 Tak, imam dobro. Tak, bo to se vsin tudi na vse zasrednji striat. No, to je poslpo. Kej strat. V potrebenih teorijstvih navah. Spročam za stabilizacije vsepunja lahko. Zato sem nemelite stojev. Zato zato smokedim. Iz njegi tega je v podstavnom viseku T, če sem bilo kvalitva protekljena, že je najzavršen vse, kako če se bo nekaj, zato je tvoje nekaj, če nekaj je kaj toh, če je nekaj, či veče, če je časnič se mene. kako je veto se evaluate, ki je tako D-6, na razredniht, ko ne imač, nam je nijakimi nekaj prameč mogu čestiti ledo z theta in infinity, theta je 0, theta je 1 in theta je sub t. In kar iz ngupča, v čarbi podvisuje za mene izamonodoroví vborg. Izminodarbitva informacija o bjermorfičke vsega, bjermorfičke vsega, bjermorfičke vsega na sveče, z dvej punčev. 0, t, 1 in infinity. Zato je to zvrstko v vsega formu. A, a, i, zb, 2 by 2. Zato je to sovrstko vsega svetlja. Izminodarbitva problema, As a monodromid deformation problem is the problem of finding the dependence of a i's of z i's. So as you move the positions of the poles, you want to preserve the monodromid data of this connection. So you want to preserve it as a topological flat connection. So it means that as you vary the parameters, and so here I chose already some gauge for the global nubism invariance, in which my points are z1 is 0, z2 is t, z3 is 1, z4 is infinity. But you can... You may choose not to work in this gauge. It's in this gauge that you will see the connection to pendeve. And so we want to make the residue of our veromorphic connection. So a i's, the residues are so they're traceless because it's an SL2. And we fixed conjugacy classes. So we fixed the eigenvalues, which I would choose to be generic, and to be equal to the parameters theta from the previous blackboard. So to preserve monodromi means that as you vary this connection, and when you vary it, so you vary not only the positions of the poles, but also you allow to vary the residues while preserving the conjugacy class. So you want this to be compensated... So you want to be able to compensate it by a geisha's formation. So to find some function epsilon, so epsilon depends on z, also in a meromorphic way, the variation is just the commutator of Nabla with this parameter epsilon. And by matching the singularities, it's easy actually to find that epsilon is, if I'm not mistaken, simply negative a i minus e minus e i. So that implies that the matrices evolve according to the so-called Schlesinger equations, which I forgot to write down. Anyway, so this is something like commutator a i a j, if i is not equal to j, a negative sum. So from that moment on, you can forget about the sphere with punctures. You can just focus on a bunch of matrices with fixed eigenvalues and study the evolution in the space of matrices. Now that space of matrices, so the space of a i's fixed thetors up to overall conjugation, so it's a constant geisha's formations. So we simultaneously conjugate all residues by the same s-autometrix. This is actually a complex symplectic manifold, and the sum of i's is equal to zero, of course. Thank you. It's because of the sum of residues. So this is a complex symplectic manifold, which is, in fact, so what I wrote here is a symplectic quotient of the product of co-joint orbits, or adjoint orbits. Co-joint orbits makes it obvious that this is a symplectic manifold by the diagonal action of the group, as so too. So each orbit is a kind of a complexification of the sphere. One should, when taking this quotient, one should impose obstability conditions, which I will not talk about for the second. Maybe I'll talk about it later. So this is a symplectic manifold, and you have some evolution equations of a symplectic manifold. So you may ask, the first question you may ask, whether these are Hamiltonian equations, and indeed they are, indeed dZi aj is a Poisson bracket, where the Hamiltonian hi is essentially the residue of trace squared. Well, what it means is just take this guy squared and compute the residue near, at the point z equals ti. This is the sum over j not equal to i trace aj aj. So these are remarkable Hamiltonians, because they obey not only the Poisson commutativity, but they also integrable in the sense that you can form a kind of a symplectic flat connection with spectral parameter, like a lambda connection. So these conditions imply, in particular, that you can integrate them to some generating function. And so this function is usually called the isomonodromic tau function. So it's function of, right, it's a function of the, on this symplectic manifold. Well, it's a function, how should I say it? So it's a function on x and the space of parameters zi. This delivers the same del i and del zi. This is d by d zi. So, but you can also think of this formula in, so you can also do the usual Hamiltonian Jacobi story, namely choose some polarization, so choose half of the coordinates on x, and just try to solve these equations in Hamiltonian Jacobi form. So then, so choose, first of all, need some Darbou coordinates on that space and write various choices. So first of all, if n equals to 4, which is the case, which I'm considering here, this is actually a surface. It's a, well, each orbit is too complex dimensional. You have 2 times 4, that's the dimension of the space here, and you do the Hamiltonian reduction, symplectic portion respect to sl2, which has dimension 3. So you get two dimensional space. It's a surface, which can be actually explicitly described. So actually it's a cubic surface in c3, and this c3 is, well, there are two ways of thinking about it. So one is to look at the residues, try to parameterize the space by the residues, and then think of each residue. So each residue in itself is, so it's a 2 by 2 matrix with vanishing trace, so I can write it in this form, so the zero trace condition is obvious, and then the eigenvalues are found by you compute the determinant, so you get, so you can think of the residue itself as of a vector in a three dimensional euclidean space, complexified euclidean space, of fixed length. Now the condition that the sum is equal to zero means that they form a closed polygon, and now the fact that we divide by the action of sl2, sl2 acting by conjugation simply rotates the coordinates x, y, and z simultaneously. And so it means that you look at the space of closed polygons, quadragons, quadragons in a three dimensional euclidean space up to translations and rotations, and so that space is parameterized by the length of, for example, by the length of two diagonals, so that's two parameters, and one choice of Darbouk coordinates is to choose the, you can choose the length of the diagonal connecting the first and the third points, so this will be essentially, and the angle, so the usually euclidean angle between the triangles delta one, two, three, and delta one, three, four. But, well, this description somehow doesn't know about the fact that we want to build the connection out of this residue, so it's too flat, and so instead you can map, you can do a nonlinear transformation, a transcendental map, so compute the monodremies of this connection, the monodremies around various pairs of points, and so here you have, it's actually three non-trivial loops, out of which you can generate everything, and so the traces of monodremies around those loops will be the coordinates in this three dimensional space in which this modular space is a cubic surface. Now, since there are only two parameters, there is a relation between these three monodremies, within the conjugacy class of these three monodremies, and so you can find another system of Durbo coordinates which is based on that description, so it turns out that that's essentially, it's a curved analog of this picture, where now instead of the flat space, the quadrangle will be sitting in the group SL2, so here it sits in the algebra, and instead you can, by looking at monodremie data, you'll get the same quadrangle in the group, and then you use a spherical geometry instead of the Euclidean geometry to describe the Durbo coordinates. Anyway, so, whatever Durbo coordinates you choose, so let me call them P and Q, so then, when you divide by, when you fix the gauge for the conformal group, and you have essentially only one evolution parameter, so the position of the second point, when the other three are fixed, then in the Durbo coordinates your equations will have the usual form, so out of four Hamiltonians, only one is independent. You can check that, of course, these functions obey three obvious relations, so out of n Hamiltonians you essentially have n minus three retrieval ones, so let me denote the one, which is retrieval by letter h, so then there is a third choice of Durbo coordinates, which is related to what is now called Schalian separation variables, in which these equations will precisely match the Pennevistik's equation. On the other hand, choose to the Hamilton-Jakobi formalism and then you will describe your classical mechanical system by generating function, which will be function of q and time, and so, and instead the equation will have the form ds with respect to dt equals h of ds with respect to dq, q and t. And so that's the form of the equation in which the tau function appears, so this s is the tau function. So this s actually depends not on, it doesn't depend on all coordinates, it depends on only on the half of the coordinates on the phase space. And finally, instead of, so as I said there are many choices for coordinates, what usually people do when we define this tau function is that they choose as the coordinate q the eigenvalue of the monodrame surrounding the point 0 and t. And so tau function in the presentation where q is logarithm of the eigenvalue of the product g1, g2. So here, this is my points 1, 2, n. I have some, I choose some base point and then define the basis in the fundamental group in the corresponding monodromes. These monodromes around the loops which go straight to the points and then make a small circle around them. So these are the monodromes of this metamorphic connection along with these base loops in the fundamental group. All right. So that's the reminder about the tau function of Penlivé. And the interesting fact which was observed in Conjecture at first, I think proven sometimes later in 2012 by Gamayoun, Jorgof and Lisové is that this tau function has a representation in terms of conformal field theory of a rather explicit form. So I should say that back in the 80s Sato, Jimbo and Miwa have already observed the connection of this tau function to some conformal field theory. I mean, they were studying. And even earlier, of course, McCoy and Wu found that the generated versions of Penlivé 6, like Penlivé 2, Penlivé 3 emerge in the studies of the correlation functions in the Isid model in the scaling limit. But this result was so explicit that it prompted some explanation. So I'll sketch the formula. I'm just quoting it from the paper, using the annotation. So the point is that this is a conformal block of c equals 1 lival field. So this is a conformal block of c equals 1 lival field. So this is a two-dimensional CFT. And so it can be represented as a sum of pairs of young diagrams. So these are some explicit rational functions of the shape of a diagram and of the parameters. So what are these parameters? So this vector theta is this four eigenvalues, which we had before. Sigma zero t is this additional parameter which is the parameter q in my story about tau function. So this is the monodrome, essentially the trace of g1, g2. So it's a monodrome around the points zero and t. Precisely. It's twice cosine 2 pi sigma zero t. OK. And so what is this conformal block? It's a conformal block for the four-point function. So in lival theory, you study the correlation function of four primary fields with dimensions delta zero. And so these dimensions are essentially, so they are quadratic expressions in thetas. And then you look for conformal block where in the integrated channel you have a momentum sigma zero t flowing. So that's the data which you need to fix. And so the conformal block depends on this four external momentum or the conformal dimensions. And the one which flows in the middle and of course the parameter t. And now what you do, you shift this intermediate momentum by an integer. You multiply by some quadratic, exponential quadratic form of this integer and multiply by an additional factor which is this ratio burns two gamma functions. Gamma two functions. So this is a four category. These are two gamma functions. So s, the s parameter is, well, you can also compute it. So this s is essentially the exponential of the double coordinate which is conjugate to q. So in this picture, now with curvilinear quadrangle in the group SL2. So sigma zero t is the length of the diagonal connecting the first and the third points. And p is the dihedral angle between the two curved triangles which this diagonal defines. So it's a function of, how should I say? So on the full modulus space of light connections this is an independent parameter. But once we sit on the trajectory of the Penelab equation with fixed initial condition, p will be a function of q. So p will be essentially the derivative of the Hamilton Jacobi potential with respect to q. So that was the formula. And it was kind of, it was a very strange formula because, well, they verified experimentally and then indeed it was proven by several groups. One of the proofs was by Bernstein and Shechkin, I think using representation theory. And another proof, I think, this gentleman with collaborators proved using relation to Fredholm determinants. Anyway, these are hard proofs. But when you look at this formula and if you come from the from the formational gauge theory and you know that these objects are actually instead of partition functions of some four-dimensional gauge theory and these Bern's gamma functions are the perturbative parts of this partition function, you start realizing that this is some, there must be some gauge theoretic explanation of that. But what is surprising about this formula? The surprise is that c equals one conformal block is something which is kind of very quantum in the language, in the sense of conformal field theory. While this tau function, so this s function, is, since it's defined by in the classical context, so it's a classical object and this is kind of quantum. So it's a classical limit, so in some sense it's a limit, it corresponds to limit of instant conformal central charge in some c of t of something which I will now recall immediately. And on the right-hand side has c equals one, which is not classical in any sense. So what's going on? So this relation between the classical and quantum is always, is very interesting. So in what context would you expect something nice to come out of the partition function of gauge theory in sum over some integers and multiply by exponentials of quadratic expressions? So where does it arise? So first of all, so the b object, in fact b times c essentially, well times some ratios of gamma functions is the supersymmetric partition function of n equals two d equals four gauge theory in this case of gauge theory with group gauge group SU2 and four fundamental flavors. So the thetas correspond to the masses of this, so we have four mass parameters and four thetas, the relation is not, the relation is kind of pairwise so theta zero plus minus theta t and theta one plus minus theta infinity correspond to the masses of 1 and 2 and 3 and 4. So this theory is usually represented by in the quiver form and even though from the gauge theory point of view there is all four flavors are unequal footing in matching with conformal field theory it's convenient to split these four flavors in two and so that splitting would correspond to the choice of specific identification, specific channel for the conformal block and so the t parameter is usually called q slash this is exponential of the complexified gauge coupling now sigma zero t is usually called a this is the Coulomb parameter so it's a scale, it's an expectation value it's the eigen value of the expectation value of the scalar in the vector multiplat and finally this partition function has two additional parameters epsilon one and epsilon two which are quverian parameters for the group of rotations of r4 and to get c equals one you need to impose the condition that the sum is equal to zero that's c equals one so when you translate the right-hand side of gamma u in the orga in this formula in gauge theory language then all these mysterious gamma factors and so on they simplify so the right-hand side is simply the sum over integers exponential n essentially dw respect to dA I will explain what w is times Z a plus n let me recover the epsilon dependence so let me call it epsilon so the masses are intact the q parameter is intact so I think that's it where w right where w oh yes sorry let me put it like this epsilon negative epsilon and w which depends on a, m, q and epsilon if you like this is a limit when epsilon one goes to zero epsilon one logarithm of this partition function when I take the partition function for general values of quverian parameters and send one of them to zero so you see there are two ingredients in this right-hand side one involves epsilon one equals minus epsilon two and one involves the quasi-classical limit when one of the epsilon is set to zero and the claim is that when you sum over these things can I am missing something something something yes well there should be so this parameter q which sorry I think I'm fine something is missing good let me put it just explicit shift by the monotomy data so this is some kind of rearranging of the terms in this formula so it looks already simpler in the gauge theory partition function you shift the Coulomb parameter and you multiply by some by simple exponential which is linear in the syndrome you saw so it's like kind of Fourier transform in this formula z both the z and w include the one loop factor yes yes yes everything yes yes so these bars functions are the one loop contributions alright and so and that the claim is that this is this so the tau this tau function is s is a function of everything so now q q without slash and everything else is equal to the sum ok so how do we understand this formula in that formula epsilon was set to one so in this formula epsilon is set to one so here the point is that this partition function is about top stairs oh oh sorry forget about that epsilon this is from a t2 two torus this is alright so let me now explain what is this tau function in what so I said this tau function is a classical object it's a classical limit of something so what is this quantum guy the quantum guy is of course the kinesiomological equation in fact I should say that there are two interesting equations in CFT so this is the equation in kind of wzw with the group SU2 conformal field theory but you can also you don't need to have a full conformal field theory, you can just look at conformal blocks so it's an equation for conformal blocks of SL2 current algebra which depend on the level of this the equation reads psi so this is the conformal block but I will write it as a correlation function for n vertex apparatus in our case n equals to 4 so this vertex apparatus they are labeled by the representations of SL2 which have some otherwise by spin which I will take to be a complex number and so this it takes values in the tensor product so that we call the vertex apparatus by v hat and belongs to the tensor product of these representations you can just take the verba modules for example you take n minus 1 so you take one verba module and the rest usual verba modules and then you look for the invariance the SL2 invariance in the tensor product so the equation is self-explanatory the matrices so the TIA the generators of SL2 acting only at the ith factor and the remark will think about this equation that it's consistent for any value of k so again it's a flat connection with spectral parameter so the right hand side you can view as an operator so this operator commutes for different i's and they also base some integrability condition with respect to zi's so in the limit when k goes to infinity so if k goes to infinity and simultaneously the spins go to infinity roughly as k times theta now these generators of so in this limit you can think of representation as being the result of quantization of those orbits which are erased and so in the classical limit the generators of SL2 will become functions ordinary functions on this quadrant orbits and taking the invariance in response to doing the simplex equation as we did before and so in the limit these equations will produce the Schlesinger system where now a's so these matrices ai the residues are essentially the quasi classical expression for the generators of SL2 so in other words if you realize if you realize SL2 by differential operators so then psi will be some kind of function of n variables so being certain invariance condition then in the limit when k goes to infinity this function will have the form so it will exponentiate so they will become the functions where Pi is the usual momentum and so this is just another parameterization of capital X, capital Y and capital Z in my previous formula so this is now ai has the form so this is yet another set of Darbou coordinates before you take a simplex equation in respect to SL2 then you need to impose the constraint that the sum of these matrices is equal to zero and so in that limit so this s tilde is essentially the Hamiltonian Jacobi potential and that's essentially the tau function of pen level 6 when n equals to 4 so the statement of Gamayun in this way is that of of the current algebra of a block so this is SL2 hat is expressed through the quantum c equals 1 we are sort of conformal blocks so these are the Liouville conformal blocks also known as we are sort of conformal blocks so that's the statement so how can I understand the statement on the gauge theory side well we need to map the conformal blocks of current algebra to some objects in gauge theory and that's has been done as a result of many many years of works of many many people but it actually goes back to the old idea of late sir Michael idea which relates the 4 dimensional gauge instantons and 2 dimensional siga model instantons so so the key is that this psi which is the conformal block of the current algebra is related to the surface defects in 4 dimensional gauge theory which in turn is a kind of a uplift so it's a 2 dimensions a higher uplift of the so-called Wilson points in 2D Young Mills theory so let me remind you that so in 2D Young Mills theory you basically study flat connections on the Riemann surface or constant curvature connections and in interesting observable you can define by allowing your connection to have a singularity at a point and you fix the conjugacy class of monogamy around this point and so instead of the equation the curvature being 0 you get the equation and the curvature is now the sum of delta functions with some residues this is for several points so these Js are what I called they relate to this A's which we had before so it's a different modular space modular space of connections with singularities but if you look at the underlying holomorphic bundles you can sort of project one modular space into another and then if you integrate the probolic bundles to ordinary bundles and so if you take the class of 1 on the space of probolic bundles and take the push forward that's the Wilson these are the Wilson point observables so the natural idea is now we add one more dimension to the problem and so we think of let's say holomorphic maps of an auxiliary space into the modular space of flat connections so holomorphic bundles on the remain surface and if we allow now singularities at points we'll get what is called the surface defect in four-dimensional theory so it means that you allow so this is a surface inside the four-dimensional manifold the the gauge field to have a non-trivial holonomy around the small circle surrounding the surface you fix so the conjugacy class of monogamy is fixed but representative is not and so you get a map from the remain surface into the orbit of the corresponding conjugacy class and so it's a kind of a coupled 4G gauge theory plus 2G sigma model so the gauge theory with gauge group G and this is a sigma model on some space with G symmetry so on G space in the simplest case this G space is just the orbit the adjoint orbit but if your theory has a major field then there will be some decorations of that orbit so the gauge group will be compact and so this G space but the G space in that need not to be compact so for example it could be a complex going to orbit on which G still acts and so that's what typically so what happens I will not tell the full story but so just so now technical so it's hard to work especially if we want to be able to compute something explicitly because usually so that's the case when we compute something explicitly is when the group G is unitary and we replace the instantons and corresponding to the homomorphic bundles by torsional free shifts and so we need to define the notion of the analog of this when in the bulk you have a shift not a bundle with connection and so the simplest way to do this is by doing orbitals so you impose some orbital structure so that on the quotient so the local model is that you take ordinary shifts on C cross C with some action let's say ZN action and ZN also acts on one of the factors so that as a algebraic surface you can project it on the quotient which is still as a morphic to C2 but your shift will have now have some proboric structure along the fixed locus of the combifold action and so that gives kind of much more explicit handle on the corresponding surface effects and one can prove rigorously and so using the theory of that the partition function of the corresponding surface defect obeys kz equation with k plus 2 being essentially the ratio of this is for general rank and essentially being the ratio of the equivalent parameters now equivalent parameters are not on equal footing because they have one direction so this is the direction along the surface defect and so here we have equivalent parameter epsilon 1 and epsilon 2 is a parameter transfers so this is in the transfer space so sorry, so I want to be rigorous so who is so first of all you are in sq2 case now or in general this is for sqn for um and what is your theory what kind of method do you allow so this is for any theory any quiver gauge theory so the theory which we need for the purposes of this talk is the theory fundamental matter become formal gauge theory do you want it to be, I mean do you need it's car formal, yeah it's asymptotically car formal well you don't need it is fewer gauge theory for example yes that's a limit pure gauge theory is much simpler and the parameters zi correspond to what so zi is still gauge coupling the new parameters which surface defect has which come from the fact that the sigma model has its own instantons which correspond to so let's call this g space y so we have additional we have additional instanton counting parameters which let me call x which will be mapped to the essentially x variables which will be the arguments of my car formal block so they come from this orbital structure that when you study the when you do the orbital the instantons become fractional so the instanton charge fractional lines so you have couplings for each representation all fundamental representations v i's here so this is so here in SL2 case these are just verma modules this is for SL2 for SLN so for SLN not every car formal block of SLN maps to something which has a Lagrange description so you choose two verma modules and the rest should be the kind of minimal so there are some minimal presentations which in SL2 case they all essentially the same sorry q should get points on sub c but it's not in space standard sorry c c c what's c q should get points z i's so that was Sasha's question so z i's so remember so if n equals 4 so we have only one parameter in finica so that's the parameter which counts instanton so this instanton counts in parameter in space standard position no space standard position no this is some kind of hidden hidden direction no 6 5 can I ask this homogeneous space that you denoted y what is it in some examples so in the case of this theory with matter it's a total space it's a total space of several copies of tautological bundle over p1 p1 being the SL2 or the orbit of SL2 so in general that's for the orbital case that's the translation so the orbital is the way of representing that story and what determines why is the way I need to specify the action of this secret group on the framing data of my shifts at infinity so I need to specify the representation of that in some vector space and then you have multiplicities in which the your useful representations of secret group appear and this multiplicity is determining the kind of partial flag variety you'll get for why so this is the question about what kind of surface defects we fix you don't necessarily fix the maximal surface defect not necessarily but here you fix the maximal surface defect so-called regular so the regular one n is the same as a rank of gauge group and you choose a regular representation for the but since we have a theory with matter I also need to specify what kind of matter represents so how the matter feels split that's how this choice of 2 plus 2 is made so you have four matter fields but you have only two irreps for Z2 so you choose two of them to be corresponding to the real representation to be corresponding to the number of representations in other words you're really working with quiver theory so you're working with maximal surface defects for a quiver theory and you're saying that for a quiver theory there's a way to leave the surface defects to matter surface so bottom line bottom line surface defect partition function base kz equation and this is the identification and so the limit when k goes to infinity which was the limit when we needed to get the plane level equation is the limit when epsilon 1 goes to 0 which was also the limit I used I needed to define this w function but it's not the limit so it's not the case for the z function so one more ingredient is missing and that's ingredient I'm going to present now so that's the so when things don't work you blow them up so so the idea is to compare gauge theory on two different on two spaces so you take gauge theory on c2 with blown up point the origin which is the origin of my rotation and so in addition to doing just the blow up it's already actually for the sake of partition functions it's already fruitful exercise to compare the partition functions of gauge theories on these two spaces and from that you can derive some identities of the form which will make it obvious now why I'm talking about the blow up so there is a bilier relation where you shift the Coulomb parameters of the partition functions by so in general if this is not as u2 at this un you have now n parameters capital n parameters here vector will be essentially point in the lattice of product so it's a bilier relation so z is equal to z star z so explanation is the following that if you think about this blown up space historically so it's a toric manifold and the toro section which descends to the toro section on the base has two fixed points on the upstairs and so the contribution of each fixed point is one of these partition functions except that now in addition to the instantons which get localized near the fixed point you also have invariant toric line bundles so these bundles which have exceptional some multiple exceptional divisor as divisor and so that's what you're summing over the lattice of abelian magnetic flux is going through the exceptional cycle so that's one relation but it's not enough for our purposes because we need surface defect so the idea is now to add the surface defect which will be extended along one of the coordinate axes and using the same ideas as Nakajima Nishiova used to derive this relation now you'll get relation of this form now my surface defect depends on the parameters which were these instanton counting parameters for the two-dimensional sigo model and so now I have a relation that psi is equal to psi star z and so this is a relation now which will relate conformal blocks of the current algebra of some generic level k conformal blocks of virocero algebra of some central charge k and conformal blocks of current algebra of the different value of k because now you see epsilon parameters get shifted so k gets transformed in some interesting way essentially gets shifted by 1 I guess and now in order to arrive at the relation by Gamayun, Jorgov and Lisuvay to take a limit epsilon 1 goes to 0 so then both psi's so psi's and then so they'll produce the Hamilton Jokobi potential I should have said that since that was Sasha's question I'm talking about the theories which are symptomatically conformal so there is a scaling symmetry under which my particular functions are homogeneous namely it says so the parameters a, m and epsilon they have all dimensions of mass so the scaling symmetry which scales simultaneously a, m and epsilon and under the scaling z is variant so then and the same applies to psi so I still will have homogeneity 1 under the scaling symmetry so I can actually reset after that I can scale epsilon to 1 put 1 here and now so the left hand side has this exponential form the right hand side has the form so the right hand side has this form plus m you see so epsilon 1 is now a very small parameter so my psi is exponential of something divided by epsilon 1 times s but the arguments are shifted by something small so that will produce the derivatives of s with respect to parameters and these derivatives will be multiplied by n epsilon 1 or by epsilon 2 because here my epsilon 2 is shifted by epsilon 1 as well and so the leading asymptotics will cancel precisely they will be identical on the right hand side but the subleading piece will give me a non-trival identity ok, I hope I didn't I haven't forgotten anything and so that's the essentially that's the identity which gamma uniorga for this we have so that's the exponential formula and it's really just a limit of the blow up equation in the presence of surface defect so I'm sorry I'm out of time and it's lunchtime so I should probably wrap up so in other interesting application of the study surface defect is the explicit expressions for the wave functions for the eigenfunctions of of quantum integrable systems like ellipsical azure and mozor or gaden but it will unfortunately also take more time to present them on another occasion so since I am the last speaker at the conference I would like to thank all the organizers for and all the speakers except for giving interesting talks there are questions there will be questions so there is this quantum geometrical corresponds which was discussed the two sides are related by modular transformation they are revert the side parameter notations of and this notation is epsilon to epsilon 1 they are revert to each other and then there is t transformation which shifts by 1 so it seems that this equality is in the pistation of this transformation which shifts the ratio of epsilon by 1 one of them yes but another one is transformed in different in more complicated ways so this is the ratio of second to the first and this one is indeed shift by 1 but this one is a combination of s and t transformations well it's a combination so it's a new situation that another generator of a modular but all these transformations are just the fact that within the two torres which is used in the localization which can choose different bases this is a particular example in the statement of invariance of some object under sl2z modular transformation i.e. not object of something what that something is well the z function the z function was invariant in the sense because that was defined in flat space but the surface defect breaks the symmetry between epsilon 1 and epsilon 2 because you choose the surface unless you so this was not the most general surface defect so you can define surface effects for singular curves and so you can also choose the surface effect for the coordinate cross so that will restore the symmetry I don't think it is a known conformal field theory interpretation I think probably frankly the kz-conformal block or correction function can be obtained by the separation variable from the kz-conformal block that's actually it was found by Zemologikov in 80s I mean it's by Zologikov it was found by explicit computation that the solution of Bpz equation which is the equation for the 5-point conformal block with one degenerate field maps by Fourier transform to the solution of kz equation for four points and that's reflexion of the relay I mean in the gauge theory no language this is the relation between two types of surface defects defined in this theory so one defect I defined I mean I didn't define it explicitly but I just said in words that by the orbifold and another one is by embedding this into the quiver theory of A2 type and then tuning the masses of these flavors so that they define the surface effect so these are the same surface effects in a different representation Can you interpret what is a very much idea into this form? Of course yes it was done in my paper with Lukyanov, Zemologikov and Litvinov that if you take you don't need to go through Schlesinger to get to Kelleve you can go directly from Bpz equation take a little b goes to 0 B be in the quantum legal theory and it will give you Kelleve on the nose