 I would like to start by reminding you a little bit about PQ-Webs as duality in type 2b and what happens when you place these three brains in the mix. And then I will discuss in detail a billion mirror symmetry in three dimensions and a stepwise approach. Can you hear me now? Then an approach which is called stepwise dualization and then if we have time we will try to generalize to the non-habilian case. So these are the brain setups I would like to talk about. Let me introduce first of all the standard honey-witten brain setups. The main players are D3 brains, NS5 brains and D5 brains. And usually you draw something like this. You have NS brains and then you have D3 stretching between NS brains. So here I explained where the various brains are stretching. So all of the theories we are discussing will be three-dimensional. So all the brains stretch along the first three directions in type 2b. Then there will be X6, will be the direction where the D3 brains are going. Then there will be NS brains will go vertically. So this will be X4, 5 and 6, 3, 4, 5. And then we can add the D5 brains which will go along the remaining three directions. So these setups if we just stop at the first three lines we have N equal 4 supersymmetry. There will be an SO3 acting on this coordinates which is the SO4R symmetry of the N equal 4 supersymmetry in three-dimension. So what is known is that if you do S duality in type 2b you replace NS5 brains with D5 brains and D3 brains stay D3 brains. You will have a different gauge theory and this is related to 3D mirror symmetry. And this part of the story with N equal 4 supersymmetry is quite well understood. What I would like to do is try to generalize to the case with half the number of supercharges. So one way to do is to rotate some of the brains. So we will have NS5 prime brains where instead of stretching along 4 and 5 they will stretch along 8 and 9. And similarly for the D5 prime instead of stretching along 8 and 9 they will stretch along 4 and 5. So once we do this we have half of the supersymmetry. So the theories which live at this type of brain setups will have just N equal 2 supersymmetry in two dimensions. In this case even drawing a generic brain setup with NS and S prime, D5, D5 prime one does not know what is the gauge theory living on it. So what we would like to do is just take one example and try to work out the gauge theory. So one example where we know is the following. Instead of having just NS5 we put NS5 and then we put many D5 prime brain, one on top of each other. So this is called the PQ web. If we just do this, this is a five dimensional theory because if we take the NS and the D5 prime you see that they have 0, 1, 2, 4, 5 in common. So they have five directions in common. So this is a five dimensional gauge theory. In this case it's an easy gauge theory, it's just free. But what we would like to do is to add a D3 brain going like this, transversely. So this would be now the sixth direction. So if we, this is known how to do, let me just do the case where we have two D3 brains arriving on the PQ web which is what I drew here. And then we also add the two NS prime brains just to stop the D3 brains. So what is the gauge theory over here? We have an S brain, an S brain, and then we have D3 brain. So this will give us two U1 gauge theory if we just put one D3 brain. So I draw it in this way. The circle means gauge node in the theory. Then we will have, by fundamental matters, which will be something like this. These are massless strings, which will get, these are strings which have zero length. So they will give rise to massless state, massless multiplets. So this will be chiral multiplets. I use arrows. There are two of them. And then I will add, I have to add the D5 prime. So if I add just the D5, this will give a fundamental field for this gauge group. If I add the D5 from this part will give a fundamental for this. But if I put the D5 prime on top, then I have many massless states going from D3 to D5, which will be fundamental. There will be some fundamental of this group and some fundamental of this group. Then the result is this. So we have K arrows going from one to this global symmetry group. So the square means it's a global symmetry group. So I have an SUK global symmetry, another SUK global symmetry. And they have all these type of lines. This was well known. This was known from the 90s. Now there is also super potential here, which is a cubic super potential. I just wrote it down. It's related to loops in the quiver. Now the question is what happens if I do an S duality? What happens is that instead of having a PQ web with one NS and many D5, I will have a PQ web with many NS brains and one D5. In this case the answer is not known. What is the gauge theory living over here? It's probably going to be some billion gauge theory because I have just one D3 brain. But the answer was not known. So the purpose of the talk is to work this gauge theory out and along the way we will find some interesting things involving monopole super potential, which means we have gauge theories where in the super potential there are monopole operators. These are the monopole operators which were discussed yesterday. But in the case discussed yesterday it was N equal 4 supersymmetry and the super potential was never involving monopole operators. But with N equal 2 you can add monopole operators to the super potential and you don't break the supersymmetry. So first of all let me review again 3D mirror symmetry. So the simplest example is just one gauge group U1 with one flavor, one anti-flavor. And let me add also a singlet. So this is P and this is Q. And then I have the super potential which is sigma PQ. So this is actually N equal 4 super young meals, U1 with one flavor. And this is the super potential which ensures me that we have N equal 4 supersymmetry. The mirror here is just a free hyper. So this is the simplest example of 3D mirror symmetry and this was given by Interligator and Cyberg I think back in 96. If you draw the picture, the brain setup over here we have 2 NS5, 1D3, 1D5. When I do the S duality I get an S5. So this is the free hyper. These are the massless things that give the free hyper. And this is U1 with one flavor. So we can read off the duality very easily from the brain setup. What is important here is that what is the mapping of the operators. So here we have two free fields. Let's call it P and Q, three chiral fields in N equal 2 notation. And what are the fields which are mapped over these two free fields? These are the monopole operators of this U1 with one flavor. So if we call them N plus and then minus, we have that this is mapped under the mirror symmetry in this way. We also have another gauge invariant operator which is sigma here. So where is sigma mapped? If you remember from the talk for yesterday, sigma satisfies some quantum relation which is the following. Sigma is equal to N plus and minus. So sigma is mapped to the product of P and Q. Okay, are there questions about this? Okay, so let's try to do... This is the very basic example. The next example would be U1 with N flavors. But then we can do it for N equal 4. And then we will have some generalization. We will have sigma to the k equal to plus and minus. This is what was explained yesterday. So first of all, let me modify this duality a little bit to go to an N equal to dualities. What we do is that we can flip some field. What does it mean? Flipping a field. Let's say we have a gauge theory with some operator O. We couple this operator O to a new field, sigma. And we get... So we have some theory with some super potential and some operator O. We change the super potential in this way. And we add a new field. So in this case, I want to flip this operator sigma. So the super potential will be sigma PQ. This will go in sigma PQ plus sigma times... Let's call the flipping field X. So in this case, you can see that sigma and X become massive. So I can integrate them out. So I have to use the question of motion of sigma and X to integrate them out. The question is that I get W equals 0. And I lost most of these fields. So I started from U1 with one flavor in N equal 4. I flipped this singlet. And I just get U1 N equal 2 with one flavor and W equals 0. So basically I lost this singlet. What happens on this side? I have to flip sigma. But sigma is P times Q on this side. So the duality now, the mirror duality becomes a theory with three fields, which I can call sigma PQ. And W is equal to sigma PQ. So this is an N equal 2 duality, which is the simplest one you can have. And it's just equivalent to the N equal 4 duality. Just you can go from one to the other flipping at one field. This is called usually X, Y, D model. It's a Vestumino model. There is no gauge symmetry over here. But now it's duality between two interacting field theories. The simplest one. So now we can try to do something more general. We can go and do U1 with K flavors. So we start to get something similar to this. You see, now here we have the U1 with K plus one flavor. So let's try to do U1 with K flavor. One way to do it is to use this duality in this direction, or the duality we had before in this direction. So we have P and Q, which are two free fields. And we can replace them with U1 with free fields and the super potential. But now I want to do it for each flavor over here. So I will pick the first, let's call this small. I want to pick the pair P, I, Q, I for each I, and replace it with the U1 with the flavor and the sigma I. So once I do this, what happens over here? I have a U1. This is what I call that wise dualization. This type of approach was advocated by Kapustin and Strassler, always in the 90s, where now the super potential. So I replace every pair of fundamental fields with the U1 with one flavor. I do it this K time. So I have K object like this. So now this theory is not the U1 theory anymore. It's the U1 to the K plus one gauge theory. And there will be a super potential which has K terms, sigma I, PI, QI. But now there is something missing in this picture and is the following, that when I replace this fundamental field here, these fundamental fields become the monopole operators. So these fundamental fields were charged under the U1. So this means that the monopole operators of all these K gauge theories must be charged under this original U1. The way to do is to add the BF couplings. So what is a BF coupling? This is like a Champ-Simon coupling. So a Champ-Simon coupling is usually for a billion theory, something like ADA. But if I use different gauge groups, gauge fields, then I will have something like ADB or BDA. Let me put it like this. But DA is usually called F, so this is a BF coupling. So I started from a theory which is U1 with K flavor. Now I have a theory with many U1s, the same number of charges multiplets. I add the K singlets, sigma I, and then I have to add many BF couplings in the Lagrangian. These are equivalent. I just use the basic mirror symmetry K times. But now something happens here. We see that this U1 doesn't have flavor anymore. This means that I can do exactly the path integral over here. So once I do this, this path integral gives me a delta function. Because this gauge field, I can exactly integrate over this gauge field U1. This will give me a delta function that tells me that the function of delta function, that the sum of all these field strands is zero. So I can think of this theory as the product of K U1 with one flavor, but then there is the relation that the sum of all the field strands must be zero. So a different presentation for this theory is something like this. Take a linear quiver. Instead of K gauge groups, I have to put K minus 1. And then I have many singlet fields, which are... I have many singlet fields, which are flipping to zero with all these bifundamental fields. So this is the statement. Now this is the statement about 3D mirror symmetry for a billion gauge theories. U1 with K flavor is dual, is mirror to a linear quiver. What we did is just assumed that the original symmetry was correct, and then we proved this. So if you believe the first original duality, then this is just the... this just follows. One way to see this through brains is the following. We can start from here that we have an NS brain, a D5, an S prime brain, which is rotated. So we have N equal to 2. Then here we put many D5. And then we do the S duality, whatever. The D5 become an S. And then here we have a D5 prime. And here we have a D5. So we see that also in this picture we have a linear quiver. So many U1s, one connected to the other. And then there is a flavor for the first one and the flavor for the last one, which is precisely this. So again, all this goes back to 97, I think. But this type of procedure is some kind of procedure that we can try to apply to this theory. So here we have K plus K plus 1 pairs of fundamental fields where I can apply this basic step. When I do this, I will create many U1s. So we can already see that here we will have a theory which even if it looks like there are only, you know, the NS brains are all on top of each other. We will get a theory with many U1s, which is kind of strange. Anyways, let's first do the case K equal 1. Because if the case K equal 1, you already know the results. And also we learn something. So the case K equal 1 we have from one side. We have this theory, U1, U1. There are six fields. And from the other side we know what is the theory here because we have just one NS5. It's basically pretty much the same theory, but it's not gauged. So there are six fields, which you can see them as massless strings going from D brains to D brains. But there are no gauge fields because there is just one NS brain in order to have gauge field you need two NS brains and these three that can slide upon them. And we have this type of super potential, the cubic super potential. So this would be S and S, D5, D5. And here we always have the same 1-1 PQ web which upon S duality goes into itself. So this is the same on both sides. Okay, so we can try to understand this duality but one way to understand this duality, first we would like to prove it, usually always using the basic Abelian mirror symmetry. So all this talk is based on a paper that I wrote with Sara Pasquetti, which is 16.05.02.675. And in the paper we prove this duality but along the way we found the third theory which has just one gauge group and so this duality actually becomes a reality. So one way to find a theory with one gauge group is to basically start from here. So let's put here K equal 3. So you see that the matter content is the same as this. We have a linear query 1 cross u1. The difference is that here we have three singlets with this type of superpotential. Here we have zero singlets and a different superpotential. So here the superpotential is, if we call this P and Q, the superpotential is A P Q plus A tilde, P tilde, Q tilde. So what can we do is that we can try to modify these to arrive here. But we know that there is a mirror in this case which is u1 with three flavors without superpotential. So let's try to add here, we can deform and add sigma i xi. So we have three singlet fields and we flip the sigma i. If we follow the map, the mirror map, the sigma i's where, so this is like P i Q i. This was where like P i Q i. So there are three sigma's and here there are three mesons that I can write. So what it means is that here I should add a superpotential which is xi P i Q i. So three terms in the superpotential. So I can write, let me write this differently in this way. So now I break the su3 global symmetry to u1 to some power. And here again I can integrate out sigma and x and I get w equal to zero. So this is a different presentation of the basic duality but this goes, now we are getting something similar to here. Now we have to add this type of superpotential, apq. So what was the map before? When we had u1 with k flavors, the map of the mesons, so we had P i Q i is a matrix of mesons. So let's do the case of three flavors. Then here I have P i Q i P 3 Q 3. And this is a non-trivial aspect which is not discussed usually. So the diagonal one, they were going to see to the sigma's in the basic ability of symmetry. The off diagonal one, they are going to monopole operators. So for instance P i Q i goes to the monopole operators which charge one and zero. So here I have two gauge groups. So when I say a monopole operator I have to assign the charge under both gauge groups. Then I can have the monopole operators which charge one and one. And this goes, this is map to P i Q 3. Then I have charge zero and one. And this goes to the other one. On this triangle is the same but with negative charges. So this is the map for the basic mirror symmetry for you one with three flavors and it's possible to generalize. Here we also have mesons. M with charge plus one here is mapped to the operator product of all the P i's. And M with charge minus is mapped to the operator product of all the Q i's. So we have a linear quiver. We take the product of all the operators going in this direction. We get a gauge invariant. This gauge invariant is mapped to a monopole on the other one. On the other hand here we have mesons, quadratic, gauge invariant. These are mapped to a monopole on this. This is a generic feature of mirror symmetry which is mapping normal operators to monopole operators. So if you have some relations which are satisfied by normal operators on one side they will become quantum relations for monopoles on the other one. OK, so here we want to add the APQ to the story which is the product of all the P in this direction. So we want to add the super potential. This is P1, P2 and P3. We want to add to the super potential P1, P2, P3 plus the other P1 tilde, P2 tilde, P3 tilde. So we get, now this theory, so if we do this becomes the theory describing the APQ web. But we saw that what are these two terms mapped? These are mapped to the monopole. So here we have to add monopoles plus one, plus monopole minus one. So let's say it again. This is a U1 cross U1 gauge theory with some super potential which is precisely the one that we know describes this PQ web. This is due to a U1 gauge theory with three flavors, three singlets and monopole super potential which is not obviously related to the brain picture. But this is a duality. Now it is also possible to show that this duality, one way to do is, I'm just going to sketch the discussion, one way to do is to think about this theory as U1 with two flavors. So we have two incoming arrows and two outgoing arrows, but then I gauge these two U1s together. So basically I take U1 with two flavors and these flipping fields, then I gauge and I get this theory. But if instead of starting from U1 with two flavors and the flipping fields, I start from its mirror, which is just U1 with two flavors, I get some theory which if you inspect a little better, you see that actually it becomes, so from one side I have this, from the other side I have U1 with two flavors. This is the mirror symmetry for U1 with two flavors. Now I gauge something over here and I get that theory. So these are U1 cross U1 gauge theory. When I do the same gauging here, I get the product of two separated theories, which is U1 with one flavor two times. But this again is equal using the XYZ, this is equal to XYZ model and this is equal to another XYZ model using duality again. So I see that the product of two XYZ models is equal to this theory. But this is exactly what we have here. We have six fields with two superpotential couplings which are XYZ. So basically what we found is that there is a reality here. This is one presentation, this is another, and this is obviously related to the brain system. This is another and this is related to this dual brain system. So what we want to do in the next half an hour is to generalize this reality in two different directions. One is to add more NS5 and the other is to just look at this. So one is to generalize this part of the reality and one is to generalize this part of the reality. Here we have U1 with three flavors and here we have just the vestumino model. So what we're going to do is that here we can replace U1 with three flavors, with UNC, with an F flavor, generically. Add the monopoles and see what happens. This will be a Haroni cyber type of duality. But for the moment, let's just try to generalize in this direction. So there are questions. Okay, so let's apply this stepwise dualization to this theory. So here I have many, I have the two U1s, then instead of taking A and A tilde, which are despite fundamental, I just replace it with the usual big A, big A tilde and sigma A. Then instead of having PI, PI tilde, I replace it with U1. So I have sigma I, PI, big tilde, big and so on. So I have K of those. Same story from the other side. BF couplings, which are connecting me. And then I have a BF coupling like this. So I just use the 2K plus 1 times the basic abelian bill of symmetry on this case theory. And I get this strangely looking object. Now I can do the integral of the two U1s. I get two deltas now. When I implement the delta, I get a linear quiver exactly like the beginning. Then I get this U1 in the middle here. So these are A and A tilde. Here I have K minus 1. And then I get another linear quiver from the other side. I have all the singlets. I'm not drawing the singlets, but there will be many singlets. 2K plus 1. OK, so we see that what we are getting here is a theory with many U1s, which should describe this brain system. We should be careful about the super potential, because here we have all the singlets and all this type of super potential, which will be sum of sigma i, pq, pi, qi, pip and so on. So we have all this type of flipping super potential. But then we also have the super potential coming from this cubic. So what is this going to become? It will become some monopole operator super potential. But the monopole operator here is not local on the quiver. So for instance, the terms pi, qi, a will be a monopole. So the super potential here will be this. Then we have sigma a, a, a tilde. Then we have something similar to this with the tilde, with the qs. But then we have a monopole. A monopole will be something like all zeros, one, one, one, all zeros. This will be one monopole term associated to p1, q1, a, p1 tilde, q1 tilde, a tilde. This is just the first half. Then we have the same with the minus, all zeros, minus one, minus one, minus one and so on. Then we can do p2, q2, a, and this will be this object. Zero, zero, one, one. Then we have one in the middle. So this notation is the charge. These are all the charges under all these ones. This one is the one in the middle. Zero and the same with the minus. And the last one will be m with all one all the way. So we have many monopole operators, 2k monopole operators, half with the plus sign, half with minus sign, but they will involve groups which are not attached to each other. So this is not the local in the quiver. Usually everything was, when you have a quivering with n equal 4 supersymmetry, every node will just talk with the nearest neighbor. This is a little bit more complicated. Another thing we can do, we can use the result from before and replace this one with three flavor in the middle with the square. So before, if you remember, we had this duality. So we can apply this duality inside here and we get a theory with one less gauge group, this is one gauge group, this is zero, and we have six fields. So this is a presentation which, in the case k equal one, reduces to what we know. The other one didn't reduce. So now we have four different theories. We have this theory, then we have this one, which would be the dual of here, and then we can also use this duality inside both to change the presentation a little bit. If we use this presentation, we lose this term, the sigma tilde a tilde, but we get, if we call this is p1 q1 p1 tilde q1 tilde, and this is xx tilde. We have a qubit term, px p1 q1 plus x tilde p1 tilde q1 tilde. And here we lose the one in the middle, and it's just like this. So again, we have many superpotential operators. We have the cubic terms, and then we have flipping terms. So this will be the result for this question mark that I drew before. And the interesting thing is that there are many ones more than you would get if you spread this k NS5 brains, because we have k NS5 brains, so in n equal 4 you would expect k minus 1 in one, but here we get 2k minus 1, so almost twice as much. And then we have all these monopole superpotentials. So this is the result. So I think for the one thing one can do is to map the chiral rings from here to here. This is not easy, we just mapped the generators of the chiral rings because it's easy to study the mesons, it's not easy to study the whole set of monopole operators. So for instance here we have many mesons which are mapped into many monopole operators that you can construct here. And then we have a few monopoles, this is the one you can construct out of these two U1s, and this are mapped to many mesons or many type of normal meson operator you can find over here. But you can check that at least at the level of the generators of the chiral ring there is a perfect matching between the two sides. Okay, if there are no questions I will go back to this duality again. So let me write it again. So from one side we have U1 with three flavors and the superpotential is, let's call it this Pi, Pi tilde. The superpotential is there are three flipping terms, Sigma, Pi, Pi tilde. And then there is the monopole with charge plus one and the monopole of charge minus one. This is this side of the duality. From the other side you just have six fields, A B C, A tilde, B tilde, C tilde and the superpotential is A B C plus A tilde, B tilde, C tilde. So this presentation of the duality is not SU3 invariant. So what we can do is to get rid of all these three singlets in such a way that we have SU3 times SU3 global symmetry. What we have to do here is to add the Sigma i xi. Now we have to look at the map. So here we will be introducing some more cubic terms because the Sigma i are mapped to quadratic operators over here. The end result is the following. We have U1 with three flavors and just a simple monopole M plus one plus M minus one is dual to this object. This is X ij. This is a bifundamental matrix of SU3 times SU3. So there are nine fields. We started from six fields. We added three fields. And the superpotential is just the determinant of X as a matrix, as a three by three matrix. So there are nine terms in the superpotential. In this way you can see immediately that the SU3 times SU3 global symmetry here is preserved on this side. One thing is that the duality, this duality we discussed at the level of the field theory, but if you put the field theory on S3, so we take n equals, this is what in the previous talk we heard about this S3 partition functions for n equal to n equal to on S3. You will find some partition function on both sides and this will be something like dx. And then we will have three times, we have Sb of X plus Mi Sb of minus X plus Mi tilde. So this will be a U1 gauge theory with three flavors and three entry flavors. And from this side you will see that there is just a product of nine singlets. So basically you are projecting down the duality from the level of the full gauge theory to the level of S3 partition function. You will find some identity. This is the identity, it doesn't really matter. But this identity was already present in the literature. So for instance already in 2002 there was a paper that was discussing this identity. Of course they didn't know about the relation with gauge theories and it was called Ultimate Integral Identity because apparently for their purposes which was some integrable system story this was a very basic identity. Otherwise it was also known as New Pentagon Identity. So what we did in a paper which we didn't publish yet with Francesco Benin and Sarapasquepis to try to generalize this story to the case where you have UNC with NF flavor. So I just tell you the result. The result is you have NC, then you have NF flavors. And the superpotential here is just M plus plus M minus. So these are the two basic monopole in UNC with charge. So M plus is the following. The charges are if we go on the carton of UNC there will be plus one and all zeros. This is M plus. M minus will be something like all zeros and minus one. So these are two different monopole operators. We have to add both of them to the Lagrangian. And once we do this we are left with the global symmetry which is SU and F square. Because the monopole operators if you don't have Lagrangian here you have SU and F square but then you have a U1 topological symmetry and then U1 axial symmetry. But these M plus and the minus are both charged under both of these two. You want symmetries with different charges so you're breaking both of them. And the dual is U and F minus and C minus two with two n-flavors, with the n-f-flavor, sorry. And the superpotential is the usual cyber superpotential so some i and j, M ij, qi, qj. So this is very similar to their own duality but their own duality doesn't have the minus two. And then here we have the other superpotential is again M plus plus and minus where these M plus and the minus are the monopole operators of the dual gauge group. And here is another difference. Their own duality, there is a flipping of the two monopoles. Here we don't have any flipping but they both appear in the superpotential. And again here the global symmetry is SU and F square. So one way to argue for the symmetry one can check that this duality makes sense for instance one can compute the modular space of acqua on both sides and it matches. Another way to argue for the symmetry which also provides a ultraviolet completion for the theory is that because the one question you can ask here is whether starting from just U and C with n-f-flavor and adding this superpotential is relevant. So generically if n-f is too big the superpotential is not relevant. The boundary case is n-f equal to 2 and C plus 1 anyway. But we can start from a duality in four dimension so this is a 3D n equal to duality. We can start from a duality in four dimension which is a USP duality. USP 2 and C with n-f-flavors usually it's called 2 and f with 2 and f-flavors. So this is not anomalous in the four dimension and there is a dual which is the interligator-pullo dual which is USP 2 and f minus 2 and C minus 2 with 2 and f-square-flavors. And here there is a superpotential which is mqq. This is very similar to, can you read here? So basically this is like the cyber duality for USP gauge groups. If you take nC equal to 1, USP 2 is SU 2 and so this is the notation I'm using. So this you can reduce to three dimension. You find the same duality except that when you reduce to three dimension you are turning on a monopole superpotential. So in 3D, let's just go directly to 3D, you have this duality but now the superpotential here is the monopole which is there is just one monopole for USP and again here we have the monopole. Now there is a way, so this theory is SU 2 and f global symmetry. There is a way to turn on real masses in such a way that the USP is broken down to U and C and the 2 and f is broken down. Half of the flavors become massless, make up massive half of the flavor. But now the superpotential gets doing this. Again you are going, you are giving some expectation value and giving some expectation value. This is a mechanism which was discussed by Polio back in the 70s. Generates monopole super, can generate a monopole superpotential in the infrared. So here we get monopole plus plus monopole minus, so we get precisely this. And if we do the same on the mirror, on the dual, we get precisely this dual. So this is another generalization of this reality I was discussing at the beginning. One nice thing is that if you plug nf equal 2 and c plus 2, which is in some sense the self-dual point where there are charges for instance of the basic fields will be exactly one half. So if you are familiar with the cyber duality in four-dimension, if you have SUN with two n flavors, it's a similar situation. This theory lives at the s-duality wall of a four-dimensional gauge theory, which is SUN with n equal 2, with two nf flavors. So this is n equal 2 in 4D. So this theory can have a s-duality wall. These were discussed by LeFloch. And he found some sort of duality like this, but he didn't show that there is a monopole superpotential. So some things are not really working well. So anyway, the last comment I want to make is that if you start from this s-duality wall, you have a co-dimension one. You have a wall in four-dimension. On the wall, there is a three-dimensional theory which is precisely this type of theory with the monopoles. So this is another example where going from higher dimension to lower dimension, you generate monopole superpotential. The other example was when you compactify on s-1, you generate one monopole superpotential and also here. So I think I can stop you.