 Thank you, Jan. So I would like to thank the organizers for inviting me here. It's a great honor, especially on such an occasion. And like many of us, I was greatly influenced by Maxim. So I'll talk today about a connection between math and physics. So I'm a physicist and I'm interested in physics components of the story. But it's a nice bridge between what I'll hope I'll explain to you as very rich mathematics and very rich physics. So every talk is a delicate balance between telling the truth, meaning presenting all the details, proving all the theorems, explaining everything. But then there is a danger that will be bogged down on those details and we'll never see the big picture. So I'll try to make this delicate balance between presenting big picture with lots of pictures and also some of the concrete results and examples. So I'll try to emphasize those points where I'll have at least two concrete results in this talk. One is going to be concrete new results for mathematics using this correspondence that I'll explain and one will be actually new result for physics. Mathematicians in the audience probably don't care if there is a new result for physics, but I certainly do. And of course, all the details that I'm going to miss in trying to maintain this delicate balance could be found on some of the papers I mentioned here with two great guys, Abhijit Gadi and Pavel Petrov. So what we'll really do in this talk is we'll study, well, as this physics guy says, five brains on causative four manifolds inside the G2 halon in the space. And that's actually what we're going to do throughout this talk in this hour. And there is a famous cartoon which shows that there are different versions of it. This is for IT guys. This is you tell dogs something and they hear, okay, ginger, blah, blah, blah, ginger and so on. So similar in this opening line, there are some words which may be very familiar to you, but there might be some words which are terribly mysterious. For example, I imagine to mathematicians the word five brain is very mysterious and everything else probably is more or less familiar, maybe not too well, but at least it's from math. For physicists, actually the situation is reverse. I have to say, physicists really understand only one word five brain and all this causative four manifold G2 halon in the space to physicists actually that's even worse. So anyway, I'll try to explain what the five brain is. So since I'm aiming this talk more for mathematical audience, what are five brains? These are very rich interesting objects in 11 dimensional M theory, but and they have a lot of rich structure, but at the zero level of approximation, they're just six dimensional sub manifolds. So these are certain gadgets supported on six manifolds inside 11 dimensional space time. That's all you have to know. They do carry some additional structure. For example, beyond the zero level of approximation, you can have multiple such sub varieties and then multiplicity gives rise to symmetry, which is typically UN or SUN. That's what we're going to consider in this talk most of the time. So there'll be G, which is either UN or SUN, some group compact group of cartoon type A and and here we'll refer to multiplicity of those six dimensional subspaces in 11 dimensional space time. So here this picture is illustrating this is cartoon four for such embedded hyper surface and big space. And now the idea, which goes back all the way to color, say client, but have actually for each and consequences, especially for us, is that what we'll do, the kind of game we'll play is that we'll try to take this six dimensional object, six dimensional five brain and split its six dimensions into compact manifold mn, which is what we'll aim to study. And remaining part will be just a flat Euclidean space of dimension six minus n. So here I'm using kind of obvious, if you wish, I'll refer to this kind of decomposition later on as six equals n plus whatever is left six minus n. The consequence of this, and again, physicists can tell you more and more about it, but this will be not just an extra talk, it will be whole big course, how some theory, this extra data, which leaves in six dimensions upon wrapping the six dimensional big space time on some manifold of dimension n gives rise to some effective field theory in six minus n dimensions, which is labeled by space on which you compactified by this compact manifold mn. So this idea is fairly simple and again, it's 100 years old, but its implementation for us will be fairly interesting and quite rich. So just the very fact that five brains exist makes all these theories, lower dimensional theories in Euclidean dimension six minus n labeled by other manifolds exist. So that's already basic fact, which is kind of punch line of the talk, if you wish, I'll of course put lots of bells and whistles and that will explore it, but that's the idea and it's fairly simple. So by the way, if you have any questions at any point, feel free to interrupt me. So six, conformal field theory six dimensions, which also they're labeled by ADP. Yes. So that's essentially the same story. So that's a six dimensional conformal field theory which leaves on this five brain. So these terms are completely interchangeable. And for me AD will be always of carton type A. So that's the same story indeed. Yeah. So the good point about this lower dimensional theory, which will be obviously in dimensions less than six, is that such a theory upon compactifying the six D theory on a manifold mn will actually depend on topology and also on geometry of the manifold m of dimension m that you choose. So this is, if you wish, a functor which sets n dimensional manifolds into six minus n dimensional theories, physical theories. That's a kind of funny functor, very strange object. But we're going to explore its consequences. And I'll start with something that's by now fairly well understood, at least many aspects of this story are well understood and we'll play the game where we're going to split this six dimensions in different ways. So first here, I'll split six as three plus three. And then of course, we'll come to four manifolds and split it into four plus two. So again, this is kind of simple algebra to keep in mind. So if you choose a three manifold, this little n equals three, then complementary dimension is also three. So what you get is a three dimensional theory, I'll call it T of m three, which is labeled by three manifold that you want. It carries certain supersymmetry. Again, there are lots of bells and whistles. But in the first round of approximation, I'm going to skip that. Now, you can view this construction, and that's the modern perspective on things, as constructing invariance of three manifolds and four manifolds, which are of completely different type that are not very familiar that we don't usually think of. When we think of invariance of three manifolds, four manifolds, we typically think of numerical invariance. In recent years, people start talking about invariance of, say, knots and three manifolds which take values and vector spaces. These are a categorification of numerical invariance. But this is something even more esoteric or weird in the sense that to a three manifold or four manifold, we associate a physical theory. And theory in dimension six minus n would actually be of high categorical nature. It will be something that is not just a number, it's not a vector space, but contains a lot of information, a whole load of information, a lot more than just a number. Do I have three? Is it one part or it calls a boundary? A boundary of which? Oh, three manifolds, for instance. You can develop the story where you have three manifolds with boundary, without boundary, so it's a good question. In other words, in this context, it's a perfectly valid question, and boundary, of course, will be important if it's present. If it's not, then it's not. Good thing is that you can recover some of the more familiar invariance of three manifolds, four manifolds, by trying to take this physical theory and ask a question, okay, what is the partition function of that theory? That, of course, is going to be number of function. You can ask about correlation function, something that Nikita considered in his talk, or various other gauges. For instance, simple question is to ask about supersymmetric vacua. I try to meet all the words infrared, because I thought mathematicians don't know them, but now they do, so therefore, this is really about infrared physics of such theory. You can ask, say, what a supersymmetric vacua. Many of such theories, even in three dimensions, can be described by Landau-Ginsberg type potential function w. That's a holomorphic function, exactly of the same type, and these are close cousins of two-dimensional theories that we heard in previous talk. This will be certain space. It may consist of isolated points. It may consist of continuous sub-variety, modular space, but in any case, what happens is that this space of supersymmetric vacua is nothing but space of flat connections on a three-manifold m3. If you start with ad-5-brains of this ad-type, which for us will be either SUn or Un of type A, then this will be complex flat connections for the same complexified Lie algebra. At least, this theory, it's not some stupid functor which takes three-manifolds into some mysterious theory. It actually knows about, say, flat connections or character variety on a three-manifold. That's one statement. Once you start exploring this correspondence, basically, the rule of the game is that you explore different invariants of this theory in lower dimensions and six minus d dimensions and ask for, again, different gadgets and field theory language and try to recover different invariants of three-manifold, four-manifold, and so on. Here I give an example. Say, m3 is a land space. Say, s3 mod zk, and fundamental group zk, basically complex flat connections just like real flat connections are parametrized by homomorphisms from fundamental group in this case, zk, into the group g, modular conjugation. This is a finite set. A nice interesting fact is that this finite set happens to be labeled by the same type of label, which labels highest weight, integrable, or representations of the loop group at level k. This may be viewed at this level as a coincidence, but, in fact, you can try to work out what this three-dimensional theory should be for land space, and then you're going to find, so I'll erase this and save it for later, so theory associated to land space, say, k1, is Schoen-Simons theory. In fact, certain supersymmetric extension of Schoen-Simons theory, but that won't be important to us at level k, because if you ask yourself, have I seen this somewhere, this indeed is the space of ground states in Schoen-Simons theory, either supersymmetric or not. So, if you think of vacua being flat connections, then already these basic facts suggest that the theory associated to land space should be some version of Schoen-Simons, and it will play a role for us later. Now, you can ask about three manifolds which do have boundaries. So, for instance, if you ask about not-compliments, which is another nice class of three manifolds with a total boundary, either not-compliments or link-compliments, then in this case, complex flat connections will come in families. In particular, if you talk about not or link with a single component, there is this famous A polynomial whose zero locus defines for you the character variety. And this is exactly the space of critical points for three-dimensional theory, which is associated to not or link complement. And again, this is a nice class of examples. By studying this correspondence in present context, you can derive various versions of so-called volume conjecture. One of the versions says that there is an operator which, in physical theory, on this side of the correspondence is some sort of word identity, very much like a Nikita stock that annihilates partition function. So, and then there are various upgrades which even get to homological invariance that I'll mention very briefly in a second. So, this partition function, Z of m, what is this partition function? Well, you could think that if you already have access to space of classical solutions in trans子imons theory, namely flat connections, there should be a way to quantize it, namely to construct the full trans子imons partition function of a three-manifold m3. So, now trans子imons appear on this side, whereas moment ago I was talking about trans子imons on this side of three plus three split. Now, yeah, there is such a, such a gadget and there is a question, about three-dimensional field theory T of m3 such that answer to that question is actually trans子imons partition function on a many-fold m3. The corresponding question here involves turning on omega background very much like a Nikita stock and counting instantons except that the problem here is lower dimensional. So, Nikita studied this omega deformation of four-dimensional gauge theory where you have four dimensions of spacetime, both of them are considered in equivariant context. This is a baby version where instead of instantons one has a sum of vortices very much like in the work of Taubes and so on. And again, one considers equivariant vortices, meaning they leave on the plane and we are trying to look at their representation with respect to rotation group in two out of three dimensions on this side and that's a very well-defined quantity. Physicists studied and finding enough they recover complex trans子imons partition function over three many-fold m3. So, that's the relation here. So, it's a vector in if now in the case of three manifold with boundary this will be a vector in the Hilbert space of associated to the boundary. Yeah, exactly. So, in particular it in this context it depends on parameters and well these are some of these parameters and on one side of the correspondence here this operator equation this word identity is known as quantum volume conjecture or some sort of aj volume conjecture. And on this side of the correspondence this is a statement that quantization or a certain operator relation associated with classical modular space because a the zero locus of a was the classical modular space of vacuum for three-dimensional theory here annihilates this vortex partition function. So, that's it this statement has a meaning both in this world and this world and luckily they agree. So, again there are lots of papers on the so-called 3d, 3d correspondence by now this is fairly big industry and I'll just mention a couple of such applications but my goal will be eventually to actually move to the next level and try to develop the same sort of dictionary or industry for studying theories labeled by four manifolds because this is much more mysterious both in physics and math and to my mind this is much much cooler. Finally, I kind of suppressed for a sake of presentation the fact that there is some supersymmetry in this three-dimensional theory which is labeled by three manifold and this didn't play much role even in the example I mentioned here but it is supersymmetric and as such it has supercharged so therefore you can ask for invariant or object associated to this theory which is neither space of vacuum nor some number such as partition function of a certain kind but rather a space namely space of states which are annihilated by supercharged q but not q-exact so in other words we can ask what is the q-cohomology in this theory which is labeled by three manifold. If theory itself is defined by three manifold and we ask for certain question about it the answer should also be something that knows about three manifold and again if we talk about not complements what's interesting is that this q-cohomology recovers mathematically defined Hovanov homology and its variance so that's that's an interesting yeah so here on the left hand side H actually stands for SLN version of Hovanov homology in fact decorated perhaps with various representations basically categorification of quantum group invariants doubly graded space and in physics literature such elements of q-cohomology are sometimes referred as BPS states BPS states are so-called supersymmetric states these are states in the theory which are annihilated by q but not q-exact so that's a definition so sometimes the statement is mentioned as not homology being being interpreted as a space of either supersymmetric or BPS states. Now until so far I talked about the six-dimensional theory and we didn't even consider where it leaves it actually leaves in big 11-dimensional ambient space and that embedding of six-dimensional sub-variety in big 11-dimensional space is not arbitrary basically the picture is as follows the 11-dimensional space consists of r5 five-dimensional vector space and Calabria which for most of the discussion we can take to be just total space of the cotangent bundle to three manifold here after all neighborhood of any Lagrangian inside Calabria looks like that so that that will be good enough and we'll split some of the dimensions of the vector space which are not used or not curved in the following fashion r will be singled out and usually is interpreted as time think of time as in fluorotherapy and it will be useful in a second and the rest is subject to this equivariant omega deformation which gives rise to q-grading and homological grading in application to two not homologous so again total world volume or support of this five brains is equal to six so this is our three plus three split this manifold is interesting curved has interesting topology and geometry that we want to detect here in this r3 leaves the three-dimensional theory I was discussing on the previous slide and it's embedded like this in mb and 11 dimensional space I want to point out already here that there is a four manifold inside which will may which will be much more interesting in the rest of the talk namely if you combine the time which didn't participate in rotation symmetries that we used and three manifold here this is basically indeed like a floor homology picture where you have some non-trivial slice and across time floor time and together they make a four manifold or cross m3 so that's the first remark and passing we'll come back to it in a second second remark I want to make about this embedding is that you can look at this picture from many different view points that's very typical in physics you can study the system from one perspective from another perspective you can look at it for instance from this point of view of r3 here which gives some three-dimensional theory labeled by three manifold or you can look at this picture from the viewpoint of mb and spacetime in particular from the viewpoint of the calabi out here and then it gives rise to certain enumerative problem what we call by this a space of bps states in that context is more appropriately known as a space of so-called open and refined bps states so there are many ways to approach this problem now what i'm going to do later is something that to which i alluded before is consider this r cross m3 and not just let it be a direct product but rather consider evolution of a three manifold along r in other words considering on trivial co-boardisms for instance here is a cartoon which represents co-boardism between say two link complements and this will relate this will give rise to a function between different link homologies now such functions are all very interesting and again if you ask yourself a question in this embedding of not homology in the physical setup how would uh co-boardisms play a role this is this is how they would appear they would take one of the dimensions of this 11-dimensional picture which has not been used and combine it with m3 into a four manifold so if you want to study functors acting on homological not invariance which is something that i've been studying in past several years you are inevitably coming to to the picture of four manifolds which are which are co-boardisms another interesting fact i want to mention here is that among all such co-boardisms there is something special in particular among all nots uh unnot plays a very special role uh for the following reason so suppose you have a knot such as uh say this trefoil here and suppose you take um a copy of unnot very much as as shown over there and we want to consider co-boardism from here to their connected sum so unnot a special in the following way that if you take a connected sum of unnot with any other knot then it doesn't change the knot so gluing this or reconnecting it in a way like shown here basically gives me back the trefoil knot which was what we started with so as a result we get the following statement that co-homology of the unnot even from the knot theory perspective is an algebra that acts um on co-homology of every other knot say in this case trefoil and this this is a fairly general argument which again might be very tricky in several versions of various versions of knot homologies but this is the basic idea that's the origin and another uh interesting observation is that it happens that um in the realization here the co-homology of the unnot which is this algebra is also algebra of so-called closed BPS states in this context they just coincide whereas um their origin is rather different this comes from knot theory and can be computed by by known techniques in low-dimensional topology this has enumerative origin and was studied by Nagawa and Nakajima in fact both of this dependent choice of chamber and so on but they just coincide at the level of answers and in fact um what this will suggest then is that so-called closed BPS algebra acts on the space of open BPS states this is a more general statement and in the context of closed BPS states this goes back to work of Harvey and Moore who conjectured that closed BPS states form an algebra in the language of my previous speaker where if you have something finite something infinite you combine them you get something infinite here I could say that if you have closed and closed you combine them you still get something which is closed however this is no longer true if you have closed and open when you combine them in a mysterious product operation that physicists call forming bound state you get something which is still open so closed times open is actually open and for this basic reason we conjectured with Marcus Tossage that um open BPS states form a module or a presentation of the algebra of closed BPS states and of course uh this this um right so um this this is uh right um so there are different completely different versions of uh algebra structure on on the space of BPS states and um in fact from you and Jan I've been learning a lot about even different connections including say motivic homological whole algebra so I don't know which one of them will be uh proper mathematical home for for this picture but at least again at the physical level that's that's that's the story I would just say that that's the same algebra is Harvey and more used and I agree that's a good question for me again this was kind of the aggression about BPS states what I was trying to illustrate really is that given a theory there are many questions you can ask to pass from big full-fledged theory which probably should be described by categories high categorical structure to numerical invariance and vector spaces just like this cuckoo homology so there are many questions you can ask to get from very complicated esoteric field theory down to numerical invariant if you wish so now I'll try to come back to this previous setup and really push a little bit further um this uh co-boardisms between say different three manifolds which as I mentioned earlier naturally lead us to the world of four manifolds so I'm going to take this r cross m3 and replace it by a general four manifold m4 could be co-boardism then in order to preserve supersymmetry and for other reasons what's going to happen is that um on this upper part of the slide a slide the spacetime r which was unused will naturally combine with kalabiyaw geometry into something seven dimensional and this seven something seven dimensional here is a very boring version of g2 manifold but more generally if this is non-trivial four manifold then it will sit as co-associative so-called co-associative cycle inside much more interesting g2 manifold which will be locally the total space of self-dual two forms on m4 so and this is the cartoon which represents this co-associative four manifold interpolating between m3 and m3 prime and the ambient space is a g2 halonomy space of dimension seven which has two boundaries kalabiyaw kalabiyaw prime associated to to the three manifolds here now in a physical picture again I'll start coming back to to using this uh five-brain theory and asking okay if uh three manifold before gave me a three dimensional theory then what about this four-dimensional co-boardism and then later on more dimension more general four manifolds what do they correspond to well because total dimension has to be six the four-dimensional co-boardism will give rise to something two-dimensional because two plus four is six and it has to interpolate or in this case be the interface of three-dimensional theories which are labeled by the corresponding boundary components that's a general principle that you can just get from dimension counting so again uh this was a long introduction or motivation if you wish to very basic principle which I could have started after the second slide said instead of going through this three three three-dimensional three plus three correspondence I could have said okay let's just do two plus four so I'm going to play the same game here but this detour or digression in three-dimensional three plus three correspondence is actually going to be very useful is actually going to be very useful because we'll consider four manifolds with boundaries and we'll be gluing four manifolds as well okay so given a four manifold you get some kind of two-dimensional theory which is labeled by four manifold and by the same general principle it will depend on geometry and topology of m4 and this is where the story starts and that's where the interesting the interesting dictionary unfolds the reason is that four manifolds are much more interesting than three manifolds and for many years I've been studying not homologies and the context of physics bps states and so on but mathematicians always ask me why you're doing this you should be we are studying knots because this is beginning of four manifolds for us and I constantly resisted because knots were much easier and four manifolds are much much more complicated so this is what I mean by saying that it's very rich math and if you think about it then this two-dimensional the class of two-dimensional theories that you get by compactifying 6d theory on or six-dimensional five brains and four manifolds it also has very rich physics so presymmetry here is very low so there is very little control and a lot of kinds of things that can happen in physics happen in this context so even things like dynamical supersymmetry breaking which is something that actually occurs in real world possibly yeah so if you take this two-dimensional so you do some kind of half twist or something how did the geometry of the four manifold does it I'll answer exactly this question in two slides excellent question so now we'll play the same game we have a complicated functor which takes four manifold and associates to a two-dimensional field theory luckily this two-dimensional field theory is not something general it's it's actually very it's going to flow to conformal theory in the infrared now mathematicians know what these words mean so in particular in two dimensions can feel theories are not that mysterious conformal theories for instance are vertex separated algebras so there is probably mathematical framework for what I'm going to tell you because as dimension of this mn increases dimension of this theory dimensionality decreases so we get to more and more conventional things now we're in dimension two now there are very several things you can ask about two-dimensional theory one thing you can ask is it's elliptic genus which actually already appeared in one of the previous talks in fact your chink will mention that and this amount of supersymmetry is just barely enough to allow for for existence of elliptic genus so you can ask what is such elliptic genus or I'll ask a refined question what is equivalent elliptic genus because this theory turns out to have certain global symmetries similar to what Nikita had in his talk so one can refine this notion of elliptic genus to keep track of those symmetries as well so it turns out well elliptic genus very roughly if you've never seen it before is kind of euler characteristic or refined version of euler characteristic it turns out that if you use this dictionary and ask what does it capture about four manifold it captures euler characteristics of modular spaces of instantons on a four manifold so here n is the second churn class this c is the first churn class of the gauge bundle you solve self-duality equations in each case for fixed values of second churn class and first churn class you get some modular space you take it sort of characteristic you arrange everything in the generating function and claim is that that's exactly the elliptic genus of this theory which is labeled by four manifold yes you do and that's that's this uh equivalent extension right on the on the other side of the correspondence now non-trivial fact or check a few wishes that elliptic genus is famous because it has some modular properties for certain obvious reasons which i'll show in the next slide and much less of this is the fact that this generating function of euler characteristics also supposed to have some modular properties so this two-dimensional theory could be very boring and variant of a four manifold but this already shows that it's not that it knows something about euler characteristics of modular spaces and here modularity has to do with work of waffenwitten who studied this this this function and their path was in fact not so different from what i'm considering here they consider the six-dimensional five brains on two-dimensional torus cross our four manifold of interest then if you assume that m4 is kind of curved small manifold and try to forget about it what you get is a two-dimensional theory on a torus which is labeled by m4 if you go the other way what you get and compactify on a two torus first you actually get four-dimensional super angles because you go from six dimensions to four dimensions reducing on a two torus and this symmetry of sl2z symmetry of t2 becomes the famous electric magnetic duality group of young meals on four manifold and that's some topological version of the same meals because m4 is curved and they conjecture that this object has to have certain modularity properties so unfortunately well this would be fantastic and even at the level of euler characteristics we would love to have some access to modular spaces of instant tons on general four manifolds unfortunately it's very hard to construct this modular spaces explicitly unfortunately we have very little access to even their euler characteristics so if i ask a physicist or mathematicians in which cases do we know this generating function of euler characteristics of modular spaces of instant tons on some four manifold the answer in this big space of four manifolds will be just here and here and maybe a couple of other points so this point is k3 surface this is what sometimes referred as del petso nine or half k3 and then probably a couple of del petso's above here would be the only examples for which until so far we can write down this generating function so that's very bad because as you see this is big geography and um botany of four manifolds and part of my motivation is to explore this territory way beyond this poor points even at the level of compact manifolds and what i'll try to do is i'll try to present to you some new results which allow to construct this euler characteristics of modular spaces of instant tons and gain some information way above this example so kind of push it way beyond what was no now you could ask okay what about instant tons on non-compact four manifolds uh well there is a famous construction a dhm construction at e drain filter chin and manian of instant tons on r4 and nakajima provided what could be viewed as generalization of that construction by looking at instant tons on le space by the way i le spaces which will play some role for me later are bounded by three manifold which is precisely the land space of the type that i mentioned as one of my examples and that's of course not an accident i'm trying to be efficient and use every information that appeared on the slides now nakajima looked at this generating series again of early characteristics of modular spaces of instant tons sum them up together so we have considered bundles of different topology and some of the results all together and what he found is that the result is actually a character of a fine cut smoothie algebra where the algebra itself is determined by topology in this case by the dinking diagram of carton type a whereas the level of the representation here whose character we obtain is determined by rank of the group uh if you study u n or s u n instant tons then this will be the level of representation so the role of the the geometry of the four manifold determines the uh cut smoothie algebra and the rank of the group on the instant on side determines the level of this representation okay so that's that's uh essentially the only result which again was available until recently about such early characteristics in uh non-compact context so you can view uh what i'm going to tell you next is attempt to generalize this my goal will be to use this big technology to generalize even this statement for us in this correspondence this character is interpreted as elliptic genus of some theory which is associated to a k manifold and uh ad type uh u n or s u n now i'll show you the first um new result which was derived using this correspondence between uh four manifolds and two dimensional theories but in retrospect there is nothing that prevented us from saying or somebody saying this back 20 years ago and this is really very simple so the idea is the following suppose you have a k minus one manifold this is a four manifold uh whose second homology group has intersection form of type a k minus one and suppose you want to build uh a k manifold by attaching to it another uh piece which contains one two cycle and again i'll use proper mathematical language in a second uh so that all together you have a k manifold now we already know if if this is a t q f t then um we know that partition function of the big thing a k should be roughly speaking sum over boundary conditions i'll denote them by row on the uh gluing along this three manifold of partition function for a k minus one and partition function for this uh cowardism i'll call it b okay so this is again a basic fact about essentially any t q f t and waffa witton tell us that this partition function comes from a t q f t so if it really behaves as to q f t or even some relaxed version of this you would expect that this formula should be true now let's combine this with what nakajima told us 20 years ago he said that this is a character chi of a k minus one this he said is a character of a k and what i'm trying to do is i'm trying to determine what is the z of b the partition function associated to cowardism so the statement that is my first new result here is that this partition function z of b which is the sum of earlier characteristics of modular spaces of instantons now on board isn't b is something that's called uh branching function of a coset model so branching function over coset theory it's a certain object which arises precisely as the composition of characters of one group or algebra g into characters of a smaller algebra and here is the trick that the lee type of algebra a fine cut smoothie algebra in this problem was determined by topology of the four manifold so as we're trying to make the four manifold bigger and glue some pieces which are this bordism or co-bordisms we're basically changing the type of this lee algebra and nakajima result and if you ask a physicist or in fact i think many mathematicians from again vertex algebra perspective know this how characters decompose they do decompose like this so there is a summation over say row prime this will be row and this guy will be labeled by row prime and row and the only unknown ingredient here that co-bordism b then is precisely the branching function so that's um that's a very well known fact and therefore consistency of such gluing immediately leads you to to postulate that the generating function of modular spaces of instantons on such co-bordism is the branching function this c row row prime which is this object here so that's again the first new result and it can be now used to start gluing this co-bordisms in different ways to get a lot of new things just go way beyond ad type lee spaces which nakajima started to construct results for instanton modular space earlier characteristics on much more general four manifolds which you can get by by gluing such pieces together in completely different order so that's that's one result again here the slide essentially explains the same thing i just told you so again if you know uh basically partition function for big four manifold for its smaller piece then you can extract this coefficient which is supposed to be labeled by boundary conditions in this case uh boundaries or lens spaces lk1 lk plus one which i call row in row prime and your summing over boundary conditions here and then again very very general argument will immediately tell you then that should be branching function so it's a very well defined concrete object now again i'm playing this sort of game where we start with the four manifold we construct some theory which is labeled by four manifold and group g which is either u n or s u n and then the point is any question that you're going to ask about this theory two-dimensional theory will be some sort of manifold four manifold invariant here i asked the question about elliptic genus of this two-dimensional theory and as i try to explain to you in uh this few minutes it captures interesting information about four manifold namely it captures information about earlier characteristics of instanton moduli spaces which is probably not too surprising because elliptic genus after all is also a characteristic morally or conceptually well you can ask question about q-cohomology this is exactly uh question kevin asked me so if you do the half twist and ask about ground states that's the same as asking about q-cohomology in this theory again there is a super charge and if you think about it uh what it translates to is a statement about donelson invariant so donelson observables and their multiplication ring is precisely the chiral ring in this half twisted theory or put differently uh it's a it's a ring of q exact sorry q closed modular q exact operators in this theory so this theory which is again a very funny invariant of a four manifold is not so dumb after all it knows about moduli spaces well it knows about the early characteristics it knows about intersection form intersection um well you can also associate uh very different type of invariance to this two-dimensional theory here we had some numerical invariance we had some spaces and in their intersection cohomology one can also ask about space of marginal couplings in this two-dimensional theory that's a very well-defined physical question it speeds out some geometry as an output and I have no idea what this geometry means for the four manifold so it's some manifold parameterized by m4 so or labeled by m4 so there are lots of kinds of questions you can ask and this allows you to build the dictionary between again four manifold and various observables in this four and this two-dimensional theory which is labeled by four manifold okay any questions all right so in the next in the remainder of the talk I'll try to say a few words how one can build such two-dimensional theories very concretely so if you want to implement this dictionary it would be useful to know how to construct this two-dimensional theory so we can take various invariants of this 2d theory and it will be invariant of a four manifold we already talked about co-bordisms in fact I try to explain that if you're studying functors acting on homological not invariant you very quickly around to problem of studying co-bordisms between three manifolds or maybe three manifolds with knots so here I want to do something opposite and consider a four manifold which is stretched along m3 so this is a neck which whose cross-section is a certain three manifold then again the same principle that total dimension should add up to six quickly tells you that well and we already know that to a three manifold you associate some three-dimensional theory but when it's kept off on the left or on the right you get a two-dimensional manifold because total dimension is six so if you borrow four dimensions in here you're left only with two dimensions in the external space and what this describes is it describes two boundary conditions for this three-dimensional theory which is labeled by tm3 and one boundary condition is labeled by this four manifold with boundary another boundary condition is labeled by another four manifold with boundary which are glued together along the neck m3 cross the interval so the basic idea in this whole program is to chop four manifolds and basic pieces and then start gluing them together so you can glue the theory associated to a closed four manifold from basic building blocks in the past work with various collaborators we successfully implemented the same process for building three-dimensional theories associated to three manifolds by chopping three manifolds and basic pieces and again gluing them together so in the world of four manifolds and now I have to be a little bit more and more concrete as we progress the basic pieces and the basic construction is usually based on handle decomposition this picture is borrowed this beautiful picture is borrowed from akbulut's book you start with a zero handle which is basically four ball then glue on top of it one handles two handles three handles and four handles the last two in good cases can be attached in a unique way also to make life simple most of the time i'll assume that there are no one handles so we just deal with such handle decomposition which involves only two handles and therefore only two cycles so simply connected for manifolds information about such handles is usually encoded by drawing pictures on the boundary of this zero handle which is a four dimensional bowl it's a boundary is a three sphere shown here and what one usually draws is a touching circles of one handles and two handles one handle is represented by a embedded circle which could be knotted could be unnotted and labeled by a bullet but attaching circle of a two handle is shown by by usual type of knot and decorated with a number integer number which is early number that describes the normal bundle of this of this attaching circle or normal or handle to handle so in the end if you're interested in say four manifolds which are simply connected and build out of handles the picture is that you have a bunch of knots very much like a knot theory except that you decorate them with integers which are called earlier numbers and that encodes the information this is how you label four manifolds so next step is to construct theories two dimensional theories which are labeled by this sort of data or start gluing them together in this case intersection form on second cohomology of four manifolds basically given by linking numbers of this knots if i and j are different or simply by this earlier numbers if you're talking about self linking so sometimes again this this was general picture if your knots are all unnot then it's much easier to encode the same information by so-called plumbing graph so if you have say a curvy diagram which consists of bunch of unknots linked together in this chain type manner then in effectively what you can do you can encode the same intersection form by drawing this quiver type diagram or a graph called plumbing graph where each node corresponds to a two handle and again it only works if the corresponding circle is unnoted if it's unnot doesn't have knottedness itself and then all you have to do is to draw these bullets draw lines whenever they're linked and again decorate everything with these integers which are earlier numbers so this is for instance the curvy diagram and the corresponding plumbing graph of the e8 manifold okay i want to point out that it doesn't always work even if you work with just unnoted circles then this is a good example of a four manifold which is bounded by four torus here the linking number between each of the two circles is zero so unfortunately you run into conflict and you cannot describe it by any kind of plumbing graph like this so it only works for some type of four manifolds now let's come back and revisit what we discussed before you can build many four manifolds very large class by coborgisms very much like we did before and now hopefully these pictures mean a little bit more to you they just mean attaching a two handle and you can build again a lot of four manifolds by this process simply connected four manifolds by attaching two handles using the so-called normal trick which isolates each particular handle and makes it disjoint or unlinked from the rest of the stuff and you start with some kind of again plumbing graph say associated to this curvy diagram it could be much bigger and then you continue attaching additional handles in this case making this plumbing graph into a tree you can even start doing loops again modular certain provisors and so on and so forth so the point is if you know what to associate to boundaries and how to do this gluing and this is a question about this three three correspondence and that's been studied very well by now we do and also if you know how to what to associate to each individual coborgism you can build lots of four manifolds so this is very constructive approach and I already tried to explain to you that Nakajima's result can be easily extended to coborgisms as well basically each individual coborgism gives you at the level of partition function some branching function of a coset model so here is big summary of all kinds of facts about four manifolds and what they correspond on two-dimensional theories some of this already explained to you for instance coborgism corresponds to a domain wall between theories labeled by three manifold boundaries gluing corresponds to fusion of such domain walls this partition function buffer within partition function corresponds to elliptic genus its value on a coborgism is a branching function and so on and so forth well we talked about this chiral ring and donals and polynomials now in the remaining five ten minutes I want to do the following I want to kind of switch gears and consider yet another application ask about various moves called Kirby moves that don't change boundary but potentially may change four manifold and also that don't change two-dimensional theories so these are items here and here okay and they'll lead to something new so as I explained before some class of four manifolds can be labeled by graphs kind of like this could be very complicated graph possibly with loops which are decorated by integers these are earlier numbers and let me play the following game this can be viewed as completely different separate chapter from anything we did before so in two slides we're just going to discover something cool I'm going to play the following game let's just even do a billion case so I'll use group you one and to every such vertex in a graph we'll associate gauge group you one with its gauge connection a and unlike what we usually do we'll try to build trans diamonds theory whose coefficient is this early number a so in this case everything is a billion and the dots actually stand for possible fermion completion which won't be important for me now if you have a link connecting such bullets or vertices labeled by a i and a j to every edge we'll associate by linear trans diamonds term taking gauge field of one of them and gauge field of the other informing this a i which d a j combination so question is which appeared completely for for independent reasons and say condensed matter literature or or physics literature how do we classify all such trans simons theories so and various people have studied it but I'm going to ask a question what are the operations or what are the moves you can do on this class of theories this quiver trans simons type here is even in the billion case which are equivalences okay and again we'll find something rather interesting so suppose you have a piece of the graph which looks like this you have one vertex with coefficient trans simons coefficient a plus minus one it's this guy and it corresponding gauge field is b I call it capital B which is linked to only one which which is linked to vertex which has only linking number with the first guy of coefficient plus minus one and that will get in our dictionary trans simons term a d a begin with coefficient plus minus one the fact that they are linked together means that there is a trans simons term bilinear in both and if you try to classify such trans simons actions and this is the configuration you have then you may notice that a appears only in these two terms again if it's not linked if this node does not link to anything else then the part of the big Lagrangian or action which contains a connection a appears only here so of course what we should do we should try to integrate it out and if we do this we solve for a it basically equations of motion for a say that it's minus or plus b and if you substitute everything back so a is minus or plus b then you substitute it to this terms you see that you get a bunch of terms of the form b db coefficients sum up in a nice way that's why I chose these combinations with little shifts so basically what you get is you can remove effectively remove this vertex which had trans simons coefficient plus minus one now what I did I did a very boring game I try to classify trans simons quiver trans simons theories for you but of course if you now come to four manifold people and ask after we translate this in this graphical language what does it mean that this graph is equivalent to this graph well this is an example of what's called a three dimensional 3d Kirby move so 3d Kirby moves are these operations on Kirby diagrams which change potentially change four manifold but don't change its boundary the boundary three manifold and in this case this is operation of a blow up which takes a part of the plumbing graph attaches an additional vertex with only one edge and coefficient plus minus one and the rule is that if you do that this coefficient also has to change by plus minus one so then you can continue playing this game this is almost complete set of three dimensional Kirby moves again you see that you have the separations of blow up and blow down where you can create a vertex for destroy vertex with coefficient plus minus one and then it shifts the other guys if you have some vertex with earlier number or trans simons coefficient zero you can just destroy it and then everything else becomes detached and here is another move so all these Kirby moves are hard to remember but I want to give you a general principle if you look at this dictionary and try to think of this Kirby diagrams or plumbing graphs as quiver diagrams for trans simons theories it's actually very easy to remember what the Kirby moves are these are some operations which typically involve vertices of numbered by plus minus one and zero and morally there are just equivalences of these trans simons theories corresponding to this gaussian integration or even something simpler for example here I give you an example of this type of Kirby move where you have something labeled by zero it's even easier see if it's labeled by zero it means that this gauge field doesn't even get trans simons term it's on trans simons coefficient is zero so it's only linked to the other vertex which I labeled by field b its corresponds to field b with coefficient trans simons coefficient a and a appear is only in this bilinear coupling which has to do with the edge so therefore if you try to integrate it out this a appears as a Lagrange multiplier and integrating it out basically forces b to be pure gauge basically because of this term and that's why everything that involves b disappears so that's a very simple again field theory proof if you wish or explanation of 3d cubicle just exactly the same argument work in the non-Abelian case a non-Abelian case is a little bit more interesting so unfortunately not unfortunately it's well no the conceptual part does work in the same way you basically ask I classify quiver trans simons theories and again you recover this 3d Kirby moves so when I presented this to Mike Friedman during this year he said that it's actually a very nice way to remember Kirby moves because everybody constantly gets confused if you work with them you all you know that it's hard to remember which way you're shifting this early numbers up or down so this gives you a machinery how to remember it basically it's classifying trans simons the squiver trans simons theories and again that's one application of this correspondence in the remaining two minutes I want to mention the application in a completely different direction which came as a surprise so in this part of the dictionary which involves four manifolds we asked what about four d Kirby moves which correspond to so-called handle slides these are moves which are supposed not only to preserve the boundary but also preserve the four manifold so this should be some kind of equivalences of two-dimensional theories and this came as a big surprise because corresponding symmetries or equivalences of 2d theories were not known at that time and therefore this is where mathematics gave some prediction about physics using this 2d4d dictionary in particular even in the nabilian case it gives rise to identity this is from contour integral for which which represents this equivalent elliptic genus I mentioned earlier that basically packages instant on early characteristics the characteristics of instant on modular spaces and if you translate it to physics again now going backwards so in the previous example I went from three-dimensional physical theories recovering Kirby moves here I'm going backwards I start with Kirby moves and try to interpret what it means for physics for physicists it means that this integral integration has to do with u1 gauge symmetry and it contains some theta functions numerator denominator which correspond to various types of carl multiplets on two dimensions boils down to just the product of theta functions so there is no integration on this side it means that this two-dimensional theory has no gauge group and has only matter multiplets which contribute to elliptic genus as factor of a theta function so what it does it says that there is an equivalence between two-dimensional super QED a billion theory with bunch of charged multiplets and some free multiplets and even this statement was actually not known back at the time so that's a non-trivial duality between two-dimensional theory so I have to put certain decorations on it in particular there is some kind of twisted super potential which makes it non anomalous but this phenomenon is already quite interesting at a physical level because if you look at it carefully what happens is that these three multiplets carl multiplets on this side are basically mesons which are made of field phi which carries charge minus one and field psi i which carry charge plus one and general principle same as confinement which is what happens even an abelian theory in two-dimension says that you can only have combinations of total charge zero and in two dimensions this is especially a strong condition which happens also even an abelian theory what's cool is that then you can push it to non abelian theory this is kind of analog of question that Ezra asked me a moment ago and here you get something completely insane by physics standards because until this point there was no single fact in the literature about non abelian two-dimensional theories with this amount of supersymmetry so again mathematicians may not be excited about it but in this whole story as a physicist I'm most excited about this fact because these theories are of very special interest to physicists they play a very important role in string theory and nevertheless until last year this there was no single statement about non abelian theories of this kind and this is a completely new non trivial duality which again to me is the most interesting prediction that comes from these handle slides from four manifolds if I wasn't studying four manifolds I would not come up with with this duality now everyone talks about a fantasies Yuri Vanovich spoke about fantasy of penrose and then we all mentioned words gravity and speculation so I'll finish with one speculation which hopefully will help to remember what what we talked about I talked about graphs so four manifolds in this context are labeled by these plumbing graphs and plumbing graphs also come up in study of a model in particular these are very nice papers that influenced me when I was younger so work of Maxine with Yuri Vanovich in particular and Nikita's work on which which mentions trees in a very very explicit way so of course when you have a toric target space X then studying rational curves and maps into that space boils down to combinatorial problem which is essentially sum over trees and graphs and if you think again in a very speculative way of these trees and graphs being plumbing graphs of four manifolds such problems basically boiled down to summing over four dimensional topologies or four manifolds labeled by plumbing graphs so therefore one might think whether four dimensional gravity which has this bubbling fluctuating structure and sums over different geometries is some sort of a model so that's a wild speculation and on this I would like to wish Maxine many more years of inspiration to people like me. Thank you for this nice visual talk. Any questions? So for headless lives there's something like a pentagonal relation if you look several of them which is kind of similar to the probably relation to the logarithm and other appearances do you get something like this if you like to propose one kind of slide over the other over the third there's some kind of no no I don't I know that there are some unfortunately we had other this in 3d 3d correspondence the there was indeed a very nice dictionary between Pochner moves of triangulations of three manifolds and the corresponding operations and field theory but even in that case the story breaks down because unfortunately those triangulations don't capture all flat connections and again in the end it doesn't quite work so even that story is is poor in four dimensions it gets worse because not all four manifolds are triangulable and PL structures are very different so that's already a bad sign but so I only thought about triangulations of four manifolds in this case I don't know this operation that you're referring to where you have something like pentagonal handle slide I mean I would like to hear more about it. Okay yeah I think it's mixed to the last slide yeah it's easier yeah yeah but here yeah with the relation of total section yeah I remember this was it's some kind of primary diagonal calculus which was never interpreted in some field theory it was to get some iterabates and now he says it's for the manifold support. Yeah so here I'm suggesting that maybe this Feynman type calculus also gets connected to plumbing graph yeah as interpretation yeah certain decomposition of this Feynman graphs make sense because this is basically gluing over four manifold so and yeah I think it's a natural question again for at this level just a speculation of course. So you had these Kirby moves for these Chairman's theories is is it possible that they could be changes of Lagrangian and BB formalism or is that too much to hope? Because what you did in the Abelian case looked a lot like a change of Lagrangian in some ambient BB. Yeah and maybe it is I want to emphasize that this is for instance not the same like a politician I'm going to answer slightly different question so when I first thought about it and again people like Mike Friedman and others say oh wow this is great way to to encode these Kirby moves which we never can remember but then I was thinking whether this is really a property of quadratic forms or Lagrangians or physical theories and I think it is actually indeed the right question to ask because after all this is this are closely connected and I was doing baby operations which you can definitely write in terms of quadratic forms but unfortunately it's not quite because what happens here is that even the basic fact that you can add in this relations you can just add a vertex which is completely isolated labeled by say plus one or minus one. That's a trivial trans-heimans theory because it's Hilbert space of you want trans-heimans theory at level one is one-dimensional so it doesn't affect anything but of course at the level of quadratic forms it wouldn't be equivalent I mean we're changing the rank and in a not trivial way. So again I'm giving you slightly different answer to slightly different questions saying why mathematicians may not have thought about it because it's not just equivalence of quadratic forms which would be cool but no it's the equivalence of trans-heimans theories even on a billion ones. On one side of your picture you were considering gluins of four-dimensional manifolds bounded by three-dimensional but on the other side you're looking at two-dimensional theories also supposed to be glued through three-dimensional theories. I'm doing this computation six minus n equals n. Right yeah that's correct that's that's that. Do you know how to compose say two-dimensional theories through three-dimensional theories? Yeah yeah so these two-dimensional theories are kind of like interfaces and or physicists call them domain walls and the composition the process of composition is basically composition of domain walls so literally you can imagine three-dimensional space with a bunch of planes sitting next to each other and then you ask once you collide them together or put this like a sandwich right what is it so and there is a very well defined process how to study this so for physicists this is already a good enough answer. The walls are three-dimensional between two-dimensional objects. Well the dictionary is that the dictionary is perverse so if you have three manifold you get a three-dimensional theory if you have a four manifold which say could be co-bordism then you get a wall between two phases of three-dimensional theory so they all together have they always have to sum up to six that's the idea that's that's if you forget everything else from this talk that's the only thing you have to remember.