 of vertex operator algebra associated to a four-manifold and a Li-algebra E1. So in general, there was some kind of prediction from physics that there exists for any smooth-vented four-manifold and G, which is a simply laced or U1. One can construct vertex operator algebra, which should depend only on topology and possibly smooth structure of four-manifold. So in the case when the G is U1, and so again here we assume that for simplicity that it's simply connected, but it possibly can have some boundary in general material, and so it can be constructed as follows. So it can be understood as a Heisenberg, so first you take a Heisenberg, VUA, which we defined before, for the real vector space, so Heisenberg VUA, the input is a real vector space with a symmetric bearing. So in this case this real vector space is a homology with real coefficients in degrees 0 and 4, and there is natural bearing here, even by intersection, between 0 and 4 cycles. And then we take a lattice VUA for the lattice lambda, which is a second homology lattice, which by the following, by the first by Pankarelev's duality it can be related by relative co-homology lattice, and this can be understood since there is no torsion, this can be understood as a lattice inside a real vector space of second co-homology with real coefficients. So this is the following vertex operator algebra. Before we proceed, let me make the following remark. A general remark about representation of the vertex operator algebra. So the modules of a VUA form a braided monoidal category with respect to what is called fusion tensor product. So this is not the usual tensor product of vectors. The module obtained by this fusion is not a tensor product. As a vector space, not just a tensor product, but there is a certain notion of tensor product, which is sometimes called fusion. And moreover, so under some assumptions, which include semi-simplicity, so this category, this category semi-simple, and there are a finite number of objects, of simple objects, and moreover, there are some non-degeneracy conditions. So this has a kind of more rigid structure. This has structure of what is called a modular tensor category. So in general, this doesn't have this kind of more simple structure of model tensor category. But in many cases, even if the kind of the category, the representation category, like the full representation category of VUA can be rather wild, but sometimes there is an MTC, can be a certain subclass of representations, which are closed under the fusion product, and they can form a modular tensor category inside this larger, the whole category. And this kind of structure of modular tensor category is important in constructions of invariant of three manifolds by relationship to write constructions. So in principle, any model tensor category can give you an invariant of three manifolds. So in this case, we can indeed consider a certain class of representations which will form a model tensor category. So these will be the following models of this VUA. Let me know by M mu. So we already kind of already seen them. So they will be labelled by the elements of the cassette, the lattice divided by the dual lattice. And they will be just given. So here I take, first I take just this Heismberg VUA by itself. This is the first part for H0 plus H4. And here I consider the module of this VUA, which as you remember it was constructed by the sum of, sorry, I do a sum of elements of the lattice and I have shifted by this element mu of Fox space labelled by lambda constructed from this. Constructed is Fox space for the Heismberg algebra constructed for this vector space in the bearing. So essentially here I take a kind of provocium module and here I take a module of this lattice VUA. And so since we know that the, so lambda is, can be applied with the second cohomology. Well, let's say less second, second homological lattice and lambda star then can be identified with just H2 and 4 with integer coefficients. And the quotient is just from the long exact sequence of the, of the relative cohomologists you can see that this is the same as, you can try this H1 of M3, the boundary of my 3-manifold, of my 4-manifold, which is the 3-manifold. So you see, so this is the, at least the kind of, so the, okay. So let me make the form, so you see that the kind of, already the set of this label, they depend only on the boundary. So in general, so I can make more kind of a stronger statement as a category. So modules, okay. They also, in principle, they also have this label U1 from 4 form a module of the category which we've denoted, kind of module of the category labeled by U1 and M3, which depends on topology of the boundary. So in this particular case when it's U1 it actually depends only on homotopy class from 4 till it depends on the kind of, as the kind of the fusion, the tensor product structure of this category will depend on, in this particular case when this is U1 will depend on the kind of some linking part, pairing on the, on the each one. But this is something which should be generalized for any G. So you can construct, there should be some vector super algebra associated form manifold, but it's the category of its representations. Maybe we can, in general one should restrict representation of some nice subclass. The category of representation should depend only on the boundary of the, of this form manifold. It's quite restrictive. So usually, this is something like called rational vertex operator algebras. But it's, well this is the case, usually most studied. Usually the most simple examples of vertex operator algebras are this rational type. Well, yeah this would be definitely not but here, so here we consider, I mean, since M3 was simply connected in this case. Yeah, for example, if you, like in some examples when the boundary is something like this, two times this one and there is a representation, it's not a rational, you don't get a rational vertex operator algebra. And so this kind of, the fact that the representation, representation theory for this vertex operator algebras to form manifold only depends on this boundary. Nicely reconciles with the fact that we can glue kind of, there is some gluing prescription for vertex operator algebras to form manifold. So consider a case, for example, not for technical, consider the case when we want to just consider two kind of four manifolds, M1 and M2, such that the boundary of, so the boundary they have a connected, the boundary just has one connected component and so the boundary of the first is some three manifold M3 and the boundary of the second is the same thing with opposite orientation. Then there is a statement in this case and suppose, so M4, let M4 be M1 glued with M2 along M3 and so it's not, it's not going to be just a tensor product. So if you, if you glue this, it's not going to be just a tensor product of vertex operator algebras associated to these pieces, but it will be the following sum of tensor products. So, so as you, as you saw, one can lamble the modules of vertex operator algebras for both basic pieces by H1 of M3 with integer coefficients. So here in this case, this is the finite group and we take and it's easy to verify that this is indeed a, I mean, this tensor, this kind of combination forms a nice forms a, forms a vertex operator algebras. So in general, you cannot just take a, like take a direct sum of a bunch of modules and say this is a new vertex operator algebras, but there's particular sums which you can do. Any questions? Well, here, yes, in the, in the U1 case, you only use caromology, you use intersection but in the U1 case, it's something very simple, but it's a nice kind of example where you can consider all the structures. Okay. So now I want to consider a little bit different things. So, so I want to consider what is called a multi-monopoly invariance, but which are kind of generalization of the hyperquitin invariance. So, before, before doing this, I want to review what are the hyperquitin invariance and so they already appeared, for example, in Lotus lectures, but so I really, I want to go in a bit detail because I don't need this, those details to consider this generalization. So now let me assume that so now M4 will be closed. It doesn't have any boundary simplicity. So let me remind you what is this spincy structure, spincy structure on M4 to ease the lift SF4 SF4 SF4 bundle which is the principle bundle of abnormal frames abnormal frames in the in the tangent bundle to spinC4 principle bundle with respect to projection with respect to map spinC4 SF4 so so spinC4 is by definition is a spin4 times C1 divided by diagonal Z2 but in the particular case when this is we have four here this is also can be understood as SU2 times SU2 product of took a piece of SU2 which will denote by minus and plus to distinguish them and okay and so this this whole group can be mapped to as I can just projected to U1 divided by Z2 which is isomorphic to U1 itself and but I can also project it to SU2 times one of the SU2 times U1 times Z2 and project to SU2 plus minus plus or minus times U1 divided by Z2 which is some U2 group so then let me denote by L it's a line bundle it's a complex line bundle associated to the U1 principle bundle which we obtain by this projection and W plus minus this will be complex rank two bundles associated by fundamental representation to U2 plus minus principle bundles so this this is often called determinant bundle and these are wild spinner bundles so now so again we want to assume either we want to assume kind of that for my fault let's assume that a kind of a weaker condition that's not simply connected but the co-homology the first co-homology with inter-coefficiency is zero then the set of spin C structures is on M4 which is denoted by can be denoted by spin C of M4 can be described as follows this is set of elements of second co-homology lattice with the conditions that lambda is the same as second C-fil-vitin class of the tangent bundle and so this lambda has a meaning as the first term class of the line bundle of the determinant line bundle so we also want to we want to use the following relation if I take a tensor product of W bundle with W bundle this will be this can be decomposed as a the bundle of self-dual two forms on a form manifold of tensor with C so this is some rank three complex bundle plus the trivial rank one complex bundle well they are associated to certain representations so this is a determinant representation this is associated to two-dimensional representations which you obtain by you can there is a map homomorphism from spin C4 to U2 two different homomorphisms which can be used to construct this two-dimensional by using the fundamental representation of this U2 you construct this these W well if if there is a yeah so well at least the other way it works if you have almost complex structure you can there is a canonical spin there is a canonical spin C4 so now so now we want to consider so before we consider C4 let A to be a connection on line bundle L so locally it can be given by some one form and so this so this defines this also defines connections W plus minus by also using the kind of we choose some metric and choose leverage with a connection so together with A this defines connections on W plus minus and by Psi let Psi be a section of W plus so here I forgot to write you take a conjugate complex conjugate then the W equations the following equations so you write it as follows so ok let me explain the part so this is a Dirac operator in general Dirac operator can be considered as a map from W plus or minus to W minus plus and so this is a cell double part of curvature connection A so this equation so this equation is equation in sections of gamma W minus so because Psi was a section of gamma W plus and this is equations in a section of lambda 2 plus so again here so in general as I showed you before so if I take sections the easy so this tensor product will be will be a section of in W plus tend to W plus bar which we can decompose into this bundle plus material bundle so and here the plus means we just take a projection to this bundle then the so we can see the solutions to this equation we know by B there will be a subspace in connections L times sections W plus and but let me so this is a technical subtlety so I want to a subspace a certain subspace in the space subspace where the psi is not identically 0 solutions where psi doesn't vanish the section that is not 0 everywhere then there is a on this space kind of on this subspace of this larger space there is a free action of the gauge group gauge group can be answered the maps from M4 to U1 then you can see the following quotient M so M will depend on lambda which is the class which is the first-gen class so we fix a line bundle L first-gen class L is a quotient B divided by and so maybe I shouldn't raise this completely and there is there is an important theorem in this theory of the algorithm why is that for generic for generic metric and if B2 plus is strictly greater than 1 this is actually a manifold is a manifold oriental more or it's orientable manifold of expected dimension which can be given by the index theorem it's given by the following formula lambda square minus 3 sigma plus 2 kappa divided by 4 but okay let me make sometimes it's not a convenience to work with generic metric so often people consider instead of generic metric people consider a certain perturbation what is called perturbation so they add some generic instead of generic metric they add a generic perturbation which is some some dual to form well there are some other topological information there are some classes you can calculate which will be actually this environs well it's a yeah it's a compact it's it's not done it's it's closed yes more there is there is a conjecture that it's it's actually non-empty when it has dimension dimension it's only non-empty when its dimension is zero so it's just a bunch of points well there is a conjecture this is a conjecture this is a conjecture yes and more so it's kind of the same is that if you change a metric the m lambda in this larger space where it's kind of embedded this quotient with larger space divided by the gauge group will change will be will change there will be some gobertism between the different amounts for different metrics inside inside the ambient space which I raised so okay so how one defines this ubiquitin variance so there is a universal line bundle where this product of m4 and lambda such that if I restrict it so it takes a particular point in the modular space and there so this bundle on m4 is coincide with my line bundle the German line bundle L and one can consider a projection here to the modular space and then I can consider the class U which is the push forward with respect to projection the class in co-homology of for manifold punctual to a point a product with a projection class of lambda here so this this gives me element in co-homology of lambda integral coefficients so this has degree 4 which I integrated over m4 this gives me a class here and then so then the application variance so it's a map from a set of spin C structures for manifold to integers and is given by its value is given by the following integral and so there is a so called simple type conjecture that this modular space is there empty unless the virtual dimension of lambda which is coincides with the actual dimension is 0 well again this is a when B2 plus so you don't actually so if you are like according to the simple type conjecture you don't actually need to integrate any classes you just count with some signs a number of points so there is some natural orientation on this on this modular space and you count just a number of points with signs okay any questions yes okay so now so now we want to generalize it to what is called multi-monopole invariant so as before so A so the setup will be essentially the same as before so A will be still the connection on this the terminal line bundle L but now instead of the single field psi we take N sections W plus bundle and we can see the equations which are multi-monopole equations this is going to be very natural generalization so if you take just a sum here and we have a Dirac equation for all from 1 to N on each on each of these so from physics point of view you kind of modify your matter kind of if you consider this coming from the topology with Vistit n equals 2 series with U1 gauge series so the usual then be written invariant has just a single charged hyper-multiple and this has N charged hyper-multiple so again when you can see the solutions there will be a certain subspace connections again let me restrict it on a subspace at least one psi I for a certain index is not identically zero it's not zero everywhere it's not a zero section and so one can see the corresponding model space which is a quotient the gauge group as before so now it has the expected dimension a bit different but it's also can be easily calculated by the index theorem as the following formula so it's before chi and sigma are electric sorry I didn't mention it explicitly but I see everybody can't know so chi is electric characteristic of m4 and sigma is a signature and lambda square is a intersection self intersection of this class lambda so although it's not explicit so this is actually always an integer and now there is a problem that the space is non-compact even if you make some assumptions like this so in particular this the integral wouldn't be well defined or whatever you want to integrate or the space is not defined so how one can kind of see non-compactness where does this non-compactness come from and characteristic so well I'm not going to consider so consider some special at least one can see the non-compactness is in some special case so let for example m4 be a spin and with negative signature so there then one can show that there exists there is a harmonic by the index here there is a harmonic spinner xi which is a section w plus and so let me take so the if the if the four-manifold is spin one can take line bundle to be this line is determined line bundle to be trivial if it's spin sorry and so the harmonic spinner will satisfy just the usual ordinary Dirac equation Dirac equation zero and then one can start the following solution by taking so so I want to assume that n is great non-compactness should happen when n is at least two so I I take my few psi one to be some complex number times this xi and psi two to be the same thing times j acting on xi where j is a certain element of u2 group well, namely, it's something like this so just squares two minus one and one can see this is a this is a solution of this this this will satisfy my equations for any alpha so this will be one so this will be zero and this will be also zero and so all other all other guys when i is greater than two I take zero so at least you see there is a one non-compact direction at least one non-compact one direction where you can go to infinity I mean, I want to make a more detailed argument about whether you're here so at this point I kind of finished finished the review kind of the review part and I started talking about joint work with J. Deschenko yes presentation of some group meaning representation of some group right yes what is this group so this can charge hyper-multiplets do they how do they mix between each other I guess this is the standard thing for the physicists but I or what do I mean well there so you have now well there will be some what is called flavor symmetry group which is actually will be important but what is this group this is SUM which are so you can you can find many viral details in the paper okay so this is this was kind of very quick argument for non-compactness and so indeed so it's so one can notice there is so once we add so in some sense so once we add the editor in your field side we have this problem of non-compactness but from the other point of view we kind of get something nice in return which will help help us is that there is so we consider it so that our fields were in this space and there is all divided by G and there is natural SUM action on the space we just act so I can understand this clearly as a sections of gamma W minus tensor at v c n and this there is action of SUM here so I don't consider UN action because UN action you want the the diagonal part of UN acts trivially because there is quotient with respect to G so there is no so as a and so I can use the symmetry to to define to do to do integrals a covariant so in particular one can do the following thing one can consider what is called a co-variant cyber-critical invariance of n n-monopoly invariance by considering so let T be a maximal tourist in this SUM and I can see the T-quarant integration of this model space so in principle I can integrate any a-quarant characteristic class but there is some kind of one canonical choice just to integrate one so then so in general whatever I have here one can one or anything else or any other a-quarant characteristic class so I can use a a-tablet localization formula for well normally a-quarant the version where you take 6 to 0 appear before so ok let me so what so first let me I forgot to mention what does this thing depends on so it depends still on lambda and it will be on that which will be elements so the they can institute that so is the T-quarant homology of a point is generated by parameters so you can there is natural generators you can choose n generators but such that with the relations at the sum up so your environment will depend on on these generators of a a-quarant homology of a point and so the a-tablet localization formula gives you that you can reduce it to the sum of fixed point subsets of of our model space and for each so you and for each fixed point subset you do integration of one over a-quarant early class of normal bundle to see well if if you assume simple type conjecture this will be always point so the fixed local will actually be in general the one can argue that the components isomorphic to just m1 the model space of a single instanton so you can easily see this because well for roughly you can see this by things that only so this is the fixed a-lo-cos fixed point lo-cos is of course when so if you if you can see the vector of these sections only is given by configurations where only one of the fields only single psi i is not identically zero otherwise so if otherwise this is not the SUN group even but for a single guy it will multiply it by some which is by their face which the face but once you consider the quotient this is the same this will be identified well you have your group acts on the normal bundle and then you consider well the kind of you consider the splitting principle and for each kind of this the normal bundle from some representation of the group multiply all the rates of this representation yeah this is yeah this is what you can and so in particular okay I don't have like five minutes left so let me so assuming a simple type conjecture so again so okay I said it but I didn't practice so the statement that it will be so each the simple the fixed components due to this argument the simple components are just will be isomorphic to single monopole model space which we denoted by just m lambda and so assuming simple type conjecture you can explicitly do this calculation well there are some subtleties of kind of with what with what orientation they appear here you can deal with the subtlety and so the result is that so there will be it's just proportional to be to the the ubiquitin environment of lambda times the following rational function of equivalent parameters here is that so this is essentially the product of rates of the action on the on this side well here the normal point will be just whole tangent space and the product of g not equal to i g i 1 lambda squared minus c okay this is the result so what so I promised in the beginning that this should have something to do with vertex operator algebras so you can already see this is kind of looks like similar it has structure similar to the correlator in vertex operator algebra which I wrote before so this is consistent with physics prediction namely realization of invariance here 6d SCFT on so this multi-monopole variance on 4-manifold times some punctured sphere so what kind of mathematically this means that we want so kind of physics predicts that we should do this result with some observable here in 2d which explicitly given by the following using the notations which I introduced before so in terms of this voia u1 can be written as follows this is a dual to the highest weight vector associated to lattice element lambda from physics you expect the following expression and where s s plus are certain elements of this voia of u1 m4 so here they are promoted to fields depending so here it's j promoted to fields using state fill correspondence so as usual a of z it just means the value of this and so more okay maybe I shouldn't have raised this metaphor so more explicitly this s of z is given by the following sum over so you sum lambda over all spin c structures of m4 which can be understood as a kind of subspace in the lattice lambda given the subspace given by this parity condition of the vertex operator lambda depending on that times the line lambda and so and s plus such that it has the following if I multiply it to the vertex operator algebra depending vertex operator depending on w I get 0 with the following singular part lambda square minus sigma plus and the rest will be a normal ordered product of these guys so this kind of the vertex operator algebra they kind of provide relation kind of natural setting natural setting how to how the the single monopoly and multi monopoly are related to them but kind of and some from other point of view you can see this relation by doing the just the synchronized localization okay my time is up stop here