A relationship between higher chiral algebras and gauge theory
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Transcript: English(auto-generated)
00:16
Thank you very much for the chance to speak.
00:20
Yeah, so today I'm going to talk about a project that is an extension of my work with Owen Guilliam in his last talk, but maybe focused on a particular example and I think a relevant example for maybe some physical applications. So let me just kind of remind you of a big picture that came up in Owen's talk, which is that the
00:45
observables of a quantum field theory define a factorization algebra.
01:02
So he talked about kind of a very nice, simple quantum field theory, the beta-gamma system in two complex dimensions and studied its factorization algebra. And we saw that, so for instance, the beta-gamma system, and we saw that we were thinking about
01:25
symmetries of a QFT, which themselves were also measured by some factorization algebra. This is what he called the quantum and classical currents. And symmetries of a QFT implemented themselves as a
01:41
map of factorization algebras to the observables of a factorization algebra like that. And this was this version of the nodar map here. So in my talk today, I'm going to show how these
02:03
higher dimensional current algebras, which we choose to represent as factorization algebras most of the time, actually appear themselves as the observables of some quantum field theory.
02:24
So these appear as observables. So I'd like to distinguish this from their appearance in the previous talk, in which they appeared as actual symmetries. Now I'm saying they appear as honest observables of a quantum field theory. And I'm going to choose to focus on a very
02:40
specific example, but hope to maybe give a general picture as well. So first I'd like to maybe state the kind of object that I'd like to focus on. And this thing kind of mysteriously came up in Owen's talk, but maybe I'll say a few more
03:01
words about it. But for any dimension, this is a complex dimension. So this is over C, say D. And any element, theta, this is going to be a
03:25
symmetric polynomial of order D plus one on G. And it's also going to be G-invariant. One can define a non-trivial extension of
03:46
Lie algebras, actually DG Lie algebras, of the following form. So you take your Lie algebra G, and you consider tensoring with the commutative algebra that basically looks like the space of
04:01
sections of punctured affine space. So this is punctured algebraic affine space in D dimensions. So we have some non-trivial extension of this form. I'll label it by D and theta, like that.
04:21
Except you need to do something really, really important. You don't just want to consider sections or functions on derived affine space. So these are sections of the trivial bundle, like this. Just functions on punctured affine space. If you just leave it like this, there's no interesting non-trivial extensions. What you really need to do is look at the
04:41
derived space of sections. So this Lie algebra was first considered by Owen mentioned in the work of Ferrante, Penion, and Kapranov. And they talked about a lot of really nice relationship between this extension and the moduli of G bundles in arbitrary dimensions.
05:03
And for me, I really want to think about it as kind of the right higher dimensional analog of the affine algebra, affine Kaczmaty algebra in CFT. So this algebra will come up and play a fundamental role in my description of certain algebra of observables. But I just wanted to kind of recall this
05:22
object was floating around in the background during Owen's talk. And maybe I should write down the formula. So the formula for this extension definitely
05:41
came up. So the corresponding two-co cycle, the two-co cycle just took a tuple of elements inside of
06:01
this algebra here, this Lie algebra. This is a DG Lie algebra now. And mapped it to this higher residue-like looking class. So it looks like you use the Lie algebra polynomial to pair off the Lie algebra parts. And then you apply this higher dimensional
06:20
residue along the 2D minus one sphere to that form there. So we have a really explicit model for this DG Lie algebra as an L infinity algebra, if you like, where the D plus one bracket, D plus
06:41
one operation is given by this formula there. So I'm going to tell you how this DG Lie algebra actually appears inside of some higher dimensional gauge theories. But to give you a flavor for the style of higher dimensional gauge theories I'll talk about, I'll first give an example.
07:00
So these higher dimensional gauge theories, or higher dimensional theories in any dimension are what I want to call holomorphic field theories. So just as an example, there's a very natural
07:21
field theory that I want to call holomorphic. And this is holomorphic Chern-Simons theory on a Calabi-Yau threefold. Call that Calabi-Yau threefold X here. And I'm going to let omega be the non-vanishing top form, holomorphic top form.
07:43
Then the fields of this holomorphic Chern-Simons are just going to be 0, 1 forms on X with values in some Lie algebra, G. So it looks very similar to Chern-Simons except I'm not just looking at all 1 forms, I'm looking at 0, 1 forms.
08:02
And the action, the action is very familiar as well. So I take my ordinary kind of Chern-Simons action write it like this, one half a d a plus one third
08:24
a bracket a a, like this. And I wedge with the holomorphic threefold. So here I've chosen a non-degenerate pairing on my Lie algebra G to write this down, just like you do in Chern-Simons.
08:44
So why do I want to think about this as a holomorphic theory? Well, its solutions to the equations of motion generically depend on some interesting complex structure on the underlying objects involved. So for instance, this example, the equations of motion exactly pick out the holomorphic G-bundles.
09:07
In this perturbative description, I'm just describing deformations of the trivial holomorphic G-bundle on X, but you can do this near any holomorphic G-bundle as well.
09:25
But this is the general kind of flavor of theories that I have in mind when I say holomorphic field theory. We saw another example. So there's a slew of examples in physics, and many will come up during the talks this week.
09:41
But in physics, holomorphic field theories generically arise as twists of many supersymmetric field theories. This is kind of a generic fact for any
10:02
supersymmetric field theory. And Owen actually did an example of this. So this, for instance, this beta-gamma system he talked about on C2, say, where the complex manifold is C2.
10:21
This came from a really simple supersymmetric theory that's n equals 1 chiral multiply, free chiral multiply. And the example I'll focus on today also arises in this way.
10:52
So maybe I'll state the main result, and then we'll get into the details of the theorem.
11:03
So the type of supersymmetric theory I'll consider is 5D n equals 1 super Yang-Mills theory. And the first claim that I won't spend too much time on is that this theory admits a twist to a 5D gauge theory.
11:27
Maybe I'll say holomorphic, put holomorphic in quotation marks for now, gauge theory on C2 cross R.
11:46
And I can actually put this gauge theory on this manifold C2 cross R bigger than or equal to 0. So now this is a 5-manifold boundary. And if you don't know anything about supersymmetric field theory or super Yang-Mills, you could just start with this description of this 5D holomorphic gauge theory.
12:01
So itself has a nice mathematical description. There's some natural objects that pop out of the equations of motion here that I'll talk about in a minute. And by holomorphic, I'll just say it's kind of holomorphic in the maximal sense here. So it depends holomorphically in this direction, the C2 direction, and then it has some topological direction in the transverse direction.
12:21
So that's kind of what I mean by holomorphic. So it doesn't exactly fit into the style that I showed here. But what I show is that there's a boundary condition. So that is a theory on C2 cross 0. It's a boundary condition of this 5D gauge theory. So now it exists just on the C2 guy
12:43
whose boundary observables, so the observables on C2 are equal to the higher-dimensional Cat's Moody current algebra
13:06
that Owen introduced in his talk. So this higher-dimensional current algebra, as we just kind of recalled up there, this depends on for some element.
13:23
So to really specify what I mean by Cat's Moody, I need to specify some degree 3 polynomial, some 3G dual inside of G there. And I'll say what it is for this example. So you can think about this as being a higher-dimensional version of the level in ordinary Cat's Moody.
13:40
So what this result is saying is that at the boundary to some 5D gauge theory, there appears a higher-dimensional Cat's Moody with some non-trivial central extension labeled by this theta. And I'll say what that theta is for this example later in the talk.
14:01
So any questions on this statement so far? So that's the general direction we're going to go in. Yeah, but if it's the usual topological in dimension 3, in the boundary, it also uses holomorphic structure. Yeah, so I'll recall that. The second set, yeah. What's that? But in a usual kind of classical story,
14:22
besides there's no complete direction at all. So for a topological set, if it's in boundary, it's holomorphic. Exactly, yeah. Yeah, so here it's not even topological. You're right. Yeah, there's still some holomorphic. But even in Chern-Simons, there's kind of a critical version of Chern-Simons that is not topological. And even there, you do see kind of critical level
14:41
of Katz-Moody. So I'll say a word about that. Yeah, we all recall that. So I do like to think about this as the kind of higher dimensional extension of a very well-known correspondence between Chern-Simons theory and WZW, chiral WZW. And I'll say a few words about that now just to set ourselves up.
15:05
So example. So this is Chern-Simons. Well, Chern-Simons makes sense on any 3-manifold. And I'm going to do it on a 3-manifold with boundary of the following form. So here, sigma is some Riemann surface.
15:28
And of course, we note the fields with respect to this decomposition I can write as the following. So a looks like az dz plus az bar dz bar.
15:42
So these are just local coordinates on the Riemann surface, z and z bar. And I'll write t for the topological direction, or for the R direction, plus at dt, where all the components are just smooth functions.
16:01
And at t equals 0, at t equals 0, there exists a boundary condition. And what it does is it takes a. So when I evaluate at t equals 0, of course this component goes away. And the boundary condition specifies, it projects onto one of the two components here.
16:20
And the one I'll choose to project to is az dz. So I'm only remembering the kind of holomorphic component of the connection on the boundary. And it's this boundary condition
16:44
that gives rise to the chiral wzw model sigma.
17:08
So maybe I'll just say a very, very modest extension of this, which is the formalism that we like to work in. We can kind of reformulate this, reformulate in the BV formalism,
17:22
which doesn't really buy you much new kind of structure in this 3-2 example, this 3D-2D example that is very relevant for the five-dimensional 4D example that I've stated in my theorem. So I'll just say it in this Chern-Simons case. The fields in Chern-Simons,
17:44
well, they just look like the Dirac forms on the 3-manifold with values in the Lie algebra. And I can write that with respect to this decomposition in the following way. So maybe I'll just write that. So it just looks like star forms sigma cross r
18:04
bigger than or equal to zero with values in the Lie algebra. And since I really want gauge fields in degree zero, I need to shift this down by one. And then with the complex structure on the Riemann surface, I can write this as zero star forms in sigma tensor
18:22
with star forms just all forms on r plus tensor g. So that's one component. And then I have an extra component coming from the holomorphic part in the Riemann surface direction. So that's one comma star forms on sigma, tensor forms on r, tensor g, like that.
18:44
And of course, there's a connecting map here that's just given by the holomorphic Dirac differential on the Riemann surface that maps a zero star form to a one star form. So I can rewrite the... I'm just rewriting this Dirac complex in terms of this more complex notation, this holomorphic notation.
19:03
But in this notation, if I write... So I'm going to write my new fields as... Let me write them as alpha zero comma star and alpha one comma star. So I'm writing zero comma star for no limit up here and no limit down there.
19:20
The boundary condition that just extends that boundary condition to this full BV space of fields is really simple still. So it just says that alpha at t equals zero... So alpha I'm just writing as this two component object at t equals zero is exactly alpha one comma star.
19:52
So locally, which if you want to look at, say, the local operators of chiral WZW, this isn't really telling you anything more. This BV formula doesn't tell you anything more
20:01
because locally there's no interesting higher cohomology for the local operators. So everything is still a constant degree zero and we just have functions. The operators in the boundary just look like functions on this holomorphic connection there. But we'll see in the higher dimensional example, it's really necessary for me to remember the derived directions,
20:20
so the higher double directions on the boundary. Any questions on this? So maybe just some remarks on this perspective.
20:42
Just some remarks. In this BV approach, in our approach to QFT,
21:01
developed by Cammie, Castello, and Owen-William, as I recalled in the beginning of this lecture, the observables form a factorization algebra. So in particular, the observables on the boundary
21:22
of Chern-Simons form a factorization algebra on sigma. So we get some factorization algebra on sigma. And locally, locally on sigma equals C,
21:41
this recovers the affine Kaczmudy vertex algebra. And the factorization homology evaluated on closed surfaces recovers the conformal blocks of the vacuum Caczmudy.
22:02
So it's kind of what you would expect from this Chern-Simons WW picture. We're not getting anything really new there, but it's nice that it fits into the framework. And there's also the issue about level. So kind of a subtle issue is that
22:22
if Chern-Simons is level K in the bulk, so classically, if you assume that we have a fixed invariant pairing and we're fixing the level to be K, then the Caczmudy algebra has level K plus some shift,
22:49
call it KC, and this is the critical level. So on the boundary, naively, we'd expect just to see Caczmudy at that same level.
23:01
Caczmudy is also an object that depends on some level. But quantum mechanically, we don't see just the level K, but we see some shift in the level. And this is something that's generated really. If I was being careful here, I would put an h-bar in front of KC. So this is really something that's a one-loop effect. It's generated a one-loop. And there's kind of a nice diagrammatic picture
23:21
implementing this phenomena. And that's to consider the operator product. So if I put two local operators in on the boundary, this is my boundary Riemann surface. And we have this real direction, R bigger than or equal to zero. This critical level is generated by some one-loop diagram,
23:42
and it's of the following form. So what you do is you flow it to the bulk via the propagator, and there's some one-loop diagram that's really simple. It's a two-vertex wheel. And you show that this thing is proportional to the critical level
24:02
times some local function on the field that exactly incorporates the central extension of the Katz-Moody algebra. So the sense of this thing is one loop usually, right? We keep track of h-bar if we're careful. So this critical level is a quantum shift.
24:21
I really want to stress that, because there's a similar shift that happens in my example that I like to think of kind of the critical level and appropriately contextualize in higher dimensions. So my last remark is that there's also an extension of this boundary condition
24:56
to what we call chiral boundary conditions
25:01
chiral boundary conditions for a wide class of 3D topological field theories that are labeled by a geometric object called a Krone algebra.
25:29
So these 3D TFTs will be talked about. So by the way, this kind of work here, thinking about generalizations of this boundary condition for more general 3D TFTs is joint with Pavel Safranov.
25:45
And Pavel will talk about this example in much more detail in his talk. But I should say that the boundary observables in this case recover a lot of other well-known vertex algebras or factorization algebras,
26:01
most notably the chiral differential operators and chiral derang complex, as well as many others that are kind of variants of those vertex algebras there. So I won't say too much about that, just to stress the fact that
26:21
in this kind of degenerate limit where I take a... So a Krone algebra is something like a vector bundle that lives over some manifold. And you can think about the 3D TFT as basically labeling maps from a 3-manifold into X, together with some linear data, like sections over the pullback of this bundle, roughly speaking.
26:42
But in the case that X degenerates into a point, these boundary conditions, like the chiral DW, exactly become these chiral boundary conditions here. So it gives a nice kind of systematic relationship between CDOs and the Kac-Moody algebras, if you like.
27:04
So yes, Pavel will talk much more about that later. I just want to say a word about it, how it fits into this setup.
27:34
So now I wanted to move on to the main part of my talk, which is to introduce this 5D gauge theory
27:42
that witnesses these higher Kac-Moody algebras on the boundary. So the input data I'm going to start with is X. X is going to be a complex surface,
28:02
which I'm actually going to assume, just for simplicity, is Calabi-Yau.
28:21
So I have a top form, a non-generate top form. This condition is not necessary. It just makes the theory easier to write down. You'll probably see an obvious way of getting rid of this restriction.
28:45
And the fields are going to look very close to a Chern-Simons theory. So I'll write them like this. So there's two components to the fields, A and B.
29:03
So I'm going to label X locally as Z1 and Z2. Those are going to be my holomorphic coordinates. And A is going to look like a 0, 1 form. So A Z1 bar, D Z1 bar, plus A Z2 bar, D Z2 bar.
29:22
And then there's a third component that labels what you can think about for the time direction if you like, but some other real direction, T. So here all of the A Z bar I, AT. So these are all fields. This is a five-dimensional theory. These are all functions, C infinity functions on X cross R.
29:45
I could choose any real one manifold, but I'm just going to look locally in the transverse direction for now. And then B is very similar, except these things are all proportional to the top form, like this.
30:00
And then similarly I have a decomposition like this. And the B C bar I's and B T's are also just smooth functions on X cross R.
30:24
The only way B's are different is they have this extra factor of the Clavier form. And then the action is really easy to write down. It's of Chern-Simons type. Looks like this.
30:50
So there's a kinetic term that looks like... So again, so G here, X is a complex surface. G is the same kind of data you get in Chern-Simons, so it's a Lie algebra, ordinary Lie algebra with an invariant pairing.
31:06
And I use that invariant pairing just like I do in Chern-Simons to write down the action. So that's the kind of ordinary kinetic term. D here just represents the Dirac differential on X cross R.
31:33
And the interaction part just looks like B, so it's linear in B in the second component and brackets A with itself. You don't use cubic planar at all, no?
31:50
It's cubic in the total sense. It's cubic as a function of both B and A. Yeah, B and A kind of play similar roles here. So I want to...
32:00
So it's double the algebra and get cubic invariant cubic summation. Yeah, that's right. Does it transform the same way in the gauge transformation? I'm just wondering.
32:21
So the gauge transformations are exactly... So I should say that the equations of motion, what does it say? You can work this out. So for A, for A at least, what it tells you is that you have a holomorphic G bundle on X,
32:53
and together with a flat that is flat in the R direction.
33:03
So if you think about this as saying I have a flat family of holomorphic G bundles on X. So this R direction just coincides with an ordinary trans-simon's direction. It's flat. It gives a topological direction in the field theory. So it's like saying I have a flat family, but I have some interesting G bundles on the X direction.
33:21
So the gauge transformations you can understand as saying, well, I can always perform ordinary gauge transformations in the R direction just by a flat connection, the ordinary gauge transformation of a flat connection, but then I can perform holomorphic gauge transformations in the X direction. Does that help? And similarly for B as well.
33:40
So it's very important to get one-dimensional relation transversal homomorphism. You can think about it like that. Yep, absolutely. Question? Yeah. So if the variation over B gives a situation for motion for A and for... For A it's for B and for B it's how we can contribute to it. Yeah, B it's not so easy maybe to interpret.
34:02
So for A it's this semi-flat. Can B be DA plus commutator A in the action? Isn't it one-half in the second term? Oh, this is one-half. Thank you, thank you. Yeah, thanks. So it's like B-half theory. And this is not. This is not, yeah.
34:21
I just mixed up my factors, thanks. Yeah, this is like a B-half theory. What if you don't consider like separate fields in B but write the same action when just using the A field? Like if you say that B is only at which A? This is a functional field A. So writing it that way you can arrive at more interesting...
34:42
There's like more interesting deformations you can write down in this theory. Yeah, that's easier to interpret in that language. But for the boundary conditions I write down it's a little easier to think about as like two components like this. Yeah, there's certainly something you can do. I'm not sure it'll play such relevance in my talk today.
35:08
So Owen mentioned kind of a nice computational result of these types of holomorphic theories that I've considered. Now this is not exactly a holomorphic theory. It's really like holomorphic plus one theory.
35:22
I have this holomorphic structure in two directions, but then I have this topological structure, but this kind of normalization result still applies, so I'll just write it. This follows some calculations I did in my thesis, modest extensions thereof. There exists a one-loop actually finite quantization of this theory, of this 5D theory, at least on flat space.
36:09
The main result I'm going to state today is a property of this one-loop quantization. And by finite I mean, as was mentioned in the last talk, finite I mean
36:20
the finite items you write down are strictly finite without the introduction of counter terms. So there's a really nice kind of explicit quantization you can write down for this class of holomorphic theories. Even if they're not exactly holomorphic, but kind of maximally holomorphic.
36:51
So the next thing I want to do is to consider this 5D theory, not just on this open space, but now on a five manifold with boundary.
37:07
Actually, sorry, maybe before I do that I'll just remark then, there's an extension in the BV formalism, so this is going to be really important for my result.
37:25
So here we have two fields, alpha and beta. So here alpha, the degree zero part of alpha zero is what I was calling a over there, the degree zero part of beta is what I'm calling b. So I'm just going to write down all the ghosts and anti-fields together in this BV language.
37:42
They look like this. So you have a zero star form on x. So this again works for any complex surface, but I'll really be focusing on C2. So it looks very similar to this Trent Simons that I wrote down in three dimensions.
38:23
And there's a nice way of thinking about this in the BV formalism. Well, this top thing I'll never forget about the shift. That's just a D.G. Lee algebra. G is a Lee algebra and I'm tensoring with a commutative algebra. The differential is this d bar plus this derangive differential in the r direction. The bottom line is clearly a module for that D.G. Lee algebra,
38:41
where I just act by the adjoint action and then by the wedge power of forms. So when I write down the action in the BV formalism, I just use that structure, that of a D.G. Lee algebra in the top, and the bottom thing is a module for it. So that's why it generically has this BF type formula, why the action looks like that.
39:08
And maybe I'll just claim, I'm not going to go into too much detail here, but the claim is that the 5D super Yang-Mills n equals 1 super Yang-Mills
39:24
emits a twist to this 5D theory on C2 plus r. So if you like, I'm describing the holomorphic twist of 5D n equals 1 in this BF language.
39:42
This perspective is not really important for my talk, so I just wanted to state that as a kind of motivation maybe for considering this class. When you say 5A equal to 1, what's the... So you just write it in a Poissonic field, and it's in some fermionic... So something happens in the twisting.
40:01
The fermionic direction gets twisted into a comological BRST direction. So a lot of the fermions that were present in this n equals 1 theory actually became ghost and or fields and anti-fields in this theory here. So there's no Z2 grating in this. There's no parity here. There's just a total BRST grating. So this is 5D n equals 1 with true matter?
40:24
No matter, yeah, just pure. And if you add matter, is there a similar twist there? Yes. We'll be throwing in some representations down on this side. So maybe I'll write down...
40:41
So there is a boundary condition at t equals... So for the theory on C2 cross R bigger than or equal to 0 now.
41:03
So I'm putting the 5D theory here. I'm looking at the boundary, t equals 0. And what I do is I take this two-component field at t equals 0,
41:20
and I just project out to one of the components. So I just project out alpha to 0 and then beta, the field beta at t equals 0.
41:47
I'll keep that up.
42:03
So what do I... Some really basic things I can extract from this. So clearly the operators then on the boundary, the boundary operators,
42:23
the boundary operators are boundary observables. Well, they only depend on beta. There's no alpha terms by the boundary condition there. We can write down a generating basis for the local operators in the following way.
42:45
So I'm going to label these things. I'll call it O for operator. It's labeled by x, which is going to be an element of the Lie algebra. It's going to be a vector, integer vector. What am I calling it?
43:02
n, like this. So n is just a two-component vector, n1 and n2. And integers, positive, non-negative integers. And then w is just going to be a point on C2. So it's a local operator, so it's supported at a point.
43:22
That's where the observable is supported at. And what it does is it just takes beta t2z. So this is something now that lives... This is not in BRST degree zero. It's in some non-trivial BRST degree, according to my conventions.
43:42
I'll say what I mean by that in a minute. But it takes something like this. So here beta is just a function on C2. And it maps it to the following. So I take beta, I pair it with x.
44:03
Sorry, so beta is an element. It's a Lie algebra value function on C2. So I just pair it with my Lie algebra pairing. Now this is just a function, and I take its derivative. d by dz1, n1 times, d by dz2, n2 times, and evaluate it.
44:39
So notice, really important,
44:43
it was really necessary for me to be working in this BV setting, or else I wouldn't have seen anything interesting, because these all have degree... This has BRST degree, degree plus one.
45:03
Because beta d2z, if you look back to my formula there, that thing lives in minus one of level of field, so the operator is degree plus one, it gets flipped.
45:23
So I want to check, as my first check, for my main result to have any chance of being true. I want to check that these local operators that I've given here
45:41
agrees with the description of the state space of this higher dimensional Cat's Moody algebra that we wrote down in the last lecture here. So remember, for the higher dimensional Cat's Moody algebra,
46:03
we're going to have a factorization algebra on C2. Let's just recall what it is. Well, it takes any open set inside of C2, and it maps it to some Lie algebra homology-looking guy. So it looks like the Lie algebra homology of 0, star forms
46:27
compactly supported, that was really important, on U with values in G, like that. Classically, this is exactly it.
46:40
There's also, which you'll see quantum mechanically, it gives rise to an extension of this. We don't just look at least at the algebra, we look at an extension of it. Let me just work classically for a moment now to see that the operators agree. So on U equals all of C2. If U is just all of C2, we can just calculate
47:05
compactly supported functions on C2, values in G. So this looks like some big symmetric algebra on compactly supported double forms on C2. Tensor G shifted down by one,
47:22
and then there's some non-trivial differentials. There's the linear part of the differential that's D bar acting on the 0 star part, and then there's some Lie algebra part, Shevla-Annenberg part, acting there.
47:48
So why does this look good? Well, first is 0 star C on C2 has
48:12
cohomology only in degree plus 2.
48:23
So it only has cohomology in degree H2, D bar of C2. So this thing's actually, this thing's quasi-somorphic to this, compactly supported.
48:41
If you like, this is some sero-duality. But I want to choose to identify this. I want to identify inside of here a really nice subspace, in fact a dense subspace, via a higher dimensional version of the residue pairing. So if you like, you can think about this as the residue pairing.
49:02
And the space I'll write down is the following. So you look at two holomorphic 2-forms on the disk, on the 2-disk dual.
49:23
So why would you expect this to be a higher residue pairing? Well, what I do is I take a, sorry, maybe I should. So I want to think about this as being isomorphic
49:42
to the following Laurent polynomial-like looking space. So I look at the purely negative Laurent tails in polynomial variables z1 and z2 inverse. And then all I do to make this dense embedding
50:00
is I take a polynomial, say it looks like z1 minus m1 minus 1, z2 minus m2 minus 1, and I map it. I map it. So maybe I should say, maybe I'll describe this one first.
50:22
Sorry, this is getting a little... So I map this to some compactly supported Daubot form, and I do this using the Bockner-Mardinelli kernel. So maybe I'll say what I mean by that in a minute, but this Bockner-Mardinelli kernel plays the same role as the residue class inside of one-dimensional complex geometry.
50:54
So it sends it to the following. You look at d bar applied to z1 minus m1 minus 1,
51:05
z2 minus m2 minus 1, all times the Bockner-Mardinelli kernel there. And I need to choose a compact support. So I choose any where f is any compact supportive function,
51:35
smooth function.
51:49
If you like, this identification here is more kind of fruitful to think about, so maybe I'll just say a word about this. In the function, which is identity near 0,
52:00
equal to 1 near 0? Yeah, so centered around 0. So I want it to be radially symmetric. Maybe I'll say, so this identification is a little easier. I should have started with this. But all I'm doing is taking a polynomial of this form, so z1 minus m1 minus 1, z2 minus m2 minus 1.
52:26
This maps to a thing that takes in a. So eta here is any polymorphic 2-form on the disk, and it maps it to the residue around S3.
52:42
So it's this higher dimensional residue of dz1, m1, dz2, m2 of the Bockner-Mardinelli kernel times eta.
53:03
So this is the kind of unique thing, this residue of around S3 of omega dm with any function fd2z.
53:21
So if I choose a, let me just choose a basis for holomorphic top forms. This is just the value at f at 0. I'm just kind of floating around a lot of variables here, but the kind of thing I really want to identify is these purely negative Laurent polynomials, which I'm identifying via a residue pairing
53:42
with the kind of linear part inside of this factorization algebra here. So keeping those formulas on the board above,
54:04
we see that there's a natural identification with the local operators I wrote down for the boundary in the following way. Oh yeah, that'd be great.
54:24
Is that good? So I just take a, I just do kind of the natural thing, c1 minus m1 minus 1, c2 minus m2 minus 1
54:43
with values in x. So this is an element inside of, I'm viewing it as an element inside of compact supported forms on c2. So it's a 0-2 form, the way I'm identifying it, tensor g.
55:01
And this is placed in degree plus 1 as well, right? Because this thing was concentrating in the homological degree 2. So 2 minus 1 is 1, so this is in degree plus 1. And it sends it to this operator, o x, at 0.
55:21
I've chosen to center everything around 0 for this example. So this shows that, at least classically, the local operators coincide.
55:44
Now at the quantum level, it turns out to be simplest to compare a piece of this. So I haven't actually talked anything about the factorization product structure here. I've only evaluated on a disk and shown that they coincide as vector spaces or really as coaching complexes.
56:03
So I'd really like to say that the factorization algebra structures are compatible on both sides. That would be the strongest version of this result. Unfortunately, there's not really an efficient way to write down the full OPE structure for a higher dimensional holomorphic factorization like this. This would be like the idea of generalizing
56:21
chiral algebras to arbitrary dimensions. So a full definition has not yet been written down in totally algebraically satisfying terms. But there is a related object that gets back to the first object I wrote down here. We can study, we can compare what are called the mode algebras.
56:42
Mode algebras of higher katsumuni and the boundary operators. So what do I mean by the mode algebra? Before, we were looking at C2 sitting inside of C2
57:04
cross r bigger than or equal to zero at the boundary. What I'm going to do is not look at all of C2, but just look at C2 minus the point.
57:20
And if you like, in the 5D theory, what I'm doing is looking at C2 minus the point cross r. And this map came up in Owen's talk. What I'm going to do is take the radial projection of the C2 minus the point. So just look at the modulus. So this produces some real number.
57:43
And then in here at the level of the gauge theory, I can think about this as doing, I'm compactifying the gauge theory to r bigger than zero cross r bigger than or equal to zero. So this really looks like some, this is like a compactification along S3.
58:07
So since I have some holomorphic factorization algebra here and there, when I push forward, I get some factorization algebra that turns out to be,
58:21
or at least has a dense subspace, just as in this description right here, to actually be locally constant. And I can state my main theorem in more precise terms. So this locally constant, this locally constant
58:46
1D factorization algebra defines an associative, or really it defines an A infinity algebra. Algebra, I'll call it, let me just call it A.
59:06
And we have the following theorem. There's an isomorphism, or there's a quasi-isomorphism,
59:22
isomorphism of A with, well first, there's a quasi-isomorphism of A with, I do the same exact construction, not just for the boundary theory, but I can do this for this higher dimensional, this higher Kaczmude algebra on C2.
59:45
So the first thing is that there's a quasi-isomorphism there where I'm taking the same construction, I'm looking at the S3, backflowing along S3. And this has some...
01:00:00
with central extension, extension given by theta equals, now I'm going to write down some explicit polynomial. Well, you look at G in the adjoint, you take its character and project under the third component.
01:00:20
This is some element inside of Sim 3, G dual G. So precisely we see the higher CATS community at this extension, at this level here. And in turn, this is related, this is work that Owen was talking about in his talk. This is related explicitly to this algebra of
01:00:41
Ferrante-Penihana-Kampranov. This is given by the enveloping algebra of this G-hat 2, theta, which extends this derived version of functions on the punctured affine plane with values in G.
01:01:01
So explicitly, we've identified then the algebra, really an algebra of observables. It came from just a piece of the observables, namely the S3 modes, but we've identified a piece of it with this very explicit A infinity algebra like this. So maybe I'll say one word because I wrote down the picture earlier.
01:01:21
So the picture that you should have in mind, why this extension comes from. So it doesn't come from, so I have this big kind of space here, the C2 in some real direction, but I'm projecting, it's punctured. This is C2 minus the point.
01:01:40
I'm projecting onto just the line, right, R bigger than zero. And I think about the book as being labeled by another copy of R. And the way this extension arises is not from, like in the ordinary cat's moody, was a correction to a two-point function, this 2.0 PE.
01:02:01
It arises as a correction to a three-point function, which is really like an A infinity structure set in these more homotopical algebra terms. And the three-point function arises as computing a diagram where you put in local operators in this one-dimensional direction, which correspond to sphere operators up here, computing their three-point
01:02:21
OPE, or factorization algebra structure maps, and it's corrected by some Feynman diagram that looks like this. So it's very similar to this 3D, 2D correspondence where you see a two-vertex wheel, this time you see a three-vertex wheel labeled by the propagators like in Owen's talk. And this thing you show is basically proportional to
01:02:41
this CH3 of G in the adjoint. So maybe I'll stop there. Other questions? Do you know what the E2 algebra is acting on this?
01:03:02
Which E2 algebra? The one coming from the bowl? Or maybe it's not E2, but is there some kind of... No, it is going to be E2. No. We should be able to work it out.
01:03:22
Other questions? Question. If you start with this invariant cubic polynomial in the algebra, you can also build a five-dimensional pure topological theory, an angle of Chern-Simons. Just integrate, imagine five-dimensional boundary, six-dimensional... So add a full Chern-Simons term to the...
01:03:41
That's correct, yeah. Yeah, it looks a little bit more than what you considered, but I think nobody knows what to do here in this theory because in trivial connection you get no quadratic term at all. Yeah, it's like integral... In cases, integral one form, A times dA squared. Oh, I see, yeah.
01:04:05
It's also kind of five-dimensional. This theory should kind of degenerate into that theory maybe. Yeah, it's pure topological. Oh, I see. That's really interesting. Yeah, I'll keep my eye on it. Thanks. Yeah? So what, if anything, goes wrong with these theories
01:04:21
in the boundary condition? You still have to find if instead of having a surface, you have some higher dimensional defaults. And this proposition, I see, would still have been truer, seeing maybe some kind of logical condition on the... Yes, yeah. So does something go wrong with the analog if you try to create the analog of this theorem? In higher dimensions?
01:04:41
No, so I know an example in 7.6 works as well, where this kind of six-dimensional version of the higher cats mini pops up. But it's very similar in structure to this. Nothing is really... Does something go wrong if you just try and sort of write the same proof that's replacing 2 by d and doing the same? Is there something that you have to...
01:05:01
No, just interpreting as like... So I should say, selfishly, the reason I care about doing this is because it gives kind of insight into maybe seeing some dualities in physics. So I really care about this 5-4 correspondence for reasons like that. But there's nothing that stops you from doing this in arbitrary dimension if you don't care about kind of relevance to other maybe interesting physical theories.
01:05:21
That is also relevant for... Yeah, that's why I've done that one too. Yeah, I don't think anything should be wrong in the system. I don't know how relevant it is to other... Other questions? Can I ask about matter again? So this main theorem here, does it have an extension decision because of matter? Yeah, so you'll see some BRST reduction of the
01:05:41
matter in the boundary. So I mean, instead of cosmodial algebra... Yeah, you'll see the higher chiral version of BRST reduction of that module, a representation that you chose. Is it easy to write it? Yeah, I could give a description. I don't know if it would be useful at all.
01:06:01
At least I could describe this algebra really explicitly. The A infinity algebra is really easy to write down.
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