 Thank you very much Anton. So first it's a great honor to be invited to this conference. Of course Maxime has had a huge influence on everything I know, everything I understand since I started Mathematics. I remember in my first year in graduate school my advisor told me that I should read all of Maxime's papers. This was not easy in Gronowski. So this took me the next several years. But you know somehow this is how I learned everything about homological algebra and operals and the BV formalism and Trincyman's theory and mercimetry and infinity algebras and the modular curves and all this stuff. So I think it seems this great paper from 1994 in the European Congress of Mathematics and I spent an incredible amount of time reading this paper. It was great. But anyway, so today I want to talk about something that's long been a dream of mine. Which is something about understanding some piece of string theory in mathematics. So one of the string theories physicists would like to consider is 2B super string theory, which is something defined in ten dimensions. And then one of the important ingredients in 2B super string theory is adiast CFT correspondence which conjectures some relationship between open strings and closed strings. So what I want to the aim of this talk is firstly understand a small piece of 2B super string theory. So it's going to be some twisted form. And this is you know something which is maybe familiar to many people. Maybe some aspects of this since Witten's work in the 1980s you know for example Witten tried to study n equals to supersymmetric gauge theories and he argued that we're going to understand a piece of that n equals to supersymmetric gauge theory by thinking about Donaldson theory and that was an incredibly fruitful idea. So I want to apply something similar to string theory and supergravity and also kind of prove a very weak form. So some of the version of adiast CFT and am able to access is not super not incredibly powerful but it's it hints that there are more interesting non-perturbative results which can you can see by thinking about this method which I hope yeah should be interesting. Okay so what's what's the basic idea of adiast CFT? First thing I... that's weird. Some kind of gravity is gone. Supergravity, yes. So remember just some kind of basic story of string theory closed strings in type 2b become in some limit 2b supergravity. Kind of if we have say if we're on kind of say on say x for example a 10 10 manifold and have y inside of x for this even dimensional on y I can consider open strings ending on y and this becomes some kind of supersymmetric gauge theory on y. This is the kind of very basic string theory thing and you know very many possibly all supersymmetric gauge theories can be realized in this way in even dimensions. So I want to talk about twisting this story and when we twist the story we're going to get something which is more accessible and more familiar to mathematicians at least conjecturally. So the first thing to talk about is twisting supersymmetric gauge theory. So just say on R4, say or n. You have a supersymmetric gauge theory on or n then it's acted on by definition. This is what it means to be supersymmetric by some kind of a superly algebra. If we choose some q as an odd symmetry which satisfies q squared equals 0 then the twisted theory as in Sergei's talk is is a sense is equal to the q commod you the original theory. So I'll give you a better definition in a second because you might ask what does it mean? What does this really mean? Um better. Well field theories you know if you if you're like me and you like things like the BRST and BV method of considering field theories then everything in your field theory is a homological object anyway. So for example the Hilbert space of the theory is a co-chain complex. H with some differential of BRST and the twisted theory with some kind of new complex. So we just add on this q to the differential and this makes sense because the fact this was a symmetry means that these guys commute and this because because q squared to 0 this still squares to 0. So notice that there's there's going to be a spectral sequence relating this guy to this guy. That's how they're related. So this is something you can compute. This may seem like you know the way I've written it down the procedure is a lap strike but it's something you can really sit down and compute at the classical level by messing around with various forms of various types and spinners and seeing and may various maps between them. Let's see what it means for supergravity. I could be wrong but I as far as I know this is not appeared in the literature and this is a bit more extreme maybe. Let's consider kind of say supergravity on on or 10. Now the difference between supergravity and ordinary supersymmetric field theories is that here the supersymmetry is gauged. That means we the fields here might be a metric and a bunch of spinners and the gauge symmetry is the ordinary diffeomorphisms plus also some fermionic symmetries whereas here it's just a symmetry we're not quotienting by it. So here supersymmetry is gauged. Now the way one way to model this gauging by some symmetry algebra is that we is to introduce ghosts as introduced by Feynman and BRST so that the supersymmetry algebra, sorry this may be a bit obtruse for non-physicists, but you can there's a math way to say it too but maybe I won't have time to do it. Now the way the ghosts work so ghosts what ghosts do it's a way of modeling the quotient by something because it introduces some Chevalier algebra complex based on the symmetry algebra. So the way it works is that if I have an ordinary bosonic symmetry then the ghost is some odd field but if I have a fermionic symmetry then the ghost is bosonic. So fermionic symmetries, symmetry q, q becomes some kind of bosonic ghost then twisted supergravity is by definition supergravity in a background where q is constant. So note that this makes sense q squared is zero and so q preserves something like the background metric. Okay so I hope this wasn't too confusing to the mathematicians. This is just the vector field? Yes it's a vector field. So it's always q squared? No because it's an odd vector field. Think about the Dirac differential that's a derivation of the Dirac algebra. By constant do you mean like a killing spinner? Yes. Okay I said an or 10 but if I did in some other manifold it would be a covariant constant. Yes so the fact is killing spinner that means it preserves the metric. Okay any other questions? We're doing like perturbative supergravity, perturbing around this background. So there's a bunch of fields so I'm saying we're going to perturb around some supergravity background where all of the fields are zero except for the metric and this field. Yes and if you think about it the reason this is a good thing to do is that if I could couple supergravity to some other field theory and then I put the other field theory in this supergravity background that would do what we've said above. So that's why it's a reasonable thing. Okay so now we have some physics and hopefully we're going to get to more mathy things in a minute. So the conjecture is that the twist in this sense of the B model of sorry of 2B string theory say on for example X which is some collabio five-fold sorry in X could be or 10 for example and the reason you want it to be collabio is what Ezra said because that guarantees there to be a covariant constant spinner is the B model. So what do I mean more precisely? So the B model will come from open strings but as I did gauge theory it will come from open string B models and B brains. So that's so I'll explain that in a second. Was it there or yes there are. I wasn't sure if people can see them though because of the angle. Can people see these these blackboards? Yes yes okay. Can you resolve the kappa symmetry? The which symmetry? Which is the kappa symmetry? It's a covariant poison of string theory. Oh yes so I don't I don't know how to do this. I don't know how to do this in the world she point of view. I'm only knowing how to do this in the super gravity point of view and the gauge that we point of view. The reason is that from the world she point of view the space-time supersymmetry is kind of not obvious and I don't understand I mean this is how you should prove it right. I understand space-time supersymmetry of the world she point of view and just calculate but I don't understand that. So why did the gauge replace string theory with super gravity? Well you're absolutely right this is only approximations this is this is I mean you got to hope that there's some better version of twisted string theory which I don't understand at the moment but you're absolutely right this is a twisted form of this approximation to string theory. Because the usual form is not about super gravity, it's about string theory. Yes. Why should it hold on the nose? Well it does hold somehow I suspect the reason the reason it's going to hold is that you could you could further guess that the you know the twist of the actual 2B string is the B model string and the B model string has no instantons so therefore the super gravity representation is a very good one. So the B model again is two pieces it's the open string it's again like X is some calabiow say five-fold and Y inside of X is a complex manifold I can do the kind of holomorphic transimans type, type field theory which you get from thinking about the open string gauge theory which has fields or HOM, OY, OY, tensor GLN, shift of one it's well it's a chiffon it's a coherent chiffon X you resolve it as a perfect complex you take a causal resolution of this guy so for example if if like X is equal to vector spec V a vector bundle on Y and Y is equal to zero section in this case you can compute the ORHOM by a causal resolution and then the fields the double complex of Y with coefficients is this this make you happier yeah I mean all examples will be of this nature anyway that I care a bit okay so you can ask just as a check of the conjecture you can ask do you mean we shouldn't care about other examples or we haven't got to them yet other examples are interesting although if you want to construct things of the quantum level it's much easier to do this kind of example and and to relate to actual supersymmetric gauge theories that's much much easier to do these computations on flat space the conjecture is two parts firstly that the the twist of type 2b supergravity is BCOV theory the second part is that if I take for example holomorphic flat submanifold of OR10 a linear submanifold of C5 then I can twist the type 2b type 2b d-brain gauge theory in the usual sense and what I get is this kind of holomorphic trans diamonds theory that you can actually calculate and it's true okay so this so in the case of say y is equal to C2 inside of C5 you expect from the physics this conjecture should say that this field theory tensor gln is a twist of n equals 4 young mills this you can calculate explicitly and this is true I wrote a paper calculating this several years ago 2011 and there are many other examples where it really is correct so so in what sense is this a precise conjecture yes so there are two parts so the first conjecture is that the twist in the sense of type 2b supergravity just at the classical level for now is BCOV theory and what's because BCOV theory is that what you get by the supergravity theory associated to the b-model so I was about to say that in a minute on x on a collabia 5-fold I was going to say that one more detail in a second and then the open string version is that so here if we um so more generally so the open string version is that the twist of the D-brain super symmetric gauge theory is this kind of holomorphic trance simons based on this kind of gadget yes yes I don't see it here so we're all right but I'm so what we're saying is okay the kind of diagram I want is like open closed EDS CFT and then there's a b-model if open and closed I was about to say in a minute what the closed b-model is but this is going to be so much holomorphic trance simons and BCOV this is twisting this is twisting and here it is CFT and this is I'm able to kind of prove some shadow over the statement I haven't talked about this yet so we expect again that the b-model closed string for example on x a collabia 5-fold should be equal to a twist of 2bc book out this is the conjecture so let's see if I can make this I need to explain to you so one one one we when I'll talk about ADS CFT in this b-twist we first need to understand what the closed ring b-model is oh man that's really silly I love the same blackboard I just pushed up so yeah so I need to explain to you what the b-model closed string it looks like so fields of BC of e theory well you can write them in two ways it's going to be the cyclic co-chain complex of the category of perfect complexes on x and then you can use the Hotfield-Costin-Rosenberg term to write this more explicitly as poly vector fields on x adjoin some parameter t with some differential plus m this is the usual d bar operator such as you to explain briefly poly vector fields on x is the Dalville complex on x of the coefficients in the exterior algebra the tangent bundle and the del operator is is a divergence operator corresponds to the con b operator and a maps omega i x wedge j t wedge j minus one t of x yes exactly yes this is isomorphic to omega 0 i x wedge 5 minus j t star of x and here it's the usual drama yes exactly this this correspondence is all going to work in some non-compact x since I've erased this already yellow so the questions we want to address of the following yes it's the way we're going to treat it is it's a field theory so it's a field theory the bv formalism so it's going to be described by a shifted Poisson manifold like in what Tony's going to talk about this kind of thing however this is a kind of degenerate theory normally it's described by a symplectic manifold this example is not some practice but Poisson so I need to describe you what the the dg manifold structure is and what that Poisson by vector is so the dg manifold as I learned if you remember reading Maxime's papers in the early 90s dg manifolds the differential is given by corresponds to some lead bracket on some space and so it's controlled by the differential graded the algebra differential graded the algebra which is poly vector fields on x t shifted by one with the shoot and bracket you have you with this it's going to formalism and under so to give a dg manifold I need to give you it it's the spectrum of a dg wing got a formal dg manifold so the ring of functions co-chains is Chevalier-Almer co-chain complex so really the dg manifold is equal to some kind of formal formal extended modular space of deformations of x and you can also think of it as the modular of deformations of category of perfect complexes on x as a clavier category degree t is equal to yes exactly yeah oh when I say something like this this is the content of Maxime's famous theorem but formality this equality is this kind of very hard theorem kind of formality there okay well what's the Poisson structure so the Poisson structure Poisson kernel is let me just finish the sentence I think I take the delta function that angle applies delta operator that's the Poisson kernel so sorry you want vector fields to be in degree zero oh yes yes absolutely sorry with that yes and the quantum field theory I will explain in a minute that you can quantize it it's extremely not obvious yes and it's also it's if you think about it it's a non-normalizable so but there's and only in the people case yes but so that was actually the next next topic I wanted to explain is when one can you quantize this and what does it mean yes so when you take co-chains it's a topological vector space okay so when can you quantize and what does this mean so firstly for the closed string you see the answer is that you can always quantize so why why would you expect this so I'll suggest two different ways to think about it so one is one way you think about topological field theory topological field theory in the sense of Max Maxine I didn't work on myself this produces for you something based on a clabio category yeah so that's why it's not really going to be good enough for what we want to do and I suppose you could always think with the compact supported sheeps but I think it's not going to be really strong enough for what we want what the second point of view on this it's my collaborator and myself showed how to do some quantization quantization in perturbation theory no but if you want to get this the kind of thing we're wanting that choice does not appear for what I mean by quantization because you want some solution of the master equation the master equation you don't need to extend to the boundary of the Morford space okay so maybe I'll say a little bit about our argument in a second because our argument kind of similar to the TFT argument Maxine introduced involves the open-close theory so fix a sheaf say e on x can we quantize the couple theory three for like that open the couple open-close theory and the answer is if and only if the total shunt toss of e is zero so the what's that this has to do with you know constant yes and it has to do with something in the a model which you're very very familiar with I'm sure if you want to kind of count curves with boundary on some Lagrangian then in this count there's a bad there's a co-dimension one boundary when the circle shrinks to a point so there's this argument of Dominic Joyce that says if this is the boundary of some chain then you can cancel this anomaly by thinking about some kind of closed strings so it's exactly that kind of thing so in that case you get one higher dimensional model space some choices of conservation and you have yes yes but if you work in a but if you take something which is a kind of canonical trivialization so for example the structure sheaf plus the structure sheaf with a shift then it's going to be canonical so for the structure sheaf on x plus all weps of one there's a canonical quantization of the open-close theory and why is this well the argument cc and I have I think I don't it's kind of a pretty argument but I don't have time to explain in any details given I'm running I have 15 minutes left is that right yeah so I don't have an ascending about ADSEFT so so there's some kind of cancellation between kind of homology groups controlling ambiguities of quantization for the open theory and the final thing you want to ask is when can we decouple the open theory from the tft picture this comes from a kind of co-dimension one boundaries like Maxime mentioned a minute ago I have some surface like this and I might have some cycle there I might have some cycle which shrinks and as Maxime said that the issue here is in the B in the a model this is not a problem but in the B model involves a choice of trivialization of the hodge filtration and you kind of don't want to choose a trivialization of the hodge filtration in general because that's not something you can do locally on a manifold it's kind of a global thing so so the answer is you can do this this isn't money in the following situation some y is an even dimensional club yeah e is a sheaf on y I take y inside of y cross 0 inside of y cross c and then we consider a kind of theory on y cross c with the sheaf I star V well so if you think about these these modulites we're gonna have some solution of the master equation coming from the modular space integrating over top chain to the modular spaces and the boundary the boundary of the modular space that corresponds to various terms in the master equation so if you want to have a solution of the master equation just for the open string piece you want to arrange so that the boundary of the modular space only knows about kind of open string degenerations or not this kind of one okay so I think I'm gonna slap so I think I'm not gonna get to explain much about ADSEFT so I just have to briefly explain what kind of thing we find when we quantize and then a little bit about ADSEFT what does quantization give us and firstly the closed string if I take some kind of open subset of X we have this sheaf of the algebras I can take Chevalier algebra co-chains of this guy and we think of this is from the classical observables so quantization produces some co-chain complex which looks like the same kind of thing with the h-bar dependent differential so the h-bar dependent differential comes from you know things about higher genus surfaces and well it turns out here we had a commutative algebra this h-bar dependent differential with the bv formalism is no longer a commutative product but we have we have product maps 5u and v where she disjoint and w we have a product map like this so this means we have what I call a factorization so how should you think of this I'm sorry that like I have to I'm writing it like a book which is like 500 pages long with one of my collaborators on this kind of factorization algebras so I'm sorry I said everything about it in like half of a blackboard but the picture is the picture is very similar to Graeme's picture but what a quantum field theory does we should think it for example if u v and w are discs that the product comes from a cubordism so that these spaces we assign to you and v they're like the Hilbert space of the theory on on the boundary of the disc this product map is the cubordism so it's the kind of thing you know people are kind of familiar with again any any questions about this probably lost everybody that's so I wanted to get this there's a there'll be a punch line in about a minute so similarly open string it's kind of similar if I have some sheaf sheaf on x which is supported on some y inside of x then I can send you this co-chains h bar the sheaf or hum co-chains of the sheaf or hum with some quantized some differential which depends on h bar and the differ what is the differential the leading term will be d bar and if heuristically it has pieces which come from counting curves like the a model analog of pieces coming from counting curves has a term from the pairing it's a bv operator plus some kind of sum this makes sense I mean just for for people who like top logical field theory and I think I know Kenny's thought about something very similar all that's going on is if you think about these kind of in the a model you think about counts of curves but landing on a Lagrangian there's some master equation that master equation tells you some differential squared is zero that's this differential okay so in the five minutes I want to explain how these things are supposed to be related so I'll explain how they're related in a simple example the example I'm going to consider about is on C3 but I make which is made non commutative and I put on a we have a poly vector field which is d z1 which d z2 so that the first the Poisson bracket is at z1 z2 is one the first two coordinates fail to commute and that we have an inside of this guy I see for the first coordinate side of C3 and my my sheaf is the structure sheaf on C tensor CN plus CN one see the first copy of C inside of C3 yes no it's it's we should really think of it as something symplectic cross C so it's a Lagrangian and the symplectic guy so we can quantize it exactly now if you think about what what do you find if the or harm in this non commutative world well for the first copy of C2 you know we know how to compute or harms in something on commutative that's just we just can we get like that around complex of C and then I join an odd parameter from the or harms in the other guy so that the fields look like C epsilon tensor GL n slash n so if you think about it this is like and the B model for the classifying space of GL GL n slash n another way to think about it is that the topological limit of young males of 2d young males I'm sorry I don't have more time to explain about this okay so one of the open strings of open strings factorization algebra it's an e2 algebra because we're dealing with a topological theory topological the closed strings if you think about what the structure you find and we have see a non commutative C2 cross C well the non commutativity makes a topological here whereas the other guy it's a holomorphic guy thing so it's like an e4 vertex algebra so let's give names for these guys we have the kind of a open string and we're going to take some kind of n as this n goes to infinity so we need to take the end of infinity limit which you can do in this situation and we have kind of closed strings which is the co-chains of poly vector fields and then the theorem that the open string guy is equal to the causal dual of the closed string guy as an e2 algebra sorry this would have been more of a punchline if I had time to explain it so but this is the the more general claim is that this is what the duality gives us each side we have something like holomorphic version of any n algebra when they're causal dual so I take the closed string guy restricted to the sub manifold and these things are casual dual so why I was going to say oh yeah so why is this kind of the very if you look at the classical level this casual duality is simply the Hoshilikoson bosngrithir Hoshilikoson bosngrithir combined with Segan's theorem about large n the algebraic homology so they're a classical level it's a kind of a triviality the semi-classical level there's a Poisson bracket this turns out to be a very hard theorem this is Maxime's formality theorem because we need to look at the Lie algebra structure whether it's the cyclic version of the formality theorem proved by Wilbacher but out of the quantum level it's some kind of deformation of this and from the point of view of physics you know what we're seeing is you know we're computing the algebra of operators and the OPE of the large engaged theory in terms of something kind of dual to this and the closed string theory okay so sorry I ran a little bit over and thank you for your patience differential H bar differential after I'm second fully differential operators yes it's all local well it's a little subtle me there there's some kind of choices but you can make it as local as you like yes yes yes so I didn't have time to explain this so the reason I chose GLN slash N for the reason that you know that means that the fundamental task can be trivialized with a cycle trivializing it also has support on C but if I chose GLN then I would need to trivialize the class of C the class on C3 and that trivialization is going to be something supported in all of C3 which is poles along this C so that means so that's where you know I it's we I don't fully understand this but what happens is that I need to remove C and then I have something which pulls on this on C and so I get something more familiar from the EADCFT picture make any sense I mean I'm still a bit puzzled by this aspect so this is going to be very vague but I'm trying to understand the meaning of what you said and if we drop back down to quantum mechanics E1 yes then we've got say the Durand complex on one side and differential operators on the other yes and somehow resolve the idea of resolution that the Durand complex resolves the constant chief is flung by causal duality into more to equivalence or something is this make any sense it's possible I mean I think it's somehow easier I mean so what we're what I mean we're not saying the Durand complex is causal dual it's why there are co-chains in the Durand complex some deformation of that so this is so at the classical level this is the e2 algebra of co-chains of GLN and at the classical level here it's an e2 enveloping algebra I mean it's some e2 enveloping algebra oh I was simply presenting a kind of a toy model I wasn't meant to be a special case yeah but I mean somehow it's just like a funky field theory analog balls I don't know I can't quite parse it but I see I see what you're saying it looks a little similar so it's a classical casual duality is mediated by by module so it's what's happening here yes yes yes that's an interesting question so another way to say it's mediated by an augmentation is another way to say it so you need to show why why is this augmented and why does that and some of you think but what the augmentation means remember we'll be saying that it's sometimes hard to decouple the open string guy so the obstruction to augmenting this guy is precisely the obstruction to decoupling anything another way to say it is if you look at the couple theory looks like this the harshal comology of that whole string guy and that's what you expect from something to answer your question