 I love it here and every time I come I feel jealous of people who get to work here. This is really wonderful. So what I'll tell you today is mostly based on two papers. One with Chiprin Manalescu, so this is based on joint work with Chiprin Manalescu who is my colleague in California, so that's reference number one. Reference number two is work with a number of people, so it's easier to remember it by the title. It's called 3D modularity, and again it has a number of authors, but most prominently it has Francesca Ferrari, collaborators, and since Francesca is here, a local, she's right here in the audience, we can ask her all kinds of questions. If my talk is unclear, so she's responsible for everything that's unclear. I'm joking. So the story that I'll tell you can also be referred to as BPS CFT correspondence. This is the term that Nikita likes, and for a good reason, it relates BPS quantities which count some integers, Donaldson Thomas type objects, and we'll see lots of this in these two lectures, and we have already seen many of these before, and connection to CFT. So my title, for example, chiral algebras, refers to chiral algebras of some super conformal theories. What's interesting is that usually we get conformal field theory from two-dimensional physics, but today we'll see chiral algebras appearing from three dimensions, and three dimensions here means two things, either topology of three manifolds or three-dimensional supersymmetric physics. And those two, in turn, are related by so-called 3D-3D correspondence, so a longer version of the title, it's a short title anyway, but a slightly longer version could be BPS CFT and 3D-3D, or if you want a shorter version, just 3D, that's fine. Now BPS CFT correspondence itself has long history, and in fact goes back to work of Nakajimasan, and later many other people have joined, but going back to mid-90s, that's where it started, there was an observation that if you start with four manifold, in that time of a very specific kind, resolution of clany and singularities, so let's call four manifold M4, then to that you can associate differential geometric problem, so our meeting here carries the name of differential geometric invariance, so typically that means solving PDEs, constructing modular spaces, proving all kinds of cool theorems about compactness of modular spaces, how to deal with singularities and so on, so one can construct modular spaces of solutions to PDEs on a four manifold, so most famous set of PDEs involves anti-self-duality equations and therefore one can construct modular space of solutions with instanton number N, second churn class N, and as one of their simple invariance you can try to take a characteristic of such modular spaces, so you get something that depends on M in four dimensions on a four manifold gauge bundle is topologically classified by second churn class, so we have actually lots of these spaces, and you can then form a generating function by summing over N, so this of course is not such a modular space, and the problem should be replaced by much better descriptions, so in fact most of the time when we can compute something we use algebraic geometric descriptions, and the corresponding better behaved object is what we now call Waffel-Witton partition function, so it's invariant of a four manifold and at least conjecturally, and depends on Q because we just form the generating series, and observation was that once you start your computation and topology and use a lot of differential geometry analysis in algebra to get the answer, the answer turns out to be something meaningful in algebra which has not been put in by hand, so that was a surprise, that was a cool thing back in the mid-90s when I was joining the field of mathematics and physics and to me personally this was like wow moment, you start computation in one field and you land in something completely different, and the surprise here was that this answer matches a character of a vertex operator algebra, so that was precisely the surprise or interesting moment, and now we call this vertex operator algebra VOAFM4 because it can be realized or I mean by now this phenomenon can be understood as a relation between four manifolds and two dimensional conformal field theories whose chiral algebras of course are vertex operator algebras, and this VOAFM4 is basically chiral algebra of two dimensional CFT that we call TFM4 and physics of this two dimensional CFT is labeled by four manifolds as very rich, very interesting and was one of the subjects of Pavel Putrov's lectures, so now this phenomenon or this perspective allows to generalize this relation between VOAFM4 partition functions and characters in various settings, but what I want to tell you is basically a 3D analog of this story, and there will be or there is an analogous surprise relation to chiral algebras which unlike this one we don't understand very well yet, so this had to wait at least until this series TFM4 came to the scene to be kind of controlled so you can predict for example what the central charge is and various other characteristics, so here in three dimensional story that I'll tell you it's still at the level of this mid 90s surprise where we cannot predict central charge, so it's checkable but it's a much earlier stage of development. So the statement and the surprise of appearance of chiral algebras is that if you start with a closed three manifold or in fact any three manifold which I'll denote M by M3 you can associate to it also some sort of partition function, the partition function will be called Z-head, it's a partition function of a three manifold just like VOAFM4 partition function is partition function of a four manifold. In turn it will be defined as a certain BPS quantity that we introduced in the work with Abhijit Gada and Pavel Putrov so sometimes this is called homological block or half index so there is this GGP half index of a theory T of M3 which is analogous to how theory T of M4 appears here so there is some physics in there and I'll try to avoid physics at least in the first lecture and surprise is that the result turns out to be character again of vertex separator algebra of a chiral algebra but of a much more interesting nature so first of all in examples that we have seen so far this vertex separator algebra is logarithmic and it depends on the choice of three manifold just like this chiral algebra depends on the choice of four manifold and this statement here is precisely the statement of a conjecture that can naturally be called 3D modularity conjecture first of all because it's about modularity of these three manifold invariance and secondly because it appears in the paper by the same title so my goal therefore today is to introduce all the objects on the lower part of the blackboard namely tell you about first of all this GGP half index which can be constructed in any 3D and equals two theory so this will be important for certain things that I'll try to say in the second lecture then I'll introduce this invariance that had for three manifolds and hopefully give you examples or enough examples where we'll see that there is a connection to chiral algebra so the status is very different of all both of these three parts so this is physics and sometimes if I want to emphasize that my discussion is physics and mathematicians can tune out I'll label it by P P for physics it's very concrete very rigorous you can do it I mean there is a lot I can tell you about this half index and I will in the second part this discussion of that hat will be the subject of mostly first talk so this will be framed for mathematicians mostly and with theorems and conjectures and concrete statements and this relation here should also be called M M for mystery or this is something that's interesting I mean it's a conjecture but kind of conjecture again that to me puts this roughly at same level as we had this observation in mid 90s something that happens is checkable but for instance if I can predict what the central charge of this guy is in 2019 I cannot predict what the central charge of this is and therefore this is more like a challenge so this is homework to young people and the audience to to figure out and complete the story okay and of course at any point feel free to ask questions and trap me and stop me so one of the obvious goals so the starting point is to have construction of this object that had four three manifolds and I want to point out so this is therefore a 3d to QFT or physically it's expected that it's a 3d to QFT and what it does is the following so to to a closed three manifold so I should say TQFT in the sense of a T seagull so a T seagull proposed set of axioms for how to define topological quantum field theory and it's supposed to be a functor so for me the name of the functor will be that had such that it associates to a closed three manifold a number so for us this this number will depend on additional data that corresponds to a choice of a root system or you can think of this G as a kind of gauge group and also Q which is complex number so in entire discussion today G will be SU2 and again Q is going to be in general complex number most of the time I'll later assume that it's inside the unit disk and then continuing with this two manifolds of co-dimension one two and so on one should associate a space of states so for example two manifold sigma one should associate Hilbert space H of sigma which also depends on this choices of G Q and so on and so forth and if you want so-called extended TQFT you want to continue but I'm not going to continue further down today in fact I'll try to focus mostly on the first line so today we'll talk about this line here where we'll talk about closed manifolds and in order to have a TQFT structure what's important is to be able to do cutting and gluing of your three manifolds so M3 plus M3 minus along various surfaces and in general you can cut along say genus G surface sigma but for my discussion today I'll only restrict to cutting and gluing along Torai so in fact we'll barely talk about the second point and when we do we talk about cutting and gluing in genus one so extension to higher genus is a good problem to work on yes it's not an operator from H to H it's a functor from co-boardisms between two manifolds into in the linear operations from beyond H yes yes so so my goal of this entire two lectures is to describe to answer this question essentially so the entire two lectures will be devoted to understanding how G comes well not so much how G comes in so today it will be set to ask you to unfixed once and for all I don't think I'll discuss any generalizations for a sake of time but Q how that enters will be prime subject so the whole two hours that will follow will be addressing this question so there are several things I'll try to tell you or or what will follow will fit into so first of all therefore the goal is to construct this TQFT and like I say I see I'm very honest about open questions and challenges such as understanding for example this log VOA is a good challenge and open avenue that is good to work on likewise constructing this for higher genus the scotting and gluing but luckily for our discussion today the case of genus one is going to be enough because there is a theorem due to licorice Wallace and Kirby and others which says that every connected oriented closed three manifold arises by performing an integral Dane surgery I'll explain what this is in a second a link a link L is a set of embedded circles in a three sphere so in three dimensions they can be noted so that's an example of a link which has two components it has two S ones each embedded in a three sphere in such a way that they don't touch each other or that's it's not an immersion and a surgery operation is the following we take such a link in a three sphere and then you cut out a two-bill a neighborhood of each link component as a result what happens is that neighborhood of a link consists of disjoint union of copies of S one cross a two-dimensional disc and since boundary of S one cross a disc of each copy is a two-dimensional torus on which SL to Z acts as a mapping class group you can try to do the following can remove this copy of solid solid torus and glue it back using some element phi of SL to Z again mapping class group of a two-dimensional torus which tells you about non-trivial different morphisms not connected to the identity and as a result once you use this this gluing you can produce potentially a different three manifold so claim of this theorem is that all three manifolds can be obtained by such surgery operations so in general they're called rational Dane surgeries because element phi that I'm writing here so maybe I'll avoid writing on the lower part of the blackboard can be brought to the form pq something something and then this operation is called p over q rational surgery but the statement of the theorem is a little bit stronger and is a little bit nicer it says that instead of talking about general p over q rational surgery you can always restrict yourself to the case where q is equal to 1 so that's called integral Dane surgery so Dane surgery is operation I explained a moment ago and integral means that you can restrict yourself to integer coefficients rather than general p's and q's so that's kind of nicer what it means in practice is that to perform this operation if I have a link say link with the two components I have to decorate each component with an integer number that's basically the value of p so I have to assign to it value p say p1 here and p2 here which have to be integers and they will tell me what kind of surgery I'm performing is this clear exactly so right so then there is a whole machinery that's where mr. Kirby Rob Kirby comes in so first of all maybe before I comment on the statement I should say that what he proved is that any two different presentations so there are many different ways to construct the same three manifold starting from different links and not and what he proved is that any two such presentations are necessarily connected by so-called Kirby moves we'll see examples in a little bit and in particular using this Kirby moves one can undo rational surgery into sequence of integer surgeries so that's that's one way to understand why integralities sufficient essentially yes yeah yeah sometimes it's it's a little bit more complicated because I mean you have still you need to mention this this Kirby moves because if you're doing it on components which are on knots things are really easy but if they're noted then it's not just algebra it's also a little bit of topology any any other questions so the upshot is that any three manifold can be constructed by such surgeries on links and not not is basically a link with a single component so therefore in order for me to construct this functor this topological invariance at head I'll need either construction for three manifolds or construction for not complements and crucially ability to glue to do the surgeries in other words not just have another compliment but rather to glue it back so therefore there'll be two parts in my story one is how to construct this Z-head for not complements and second is how to do the gluing these are surgery formula informally so here here we go any any other questions now before I go on I want to point out that what why are we doing this so why why is it interesting or useful so there are many motivations which I'll mention along the way and part of the motivation of course is interesting connection to chiral algebras that appears and many others but from the sake of topology if you're trying to define topological invariance you want them to be strong and interesting so when I was giving a version of this talk about a year ago Mike Friedman posed the following question so he pointed out that there was a nice work by Louis Funar which I'm going to write here on the board that the name of the paper because it basically explains what Mike's question was so the title of the paper is Taurus bundles not distinguished TQFT invariance and Taurus bundle also known as mapping Taurus is a particular three manifold constructed by taking a two dimensional Taurus times an interval and identifying both ends by also element of the mapping class group which already appears on the blackboard so you take this identification means that X zero where X is a point on a Taurus is identified with Phi of X one where Phi again is that same element of SL2Z not surprisingly because we're using two Taurus and mapping class group of a two Taurus is SL2Z so the picture for this is you have a two dimensional Taurus at each point on a circle S1 base and a three manifold is constructed by fibering this two Taurus over S1 with monodramey parameter by SL2Z metrics very similarly to how we use the SL2Z matrices and surgery operation and then as title of the paper suggests Louis Funar gives examples of such mapping Tauri fairly simple three manifolds to be honest such that no existing TQFT invariance including quantum group invariance of with and Rishi to arrive can distinguish them so his point was simply that if you take M3 constructed in this way using monodramey Phi and M3 constructed monodramey using monodramey minus Phi you already get such pairs of three manifolds let me call this M3 and M3 prime such that many existent invariance are blind to this pairs so I'm not going to go in detail but I just want to give you a character of this of the statement but the question that Mike Friedman was asking when I was describing the Z hat invariance he asked a very natural question what about the Z hats are they any better than existing invariance existing as important word because existing at 2013 and the story of that hats roughly starts around the time so I claim that this is probably the last year when a paper with such title can be written because that has one of the claims will be that this function that we're going to describe does distinguish this pairs so it's it's a for topology sake it's a pretty strong invariant more over for some values of Phi which are supposed to be this SL2Z matrices the construction I gave you here is closely related to construction by surgeries so we're talking still about the same roughly same class of manifolds for example if Phi is minus ST times ST using standard S and T generators of SL2Z by 2 by 2 matrices if you multiply it out it's going to be 1 1 minus 1 0 in this case your three manifold is a surgery surgery with coefficient p over q which is then general Dane surgery on a not K is usually denoted in this way you say that you work in a three-sphere case you're not and p over q is the surgery coefficient so in this case you can realize this particular manifold which is a torus bundle over a circle as a surgery on a not in fact on a rather simple not cold trefoil so not 3 1 looks like this it's a not with three crossings that's why three is part of the notation and surgery coefficient is actually zero so it's a zero surgery on the trip for likewise if Phi is in this construction is minus ST ST inverse then what you get is monodramy matrix 1 minus 1 minus 1 2 and in this case the manifold that we're talking about is also a zero surgery but on a different not it's a zero surgery on figure 8 not and primes are the same where monodramy matrix is minus 5 well yes I well I gave in fact I give using what I'm saying is that there are two kinds there are not surgeries on simple not as far as I know I mean that they don't have to be they can be described using as mapping Toray so this gives this definition but well it's it's a pair for mapping Toray defined with monodramy matrix Phi and minus Phi these are just examples so these are examples but I want to emphasize to make this crystal clear that if you take a general Phi which is a cell to see metrics it's not going to be surgery on any not so therefore sorry I'm just saying that these two constructions the two I gave you overlap in some cases but overlap does not have to be all-encompassing so it just includes for example these two members but but the pair where M3 is constructed from minus Phi is still a pair it just not a surgery on a not but but it's a surgery on a more complicated link and I don't know what the link is of the top of my head any other questions so either way what we learning is that in order to have constructions you have to have again two things you have to understand what this invariance are for say not complements or either link complements and you have to have a way to glue the solid torus back in other words you have to have the surgery for him so with this motivation in mind and kind of preview that these are going to be fairly strong invariance for topology of sake and some of the constructions next I'll try to give you actual construction of this invariance say how to define them so special case of of surgery which I call affectionately called surgery on unlink unlink is like I'm not or so I'm not as a is a not which is just boring circle I mean figure eight not with four crossings of this picture of the triple not obviously have some interesting topology so you can form maybe I should call this a hope flink or generalized kind of hopefully maybe that's a better name you can take bunch of copies of a not like this and if there are not if each component is not individually noted you can encode this data of surgery on such circles that can be linked to each other but each individually has no interesting linking with itself in terms of the graph so you replace each link component by vertex so in this case there are three components so there'll be three vertices and each time there are linked together you can draw an edge okay in principle you can have loops in this kind of diagrams by by having them loop around number of times for simplicity I'm saying in a special case so let me start with a special case when we'll do surgeries on oh yes simple linking and I'm not yes correct yes they just do this yes exactly so let's we'll let this in a second we'll generalize this in a second first I want to start with this and we'll generalize this to like I say my goal is to define this for any close three manifold so we'll do surgery on obituary complicated links in very good question so I'll come back to this it's one of my favorite examples I love it but obviously it's not included here yes that's correct thank you yeah well the reason I love bar I mean rings is because if you do bar I mean rings for those who know have the property that if you do zero surgery they have three components if you do zero surgery on each component you get a three-dimensional torus which is also in the class of mapping Tori with monodromy matrix one so it should be fairly simple example so we'll come back to it I love this example so it's also example where you can construct it by surgery and also as a mapping torus and of course in many other ways so anyway good and it has fairly large b1 b1 is equal to 3 so that's kind of like a random three manifold very very non-simply connected but anyway now slowly going going back so as we agreed you have to do a surgery with some coefficients so you have to specify integers on each component so for example this one I want to label by minus 157 this one by plus 3 this one minus 4 and this are day in surgery coefficients which therefore have to be recorded in this data of the graph as well minus 157 plus 3 minus 4 so what we're in this case you can encode this data of a link in terms of a graph whose vertices are decorated by integers so therefore in at least for this class I have to tell you how to construct this z hat in terms of this this data of the graph it's convenient to introduce a matrix Q which will be adjacent see metrics of this graph gamma in other words you'll say that Q has the following values in slot ij it has value of the surgery coefficient let me call it ai so ai's will be this surgery coefficients you can call them a's or p's I don't know what the best notation you prefer if i is equal to j if it sits on a diagonal it has value of one if ij connected by edge and zero otherwise so that's the usual definition of the adjacent see metrics for for a graph and then you can give a very explicit formula for for for this invariant z hat in this case so definition is that in cases of such three manifolds constructed by graphs and three obtained from from the surgeries on hopeful links or generalized unlinks is given up to overall power of Q and possible phase that I'm going to ignore that's not going to be terribly important for me at least in the beginning as a principle value integral of variables Xi on a unit circle such that the number of these variables is the same as number of link components in other words every time you do a surgery you introduce a variable Xi so they'll be X1 X2 X3 in this example complex variables which leave in C star and we'll define this as an integral over let's say xj's xj over 2 pi i xj then product over all vertices of the of the graph with factors xj minus 1 over xj to the power 2 minus degree or valency of the vertex j times times a theta function for the quadratic form Q so theta function is a sum over lettuce lettuce with the quadratic form Q in this case it's going to be for example three dimensional lettuce I'm going to raise this example in a second but since there are three link components we'll have a three dimensional lettuce it depends on X variables and Q it also depends on characteristics which I'll call a which take values in core kernel of our quadratic form Q and therefore the answer will depend on these guys as well so let me call this labels maybe a or B well yeah unfortunately the this there'll be lots of a's to to come so I'll have much worse version of this issue that that notations will be too similar the measure yeah so in fact when I was writing this thank you I realized I should have said product here in the beginning so yeah it's let me write it as product over j inside the vertex set dxj over 2 pi i xj yeah sorry that's that's a better way to write this so now I'm gonna erase this example and also you can quickly see that for this definition to be meaningful it's better to or interesting it's better to require that this quadratic form is definite so let's make this assumption I think in my notations it's better be negative definite but you definitely want some kind of definiteness for this integral to produce a nice Q series so statement which is a theorem formalized in that paper with cheaper in Manolesco but all the ingredients in fact are taking from the paper with dupe I will put proof sorry G P P V and common buffer and also a couple of ingredients from the 3d modularity paper is that this definition has sequence of nice properties so there'll be a total of four properties first one of which is that this object constructed in this way is convergent Q series inside the unit disc one absolute value of Q is less than one second property is that this object or second statement of the theorem is that this this gadget defined in this way is a power series with integer powers of Q and integer coefficients possibly up to overall rational power I'll call it D sub B which may take rational values but the rest is of the form C 0 plus C 1 times Q plus C 2 Q squared where all the C i's are integers and therefore this is basically element of Q to the D sub B Z of Q that's the second statement so third statement is that this object defined in this way is invariant under Kirby moves which I mentioned a second ago and which at the level of these graphs means that you can do certain operations on the graphs and coding the link data for instance if you see the configuration of two vertices labeled by a i and a j and connected to something else you can equivalently replace it so this is equivalence by configuration where you have a i plus minus one plus minus one a j plus minus one you don't change the rest of the graph but in terms of link data you create one more link component which is linked to other things so that was part of the move where you allow to change your link on which you do the surgery and claim as that three manifold is not going to change so that's one of the Kirby moves there are many others and it's easy to show well it's not entirely easy but it's important to show that with some assumptions on this graph which is definitely whole for all definite but you can weaken this assumption it's invariant under all such moves exactly yeah right exactly so I'll yeah I'll you'll stay in the world of this kind of hopefulings yes exactly I'll transition to Kirby moves which involve non-trivial knots in a second correct so these are simple illustrations so I apologize because this is not like one month course on topology I won't give you full encompassing presentation on definition of Kirby moves I'll just give you illustration of what they are in the context which we can easily see in examples so it will be incomplete in that sense oh yes correct correct yes exactly I'm saying what what the statement of the theorem is yeah in fact indeed this this brings a more interesting point that if you show so in this world actually if you show that it's invariant under this moves it actually guarantees that you have invariance under a much larger set of Kirby moves so I'm not gonna refer to again those more general statements but I'll come back to knots in a second then we'll see examples of non-trivial Kirby moves that change you for example that produce the same manifold from surgery on a trefoil and figure 8 and we'll have analog of this statement I'll give you an example yeah right so sometimes yeah sometimes this gamma is called I raised it is called plumbing graph yes that's that's that's another name for this no no so that's why it's important statement and that's why this theorem is going to be warm up so this theorem is warm up for the general case and more general case let's ask how it's different it's different in a way that if you have a graph kind of like for example this one you may want to take again in the context of links what what does this mean this means that you have one component linked to another link to another and then they do something else right so you may want to take to generalize it what you really want to do is to replace this component by some non-trivial knot so for example what I want to be able to do is to take a trefoil in place of this on not and perhaps do the same thing with every component so then I'll be in a completely general class so therefore what we really need is analog of this formula of this definition which works when each component is an arbitrary not so claim yeah yes yes yes exactly yes yes yes I totally agree yes yeah so we have to produce analog of this formula for links for arbitrary links yes yes exactly so therefore the statement will be yeah so it can depending what generality you want right away or in in half an hour so this is warm up for what we're gonna do next and if you want to replace one component by say not k what you want to do is to have analogous formula where remember this guy come equipped with xj the variable that got integrated so therefore claim as that was going to change in this formula is that instead of factor involving this xj minus xj inverse to this degree what you'll have so this will be replaced or there'll be additional factor that depends on that particular xj it may also depend on q and it surely will depend on the not k if you want to do non-trivial surgery on if you want to replace it by not so there will be a factor in this formula that depends on k on choice of a not and of course now you can guess what the most general statement is it will be such that there will be fl for link l with any number of components that depends on variables x1 x2 all of them so my goal is to tell you what this f's are so morally what this so therefore as I said in the beginning the story is going to be about this tqft and in order to do the surgeries I'll have to have two ingredients I'll have to have first of all what this that hat is for not complement or maybe since you prefer more general statements I'll write link complement and that's going to be exactly what I'll call f sub l that depends on many x's and q's so I'll have to have first of all definition of this guy and secondly I'll need a surgery formula I'll need the gluing so therefore it's everything I'm going to say will have essentially two versions of the statement one for close three manifolds after we do the surgery and one for many falls with total boundaries before we do the surgery this link and not complement okay but I didn't quite finish the statement of the theorem so there is a fourth statement and fourth statement says that if you take the limit of this Z hat invariance for a three manifold m3 when q goes to exponential of 2 pi i over k where k is an integer and often called the level what you get is with an Roushetian drive invariance of that same three manifold at level k and k in at least usual transimons invariant theory is usually assumed to be integer so therefore the picture that I wanted to keep in mind is that on q plane we have the unit disk where this objects that had are defined and exist when q is of absolute value less than one it may be crucial to say less strictly less because there may be diversion on the unit disk but they have interesting property that if you try to go to roots of unity you recover traditional quantum group invariance of Witten Roushetian and to write and those are only defined at roots of unity in fact Witten would emphasize they're only defined for integer k not rational k yes it depends right so Roushetian and to write that's why I said Witten Roushetian to write define them for rational k only rational k 1 k k is in q and Witten wouldn't says that you're in Simon's theory with compact age group makes only sense for integer k so anyway this is a limit this is a limit yes yes that's true except that you you have to call it Roushetian to write and that's what I said I answered your question I don't think it was clear if you insist on having arbitrary rational values then call them Roushetian to write is this clear any other questions so therefore I'm not going to in unless you give me an extra hour I'm not going to repeat all the statements of the theorem line by line in the future I'll give you several versions of the theorem for knots and links correspondingly and when I say theorem and I just tried something telegraphically it means that all items are true and it was cheaper and we found a way to abbreviate this I think which is typical in math literature and very clever so instead of saying let's assume that for example in this context before the linking matrix is definite that this quadratic form is say negative definite which was important for this to be convergent inside the unit disk so instead of saying these things the statements of future theorems you should always understand whenever make sense were defined defined as a q-series the following items will hold true and then in the case of knots and links I'll have the corresponding versions of first of all integrality statements and invariance under Kirby moves and and and relation to Roushetian to Rive or with Roushetian to Rive invariance so so then what what I'm going to tell you about this this things is three types of statements so when I'll describe future properties and generalizations how to do surgeries on one more general knots and links there will be statements of first of all reformulations of this theorem but also statements that can be characterized in several groups so one will be statements that relate this to written Roushetian to Rive invariance but in the case of links and knots analog of written Roushetian to Rive invariance as a color Jones polynomial so there'll be relations to Jones and WRT and this is an example there'll be statements how such invariance of this Z hat is related to complex trans diamonds theory in particular since we talk about as you to its complexification is SL to C so there'll be statements about relation to SL to C trans diamonds and since this is workshop or conference on differential problems and modular problems there will be modular spaces in this game namely modular spaces of flat SL to C connections on three manifolds and when we talk about not complements we'll talk about a polynomial which describes a flat connections on not complements so that will be one clue or way to see that what we're dealing with this Z hat is actually SL to C trans diamonds theory so another name for this functor if you wish is to say that morally so what this second type of relations imply is that you can think of Z hat is SL to C trans diamonds TQFT and I have to say that for almost 20 years we were trying to make sense of SL to C trans diamonds TQFT which would enjoy cutting and gluing so this goes back to even Mike in their garden work on a polynomial and then work with many other people when we invented various partition functions but those partition functions never behave nicely under cutting and gluing so they never form a TQFT in a sense you cannot do surgery and only recently with the help of young people which include Dupay, Pavel Putro, Francesca and others this was made into a TQFT structure so that's why you know I'm personally excited about it and the third type of relations that I'll illustrate so there will be relations to John's and WRT this will happen always at roots of unity when Q goes to exponential of 2 pi I over K I'm not going to assume that K is rational I'll just talk about integer values there'll be relations to complex trans diamonds theory and finally there'll be relations to to rife torsion that's in the case when manifold is closed but if manifold is a not complement which is again important for surgery to rife torsion is known as Alexander polynomial and this last aspect is perhaps the most interesting it's most interesting because first of all it gives connection to something that's readily computable so I'm started by talking about in variants of four manifolds constructed out of modular spaces and now I want to emphasize that those are hard so I myself used to work on them quite heavily but now I would not even suggest to a student to study of a written partition function of a four manifold just because modular spaces are too hard you can do one at a time and it takes you a year to complete here you can do bunch of surgeries construct bunch of three manifolds in a split second so the modular spaces involved a fairly simple so it will do differential geometry on three manifolds but it's very simple it will be much much simpler than the case of four manifolds and this last point is will provide a bunch of relations to to rife torsion and Alexander polynomial in such a way that will make this this structure very very computable so it will be very straightforward and part of it will be due to the fact that Alexander polynomials and to rife torsions are readily computable are very simple invariance of course they won't capture everything but this relations will be strong enough to help us go along the way so yeah I think this is probably a good place to stop since I think exhausted most of my time so let's let's take a break here