 The construction of vertex operator algebra associated to U1, the algebra, and some generic simple connections for manifold, possibly this boundary, and we consider it what is kind of how those guys behave under gluing, and also what roles they play. Sorry, I also discussed the relation between the ubiquitern environs and what are called equivalent multi-monopole environs of four manifolds, and what kind of role this vertex operator algebra played in this relation, so they were denoted by. So today, first I want to consider a bunch of other examples of the vertex operator algebras associated to some subclasses of manifolds, and so I consider some examples quite briefly, and one particular class of examples I will consider in more detail. So first, consider the case when the group is, the le algebra is U1, and so this is an example essentially due to an encadging. And so four manifolds will be a resolution of AD singularity, so let me label, so this is there is a, so let me denote a gamma divided by gamma H, where gamma H is a, is a finite subgroup of SU2, which, so there is no muck cake correspondence between finite subgroup of SU2's and simply laced, laced the algebras. So for a simple laced algebra which are classified by AD, we can take a certain finite subgroup of SU2 and consider this quotient and then it is a resolution. And then in this case, the vertex operator algebra associated to the four manifolds is, so I will write on this, is affine, so the VOA associated to affine the algebra H at level m, so this is a good level. So since, because of some questions during do stock, seems that not everybody is familiar with this notion, so let me remind you. So first, so here in many cases they're kind of, for vertex operator algebra there are a lot of abuse of notation, so sometimes people write the same notation for vertex operator algebra and for affine the algebra, which corresponds to, it corresponds to, so what exactly this means is that one can see there is a following the algebra, so as a vector space it is, so we start with the usual the algebra H and so we tensor it with this series T inverse, so we can see the loop algebra and we can see the central extension. So let me denote the central element by kappa and then if, so again I will use the similar notation. So the construction is going to be, everything is going to be very similar to the Heisenberg algebra since the corresponding VOAs. So again if H is some element of my ordinary algebra H, then the, by H with index I I will denote H tensor with T i and then the loop bracket is given by the following correlation. It's probably, sorry it was, so let me, so there are different elements, AB, so for any pair AB from H, so this is a key, here I use a killing form, standard killing form and so my, here I want to start with some, let's say reductive, reductive algebra and then there will be a term, another term which is just a commutator of AB and for here I take the index i plus g and here I multiply in the central element cup. So this gives you some infinite dimensional algebra and so the VOA is a corresponding VOA as a vector space is a highest weight model defined with the highest weight vector satisfying the following conditions. So the K, the central element acts on as a highest weight vector which I denote by 0 as n, my level and the i's just all act as 0 for i greater or equal to 0. So again this defines some highest weight model, this highest weight vector is 0, denoted by this 0 with brackets and so this is the representation induced by these relations and then we can equip it with the structure of the vertex operator algebras in a way completely analogous as we did for Heisenberg, Heisenberg V-ray. So I'm not going to go into detail. So this is similar to Heisenberg. So Heisenberg V-ray we had the same thing just without this. The Heisenberg V-ray was essentially the case when this is a billion, a billion realtime and in the work by the regime it was explicitly shown that this is the homology of the instanton model spaces here. They have a structure of a model of this fine algebra. And one can argue from this, the same result one can predict from the, is an agreement with the physics prediction of the computation of 6G theory, 6G SCFTO. This is so wonderful. So one can consider, let me call it example one prime. So it's kind of a small kind of a certain generalization of this. So this is considered in a paper by Gukov himself. So consider, so M4 will be, is a certain, is a bordism S3, the three manifold which is obtained S3 by the quotient, by the gamma related to some group H to S3 quotient by gamma associated to some A D V Algebra F. Such that if I take this bordism and glue it with the manifold which appeared before along the common boundary I should get the manifold from the same class but labelled by F. So you can consider some sort of this picture like this. So this is your bordism M4 and so here is this manifold or some typologically resolution of a certain resolution of singularity of some A D type and once you glue it in we get another manifold of the same class. Then the statement, determines this M4 up to the formacfism and so the statement is that this thing, so one can argue that this thing should be what is called coset VOA. So in physics it's usually denoted by some sort of quotient. I start with VOA associated to a finely algebra corresponding to the original algebra F and this level N and I quotient it with H. So this is coset. So this is notation and it was introduced in physics but it's in a sense it's wrong to write like this because you don't consider you don't quotient this vector space by this vector space. So this actually means is a commutant of H, N so you can see there are all operators in this vector superiod algebra that commute with this well the usual large space when this is of A type is what is called spherical So one of the nice examples when this is H the algebra is E8 then this is a Poincare homologous sphere. Any other questions? Another example which was considered in so let me mention the following example it was considered in Goukow-Pengen paper and I'm not sure if Serge is here if you can correct me if I said something wrong but anyway ok so the the idea is that so consider M4 is a non-compact toric surface so it can be described by the following toric form which looks something like this so the fact that it's non-compact means they all kind of fly behind a certain line and but you can consider kind of topologically it's better to understand it by some sort of dual toric diagram so it's usually you write it like this some sort of open polytope with some vertices and here you have a torus, sorry the T2 so generic point you have torus fiber and this it's degenerate so one of the cycles degenerate once you approach edge and both cycles degenerate when you approach a vertex and not necessarily but we don't want this poly we don't want this polytope to close well at least in this one can try to close this but it's going to be more complicated so let's consider this case so the there is a torus action on this which is the same which is so the idea is that locally around each vertex it looks like C2 and then one can associate something like C2 and glue these things together so first and so we want to construct what actually so we want to construct the VUA which also we will wobble it by SUN and so this will be what one can call T-coarrant so in particular it will depend it will depend a coherent epsilon2 in the notation the usual notation so there are essentially the same parameters that appear the similar parameters appear in the in-class of story, in-class function generators of a grand homogenous point so what it means here depending so if you fix these parameters to be some particular complex numbers this determines certain VUA and then one can consider the VUA of C2 and this is so let me so in general so this is what is called in general WN the VUA associated to WN algebra but not to go in details what are the W algebras so let me consider the case SU2 and this is what is called Virasura V-ray so this is in a sense a minimal VUA which contains a minimal vertex operator algebra which satisfies axioms so as a vector space this is essentially generated by the written like this where is this so again the construction is analogous of how we did, how we constructed so we started as we did in the Heisenberg Algebra or finally Algebra we started some infinite dimensionally Algebra and we constructed the VUA structure on its some high-street model and so this so of course so here when we talk about the vector operator Algebra acting on the space of on the vector space of vector operator Algebra it has a particular center of charge and this particular center of charge will indeed depend on this guy so it will be 1 plus 6 and then you want to define what do you do when you glue stuff so let me consider just the general situation as long as let me consider the case when the diagram has just two corners so the statement I think will be so let me explicitly denote this dependence on the epsilon and epsilon-2 which comes through the center of charge and this will be first well it's just acts by rotating so you one you rotate one you one C and another you one another C copies this is a one you one and another you one yes but of course we have to choose in order to do this to define this action we have to we have to choose a kind of a basis we have to choose what we call we have to choose this splitting okay but of course so here we have to we have to choose so the torus here is so suppose we fix a certain basis or a certain particular splitting of the torus into a product of you once so this you this you one corresponds to correspond to epsilon-1 and the second you one will the corresponding current parameter will be joined by E2 but then we won't so if once we identify this part with C2 and another corner C2 the the standard choices of epsilon-1 epsilon-2 here and here they will be related by some SL2Z transformation so suppose here the choice I choose just my choice is canonical here but here I have some other parameters epsilon epsilon-2 prime and so there will be something else but before I write before I say what is this so they are related by some so this is the corresponding vectors they are related by some SL2Z transformation and so we have so if you start so this the product of vertex-operator algebra gives some new vertex-operator algebra but in this special case when we have a two copies of Virasura vertex-operator algebra such that these epsilons are related by this by some SL2Z transformation this has some modules which can be added up to this vertex-operator so they have what is called non-trivial extension plus certain models so in general we cannot add modules like arbitrary modules we cannot up to the vacuum model which is the vertex-operator algebra itself so it has a non-trivial extension exactly they are related they are related between each other with this kind of formula and well you can consider a certain well this in general you can take complex so you can have 0 here in principle and there is some one can say more explicitly what you add up here what modules but what is important that this non-trivial extension exist exactly due to this relation between epsilon with inquiry parameter for this C2 and this C2 due to gluing okay so I don't want to go into detail but so what I want to consider in a bit more detail is the following class of examples is that m4 is a product of cp1 times Riemann surface with punctures this is genus g Riemann surface these n punctures in principle so you can write you can draw it like this but if you want to actually analyze if you want to analyze if you want to realize your vector space as a vector space on the level of vector space as a cohomology of the modular spaces of warp written equations then you actually need to be more precise you need to kind of you need to interpret these punctures as some sort of infinite tubes and so what is it so let me say some first general properties so the VOA of this thing so it will it will contain n copies n copies of a finely algebra associated to algebra g which I choose here with some levels which I am not going to specify but they can be explicitly written times u1 so in general there is no such for general if I take a closed for manifold of generic of generic homonymy there is no statement there should be always some finely algebra acting on this VOA but first of all in the particular case when the for manifold is scalar then you should have then there is a u1 which you can also interpret some sort of u1 action acting on the model space of warp written of warp written equations and you also have since you have a boundaries here and components for each boundary you have action of the you have action of your gauge group which can be understood as a kind of constant gauge transformations at infinity and they will be so this the ordinary the claim is that the ordinary symmetries of this model spaces they will promote it to a fine symmetries in this vertex operator algebra and how can one make explicit description well it will depend with here I'm just saying that there is this sub VOA here this is not a quality this includes the sub VOA which depends so whenever there are end punctures there will be this affine symmetry included here so and so in particular if I calculate as a trace over this VOA so I insert minus 1 to the f q to the L0 so f was my z2 grading L0 was my z grading and so I can also insert the certain element notation alpha element of an ordinary so end copies of group G such that Li of G is small g and here I take g to be compact and simply connected times u1 which is certainly u1 group corresponding to the ordinary so this is Li group which corresponds to the ordinary Li algebra embedded in this affine algebra and so then this can be understood as an element so this will be series in q this coefficients being elements of the representation ring of this group so if there was no if this was trivial then I just call it a trace of minus 1 to the q to the L0 I just get series with coefficients being integers but here I have series of this type so of course you can write it more explicitly by choosing a particular what is in physics called fugacities and then this will depend the coefficients will be some rational functions of those fugacities corresponding to maximal torque of those guys so this is some general expectation and this should be a modular modular and in a sense it can generalize your copy model form which will if I realize these guys if I go to the kind of maximal torque the corresponding coordinates of the maximal torque they play role of Jacobi like variables in this model form so this is some general expectation and now one can explicitly describe this VOA there are two ways so this can be understood as conjectures on this structure on this VOA structure on the cohomologies of the model spaces of solutions of both written equations so the first description can be given as follows suppose for simplicity so I write the statement in the case when this will be a genus zero surface then the statement that this is the same as the global sections of the shift of chiral differential operators so I will write on what is this shift of chiral differential of chiral differential operators associated to space X zero M which is the Higgs branch of the corresponding class S which is a bit naive because in principle this shift of chiral differential operators is only defined for a smooth manifold but naive if you generalize this you should expect this thing so here I wrote for genus zero if I take non-trivial genus then there will be additional data which I should include there will be a data of holomorphic vector bundle of this X and there is a generalization of this story of chiral differential operators for the case when there is also a holomorphic vector bundle over X so what is actually is I will write in a few minutes and another way to construct this thing construct by gluing basic pieces basic books so any human surface can be split into pair of puns so it should be enough to say what this is for pair of puns times Cp1 and to say how you should glue these things so in a sense you want to review I have this and a rule on the arrays for gluing so let me briefly review what is the shift of chiral differential operators so let X be a smooth so here I am assuming smooth which is not exactly true but let me anyway or give you a general definition just for a smooth case smooth complex analytic manifold so we want to define this the shift of chiral differential is a shift on X valued in vertex operator algebra so for each open subset here one needs to give it to tell you what is the corresponding vertex operator algebra so valued in vertex it would be possible to define for pair of fans for disc so for Cp1 times pair of fans for Cp1 times disc and disc is a bit bad thing to do you need to well kind of you kind of define for cylinder in a sense for cylinder that is the idea of the second approach which I also want to review if I have time but the second approach if you do this you should do not real checks you can decompose it in pair of fans in a different ways so the result should be the same but here you kind of using this approach of chiral differential operators you kind of define it globally I mean suppose you know the Higgs branch ok so so I don't have much time but so first the idea so I don't give I'm not going to go into deep theory but the idea that suppose first X take X to be just a fine space Cn and consider just a global sections what is the global sections of the shift on N the statement this is what is called N copies so this is VoA which is called N copies of the gamma system so what is this so this is very similar again to Heisenberg so what is this a single copy of so what is this VoA exactly is so as a vector space it so I start as in Heisenberg where I start with the Lie algebra given by the following relations so there will be generators beta and gamma labeled by ij going from 1 to N and as in Heisenberg VoA this would be the lower indices would be just value 10z and they will satisfy this relation times C which is central and again we can start VoA as some folk space obtained by these guys but to me more precise so we consider all possible so we start with some highest weight vector and we connect on it by all possible modes beta and gamma with a certain condition where here I allow so Ni should be strictly positive and this should be just non-negative so you have gamma 0's but you don't have beta 0's and this is a so what's important this is and again everything the structure of VoA just analogous completely analogous to the Heisenberg VoA how you construct an operator the series in operators from this guy the beta themselves they are commute and gammas they are commute this is the only material and so one can notice that this is a module over polynomial ring in gamma 0's because I can always any element of this ring I can multiply I earn some gamma 0's here and they all commute gamma 0's they all commute then for example if I now still working in Cn but if I if I want to define what happens if I remove so you can see the certain the risk open subset which is given by this so it's a complement of a set of solutions of a certain equation and I understand as a spec of C with variables X1 and Xn so this will be coordinates on the space then f is a certain element it's a certain polynomial here and I want to impose a condition that f of 0 is not 0 and then this thing will be so let me denote this in order to not carry all this heavy notation so this will be some sort of standard thing, let me denote it by just Vn, this is a Vui labeled by N so this will be Vn tensor it with the ring of normals and X where I localize with respect to 1 over f and here I tensor with itself, sorry what I mean here I need to replace this by gammas gamma 0 of gamma essentially what I, here what this means here so the space, the vector space can be understood as a space of polynomials in all these biters and gammas with different indices and here what I this was just a polynomial and here what I, if I want to go to this subset I allow to divide by 1 over f and so what I do formally is a, what I actually do is I can expand this 1 over f in 1 over f of gamma 0 in variables gamma 0 and I can do this I always do this expansion at 0 because of this condition and this will produce me a series a series in gamma 0s, sorry now I will allow a certain type of series in gamma 0s not just polynomials so this so this procedure allows you to define this on the risk open subsets of Cn and now, so here I'm kind of skipping details do a huge jump but essentially what you want to do is so you want to see some more general space X you can cover it with some patches where each patch is locally as a morphic to Cn and there will be some transition functions which I can so this will be Cn minus something Cn minus something and then so for each of these patches I have some VOA defined by this construction and then I should define a certain transition functions and this transition function should be morphisms between VOAs so the transition functions VOAs associated to different open and this morphisms defined as follows so the morphisms between VOAs so suppose here I have some coordinates X and coordinates here I will denote by X tilde so if I as usual I denote by this thing as gamma n by beta so this will have index i i minus 1 i and for example if my coordinates are related like this X i is given by some functions gi of X and then this means that gamma tilde so this is a series in that where the coefficients will be operators acting in the VOA here associated to this subset is just related to gammas in another patch by the use of this function for beta so if you have if you denote by inverse functions by F's beta tilde i is given by d tilde where you replace it by some tilde and you multiply it by you want to do the sum gi anyway you can write explicit relation between how you map operators here how you map the operator between these VOAs and so you need actually some condition on the topology of X in order for this to be in order for this shift to be globally defined and this condition is c1 equals 0 mod 2 and that the second part of TX is 0 how much time do I have so now you want to plug in this construction of the Higgs branch of classes here and you want to obtain some shift and you consider the global sections of the shift this produces some vector superior algebra and so let me mention briefly the approach with gluing so I don't have so let me so to talk about the gluing of vector superior algebra itself so you can do this but this is a bit involved so let me just mention this on the level of on the level of what happens on the level of characters so let me first consider the case SU2 well maybe I can mention so I have well let me just say some words but don't write explicit formulas so the idea is that so the general idea is that for general co-homology so if I want to calculate so suppose I start I want to start with some Riemann surface and I want to split it into two pieces so they are connected to the piece A and piece B and they are connected by some tube which I want to cut so the statement will be a certain co-homology which is known as BRST co-homology so in this particular case is what is called quantum drainfield circle of reduction with respect to this will be a reduction with respect to diagonal affine the algebra associated to the punctures the puncture this puncture in this piece and this puncture in this piece so namely this will be certain co-homology of the following tensor product so I take a vertex operator algebra associated to this piece A so now I once I cut I have here puncture tensor the vertex operator algebra associated to this piece and tensor so I need to take a tensor of two copies what is called BC system valued in adjoint representation of G so the BC system is an odd odd parity means the generator has parity 1 odd analog of beta gamma system which I described before so anyway there is some procedure but let me tell you something about the basic building block for example for SU2 this is the Riemann surface times so this will be just 8 copies so this is a combination of the beta gamma system valued C8 so there are 8 copies here there is this as promised they are indeed SU2 times SU2 times SU2 and times U1 action and for example the corresponding character here is very simple all this thing is just one of our product C to function where I take a product of all signs so C to function of any product so B is fugacity associated to U1 symmetry and X, Y, Z as fugacity associated to this SU2 and for example for SU3 you can understand this as a one way to understand this is the global sections of the shift of carol differential operators where X is E6 instanton centred E6 instanton model space and the corresponding character for example it can be explicit so here so naively on this VOA there is there are 3 punctures there is SU3 cube symmetry acts but actually this SU3 cube symmetry is actually as natural embedded in E6 and it happens that this affine symmetry, SU3 affine symmetry enhances to E6 and for example so this is for example in the work by myself, Song and Yan so this is some non Lagrangian theory but if you can still calculate it like physically the carol 2 dimensional theory doesn't have any Lagrangian but you still can calculate its elliptic genus which is a character of this VOA you can calculate which can be interpreted as a generating function for electric characteristics of wafovitum model spaces here so in particular you can explicitly see that this is indeed an element so there is explicit formula which I'm not going to write and you can indeed check that this is can be interpreted as a series it has a form of series in cure where the coefficients are elements of the representation of K6 times u1 and so for example on the level of character there is explicit integral formula how you do this okay I should stop here