero, so the epsilon parameter goes to zero, you have the usual one over epsilon one epsilon two term and the coefficient in front is what we call a twisted superpotential. That describes in a non-trivial way the infrared dynamics on the defect. And at the reason why is that because this one over epsilon one epsilon two is actually in this natural is actually a sub-leading term of the sub-leading term of the sub-leading term of the sub-leading term and two is in this approach a way of rewriting hour the supervolume in four-dimension. This is the 4x the zero mode related to the source of the instanton and he's are the super partners and is epsilon one is the 2-dimensional analytical of that. And if you look at the details of the speeders actually you see a d t bears it is actually a twisted superpotential. si pribyeveno, but it gives the same kind of information for the two-dimensional part of the theory. For example, if you take the simplest case in which you start with an SU2 theory and you introduce, defect the only one that you can introduce that breaks SU2 in to U1 across U1, then going through this procedure you obtain very explicit expression where a je vakum, kaj je vzaljene naseljnje deljevstva, kot q1 in q2 je vzaljnje dvih, izgledaj, na dveč, ali je, ki je vzaljnje 1 over epsilon, zelo je zelo izgledaj na vzaljnje deljevstva spremljajstvo. Tako, njako imaš došli in izgledaj, ko so počelji. Moje vzaljnje je, da bilo, da bi bilo pravdi, še vzaljnje uvrbi, razvrčná dynamic in nesto dvimešnje sefacer intresenje in pri mladi v conductora je bilo zelo ještje kaj Mađa. A moj pregunta je, ja da bali do dva dnevkje v skupnih početkinje in tudi pri bošnjih dovedkev. Znamenje si, da prkonej pri mladi v kostula tudi dvimešnja sefacera nato dva dnevkje ssystem. So we start with a four-dimensional theory with gauge group SUN, and we couple it to because of the amount of supersymmetry that we want to a 2,2-dimensional gauged linear sigma model in which G is a global symmetry. There is a simple example of this, which is to consider SU2 and a sigma model, the Cp1 sigma model coupled to it with SU2-flagged symmetry, which is another way of saying that we consider a U1 gauge theory in two-dimension with carol field in the fundamental of SU2. And I can draw this picture to represent such a theory. This round node here is a two-dimensional U1 gauge theory. One is because it's a U1, which has this SU2 as a flavor symmetry. But actually, this SU2 is not just a flavor symmetry for the U1 theory, it's also a four-dimensional gauge theory. So how can we describe such a coupled system? Well, if we understand that the flavor symmetry, the SU2 node, simply as a flavor symmetry, then we can use some old results and write the effective twisted superpotential for these two-dimensional gauge theories. This was worked long ago by Dada, Davis, Divakia and others, generalizing the Benetiano-Yankelovic approach for the anomaly term. And you get the effective twisted superpotential, which has a classical part in which T plays the role of the complexified Fagliei Liopoulos. This is a U1 theory. Sigma is the twisted chiral superfield, and this is the result of integrating the fluctuations, the quantum fluctuations. It's a sigma log sigma structure. It's the two-dimensional analog of the more familiar phi square log phi structure that you have in four dimensions for the effective low energy theory of the prepotential. This is a purely two-dimensional point of view. As I said, this is the twisted chiral superfield of U1. This is the Fagliei Liopoulos coupling. This, too, is because you have two flavors. And this is the ultraviolet scale. But as I told you, as you two know, this is actually a four-dimensional theory. So this has its own fields. It's not only a group. It's a four-dimensional theory with the gauge fields, scalers, and so on. So there is a possibility of upgrading these effective twisted superpotential by considering the, as you two know, that's a four-dimensional theory. And, for instance, by moving in the column branch of these as you two four-dimensional theory, namely, by giving expectation values to the scalar fields, then we can modify the effective dynamics in two dimensions. The expectation values of these scalar fields, as pointed out by Anani and Orin long ago, actually serve as a twisted masses for the twisted chiral field. But this is halfway because we are coupling a theory with a four-dimensional bulk theory, but we are treating this theory only classical. The full step is by considering the full quantum theory of SU2 and promote this classical volume expectation value to a full four-dimensional quantum expectation value. And this was pointed out a few years ago by Gaiotto, Gukoff, and Seiberg, and then used by many others. So this is the effective superpotential, which describes a U1 chiral field with Fajer-Jopoulos coupling T with an SU2 four-dimensional theory. So it has all features of the two-dimension, but also the four-dimensional quantum theory. And if we are describing a surface defect, that should correspond to the monodromid effect in the other picture that corresponds to the breaking of SU2 to U1 cross U1. Is it possible to make a quantitative check of these correspondence? And the answer is yes, and it goes as follows. We take this twisted superpotential. And first of all, we notice that we can take this classical part inside the log by introducing a dynamically generated scale in two-dimension. This is the right coefficient for the beta function. And so this is the more compact form of the twisted superpotential. And then we can analyze the vacuum of such superpotential. So we can compute dW over the sigma equal to 0 up to factors of 2 pi i. So the vacuum equation is actually more correctly written in this form. We have seen this also yesterday in some talks, which using the specific form of W, in this case, takes this. And then we can solve explicitly this equation by using the full knowledge of the quantum four-dimensional theory provided by the cyberwitten solution. So we write the cyberwitten curve for the SU2 theory, which is given here. This is the characteristic polynomial appearing for the SU2 cyberwitten theory. This is pure SU2. And this is the scale in four-dimension that goes with the beta function in four-dimension. So this is the characteristic polynomial. This is the quantum parameter that characterizes the quantum column branch. So we have all ingredients from four-dimension. And notice that we start seeing the appearance of two different scales, the two-dimensional one and the four-dimensional one. Or if you like two parameters tau here for the four-dimensional guy and t there for the two-dimensional guy. Similarly to what we had in the monodromi description. Then by taking the resolvent and doing standard manipulation, it's possible to explicitly rewrite the vacuum equation in a very simple form. And given the expression for the characteristic polynomial of the cyberwitten curve to obtain the explicit form of the vacuum that solves this equation. In this case, it's just one sigma. Because it's a u1, which has the following expansion for large value of the four-dimensional vacuum expectation value of phi. And I just wrote here the first instant on part. And then you can clearly expand it to higher orders. If we evaluate now this twisted superpotential on this vacuum solution, and therefore we find the minimum in a sense, then this is what we get. Which is exactly the same form of the one that we obtain from looking at the nekrosof-like integral. In particular, we have to identify lambda 1 square which goes like e to the 2 pi i t as q1 in the other approach. And these ratios of scales, which go with the difference tau minus t as q2. And with this, we have a 1 to 1 correspondence here. I just wrote for simplicity the one instant on term. But clearly you can check that it works at higher instantons as well. Furthermore, by looking at this structure, we see that there is a hierarchy of scales in the sense that the four-dimensional quantum scale must be smaller than the two-dimensional one, because you cannot set lambda 1 to 0 unless you have first turned off the dynamics in four dimensions. So just as a brief summary of what I told you so far, we have seen two ways of describing surface operators, one in which we identify in a four-dimensional theory a two-dimensional slice, where the fields have a non-trivial behavior. In this case, we can compute the effective superpotential for these two-dimensional part of the theory from equivalent localization, modifying the nekras of partition function. The non-perturbative ingredients are the so-called ramified instantons, which are weighted by several parameters. So we have many types of instantons. On the other side, we have seen another description in which we start from a two-dimensional point of view. And we couple it non-trivially to four dimensions. In this case, the W, the twisted superpotential, is obtained by looking at the vacuum equation of the infrared dynamics. And we see the dynamically generated scales, which are in one-to-one correspondence with the ramified instanton partition function. So there is a Q versus lambda map, so to say. And the prepotential evaluated on the vacuum solution actually coincides turn by turn in all details with the localization result. Now, what I've discussed in this example for the simple SU2 theory with U1 inside, can be generalized without any problem to a more general defect in SUN. And the resulting theory is a product of UR gauge theories in two dimensions with bifondamental matters and coupled to a globally SUN theory in four dimension, which serve as a flavor for this node, and so on, so forth. But then we gauge it in a 4D to obtain the full coupling. This quiver theory corresponds to defect in which the ranks N1, N2, and so on are related to the numbers, which I put here, by this formula. Looking it from the 4D to the point of view, by looking at the arrows and the ranks of the gauge group, it's a trivial exercise to write the twisted superpotential, which is a generalization of what I discussed in the SU2 case, and to work out the vacuum equations, find the solutions, and evaluate the superpotential on the solution, and compare with the localization site. And what we get is a precise match term by term if you follow the steps which I described. Actually, in doing this check, there is one important point, which, however, is a little more technical. That's why I haven't talked about it so far. But at this point, it's necessary to address this question. If you remember the nekras of partition function, then at higher instanton, they are usually written as multiple integrals. Similarly for the ramified instanton partition functions, they appear as multiple integrals. So when you want to compute explicitly the partition function, you have to specify the prescription with which you compute these multiple integrals. And since this is just at one instanton, so it's easy, since these integrans are typically ratios of functions, you have to look at the factors in the denominators and decide which poles have to contribute at, which poles do not contribute. In other words, you have to select the contour of integration. And in this case, that we worked last year, also these other collaborations reached the same conclusion for the type of surface operator I'm describing now, it turns out that you have to make some choices, some specific choices for this contour. In particular, for the first n minus 1 variables, you have to select the poles with plus epsilon. Namely, these are the factors that contribute. Whereas for the last one, you have to select the other factors. So the poles with minus epsilon. And if you give the epsilon an imaginary part, as it is usually done, then that means that for the first n minus 1 variables, you have to close your contour of integration in the upper half plane. Whereas for the last variable, the one which is associated to the last 4D node, you have to close the contour in the lower plane. So this is actually the precise rule that one has to follow in order to have one-to-one match between the two descriptions. These are the two-dimensional nodes and the corresponding variables for these two-dimensional nodes have to be integrated in the upper half plane. Whereas this is the last node, which has a 4-dimensional nature, and the corresponding variable has to be integrated in the lower plane. There is a very nice way of giving this prescription in terms of Jeffrey Kiwan residue prescription corresponding to a reference vector eta, which, in my notation, corresponds to introduce this Jeffrey Kiwan vector eta, which has the following form, which has a minus sign for all these variables, and a plus sign with a positive parameter, an arbitrary parameter so far, for the lowest one. Notice that these minus signs are actually correlated to the signs of the beta functions for the phi heliopolis couplings of all these two-dimensional nodes. And actually, you can rewrite these Jeffrey Kiwan parameter in a more suggestive way by introducing not only minus, but also minus psi 1, minus psi 2, and so on, where these are the phi heliopolis couplings of these two-dimensional nodes, which, given the hierarchy of scales, which I mentioned before, are actually ordered in this way. So if you give me a quiver of this form, then by looking at the ranks of the various nodes, and therefore by knowing the dimension of the various representations, I can compute easily the beta functions for the phi heliopolis couplings, and see whether the beta function is positive or negative. In this case, for this ordered sequence, all beta functions are positive. And therefore, according to this proposal, I write these Jeffrey Kiwan vector. And with that, I compute, in a nonambiguous way, the nekras of light partition function. And I find exactly the same result as from the dynamical analysis. And actually, at this point, one would like to understand why, so if there is a rationale among all these things, and if there is a possibility of understanding this sort of prescription, and what is the meaning. And in order to address this question, I introduce now another piece of information, namely duality. We know very well that a two-dimensional UR theory with some fundamentals and anti-fundamental representation has an effective theory in the infrared. And this same effective theory in the infrared can be given a very different ultraviolet description in terms of another theory in which the groups are different, so nf minus r in this case, in which the arrows are exchange, and there are even connecting lines in this way. This is a duality, a cyber duality applied to this two-dimensional system. So for example, if we start from the effective superpotential for this node UR, I just write the classical term, it's a 2 pi i t trace sigma, then the theory B has an ultraviolet description in which the superpotential is this, but these two theories share the same infrared behavior. Notice that in going from theory A to theory B, the sign of the phialiopulos coupling has changed, as it always happened in this duality. So we can do the same kind of tricks in our quiver theories, and we can take a quiver theory, like the one I described before, and apply to it one of these dualities. For example, if I am in the middle of a chain and I dualize this node in which we have both anti-fundamental and fundamental matter, then using the duality rules that I described before, if I start from this superpotential classical part, which is very simple to understand, then I get, you apply the previous rule, superpotential for this theory in which this sign has changed and also this has shifted, because the neighboring nodes in these quiver theories provide essentially masses for the twisted kind of fields. So this is what we have to use in this formula. But it may happen that I am in a theory in which a node has only outgoing arrows, so only fundamental matter. And in this case, the rules are a little bit different, even though they follow exactly from the same basic movement that I described before, and this is the result. And now, under this duality, both the coupling of the left and of the right node is affected. So we can take, for example, a fourth node quiver that corresponds to a surface defecting, which the gauge group SUN is broken into four pieces. This is of the type I described before. Then we can apply, for example, duality and go to this other quiver, then we can do a third duality, and so on, so forth. And all the way down, we end up with a quiver, which is very different from the previous one, in which the arrows are all going in the other directions. All these quiver theories are different ultraviolet description, but they share the same infrared dynamics. And as I told you, the infrared dynamics is described by the effective superpotential evaluated in the vacuum, so they should correspond to the same surface operators. They should describe the same infrared physics. So they are different description of the same surface operator. For each one of them, we can start from the classical term using our duality rules, and compute, for example, after one step, this is the classical part of the superpotential, and then all the way down until we reach the bottom. Of course, this is just a classical part, but then there is the quantum part, and then we have to treat it in the way that I described in the simplest SU2 case. And if you do that, you can get a lot of information. You can work out in all of these quivers, take any one of them, the Q versus lambda map, for example, for the second quiver in this chain, the Q versus lambda map is different, of course, with respect to what we had in the first quiver, because the beta functions are different. These numbers are essentially the beta functions for the running of the couplings. So we have different infrared scales, which, however, all map through a different, with a different expressions to the Q1, Q2, and Q3. In this case, the three nodes and the corresponding three types of ramified instanton. For all of them, the fourth ramified instanton is always related to the four-dimensional scale. And then if we, as I pointed out here, in going from this quiver to this other quiver, we change the sign in the Fajeljopoulos term of the first node, the one that is dualized. And if you buy the proposal that I gave you about the Jeffrey Kirwan vector, then I have to change the sign in the first entry of the Jeffrey Kirwan vector, because now this beta function has a different sign. And if I take this Jeffrey Kirwan vector and I use it in the localization formula, I get exactly the same structure term by term for the second quiver in the duality. So now we see a general structure emerging, in which for various quivers we can have a precise match between the two approaches. That works not only for the linear quivers that I have here, but also for quivers in which you have also loops inside. So the outline of this argument that tells us why all these different quivers lead to the same infrared dynamics, is that because, after all, they only differ because of the change of integration contours in the nekros of light partition function. So different choices of integration contours, which means different choices of Jeffrey Kirwan reference vectors, namely poles that you pick upstairs or downstairs in the complex plane, lead to different expressions, which, however, are in one-to-one correspondence with the terms in dynamical description. And simply because all results differ by a change of contours and in this case, for this non-conformal asymptotic free theories, the integrants do not have a residue at infinity. It is obvious by a residue theorem that the result is always the same for all different choices of contours. But now we understand that picking one contour of integration or another corresponds to picking a duality frame and having a correspond different ultraviolet description. So I come to my conclusions. I describe the surface operators in essentially two different ways. One with the idea of assigning non-trivial boundary conditions and singular behaviors to the fields over which I perform my path integral. The other has coupled systems in which a two-dimensional gauge linear sigma model is coupled with a four-dimensional gauge theory in four dimensions. And there is a one-to-one correspondence between the different quantities in the two types of descriptions. The monodromy vector, which tells me how the SUN, in this case, symmetry group is broken by the presence of the defect, translating to this description in the ranks of the two-dimensional gauge groups that form the quiver theory. The vacuum expectation values in four dimensions become twisted masses from the two-dimensional point of view. The ramified instant uncounting parameters from this approach are nothing but another way of writing the dimensionally, the dynamically generated scales in this quiver approach. The superpotential that is obtained from localization ala kananta cikava actually coincides with the twisted superpotential evaluated in the vacuum. This choice of integration contours in the localization approach actually corresponds to picking a particular duality frame in which step of the duality chain you are. What I described here for the pure theories, n equals 2 theories in four dimensions, actually can be generalized for other types of systems, like n equal 2 star theories. These are different because the nekras of light integrals now have also a numerator factor. So now the question of changing contours is not irrelevant anymore. Or sqcd with a fundamental matter, but even this is quite interesting, because if the amount of matter is such that this theory are conformal in each node, then, again, the issue of changing the contour integration is not trivial. We can uplift this discussion to systems in one dimension higher, and so describe surface operators in a five-dimensional theories, maybe with the transformers interactions. And they also will describe in the three dimension has three-dimensional theories with the Chen-Simons. And there is a nice interplay between the Chen-Simons parameters in three dimensions and in five dimensions. Or, and these, as to be done, extended these analysis to other types of Quivel theories with other classical groups, orthogonal simplecti. And it would be also very nice to explore the connection with integrable models. Yesterday, we had a talk in which there was such a connection between the vacuum equation X of dW or dSigma equal 1 and integrable models. It would be nice to understand all these in this context. So I thank you very much for your attention.