 So in this talk, I'll focus on the second approach to study monoproperators and the Cologne branch using one-dimensional quantum mechanics with the interface. Monopros in three-digit theory is a very interesting, not part of the objective object. The vacuum expectation value gives additional direction in the vacuum-modulized space. And also often they provide additional conserved current in the infrared fixed point. So the IR fixed point theory, open experience, not tributary enhancement. The famous example is the ABGM theory, where we have the not tributary enhancement by the monoproperators. But this operator involves very complicated quantum dynamics and strong coupling dynamics of the three-digit theory. So in general, it's not easy to analyze this monoproperators directly using some part-optive analysis. So it means it's very hard to study the Cologne branch, which is generated by the monoproperators. In this talk, I'll introduce an interesting new approach for the monoproperators and the Cologne branch. Basically, we'll focus on the monoproaction on the vortex stage. So we know that vortex is kind of the operator which creates and analyzes, sorry, monopros operator which creates and analyzes the vortex stage. So by reading of the actions, the monoproactions on the vortex stage, we can extract many important quantum properties of the monoproperators. So in this, sorry? Yeah, yeah, yeah, that's right. We'll study monoproperators in the Higgs branch and extract the Cologne branch too. Okay, that is our thing. So we'll basically compute the correlation functions of the monoproperators using the localization technique. It allows us to read off the monoproactions on the vortex stage. And also using this monoproaction, we can easily extract Cologne branch algebra involving the full quantum corrections. So our goal is to compute the Cologne branch cavalry exactly using the spousy material localization. Okay, let me first briefly introduce 3D and Euclid 4-series. 3D and also vacuum structure. 3D and Euclid 4-series will concede the gauge theory with the gauge group G number of hypermultiplanks. And any cold post theory preserves the eight real supersymmetries with Q and also the theory has the alchymetry SO4 or SO2H times SO2C. So this index A and A dot correspond to double-edged indices for SO2 and SO2C. And also alpha correspond to Lorentz index. Any cold post theory has two different supermultiplanks. Firstly, we have the vector multiplex. Vector multiplex consists of N equal 2 vector multiplex with the gauge field and the scalar field sigma in other joint representation. And also it involves N equal 2 chiral multiplex in other joint representation with the scalar field Y. There is another, so N equal 4 vector multiplex has a gauge field and three scalar field sigma and Y. There is another supermultiplex called hypermultiplex. And hypermultiplex consists of two N equal 2 chiral multiplex with the scalar field and X and Y. And with this particular superpotential, the full system has N equal 4 supersymmetric enhancements. And this N equal 4 gauge theory has very rich structure of the supersymmetric vector, which is parametried by the vacuum expectation value of the gauge invariant operators. Roughly speaking, this modulite space is a product of two branches. One is called the Higgs branch, parametried by the scalar field N, X and Y in the hypermultiplex. And the other one is called the Clomb branch, parametried by the scalar field in the vector multiplex with monopolar operators. And they can intersect and not trivial as a fixed point. And these two branches are both hyperkeller space. They have a three complex structure. And for example, this SU2H archimetry action, the Higgs branch, it will take the three complex structure as a triplet. And similarly, SU2C action this Clomb branch. Let me discuss more about the modulite space. Firstly, Higgs branch. Again, it's parametried by the vacuum expectation value of the scalar field in the hypermultiplex. And on the generic point, Clomb branch, the other scalar field in the vector multiplex, they are set to zero. And these two scalar fields are set by F term and D term constraints like this. And hyper, sorry. Higgs branch modulite space automate the hyperkeller quotient construction. It's just simply given by this hyperkeller quotient. It's the space of X and Y satisfying the term and F term occasion or the momentary constraint. Modulated by gauge transformation. And it turns out that this space given by hyperkeller quotient it receives no content correction. It means the classical geometry from this occasion is exact. So the geometry on the Higgs branch is a rather eternal line. However, in this top, I'll focus on the other branch. It's called Clomb branch. And Clomb branch is parametried by vacuum expectation value of the three scalar field, sigma and phi in the vector multiplex. And also the monopolar operator, they participate in the Clomb branch. Because of this, to understand the geometry on the Clomb branch, we have to understand, first understand the dynamics of the monopolar operators. And monopolar operators are usually defined as a singularity in the pass integral. So in the pass integral, we'll consider some non-trivial singular gauge field configuration around the point. So gauge field near the insertion point of the monopolar operator behaves like this. M is the magnetic charge. So if we integrate the field strength of this gauge field around the monopolar insertion point, then we get the non-zero magnetic flux. And M is the magnetic charge. It takes values in the weight lattice of the dual gauge group G. So the vacuum expectation value of this operator gives another direction in the Clomb branch. And combining this new direction, the Clomb branch form a hyperkeler space with a complex dimension two times the length of the gauge. And unlike the X branch, which was very robust under the quantum effect, this Clomb branch received both part-optive correction and also non-part-optive instanton correction. So the classical geometry is severely modified by the quantum effect. So in general, it's very hard to study. But sometimes we can study quantum corrected Clomb branch using some duality in three-digit gauge theory. For example, if we have some brain construction of the three-digit gauge theory, there are many dualities between the theory. We can use the duality, for example, using mirror duality and study some quantum correction on the Clomb branch by studying its mirror dual theory on the Hick's branch. However, this approach is rather indirect. So it would be very nice if we can find some direct approach for the monopole operator and the Clomb branch. So in this talk, I'll introduce one interesting direct approach, more direct approach to study quantum dynamics of the monopole operators. And also it can read off the Clomb branch geometry, more direct. So we'll consider a monopole in the Hick's branch. In the Hick's branch, there are some solitonic particles called vortices. They are particle-like objects, which carries non-trivial magnetic flux on the outer plane where the monopole particle is located as a point. So the vortex particle has same magnetic flux along its word line. And monopole operator in Hick's branch has magnetic flux, but actually this magnetic flux is confined in the Hick's branch. So it means that the monopole operator should be attached to the flux line generated by the vortex particles like this Hick's. Okay, so here the green, sorry, blue line is the flux by the monopole, sorry, vortex particle. And this red dot is the monopole operator. And we know that monopole operator carries non-trivial magnetic charge. So if the word line of this vortex passes through this operator, its magnetic flux should change like this. So we expect that this monopole operator connect two vortex solutions with magnetic charge N and the other vortex solution with magnetic charge N plus A, where this A is the magnetic charge for the monopole operator. So from the point of view on this one-dimensional theory on vortex world line, this operator can be considered as some interface working or domainal, which interpolate two different vortex solutions with magnetic charge N and N plus A. So monopole operators interface in one-dimensional vortex quantum mechanics. We can also combine several different monopole operators. Of course, they become combinations of the interface in the one-dimensional vortex quantum mechanics. So we'll try to construct gauge theory description for this combined system of the vortices with the monopole interface. Basically? Yes, what is the line? Why you restrict this equation? Sorry? Is this a generic separation? Yeah, on the hit branch. They should be attached to the flux line. And also we will define this theory such that all this line is localized at the origin on this plane, but I'll talk about it later. So without monopole operator, we know how to describe the dynamics of this vortex flux line using the one-dimensional quantum mechanics. Hanani and Tong proposed the brain construction for these vortex particles, and we can read off basically the gauge theory living on the vortices from the brain. So in this talk, I'll generalize their construction to involve not trivial monopole interface into the one-dimensional quantum mechanics. So once they have the one-dimensional quantum mechanical system, we can try to compute the partition function of this system. So we'll consider this computation again with vortex stage here with a charge n and another vortex stage here with a charge n plus a, and these two vortex solutions are interpolated by this operator v and a, and we'll compute the partition function of this system. Then it's clear this partition function will compute this correlation function of the monopole operator. And it turns out that this partition function or correlation function can be computed using supersymmetric localization. And once we compute this partition function, we can extract how the monopole operator is acting on the vortex stage. So we can extract the monopole action from this exact computational partition function. And our main goal is to extract the exact chiral ring relation or the clone branch algebra from this correlation function. I'll explain details about this one in the second part of my talk. And I'll also discuss the quantization of the clone branch algebra in our settings. So this is our line for my talk. I'll first discuss the classical geometry on the clone branch and also explain the subtleties which arise from the quantum event. And also I'll briefly discuss the x branch again and also vortex modelized space. And in the second part, I'll introduce vortex quantum mechanics constructed by Hananian Tong. And I'll explain how to incorporate the monopole operators or monopole interface into this vortex quantum mechanics. And then I'll compute the correlation function and show that we can actually derive the clone branch algebra from this computation. Questions so far? Okay, before discuss clone branch and monopole operators, so let me first explain the symmetry of the 3D and 4-gauge theory and also possibility formations. We'll mainly focus on the unitary-gauge group. So we'll consider unitary-quiver-gauge theory, but each node has unitary-tie-gauge groups and also fundamental hypermultiplane and two gauges nodes are connected by bifundamentals. And but mostly I'll focus on just a single-gauge node with a UN-gauge group. Apart from this gauge group and also archimetry-symmetry, we have the global symmetry. Firstly, the flavor symmetry, which acts on the hypermultiplane. So in our case, flavor symmetry, acting on the hypermultiplane, of course it actually only on the Higgs branch is the product of SUMF. And there are also topological symmetry. For each UN factors in the gauge group, we have the topological UN symmetry. It acts on the clone branch. And also this UN topological symmetry has a charge. The charge is nothing but the gauge charge, oh, sorry, magnetic charge of the gauge group. So the magnetic objects such as monoproloperator and vortices, they carry non-trivial UN topological charge. And this UN topological symmetry will be open to the nanobillion symmetry in the IR fixed point, but that is not today's topic. So we'll consider possible deformations of this theory. Firstly, for the flavor symmetry, we can consider method deformation. We can turn on the mass parameter for the flavor symmetry. So mass parameter, we can turn on basically three. We can turn on the real mass parameter and complex mass parameter. They are basically background the scalar field in the vector multiplex for the flavor symmetry group. So simply we can introduce these parameters by shifting the scalar field in the vector multiplex sigma by the real mass parameter and complex scalar field by, by complex mass parameter with appropriate flavor charges. And for the topological UN symmetry, we can introduce the pi parameters. We can again turn on the three pi parameters. We have the real pi parameter and complex pi parameter. They appear in the momentum constraint. So it's like shift of the momentum by constant failure TR and TC. We can also turn on, we can further turn on another deformation of the theory. We can turn on the omega deformation introduced by Negrasso for the UN angular rotation along the two plane. So this omega deformation can be easily introduced by shifting the complex scalar field in the vector multiplex by the reductive along the rotation. And with the parameter epsilon, epsilon is the omega deformation parameter. And in order to preserve the symmetry, we have a twist to this UN Lorentz rotation with the UN patterns of the Higgs-Brandt rotation. So the omega deformation is introduced by this combination of, shift of this combination for this scalar field in the vector multiplex. In our setting, we'll turn on the complex mass parameter and real mass parameter. So we'll basically break the equal symmetry to n equal two. So now we can talk about the chiral operators. And later when I discuss the Cologne-Brandt geometry, I'll just focus on the chiral ring vectors of the Cologne-Brandt. So let's discuss about the Cologne-Brandt. Okay. First example is the simplest one, Abellion theory. For Abellion theory, we know the exact geometry of the Cologne-Brandt. It's basically, we can construct the monopro operator using the elementary local field. And also the UN Abellion gauge field can be dualized to a local field. It's periodic scalar field A, which we call as dual photons. So the Abellion gauge field can be dualized into a scalar field with these relations. So this scalar field, oh, sorry. This scalar field is a periodic. And periodicity is given by the gauge coupling. Using this scalar field, and the scalar field in the back of the plate, we can construct the monopro operator in this way. This is the monopro operator with your topological charge plus or minus one. This operator carries your topological charge because basically the charge of the your topological symmetry is given by the flow, integration of the flow. So it is encoded in this periodic scalar as a shift symmetry. So if you look at this operator, this operator will carry the charge one or minus one for the your topological charge. And it is known that this operator belongs to the n equals pi times multiple. And involving this monopro operator, actually the classical Krillin branch is simply given by this relation. It's parametrized by the scalar field in the back of the plate. We have the, for UN theory, we have one back complex scalar field by. And also we have the monopro operators. They parametrized the Krillin branch with this relation. Classical in V plus one, V plus and V minus, they have inverse relation. So this classical Krillin branch is basically R three times S one, parametrized by the pi, sigma, and periodic scalar a. But this classical Krillin branch is modified by the Wonder Correction. And if we carefully compute the Wonder Correction, for example, for NF fundamental hyper-multi-plane, the exact Krillin branch is given by the scalar field in the back of the plate and monopro operators with this relation. And this geometry is a famous geometry called the A and singularity. For NF flavors, we have a F minus one singularity, but this geometry is deformed by the complex mass parameters. And indeed, this space is a hyper-calor as you expect from the specimen. So Abellion theory, we know how to compute the Krillin branch geometry exactly. Later, I will re-derive this relation using our approach. That would be the simplicity example which you can apply our methods and extract the Wonder Correction Krillin branch. However, for Nanabellion theory, it's more complicated. Nanabellion Krillin branch is still parametrized by the scalar field in the back of the plate and also monopro operators. However, we don't know how to de-align the Nanabellion gauge field like the previous example. Here, we knew how to de-align this gauge field and how to write down the monopro operators using elementary fields. But for Nanabellion theory, we don't know. However, on the genetic point, on the Krillin branch, where the scalar field phi takes the Nanjero vacuum expectation value, the Nanabellion theory will be broken to the its Abellion server group. So in the infrared, if we integrate out all the massive W-versions and charged matters, then infrared geometry simply becomes Abellion theory. Infrared theory simply becomes Abellion theory. So in the infrared, we can de-align the infrared gauge field and construct the monopro operator using its dual photon field A like this. So with a magnetic charge N, the monopro operator in the infrared can be written in this way. So in the infrared, the classical Krillin branch geometry is simply given by product of R3 times S1 divided by five groups. This description is reliable far along the Krillin branch when the scalar field phi takes Nanjero vacuum expectation value. However, it breaks down around the origin of the Krillin branch because of the quantum effect by the massless W-version and also the Nanabellion gauge symmetry is restored. So at the origin of the Krillin branch, in general it's very hard to study this geometry. So our motivation is to compute the Krillin branch chiral ring, which involves the full quantum correction on the generic point on the Krillin branch. And in order to compute this chiral ring relation, we'll study monopro action on the vortices in the Higgs branch. So it motivates us to study the Higgs branch and vortices more carefully. We'll focus on the UN gauge theory with Nf fundamental hyper-multiplex and we'll turn on the real F5 parameter. Okay. Then the theory with this to the Higgs branch. And Higgs branch is of this theory with the positive F5 parameter is a hypercalous phase. It's known as T-star grass mania and Nf. However, we are interested in the sub-manipold of the Higgs branch where we have the non-trivial vortex solution. Okay. This sub-manipold is called grass mania and Nf parameter by only one single complex color field X. And all other color field on this space should vanish. And each point on this grass mania manifold, we have the BPS vortices. We'll label each point by the letter U and we can find the BPS vortex solution which is satisfied by the following vortex equation. The first two lines are famous vortex equations and the vortex carries non-zero flux on the arc plane, arc plane is here. And flux, this is again localized almost around the origin and it takes exponentially if we go away from the origin. So we can approximate this solution as a particle solution. And also this solution with this to the each vacuum as special in paint. We'll consider this BPS vortex solution. We have a family of these solutions and they form a modular space. It's called the vortex modular space. I label them as M and U for each vacuum. And each complex manifold with a complex dimension N and it's a vortex charge times number of flavors. And this vortex modular space has isometry U1 times SN times NF minus N. This U1G corresponded to the Lorentz rotation around these two planes. And the other isometric group is the symmetry on the hexa branch. So we want to study the modular space of vortices but it's very complicated manifold. So in general it's very hard to compute, hard to study. So to make the problem simpler, we'll turn on the complex mass and also omega deformation parameters. And as I explained, they can be realized by shifting five with their parameters. Then it lifted the hexa branch and it's only finite number of isolated massive vacuum. For our case, the number of the massive vacuum is given by this number. And it's massive vacuum corresponded to the fixed point with respect to the equivalent gauge and flavor rotations. For each vacuum, we can find the vortex solution. And to lift the vortex modular space, we'll also turn on this omega deformation. Then we do this omega deformation vortex modular space will be again localized to the isolated fixed point under this flavor and gauge and also omega deformation rotation. So each fixed point or each fixed vortex stage can be labeled by vortex charge n and also the pavilion is the magnetic charge k. It's considered the n positive integer, positive or zero integer numbers. And there are some corresponded to the total vortex charge. So under omega deformation, we have the discrete vortex solutions with also massive formation. And omega deformation has a very interesting consequence. If you turn on the omega deformation parameter, it provides additional potential proportional to the angular momentum. So all the particles, they are attracted to the region. So effectively, omega deformation compactified our three-dimensional theory to one-dimensional quantum mechanics at the origin. Origin is the fixed point under this, what is it? You want jet rotation or ultra rotations. So instead of studying complicated three-digit theory, we can just focus on the one-dimensional quantum mechanics with the vortex stage. Then you can ask, what is the Hilbert space in this one-dimensional mechanics? And the Hilbert space of this one-dimensional mechanics is simply given by equivalent chronology of the modulite space of the half radius vortex stage. So Hilbert space is defined on each vacuum. So we'll label Hilbert space with the vacuum label u. And it's given by fixed point of the equivalent rotation. But for our case, the fixed point is nothing but the vortex stage labeled by integer number n and collection of integer number k. So our Hilbert space of this one-dimensional quantum mechanics is the collection of the vortex stage. And monopropylator action this vortex space, so it means it action our Hilbert space. Yes? Sorry, but you changed the, Yeah, that's right. It's just a change of the rotation. Yeah, for simple case, it's just bi-rotation, bi-rex-change, bi-permutation. Why is that complicated case if you are? For example, if you consider more generic quiver gauge theory, vacuum structure is more complicated. It's not simply the bi-rex-change, bi-permutation of some symmetry rule. In that case, Hilbert space becomes much more complicated. But in this talk, I'll discuss some simple cases. So let's discuss about the vortex quantum mechanics and the monopropylators. So vortex quantum mechanics has a nice gauge theory description with a monopropylator first. It was obtained by Hananyang Prong using the brain. So they first considered this brain system. It gives the low energy, the three-dimensional U-engaged theory with NF fundamental hypermultiplex. U-engaged theory is living on this ND3 brain and hypermultiplex are provided by this semi-infinite NFD3 brain. We want to study the Higgs branch. So we should turn on the epiparameter now. Epiparameter in this branch is corresponded to the relative distance between NS5 and NS5 prime along seven, eight, nine directions. But here I will turn on just one real epiparameter. So if we turn on the epiparameter, this computation will become this. So now the gauge symmetry that rotates ND3 brain is completely broken. And also the flavor symmetry that rotates NFD3 brain is again broken to the SUN, which rotates the N semi-infinite D3 brain and NF, we have the SUNF minus N flavor symmetry which rotates this set of D3 brains. So this brain configuration agrees with the dynamics on the branch of the 3D gauge theory. Then what is the vortex particles? It was provided by the D1 brain stretched between NS5 prime and ND3 brain. It provides magnetic particles to the 3D gauge theory. So if we want to have the charge N vortex solution, we can introduce ND1 brain. Then Hanai Tom's claim is that the dynamics of this ND1 brain describe the vortex dynamics of the vortices with the charge N. And since we have the brains, we can read off the gauge theory living on the D1 brain which describe the dynamics of the vortex particles in the gauge theory. And we have simple, this picture can be simply generalized into more complicated quiver gauge theories. For example, if we consider triangular quiver theory, they consist of several NS5 brains and number of D3 brain between NS5 brains. And from this brain construction, and we can go to the Higgs branch and again, we can introduce several D1 brains and try to read off the one-dimensional quantum mechanics which describe the vortex dynamics from the brain. So we can easily generate this construction to more complicated cases. And for this simple case, the dynamics of vortices is described by one-dimensional N equal to 2 comma 2 gauge quantum mechanics with this quiver diagram. So here we have U n, U small n gauge group with three chiral multipliers. First one is B, it's other joint chiral multipliers. And we have fundamental chiral multipliers Q and anti-fundamental chiral multipliers should kill that. They are also charged under the S U N and S U N F minus N flavor symmetry. So the symmetry of this quiver gauge theory agrees with the isometry of the vortex motorized space. And Hanayana and Tom conjecture that the Higgs branch of this one-dimensional quiver gauge theory agrees with the motorized space of the vortices as a complex manifold. So it means we can study some topological sector of the vortex motorized space using this one-dimensional quantum mechanical observables. So we'll focus on this one-dimensional quantum mechanics. Then what is the monopropylator in this quantum mechanics? So we'll consider this three-dig configuration. Now we turn on the omega deformation. So all the dynamics is localized at the origin. We have vortex solution here and another vortex solution here. It's interpolated by the monopropylator here. So in this one-dimensional mechanics, remember that the vortex charge is mapped to the rank of the gauge group. So in this one-dimensional mechanics, we can consider some monopropylator as an interface interpolating the U N vortex quantum mechanics to U N plus A vortex quantum mechanics. So it turns out that we have natural half-UPS interface in this quantum mechanics. And half-UPS more precisely preserves any closure to the spectrometry. And this spectrometry agrees. Oh, I mean, this quantum mechanics can be obtained from the 2D field theory by Dimensional Reduction. Yeah. Yeah, it's, yeah, yeah, yeah, in one-dimensional, there is no distinction. But this is just an analogy for the 2D field theory. But we have 2D-front-enical theory. We can have one comma one or zero two. They have 2D-front theory. So that's why I give this notation because I want to emphasize it. It preserves different spectrometry. So this spectrometry agrees with the spectrometry preserved by bulk monopropylator and vortex stage. Remember that vortex stage preserves the two comma two spectrometry. And the monopropylator preserves the zero four. So they are intersection preserves only zero two spectrometry. So it agrees with the zero two spectrometry of this interface system. So let me tell you the details. We'll consider a monopropyl interface, which connects m-bottices to m-prime-bottices. Let's first discuss the boundary condition. So the construction can be given by this one to three steps. Firstly, we need to specify the boundary condition for the bulk field. So in this case, we'll choose the Neumann boundary condition for both 1D vector and chiral multiplex. And it turns out that if we choose Neumann boundary condition, it induces n equal zero two, zero dimensional. Again, it's just an analogy, because in zero D, there is no chiral. So it's zero dimensional vector and chiral multiplex at the interface. So we have two systems meeting at the interface. So we have two such vector and chiral multiplex, zero dimensional vector and chiral multiplex at the interface. And we'll combine these two multiplex to some extra zero dimensional degrees of freedom. We'll add additional zero dimensional degrees of freedom psi, which is the chiral multiplex, and gamma eta eta tilde. So we'll introduce three bearing multiplexes. And this chiral multiplex is bifundamental on the U n times U n prime. And also bearing multiplex gamma is bifundamental. And eta is anti-fundamental, eta tilde is fundamental. We'll couple this zero dimensional extra degrees of freedom to the bulk boundary condition through the following superpotentials. In zero D, we have the three bearing multiplexes. So we can introduce different types of superpotentials. In zero two notation, we have j-type and e-type superpotentials. But we will turn off the all e-type superpotential and turn on only j-type superpotential in this way. And we claim that this simple construction gives the interface corresponding to the monopole operator, which interpolates two different multiplex solutions. Then how do you know it? If you look at the ground state of this system with the interface, then actually this interface provides a natural map from the modulite space of n prime vertices to the modulite space of n vertices. So there is an isomorphism between these two modulite space. If we divide this bigger modulite space by the corner of this bifundamental scalar field side, and it is known in mathematics as a HECA correspondent. So in mathematics, this map is called HECA correspondent, appearing in the geometric Langland program for the monopole operator acting on the gauge bundles in the x-ray. So our interface has exactly what we want to have the monopole action in the vortex quantum mechanics. So we claim this interface will realize the monopole operator properly. And now I'll give you another evidence that this interface is correct. So now we have the theory. We can try to compute the partition function. We'll consider the following configuration. So we put the one-dimensional vortex system on the interval between t1 and t2. And we give the boundary condition such that the system reduced to the, oh, sorry. For simplicity, I'll focus on just u1 gauge theory in three dimensions. So at t1, the system will give the boundary condition such that the t1, sorry, the system reduced to the vortex state labeled by n. And at t2, the system reduced to another vortex state with different charge n prime. And we'll interpolate these two vortex states by the interface which I explained in the previous slide. Then, obviously, this partition function will compute some correlation function for this operator with vortex state, n and n prime. And the type of this operator is determined by the vortex state. For example, if we consider n prime equals nk, then, obviously, this interface just connected the same vortex state. So we expect that this interface becomes 1. And it turns out that this interface actually becomes just simple-gauge transformation around this insertion point. And also, if we consider n prime equals n plus 1, we now have the two vortex solutions with different charges. So this operator will become the monopole operator with charge plus 1 in this case. And it turns out that this correlation function or the partition function of this tent can be easily computed using the localization. And this is the result for the simplifying to u1, three-gauge theory case. Here, omega n is the equivalent weight for the fixed point or the inverse of the equivalent volume contribution to the vortex partition function. So for the identity operator, if we plug n prime as n, in this formula, we get the correlation function of this identity operator. It simply becomes one of the equivalent weight at the fixed point. We expect that this correlation function will compute, because always now identity operator will compute the norm of the vortex states. Indeed, one of omega n, omega n is the equivalent weight at the fixed point, is exactly the correct normalization for the vortex states. And for example, if you compute the 3D vortex partition function or three-dimensional partition function lying on the interval, and if you give the Neumann boundary condition for both ends, then the theory effectively becomes two-dimensional theory. And we can compute the vortex partition function. And if we compute this vortex partition function, it's simply the norm of the Neumann boundary condition, then you get this result. And because it's a work of the Neumann boundary condition, we can expect that this partition function should be some of the norm of all the fixed points. So by looking at this relation, you can see this is our correlation function. Correctly compute the contribution of the fixed point to the vortex partition function, which is basically the inverse of the equivalent weight. And we can also compute the correlation function of the monoproprator with charge plus minus 1 by setting n prime equals n plus 1. Then our correlation function will compute these correlation functions, both for v plus and v minus, with charge plus and 1 and minus 1. And actually, we'll compute just these two correlation functions for these two elementary monoproprators. Because for U1, and actually for unitary gauge group, the elementary vortex action is enough to build other complicated vortex actions with different magnetic charges. OK, so this correlation function is enough to capture all the monopole actions on the vortex stage. And we can actually construct the Flung Branch algebra using only this correlation function. So let's compute the monopole action and also Flung Branch algebra from the correlation function condition. We know that we expect that monopole operator v will bring one vortex state to the other vortex state with charge plus 1. And also, we expect that monopole operator with charge minus 1 brings vortex state to another vortex state with charge minus 1. So we expect that this relation holds up to some unknown coefficient of c plus and c minus. And we can easily compute this unknown coefficient from our correlation functions. If you send it the final state from this relation here and here, then you can easily compute c plus becomes a simple ratio of the correlation function. And similarly, c minus is again given by ratio of the correlation function. So we can explicitly compute the monopole action using our correlation function. This is the exact result because we computed it using the localization. And furthermore, the 3D scalar field in the vector multiply, it takes an angelic vacuum expectation value when you act it on the vortex stage. So in the 3D theory, we have this coupling, phi and, for example, x. Actually, the hypermultiple scalar field and phi, the complex scalar field in the vector multiply. After deformation, it's deformed in this way, plus m i plus epsilon del g. So vacuum solution should solve this equation. And with the vortex charge n, x is a degree n polynomial. So you can see phi solves this equation if phi takes this. So bulk scalar field takes this expectation value when you act it on the vortex stage. So by combining these actions of the Cologne branch generators, it can be summarized in this way. So the monopole operator v acts on the vortex stage with this coefficient. And this coefficient is simply the degree nx polynomial of x graded by the mass parameter m. And v minus acts on the vortex stage, but it brings just to the vortex stage with charge minus. So this is the exact monopole action we computed using correlation function. And from this action, it's very easy to compute the Cologne branch generator. But firstly, I'll add one more comment. As you see, the Hilbert space, they are just collections of the vortex stage, is generated by acting monopole operators repeatedly. So all the state in the Hilbert space can be generated from the vacuum state by acting the monopole operator with charge 1. So it means our Hilbert space is the highest way for my model of the Cologne branch algebra. And what is the Cologne branch algebra? It's easy. From the monopole action, you can easily derive this Cologne branch algebra. And now the operators don't commute to each other. For example, v plus, v minus is not the same as v minus and v plus. So our Cologne branch algebra is quantized with Planck constant proportional to epsilon. So our epsilon parameter, which corresponds to the omega deformation parameter, quantized our Cologne branch algebra. And it has name. It's a spary-carbational challenge in your algebra. Can we go ahead and just run through it? Yes. It's for rank 1. I'll give you any rank in the next slide. I have more complicated solutions. But in this talk, I'll just discuss the 3D gauge theory with one gauge node, but rank 1. This case is rank 1. So we computed the complicated Cologne branch algebra using the correlation function and the vortex quantum mechanics with the interface. And if you send the epsilon to zero limit, then this relation is nothing but the Cologne branch geometry we computed using the one computation. So it reproduces our expected Cologne branch geometry. And the reason why we get the non-competitive algebra is very simple. So before we introduce omega deformation parameters, the Cologne branch operator, they can move around freely. So they can exchange their position arbitrary freely. So their algebra becomes commutative. But if you turn on the omega deformation, then because of the extra potential of this omega deformation, all the operators should be aligned along the flux line or the origin. So if you want to exchange their position, then they should hit each other. There is no way, because they are aligned on the flux line. They should hit each other, and they encounter the singularity. It means that their algebra becomes non-competitive. And also our correlation function captures this non-competitive activity of the Cologne branch algebra. Yes, yeah, yeah. It's a branch. Yeah, it's a bit. Yeah, in the. Is there a guarantee that the Cologne branch operator, they are defined in UV? So they don't care about the vacuum structure. So this structure should hold for the other vacuum and any other vacuum. Well, it's just that the Cologne branch operator has a coupling function. That's right, but we just derived the Cologne branch chiral ring. So chiral ring doesn't depend on the gauge coupling, OK? That's why we could compute it. Well, I think. So this is the approach to work for Cologne, but you can't compute all the same. No, no, no, no. As I explained, this one can capture only a whole big part of this Cologne branch geometry, not the details. So lastly, let me give you the more general example. We'll now consider UN gauge theory in bulk 3D gauge theory. So now we have the vortex state, labeled by the vortex charge N, and also abelianized magnetic charge K with N-sets. Then the monoproloperator we have again, the elementary monoproloperator we have, it's again N monoproloperator. And they act on this vortex state, and it increases the eighth element of this K-sets by 1 with up to this coefficient. And similarly, monoproloperator with charge minus 1 at I-s, can I note, not note, I-s element will decrease the I-s element of the K-integers, N-integers, up to this coefficient. And again, the vector-multiplaced color field takes non-zero vacuum expectation value at the fixed point. And this algebra was computed again using the correlation function, or the partition function of the 1-dissident with the interface. So we can compute the exact monoproloperator action on the state using our correlation function. And from these actions, you can easily derive the Cologne branch algebra. Again, it's quantized, V plus and V minus, they don't come in by the epsilon element. And if you take the epsilon to general limit, then this algebra actually reproduced the conjectured algebra by these people, Blimo, Timothen, Gaillot. So for the general case, we can again compute the Cologne branch algebra using our quantum mechanics with the interface and each partition function. Let me conclude. So we tried to compute the exact Cologne branch algebra using correlation function computation. To do that, we constructed the monoproloperators in the one-dimensional quantum mechanics and computed each correlation function. And it allows us to compute the exact Cologne branch Gaillot ring. And our work can be easily generalized to more general cases. In our paper, we also computed Cologne branch algebra and monoprolections for the triangular cubic edge theory, which appears as the theory living on the defect of the 4D gauge theory, sorry, surface defect. And if you compute the Cologne branch algebra, it turns out that it's simply, it's nothing but the finite WN algebra. And this computation also explains the finite AGT correspondence appearing in the 4D gauge theory with the surface defect. And also, as I explained at the beginning, we computed the same Cologne branch algebra using a completely different approach called modulite matrix approach. It's like this approach is basically solving the vortex solution using the Cologne beam matrix called modulite matrix. And the monoprolector acts on this modulite matrix as a column of big singular gauge transformation. And we can compute each action using similar localization. But these two approaches are completely different. And their result, of course, are greater. So in our paper, we proposed a two different approach for the monoprolectors and the Cologne branch Caillot ring. And you can also consider the following generalization. You can consider general unitary quiver gauge theory. So the Cologne branch algebra of general unitary quiver gauge theory was proposed by these people. Actually, our computation just confirms some part of this general unitary quiver gauge theory. For example, for triangular and also linear abelian quiver theory. But we have more complicated theory. But complicated theory, they have very complicated vacuum structure. So in general, we don't have the general prescription how to construct the vortex quantum mechanics and also the interface yet. But it would be very nice if we can give some general prescription for it. And we can also consider hydromanage generalization. For example, in 4D, this operator will become a tuft operator. And in 5D, it will become a monoprolectoring operator. So we can try to apply our method to compute these non-trivial defect operators in high dimensions and their non-trivial algebra between these operators. And we can also try to consider the generalization to the theory with less sparsimetry, for example, n equals 3 and n equals 2 theories. But at this moment, I don't know how to do it. And also, extension to other gauge groups would be interesting. But problem is for other gauge groups, for example, SO and SP gauge groups, we don't have F5 parameter and we don't have VOTAC. So our method actually uses the Higgs branch and the vortex stage and the monoprolections and the vortex stage. So this method cannot be applied. But probably, this modeling methods approach can be applied to get the exact chiral ring for the other gauge groups. Let me stop here.