 Yeah, so thank you for the chance to come here. I've never been here before and to talk about this stuff. Let me mention that the main results I'll be discussing are joint with Brian Williams. Although sort of the framework is developed in work with Kevin Costello, sort of broader framework. So this talk will be about, maybe I should just write over here the summary, sort of the goal of the talk. And the talk will be aimed at explaining what the terms mean. So study symmetries of holomorphic field theories in a uniform way, meaning sort of working over different dimensions, not just in, say, complex dimension one, allowing systematic generalizations of results from chiral CFT on Riemann services. So the main goal of the talk is more or less to see how there's a generalization of the affine Lee algebras, which show up in chiral CFTs on Riemann services, to higher dimensions. So I'm going to do sort of foundational background stuff right now, give some context and some techniques. And then Brian will explain some work he's currently pursuing that puts this stuff to some, has some very non-trivial applications. So I need to tell you what I mean by holomorphic field theories. We've had two talks today that sort of focused on topological field theories. So I want to talk about holomorphic field theories. I'm going to do that by example. I'm not going to give an axiomatic definition. So I'm going to study relatively simple examples. But hopefully it'll be clear how to generalize. Suppose I'll use x for a complex d-dimensional manifold. So one example, to bear in mind, is known as the beta-gamma system. There's two fields, gamma and beta. So gamma is going to be a smooth complex-valued function on x, beta is a dolbo form. So this was, if you'd like, a 0, 0 form on x. And then the action is this. So if you apply del bar to this, you get a 0, 1 form. And a 0, 1 form pairs with a d, d minus 1 form to get a top form, so you can integrate. So the equations of motion here, the Lagrange equations are the del bar of gamma equals 0 and del bar of beta equals 0. So these two mathematicians, that's a very beautiful theory. You're picking out holomorphic functions as your sort of on-shell fields. There's an obvious fermionic variant. So that's all bosonic. There's a fermionic variant, which is the BC system. And all I do is switch the parodies here. So C. But you can do other examples. You can do the curved beta-gamma system or holomorphic field theories. So you could do curved beta-gamma system. So here gamma would be a smooth map from x to some other complex manifold y. And beta would be a d, d minus 1 section of the pulled-back cotangent bundle. You could do holomorphic gauge theories like holomorphic transignments. So these are field theories to bear in mind. And then we'd like to understand how symmetries interact on them. So let me introduce what I mean by symmetries. I'm going to focus on the beta-gamma system today because it's a very easy system. And hopefully you can track what's going on there pretty quickly. So the solutions to the equations of motion are holomorphic, say in particular, holomorphic functions. So what I'm going to do now is modify the fields by having them take values in a representation of a Li algebra. And then if the Li algebra acts, of course, it'll preserve the space of solutions. So in particular, let's imagine we have a holomorphic principle bundle, which is some complex Li group, think like SLN. And V is some finite-dimensional representation of G. Then we get some associated bundle. And now I'm going to have my fields be sections of this bundle. So gamma now will be a section of coefficients in V. And beta will be a d, d minus 1 form in the dual bundle. And then I have the same action functional. But now I use the evaluation pairing between V and V dual to get a top form. And I get the same equations of motion. So now solutions are, again, holomorphic sections of V. So solutions, holomorphic sections, so on and so forth. But now I have an extra symmetry to take advantage of. There's the automorphisms of the principle bundle, which will act on these solutions as well. So let me say that there's going to be an action of the Li algebra. So this is a kind of charged beta gamma system. So I like to point out that if you have a holomorphic section alpha of the adjoint bundle, preserves solutions to the equations of motion. If del bar of gamma equals 0 and del bar of alpha equals 0, then del bar of alpha times gamma equals 0. And this is just a point-wise action. At each point in x, you're getting an element to the Li algebra, and you're acting on the fiber V. So this is an example. This is a holomorphic, what I mean by a holomorphic symmetry, charged beta gamma system. So I'm trying to give a gloss of what the talk is going to deal with. So here's holomorphic field theories, here's symmetries. And now with this on the board, I can give you the flavor of the kind of result we're aiming at. And I'll spend the rest of the talk trying to make that precise. So we will study this system, this situation, using the Battalion-Vilkovsky formalism. As articulated by Kevin Costello, we're going to use some of his renormalization techniques and the language of factorization algebras and things. And so the flavor of our results looks like this. Well, we have sections of the adjoint bundle as a sheaf of Li algebras, holomorphic sections, and it's acting on space of solutions to equations of motion. And this ends up being encoded in the following kind of information. So these are factorization algebras, or chiral algebras. And this one is supposed to be the observables of the classical field theory. So it's the commutative algebra functions on the fields with some extra structure. Now there's also another chiral algebra. This is CUR for currents, these classical currents, which are sort of determined by the symmetries. The idea is that there's a map that takes every symmetry to an observable. So this map sends a symmetry, e.g. alpha here, to an observable, depending on alpha. So this is sort of the Nerder theorem in action. Symmetries determine observables. Now in the Patel and Volkovsky formalism, when you quantize, you end up producing a deformation of the classical observables in a way that depends on h-bar. So there's a de-quantization map. This is much like deformation quantization where you have an associative algebra in powers of h-bar, such that modulo h-bar, you get back a commutative algebra. This is also a chiral algebra, a factorization algebra. And the kind of question we're going to ask is whether we can lift these symmetries or these observables in the classical, sorry, these observables that come from classical symmetries, to quantum observables. And so the answer is typically you can't lift. So typically you cannot lift j classical to a map into the quantum observables. And the obstruction is a holomorphic version of the ABJ anomaly. So one goal is to actually write this specific anomaly down explicitly. But once you find this anomaly, then you can introduce a deformation of the classical currents to the quantum currents. Let me give this anomaly a name. Let's call it theta of v, because it's going to depend on v and the way the Lie algebra acts on it. This is also de-quantization. And there'll be a map of chiral algebras from some sort of factorization algebra of quantum currents to the quantum observables. Things like ward identities can be read off from this map. OK. Do you literally mean that there's a map on the left-hand side of that vertical map? Yes, there is a vertical map. This is also mod H by H. Twist the domain of that by the chiral class line theta. How about you ask this question again in like 40 minutes and see if my answer, yeah. We're going to modify the Lie algebra of symmetries using that co-cycle, some sort of L infinity extension. And then there's some interesting factorization algebra you get out of that. It's like the enveloping factorization algebra. So one upshot of one thing you can read off of this map, in particular, by compactifying along the spheres. So there's a natural map from the punctured d-dimensional space down to the positive reels, which is taking the radial projection. So this goes to the length of that vector. And the compactification of this map along R recovers sort of results of Fanta, Hennion, and Kapronov on higher cat's moody algebra. So these three authors introduced some L infinity algebras or higher Lie algebras that include in the dimension one case the usual affine Lie algebras but generalized in the higher dimensions and showed they had a nice systematic relationship with higher versions of the affine Grasmonian and of the modulite G bundles on higher dimensional spaces. And what we'll see is that we're giving sort of the complementary physics story about how sort of Lie algebras they write down, these higher cat's moodies actually act as symmetries on observables of holomorphic field theories. So that's sort of how we're going to see the systematic generalization of results from chiral CFT. Any questions so far? So this is the target of the talk is to understand these statements. So now I said I was going to use the BV formalism, so I just want to revisit the description of the theories and the symmetries in the BV style. So let's do the classical BV theories. And in practice, this is just going to sort of mean doing everything in a differential graded way. So gamma is no longer just going to be a smooth function. It's going to be an element of the dolbo complex, dyes and v. And beta is going to be in the dolbo complex of holomorphic d-forms with coefficients in the dual bundle, but with a shift. So I should say here d is going to be the dimension of x. And typically, just for simplicity, let's always think of as just the d-dimensional space. Well, so this means we have a graded space of fields. Degree 0, we have omega 0, 0, coefficients in v. And then the beta field is d, d minus 1, because of the shift I gave. So the stuff in co-homological degree 0 is exactly the sort of naive theory I wrote down before. But I've introduced these other things. So this is all 0s in the gamma part. But down here I have omega d, 0. What I want to point out if you're familiar with BV is that, of course, these fields have their anti-fields being here. The anti-fields of this field is here. These are partnered up as anti-fields. So there's this pairing, if you'd like, on the fields, the evaluation pairing on v. You just wedge and use the evaluation pairing between v and v dual to get a form. It's a pairing of degree minus 1. We think about it as kind of providing the space of fields, this dg space of fields with the shifted symplectic structure. So let f denote this sheaf of fields. So far I haven't talked about everything being on the whole space x. But of course, I could keep track of use different opens in x and think about sections over a given open subset. So f will be a functor from opens on x. So in this case, we could say chain complexes. And it's going to send the open set u to the dobo forms on u. That's where gamma lives, plus the d forms on u shifted by d minus 1. And this is just sort of the obvious, so there's an op here. If I have fields on a bigger open set, it restricts to fields on a smaller open set. Now, if I take functions on fields, maybe I should say what I'm trying to do. I'm trying to tell you what this thing is very briefly. The classical observables are functions on the fields. I'm going to be thinking about sort of polynomial or power series style functions, because these are actually sort of dg vector spaces. And I'm totally going to suppress the function analytic issues you have to bear in mind. So they define a functor the other way to commutative algebras in chain complexes, which I'll loosely speaking say is functions on f of u. So something like the symmetric algebra on the linear dual of f of u. Polynomial functions on the fields. So I'm not using the board space very efficiently. One of the things I want to point out is that it has a certain, there's a picture to bear in mind to relate to the thinking about things like OPEs. If you have some big open set and two smaller disjoint open sets inside of it, there's this sense of field on the big open set v restricts to a field on u1 and 1 on u2. Functions go the other way. You can pull back functions from these smaller open sets, from fields on these smaller open sets to fields on the big one. That's this map from this Habs classical. But we know that the fields on two disjoint open sets is equivalent to the fields on one open set directs from the fields on the other open set, which means that this thing is actually the tensor product. So if I have an observable on u1 and an observable on u2, there's some way of putting them together. You can take the measurement you make in u1 and the measurement u2, and you multiply their outputs. And that's this composite map that defines an observable on the big open set. When we quantize, this structure gets deformed, and you can read off of it things like the OPE in vertex algebras or in chiral algebras. OK. What's something else I need to comment about? So these fields are actually a DG algebra. We had this differential, I should have drawn it, sort of del bar here, imposing the equations of motion. So Habs classical has a differential arising from del bar as well, because it functions on a co-chain complex, symmetric algebra. But it also has a one-shifted Poisson bracket, often called the BB bracket or anti-bracket, which just comes from this pairing here. So there's this kind of these vector spaces, like the fields are like a shifted symplectic thing. So functions on it should be a shifted Poisson thing arising from this pairing on F. And actually, one of the things you can check is that the differential on the classical fields is the bracket with the classical action functional. So we'll use that later when we quantize. So now I want to describe the sheaf of symmetries, which is going to be the dolbo version of the holomorphic things we said before. So let's supine script G here to be the sheaf of dolbo forms with coefficients in the adjoint bundle of P. It's sort of run flat space. Then G just looks like the dolbo forms. Tends for our Li-algebra G. So it's a nice DG Li-algebra. And G acts on the fields in a point-wise way. Because the Li-algebra acts on V, I guess way up there, any section of this, I mean, this is like smooth sections of something, dolbo forms is a commutative DG Li-algebra. And that's with values in some representation V. Of course, G acts on V, so you can extend it over this commutative DG algebra to get an action. But if you have an action on a space, then you also have an action on functions on the space, which, if you'd like, can be, this is a Li-algebra map from G into derivations. This is, of course, I've been calling the classical observables into the classical observables. So these are vector fields if you want. OK, so far so good. So we have this Li-algebra map. But I just told you we're thinking about things from the point of view of Poisson geometry. Like, we have this Poisson structure we can bear in mind. So one question we could ask is sort of mimicking what you do in Poisson geometry, there's a map from functions on F to vector fields on F, taking the Hamiltonian vector field. And you can ask if we can lift this map. Now I get to use this. So we have this map to derivations of functions on fields. We also have a map from functions on fields with a funny shift, because it's a one shifted Poisson bracket. So this sends a function F to its Hamiltonian vector field. It's the Hamiltonian vector field. And you can ask if you can lift this map, which would exactly be realizing a symmetry as an observable. And the answer is you can. And the map is very simple. So the observable for some element here on beta gamma is going to be the integral of beta. You just let alpha act on the symmetry alpha act on a field, and then you pair it with beta like you did before. And that's an observable. So a symmetry determines an observable. Note that it has the same form as the kind of flavor as the action functional. It's local on x. It's an integral over x. I want to raise a simple but important point here, which is that this formula is very simple. But you could ask, when is it actually integrable? We want it to be an observable on all fields. So there's no reason to suppose beta or gamma are compactly supported. So if alpha were non-compactly supported, this whole thing might not be something you can integrate. So we're going to have to restrict alpha to be compactly supported to guarantee that you always get something you can really measure, so an actual observable. So simple but important point, which I haven't seen sort of emphasized in the physics literature, though maybe it is somewhere I haven't seen, is that you need alpha compactly supported. So really, I'm not going to work with a sheaf, but a certain co-sheaf, or pre-co-sheaf, I should say, to the algebras, it's going to send you to the compactly supported double forms on you in this adjoint bundle. So this means compactly supported. And this is good because I want G to map to the classical observables, right? And the classical observables is a functor this way. So it's a pre-co-sheaf rather than a sheaf. You see there's an op here. So this is a good thing. So actually, instead of having G go to functions here, we have a map of pre-co-sheaves of the algebras, which I'll call J classical, and this is a map of DG-ly algebras, of pre-co-sheaves of D-ly algebras. So we're doing this sort of stuff we did before just taking into account the variance in terms of space-time support. So this map, J classical, there's two different ways to rephrase it that I think are suggestive and that will be useful when we quantize. So two variants. The first one is to rephrase in a way that looks like a moment map. So this is sort of a moment map style. So let's just recall how the usual moment map picture works with the unshifted Poisson stuff and then we'll just do the shifted version. So if M is a symplectic manifold, then in symplectic geometry, we're typically interested in G acting by Hamiltonian vector fields. So we have some map of L-algebras. Of course, you can compose with derivations on O of M and this is taking the Hamiltonian vector field. But this map of L-algebras, this thing on the right is not just a L-algebra, it's a Poisson algebra. And you could ask, well, can I extend this map to a map of Poisson algebras? And there's a very simple way to do that. You can just take the symmetric algebra on G. So it's commutative structure, just the usual multiplication of polynomials. But then you extend the Lie bracket to a Poisson bracket in the canonical way. Now there's a map of Poisson algebras into functions on M. So this is the, if you'd like the enveloping Poisson algebra, it's the left adjoint to the forgetful map from Poisson algebras to Lie algebras. But you could think about this if you'd like as functions on G-dual. So this induced map of Poisson algebras is pullback along the moment map. So this is pullback of functions along the moment map from M to G-dual. Okay? So this is just, you can go back and forth between this data and give a map out of this enveloping Poisson algebras. The same thing is just to give a map of Lie algebras. So this kind of data, they're equivalent, but this might have a more geometric resonance with you. Now I'm just going to do the same thing, but in the one-shifted context. So I'm going to tell you what this thing is now. And this map will be the one-shifted version of the map of Poisson algebras there. So to find the functor of currents, which will just be taking the symmetric algebra on this preco-sheaf. So it's going to go from opens in X to one-shifted Poisson algebras, category of one-shifted Poisson algebras, DG, of course. And J-classical determines a map which I'll also call J-classical abusively from the currents. So this shift here cancels out this shift here. So this is a map of preco-sheaves of shifted Poisson algebras. And all it's doing is turning every observable, sorry, every symmetry into an observable, and polynomials and symmetries if you want. And you can think about this as a co-moment map. So this is a one-shifted co-moment map. I find that helpful for porting intuition from symplectic and Poisson geometry into this one-shifted setting. OK. So that was one way of rephrasing or re-encoding this map that's sort of attractive. And it's told us one part of our problem. The second way is going to be more in the style of field theory, which is that map there, J-classical on alpha. That looks like a local functional. It looks like part of an action functional. So why don't we view alpha itself as some sort of background field? So view alpha G as a background field. So we have a bigger space of aqua-variant fields, which will be the direct sum of F plus G. And the aqua-variant action will define on alpha, beta, and gamma. And it will be the original action, which if you'd like, you could write. So are they only on the last line there should there also be a shift on the left? Where? Here. On the right, too. Off-seal? No, this shift. Sorry, they're supposed to be right there. Yeah, sorry. I did it here, but I didn't put it there. Yeah. They cancel out. OK. So now one way to understand what I'm doing here is as a baby version of causal duality. So this is a baby. OK. So then the aqua-variant observables will be functions on this thing. And I claim that it looks like the Lie algebra co-chains of G with coefficients in the original classical observables. OK. And the differential here, the aqua-variant version of the differential is just bracketing with this aqua-variant action. Let me just explain. Let me just seem really arbitrary unless you're familiar with this kind of construction. So let me elaborate that it's basically a version of, a baby version of causal duality. This is a baby version. To give a map from a Lie algebra G into, say, A. So let's say this is a Poisson algebra. And this is some Lie algebra map, rho. Well, this means that rho of a commutator should be the commutator of rho applied to the inputs. OK. But to give such a map is equivalent to giving a more carton element in the Lie algebra co-chains of G tensor A. And why is that? Well, if you use the Chevrolet-Eilenberg differential, write to some element rho and evaluate it on x wedge y. That is rho of the bracket. And if you, sorry, this should be a Poisson bracket. And this thing corresponds to one half of rho wedge rho evaluated on x wedge y. So up to a sine, the more carton equation is equivalent to this condition. OK. So encoding, so under this correspondence, giving this map J is the same thing as picking out a more carton element in here, which is exactly picking out this sort of this interaction term if you want to. But I want to say there's this fancy way of saying a baby version of causal duality. But basically as a physicist, you would just be thinking about having a background field determined by that. So now we want to quantize this. Before you quantize, what have you achieved by generalizing this usual node to a formula is being a complicated covariate? There is only a finite number of currents for concept charges. Now you have this full d-dimensional integral that allows you to play with the original structure but the content is the same. Yeah, the content is the same. I totally agree. Dad, maybe one thing is that this is all like that. I'm not saying that this part is particularly interesting. I just think it's a nice language for talking about symmetries. I think the quantization is sort of the interesting bit. Brian, do you have anything you would add? No. Maybe later. Yeah, it's basically just the nerder formalism articulated in this DG setting. But now let's get to the interesting part. So the loose idea of BV quantization is we want to deform the differential on the classical observables to a differential called PQ on new power series in each bar polynomials depending on the situation such that two things hold. One, this is just the classical differential but plus things are order of H bar. And the crucial condition is that on the product of observables the quantum differential fails to be a derivation for the algebra in a way that depends precisely on the bracket. So the Poisson bracket is telling us how to deform the differential much like how in deformation quantization the Poisson bracket is telling us how to deform the product. We just shift the setting of the problem here and let me remark that's the quantum observables if you can find them which is exactly the classical observables with dependence on H bar in this differential here. Still has, so this is still a functor but now it's a functor from opens to DG modules over H bar but this differential is still a derivation for the bracket. So it's not a derivation for multiplication but it is for the Liebracket. So the kind of question you can ask is can we lift this map here? This mod H bar map is a map of DG Lie algebras and you can ask whether you can lift. It's about quantizing the symmetries. So the most tractable way to approach this problem is actually by this background field method because what it does is it expresses the whole problem as what looks like a BV field theory and then we can try to use perturbative methods to actually quantize. So it's probably in some ways the least attractive one for thinking but it's the most convenient one for actually doing constructions. So now I'm just going to sketch what happens very briefly. So if you look at that, this reminds you of this equivariant action and if you're used to doing perturbative stuff you can tell that this thing tells you what your propagator is. Just if you start doing the Feynman diagrams this is telling you what the propagator is and then you have this trivalent vertex and you can ask if I try to quantize this just in the usual way what kind of graphs show up? Only very simple graphs show up. There's things that look like this. The tree level stuff gives you this kind of thing which is sort of encoding sort of an L infinity description of the same dg of the algebra and you also get wheels but these wheels only have alpha as outer legs so the diagrammatic quantization, so the Feynman diagrams, produce sort of a quantized, let's say this is the naive thing, we'll have to check the master equation in a second which looks like the original equivariant guy, the tree level stuff which depends on alpha, beta and gamma plus an interaction term that only depends on alpha so this is the wheels, the one loop terms and there's no higher terms. So the question becomes, trying to give a lift to this map is the same question as asking, sorry, just as a remark, the statement that this thing is a differential is called the classical master equation and so being a more carton element is a solution to the classical master equation and this is going to be the quantum master equation which is the same thing but replacing this Poisson algebra by the thing, the quantum observables so solving whether it's a more carton equation and the answer is no. So let me explain what the answer does look like. I'm not going to elaborate on this because I didn't give myself time but one of the nice things Brian's done is to sort of do once and for all computations that work in every complex dimension that lets you know that the one loop terms that show up in theories of this flavor don't require counter terms. So we're going to use these analytic results to say that we don't need to stick in, nothing singular, we don't have to stick in counter terms here and so the problem just boils down to checking this thing about the quantum master equation and the quantum master equation you're asking me not given the amount of time I've given myself the failure to satisfy the quantum master equation is a particular co-cycle so let me just state our main theorem the co-cycle obstructing solution to the quantum master equation meaning if I stick it in if I do dqsqequivariant Chevrolet-Eilenberg sqequivariant plus 1 half I'd like to know whether this thing is zero or not being zero would mean it's a more carton element but if it's not zero that's my co-cycle so this co-cycle on x equals c to the d is given by the following it takes in d plus 1 inputs of alpha so only the wheels that have d plus 1 legs survive as anomalies it's the trace over v so here rho denotes the map from the Lie algebra to endomorphisms of our original vector space v so I'm taking some form with coefficients in g and I'm using it to make a form with coefficients in endomorphisms and then I wedge them together I do the dels on d of the inputs and no del on the first one so the del adds a 1 makes it sort of a 1-0 form so I get d1-0 forms and then if you pick the alphas appropriately you can get a top form so this is the co-cycle this if you're familiar with the ABJ anomaly this looks like a version of that so this this is ABJ like there's no d bar just del alpha d bar forms and the alphas are elements of 0 star forms and there's no del on the 0 term? no otherwise you couldn't yeah you can only have at most d so this co-cycle determines an extension of gc as a pre-co-sheaf of L infinity algebras so let's call that g theta of v and the underlying space looks like this here h bar is degree plus 1 has the same L2 is the bracket on g but Ld plus 1 takes as inputs is exactly this co-cycle so it's theta so this co-cycle determines an L infinity extension and now this new L infinity algebras is quantum symmetries it does lift to a map I'm going to modify that in just a second quantum observables so there's this de-quantization map there's j classical the ob's classical shifted by minus 1 and there's a commuting diagram these vertical maps are modulo h bar so this classical symmetry doesn't lift but you can modify the Lie algebra to an L infinity algebras and that does act as symmetries so every element of this L infinity algebras gives you an observable in the quantum system down here we had a one shifted Poisson algebras over here we have a bv algebras and just as you can there's an enveloping just like the symmetric algebras the enveloping Poisson algebras the Lie algebra chains actually there's a factorization algebras by taking Lie algebra chains of this thing here and then a sort of quantized version using this left adjoint between the factorization algebras okay so that was the sort of main goal I was going to explain how it relates to the work of Fanta Hanyan Karpanov but I've run out of time so at least I accomplished the main goal okay thank you for listening good questions we'll get text to one dimension and we'll get this H bar for good chains we'll get some sort of following power series in the H bar the effects in flat space can you still calculate for example Brian's nodding yes Brian's the great computer out of the two of us if you want a wrong answer you can ask me to do a computation so I didn't listen to your question but can you consider X as any manifold? yeah it's just that the you can sort of take any complex manifold and you can try to do the computation it's just that there's no reason that you should have such a simple form for the obstruction it just might be hard to compute