Categorification & quantum gravity, lecture 5

0:00 Okay. So, last time, I explained that there was a, not one, but actually several complementary models for rational homotopy theory. I'm only going to discuss the simply-connected case, because the case ring of an integral pi-1 is much more complicated, they don't know so much about it. So I'm going to discuss the two main kinds of models, the Quillen model and the Sullivan model. Unfortunately, several different things are called solvents, so it's a little confusing. They sell it in models, they sell it in algebras, and they sell it in minimal models. So maybe I should just say... are contained in Sullivan's algebras, are contained in Sullivan's models. Sorry about this. It's very confusing. It's very strictly contained. It's just that Sullivan did a lot of the work in the subject. So, too many things are named after him. So, what I'll do today, I'm sorry, I hope it doesn't come out a little bit disjointed, I didn't have as much time to prepare as I usually do. I'll explain these models and relationships between them and then I'll try to explain how you could use them to construct models for quantum gravity. What I will argue is that there are at least very plausible ways to construct models for quantum gravity on suitable Quillen models and suitable Sullivan

2:30 superficially quite different, but I believe in the end of the day, they will turn out to be complementary. But that needs some doing. So, I thought I would begin by explaining the Quillman models. So, the basic point is that if you have a space X, you can replace it, but it's connected and simply connected. All spaces are connected and simply connected. The spaces are connected? Now, all spaces are connected, and you see some of them all connects. I should try to turn the same things in French. I didn't speak French, it's just a, it's just a very idiosyncratic impression of what I've ever So you can replace the space by its, this is a standard notation, it means the base loop space. You pick a point and you take all the loops, the continuous loops of beginning and end. I think this is standard notation because the letter will make it look a bit like a loop. So it's easy to remember. Well, usually it's just taken from that. It's a group. Yes, it is a group. So, you can, and then given...

5:00 Okay, so there's a puncture called de-looping, which I won't discuss. Therefore, model X by finding models around the variables, which is nice because it's a groove, so being a groove, it has an algebraic structure. Okay, now you can replace x by... We're not doing any kind of homo-deprecating, as I explained last time. So, this always means, whenever you see something in pointy brackets, it always means that it's a symbolcial complex. deconstructed as an algebra of chains of simplices. So we have a simplicial deconstruction of x-tips, is a simplicial complex. So we can construct its loop space as a simplicial space by taking basically strings of loops, strings of simplices, okay? And actually, we usually simplify this by passing to a the composition for X has only one vertex. If it's connected and simply connected, you can make a decomposition where the simplices all have only one vertex.

7:30 So you can identify that with the base point. If I didn't, then I'd have to be a little careful about which chains of simplices were loops and which weren't. It can be done another way too, but it's more technical. So anyhow, you should think of this as an algebra that's constructed. This is called the Kovar construction and it's due to atoms. So atoms originally in the 50s wrote this down geometrically as a way to construct a simplicial decomposition for Lewis-Bachs. America-Bachs simplicial You see, any time you have a topological group, and a triangulation of it, you actually get a richer structure because you can multiply symphlices, the symphlices in each dimension form a group. And you just remember a simplex is actually a map of the standard simplex into the space. So if you have two of those, you can multiply them point-wise, and the group property is obvious. So, restricting to a face commutes with this group of multiplication.

10:00 It's one of the things that's much easier to see than write down. So, um, now, um, I'm sorry, there was some confusion about the room. Yes, yes, I have a confusion. Sorry, I'm just beginning to discuss the Quillen model. I'm going to go over the basic mathematical ideas of the Quillen and solvable models, and then I'm going to talk about how you can try to use the different quantum field theories. So Quillen began by replacing the space by its base loop space. And then we only got X to use base loops. So, we only did the case where the X is connected to a certain channel. Now, the chains, you know, we've got an X for A differential gradient of algebra. So, everybody knows that the functions in a group form a Hough algebra. So, if you triangulate the group, the symphases form a triangulated, rather a simplicial group, and then form a linear happenations, form a differential gradient Hough algebra.

12:30 So this allows you to construct an algebraic model. of any space. Now furthermore, this one is great to continue to do. It's actually sort of a super-algebra. And there's a general theorem that, therefore, it is is the universal developing algebra of a differential graded Lie algebra. This can be recovered by taking the primitive elements. of the universal-enveloping algebra are formally algebra. All that can be proved. So, in fact, this is one of our many, many famous pairs of adjoint funders. This kind of health algebra and this kind of differential-graded Li-algebra. And this is, you know, it's actually super Li-algebra. That's because it's coming from topology, you know, forms and cosecals behave like that. It's very funny to do these things, supersymmetry. And this is very big in general, but it's a differential graded object that can take its

15:00 cohomology. It's a graded algebra, of course passing through the cohomology means that trivial. And this is isomorphic to... Now S is the shift operator. here. So, the reason we're shifting, I guess I need a sin or a sin. So, what do I mean by that? Well, as the vector space, it's identified with the rational part of the homotopy group of space. Now, I don't know if you noticed, it's well known among topologists, it's a classical result, that the homotopy groups of the space form a great degree algebra. There's something called the Whitehead Product. But of course now, Now, this is a classical work of Whitehead, and the construction is sort of a generalization You can construct a map from SI plus J minus one to SI one point union and it's constructed like that. Is this familiar? Yeah, okay, so every day, well, you can take me, you know, and it's foundry.

17:30 So, on the other hand, this being a cube, this is, I'm sorry, I'm just cleaning this up a little bit. So anyway, it's basically this map. You pinch down the rest of it to a point, and that's the whitehead map, and it gives me this product, it turns the glee-algebra, the differential glee-algebra complex, the graded complex of homotopy groups in terms of algebra. So anyhow, on cohomology, that's what this is. So when you go to construct minimum models, basically you're saying that you can pass to the um. So of course this comes off, but that's on the manifold. But you see, we've downshifted everything. And the reason we've downshifted everything is the... Taking the base loop space shifts the homotopy groups down one. Taking loops begins very nicely with respect to homotopy. So of course one of the really hard problems is figuring out some depth into homology.

20:00 So that makes this into a product that respects the gray, fixes that. Okay, so now, however, I should warn you that this is the Whitehead product on X. Actually, the Whitehead product on the loop space is trivial, it's interdimensional. So, it contains some of the information from X. And so this is the algebra, now the category... Thank you. So there's a rather technical construction for what we mean by homotobies and differential-graded glialgements. So it shouldn't surprise you so much, a differential-graded object is a kind of a, is a kind of a complex. So, it's like simplicial complex, it's a bit more sophisticated, so we can use it to,

22:30 we can use it to effectively describe a topology, differential, and it tells you how things do together. And in fact you can think of a different, well at least in interesting cases, you can actually take the Lie algebra and you can think of it directly as a construction of a cellular complex for X. The D is like the attachment mass for a cellular Now that's very nice, but it isn't very useful because these are very, very big things. So we would like, on the other hand, there's a notion of homotopy equivalence in the category. So we would like to restrict to more special models that would be simple enough to calculate. I mean, this is, as I've directly constructed it, it's quite huge. You're taking the space X, you're taking all simplices in it. All singular, you're taking the complex of all singular simplices in it. So, it's pretty because you're replacing the space with an algebraic object. But it wouldn't have been useful in computation if you had the same way you were reducing it. So, I get a little bit out of chronological order there. It was actually Sullivan who worked out how to construct minimal models, and then people realized that it worked for the Quillen models too. Let me see if I explain it that way. restrict to co-fibrin of L's, and what this means is that L is free. So that means that

25:00 L equals L of V, meaning a differential graded vector space. So, as a Lie algebra, it's a free Lie algebra. What do I mean by free Lie algebra? Well, you start out with a vector space, and you just form all sequences in brackets, and to impose the relations that the algebra has to solve and nothing else. It's a piece of universal algebra. It can also be constructed by forming the free tensor algebra on the corrupted space and then taking the standard bracket on that and taking the intersection of all the algebras in that that contain Excuse me. When you restrict to co-figure L, L is free. That's right. That's right. The co-fibrant objects turn out to be free. Co-fibrant means that you can pull back homotopies. And since this is an algebraic object, it turns out to be equivalent to the algebraic object being ejected. it makes a connection between topology and the, yes, that's right, that's right. So this is, so as in the algebra, there isn't very much structure, and then what there is interesting in the topology is just contained in the view. And if you tried to do homotopy types instead of rational homotopy types, you could never in a million years do this. Homotopy types are very delicate and very hard to compute and very non-linear. And you could never represent them as free objects. It's just too much to expect. But rational homotopy types are much simpler. In fact, the theorem that says you can represent them by free objects allows you to prove all kinds of things about them.

27:30 one motivation for trying construct the quantum field theory on the rational homotopy types is just Streetlight Principle. You know, Streetlight Principle. You know, if you drop your key at night, you look under the streetlight. And it's because if it's not there, you won't find it. So this is like one little spot where there's a whole lot of light. And there's a lot of very powerful tools. So we just sort of wishful. Wish for it to be there, because if it's there, we can do things. and in fact a huge further simplification is possible but I think it's better if I tell you about the Sullivan models first it's okay so those were the equivalent Quillen models. And let's just say that the Quillen models are differentially algebra. I mean it's also a half algebra. I mean actually with Quillen in his paper he wrote a whole string of models and he showed that there were pairs of edge and funders inducing hometown equivalences between all of them. That's That's part of what makes it so attractive is a piece of mathematics, they're always different formulations. So there's also the Hough algebra model, and then there's the simplicial group model. I mean, there's a co-algebra model. Now Sullivan... Sullivan did something phenomenally clever, and then he went on to do a whole bunch of other very clever things. He found a way to generalize Durand theory to

30:00 simplicial complexes. And if you have a simplicial complex, you can attach to to each simplex the differential graded algebra of polynomial forms. I mean, being a simplex, it's got preferred coordinates. It's a map of the standard simplex into the space. So, on that space you can look at the differential forms whose coefficients in that coordinate system are polynomial. meal. So, let me... Okay, okay, sure. You guys should write them. I'm sorry, I'm sorry. I'm sorry, but where's the sound? I'm sorry. I'm sorry. I'm sorry. I'm sorry, yeah. I'm sorry, I'm sorry. Okay, so it's just a set of all polynomial forms that can be written in the coordinates as polynomials in the coordinates, times wedges in the coordinates, and of course there's this relation. So you can just write down formally as a differential created algebra like that Now this is very interesting for a number of reasons One is that we can study this as a differential

32:30 created algebra over the rationals Of course if we If we took smooth functions, we'd be sum. So, I think it's quite a difference because of sigma? Yes, this is the standard simplex, yes. The standard simplex, I should have said that, that's the definition of the standard simplex. It's a little, it's part of a plane in our n plus 1, and those are very centric coordinates. I'm saying for sigma for B is what? not. You apply calculus to this, differentiate both sides of this equation. Okay, very eccentric coordinates aren't quite coordinates. There's one redundancy. so now Okay, so you pick up one form for each. simplex. I'm trying to say this simply. I could spend a long time writing this down. When you restrict to a face, it has to agree. So that would mean that if you had two simplecies that meant on the face, the forms would have to match on the face. So this is sort of like differential

35:00 sort of like so that means it resembles the Duran theory to the point of saying sort of like I'm not quite sure what I mean either it is similar to the differential forms on a manifold but of course we're not assuming and we're only thinking of polynomial forms. But it turns out, This is not easy to prove, and it's quite tricky. So, in other words, if you had a case where it was a manifold, then you see the Durand complex on the manifold is a differential gradient algebra, a skew commutator. And the polynomial forms on triangulation would also be a differential gradient algebra, so we're in the same category, and it's a model category. So it's meaningful to say that they're helping protect the equipment. On the left. On the left. On the left. Maybe it would be less increasing. So, in other words, you can calculate the cohomology of this complex, the cohomology of this complex would be the cohomology of X as a space, and they correspond to the Duran cohomology of mental.

37:30 But we're able to do everything on this side with just rational numbers, because polynomials take rationalists to rationalists. They're the only functions that do. So we're actually able to do all this over the rationals, which I find attractive. Are you going to apply to your the ingredients you started with, or do you think you made it? Oh, no, no, this applies to the term, yes, I'm sorry, I'm explaining a different way of constructing a model, But I'm going to apply these different models to the same thing, so that's right, yes. And in fact, one of the important points to make is that they're actually equivalent. All right, sorry. Okay. Excuse me. So, um... The functor, well, I mean it is a functor, establishes a homotopy equivalence between So in other words, the homotopy types of differential period algebras are the same as the homotopy types of social spaces. And in fact, there exists an anjoin. So we can go, and of course, parsimonious

40:00 that we write it like that. It's the geometric, it's called the geometric realization. So, we can go backwards. If we have a differential gradient algebra, we can write this as a superficial complex. And if you go over and back, you'll get something much bigger than what you started with. These are big, free constructions. But although it's much bigger, it's at the same homotopy pattern, rational homotopy pattern. So it's an equivalence, it's a homotopy equivalence of the two categories. So everything about rational homotopy theory can be reduced to studying rational homotopy theory of differential periodologies, which is a huge step. Now, this is the A you have here, the same as the subset of O. Oh, sorry. No, no, I'm sorry. I'm abusing contagion. This is any differential. question. Sorry, I'm caught on my algemist. Yes, but no, this is an edge on the thumb. So you get a particular structure of this type, like this, and you can go the other way if you have a structure of that type, you get it as a simplicial complex. Now, this is actually very cute, too. It's a very beautiful piece of really abstract in that case. So the n-symplices of my retometric realization are the center of all homomorphisms from A into the standard differential grades algebra

42:30 of polynomial forms on the standard symbols. So all possible maps here correspond to the symphysis, and then the boundaries of maps just come by restriction. So, that's enough to tell you that you have a great distribution function. And although it's very, very big, it turns out to represent the homogeneity of time. So if you started out with a space, triangulated it, formed this A. And then you took the synodal homomorphisms of that A into the standard thing. It would get the new specs. And it would be of the same rational homotopy type, but it would be a big universal thing. It would be much more complicated in general. And it would be a rational space. It's homotopy and cohomology, and we're sort of interaction. And actually, if you think about these two definitions, there's a very abstract, beautiful proof that it's adjoined. Because you see, if you take the set of all these guys for n, it sort of works two hats. It's both a differential graded algebra and a simplicial set. So it's a simplicial object in the category of differential graded algebras. So it's got two structures. And how many would one match you to the other? So how many one way takes you from sequential sets to differential graded algebras, how many as differential graded algebras takes you from differential graded algebras to sequential sets. And it follows by these inferiors, right now, the sense that the two functions are, the two filters are adjoining.

45:00 You have this big universal object that's generating all of this, which is easier to point it into right now. It's just grouping all of these things together and then taking all the face maps. So it's a reason to need to switch over deep or deep emotion. Well, you need it in order to get rid of it. I mean, basically, this would be a sublimely useless theory that you could throw away all that and make something simple on it. Well, it's a universal object. I mean, any time you take universal objects in mathematics, you get something big. You know, you take a set, you take a free group of it. It's a great big group. But then you read some quotients of that, and you could lose a lot of the information in the quotient. But it's just, it's all very standard abstract mathematics. advocates. Okay. So that's another of my famous series. So these two funders allow us to think, to pass from the category of spaces to the category of differential period algebra. And we just have to study the homophobic types of differential period So, now, let me say, though, that from the point of view of somebody who's trying to do quantum physics, this is a very interesting instruction. Let's say you had a space, and embedded in that space you have a simplex. So there's some region in trying to study, let's say there's some simplex embedded in the region. Now you could take the differential graded algebra of forms in the region

47:30 and you could map them to the differential graded algebra of forms in the simplex by restriction. So I want to say, morally, that the existence of this homomorphism tells me that this simplex is sitting inside its space. But now, as I've explained earlier, I believe that we cannot have an absolute point set as a foundation for a quantum theory of gravity. because there are many, you know, being at one place in one metric cannot be meaningfully correlated to being at one place in another metric. So I think that quantum spacetime should be thought of as like layered version of ordinary spacetime because the different, the different quantum geometries which actually correspond to layered academies in the same space that could correlate with them. Now that's what this does. If you start up with a space and then introduce this, back and forth and introduce this, you get some space where all possible homomorphisms here correspond to places. So I want to suggest that a space like this... Bases, what do you mean by bases? Well, okay, I mean, so suppose you had some region. Bases in the ordinary meaning, of course. Well, yes, in the ordinary meaning, but with quantum physics in the back of your head. Yes, places. You might study the radiation that came out of it, and since radiation, at least the part that's relevant to us, has basically the form of differential forms. You might study all the possible differential forms that can come out to you from this region, consistent with the Beckenstein bound. So in principle, if we had a microscope that was powerful enough to study teeny tiny regions where the Bekenstein bound was important, we could study the differential graded algebra

50:00 of all vector fields and all tensor fields and so on that could come out of it. So we could study the differential forms. And that would be some differential graded algebra. Now, what would we mean by a place in this region? A place in this region is not a place you can go. You can't go inside the region. It's only a place you can see. and the way you would see it would be by taking all the information from all the radiation and deducing that there was a place. So if you wanted to find a triangular shaped place, you would try to find some projection of these forms and you would say what values do these forms take on this triangle? all. So all possible places would be all homomorphisms of this differential graded algebra into simplices. And then they would go together and they would give, that would be a must. So that's what I think. I think that the geometric realization of a differential-graded algebra is a plausible model for quantum space-time. as I said in my paper I think I've given everybody my paper I think that quantum space time is like a piece of paper when you examine it at large distances it seems to be smooth manifold but when you examine it at small distances it's a very very complex weave but somehow that very complex weave change its home at every time. Okay, so I think it's interesting to think about using this structure to try to implement that. It's not a question of shape. No, it doesn't matter.

52:30 But in this case, it's a question of shape. Ah, well, at some point, I'm going to have to recover some new classical limitedness theory. I'm going to have to argue that when I split a macroscopic region, it can approximately be reproduced. I don't know how to do that. That's going to be quite hard. It's always very hard to go from quantum theory back to the classical description of the macro world. to it that I understand that makes some sense is the decoherence approach. And I think that will be the way we'll have to think about this. But, yes, of course if you took a very small fuzzy little piece of paper you might not even be able to say that it has, you know, if you took a piece of paper, a dimension is comparable to the thickness of the paper, you might not even be able to say what two-dimensional shape it had. It would just look like a little ball of linen. Okay. So, one of the directions I want to explore is taking these things and constructing states and models. I'll come back to that. But I should tell you a little more about the foundations of the subject. I apologize for this talk a little bit. It's a little more rushed than I would like it to be. I'm saying a lot of things going well. Okay. Now... Now, there's the issue of the minimal models. So, um... So, differentiated parallel gradient.

55:00 So I'm tempted to say a solid model is solid if, sorry about that, So as an algebra, it's the free algebra and the vector space. So once again, in these models we find that it's enough to consider free objects and again it's for the same reason, nickel fiber. Now you have to be able to write V as the ascending union of VK and And D, B, K is contained in B, K, when it's on. Yes, it's a filtration. Okay. And now, A, B, G, A is minimal If, if, A is Solomon, N. So in other words, the image of D can always be written as non-trivial product. I say lambda plus because lambda zero is just the rational numbers so we don't want that. It's a non-trivial product. And then, Karen, every solid algebra can be written.

57:30 Of course, contractible doesn't begin with the K. We already used C. So everything can be written as a product of a minimal cell of an algebra and a contractible cell of an algebra. And k is contractible if it has a basis, xi, yi, dyi equals xi, dxi equals l. So this is an algebraic idealization of canceling handles. all of these things are the idealizations of handlebodies. So when you have obviously the cohomology of anything like that, it's going to be zero. Eliminating that, see this condition eliminates that possibility. So then it's theorem that if you don't make put in this condition, you can find the basis of these and when you take that out, quotient by its left side, it's left side. In algebra. So we now have a notion of minimal octets. And then there's a theorem. Thank you.

1:00:00 Thank you. Okay. So there's one and only one, up to isomorphism, minimal DGA in each homotopy type. So we can completely reduce the study of rational homotopy to the study of minimal differential graded algebras. And this is very powerful. Now this construction, this is due to Sullivan, and the construction of the minimal object is a little bit reminiscent to the construction of Allenburg-McLean spaces. You add generators in dimension 2 to pick up the cohomology, and then you go up to 3, and you destroy any accidental cohomology you've produced, and also then add things to correspond to the 3-dimensional cohomology, and you do this inductively. And it could be an infinite process. Even if it's only a finite complex, the process could be infinite. But this is one of the nice things, an infinite process at the end of the day. It's a little bit like the cat and the hat, where it keeps moving the dirt from point A to point B to point C. And at the end of the day, the dirt disappears by induction.

1:02:30 A priori, this could be a very complicated construction, but in practice, for interesting cases, it terminates very quickly and you get very simple algebraic objects. If I were giving a series of lectures, I would stop at this point and work some examples. I don't think it's a good idea now. We don't have so much time. And now, the analogous result holds a term. It's fairly easy to imitate Sullivan's construction. In the first place, you can always restrict to covibrant models, which means it's enough to look at free Lie algebras. And then there's a simple definition of when you're minimal. Okay. So it's an analogous condition. Now furthermore, there's a pair of...

1:05:00 So there's a pair of, this is the co-bar, the bar, and the co-bar, well these are one of incarnations of the barnacle board. There's a number of other structures that go by that name. It's easier to spell. And this is like a looping. It's a differential gradient version of the, Well, it's an algebraic version of the construction that Adams invented to construct the loop space from the original space. And this one is sometimes called the D looping. It's a differential gradient version of the construction that McClane invented to construct universal, to construct the other McClane spaces. It corresponds instead of being classed by the space for the group. You can think of this as like a differential gradient group. So you get a differential gradient version of what the plane does for a group. And it gives you a delugion. This is not a blue space, this is a delugion space. It's quite easy to write down. That's an algebraic construction. And these take cofibrin, cofibrin, which isn't a big surprise because they're only the equivalences. So if you stare out with this restricted class of maps of of models, if you do these functions, you stand in them. So it's a very beautiful picture. What I believe is that in physics, this will correspond to duality. That these models here look like the loop space, which is homotopy equivalent to the causal path space. So the quantum construction on the L model will look like a, it will look like a

1:07:30 quantum theory of the appearance of things when you lens and first find the results of the lensing procedure. It will tell you the probability we're seeing various multiple images. And this one will look like a space-time picture. It will look like a thickened space-time. The quantum field theory here actually will be constructed at its geometric realization. And the geometric realizations will look like a thickened space-time, and we'll be able to construct a path integral under the kind of states, a model island that will act like a path integral, which will be, since the layers can be thought like geometries, we can think of it as a path to go over geometries. And that's what I do, but I can't prove that yet, it's still quite a good word. And I should add one more thing, that physical experiment, no, okay. So I'm getting into, I'm trying to use these things to do physics. Now it turns out that within these classes of models, there's a natural subclass where the homology and homotomy groups are both all finite. Not just finite degree group groups, all together to find out. And those are called the elliptic models. And there's a very nice algebraic theory of them. And in particular, there's a conjecture due to ANEF that says that every model can be successively approximated more and more closely by elliptic models. It's been proved in enough cases that I think that the cases that would actually come up in four-dimensional topology will fall into this. category. So I can think we can take it as proven. So what I believe is that if you have a few model of space, you want to describe it by its multiple images. It's got a bunch of teeny tiny black holes in it.

1:10:00 They're causing scattering, which is causing multiple images. Now you know, as you get the successive geodesics that go around the black hole more and more times, are more and more red-shifting. The red shift goes to infinity, because they're passing closer and closer to the area. So if you do the experiment in a finite time, you make the flash, you examine the highly curved region, you examine it for a finite of time, and you examine what you see, and you decide what you've got from your results, you're in effect doing it in a bounded region in space-time. And a bounded region in space-time is subject to the Bekenstein bound. So if you naively construct the loop space of these things, they're free algebra, they have an infinite number of images. But I believe the actual experiments would take place in elliptic quotients of these things. So once you accept that, so an elliptic quotient have to be some module over this Li-algebra. You're cutting off at some point, then you, when you cut off, you're passing to a quotient of the Li-algebra, and therefore it's a module over the Li-algebra. So I believe that there's a natural way to try to quantize that, because it's a module over a Lie algebra. So anytime you have a module over a Lie algebra, you can just take all the generations of the Lie algebra and multiply them by I, and then those are the Hermitian operators, and you have a quantum theory. So I believe that that model, when you truncate it appropriately, you see, there's only a quantum theory if you include the observer. You include the observer by choosing the appropriate truncation. So there's probably some way of thinking of this, a topos of observers or something, but I don't want to get there. The category of representations would be sight. So you could think that actually it's a sheaf over that, but it's just a one. One is funny hard things like, you know, in quantum in a module over L of V, V I, V I bracket V J, V I bracket V J bracket V K

1:12:30 essentially would be operators. So what I believe is that these would contain the geometry. These operators would tell you something about what the time delay was when you went past a black hole, and then the relationship between two black holes, and then relationships between And suddenly they just need to be interpreted as a sort of quantum geometry. So it ends up a crisis to think that if you have the right quantum topology, the quantum geometry is already there. You don't, when you do classical relativity, you add the layer structure. You have the manifold, and then you pick the metric. And the classical theory depends on the choice of the metric. When you do quantum theory, you don't make a metric. You study all possible metrics. So somehow the quantum theory has to be a property of an animal. Okay, except that I don't believe you have a good quantum theory for closed universes. I think that's one of the things that we've been sticking at. You only have to have a sensible answer if you ask a sensible question. You can only ask a sensible question by describing the observer, how be it to them in an idealized way. So, when you include this extra structure, if you take an elliptic approximation of the thing, you will get something that naturally acts like a quantum theory. And so in some limit, when there's an extremely large number of generators, we would have to interpret the relationships between these things as geometric relationships between the little teeny tiny black holes, and we'd have to show that in some way we could have the state's equation back.

1:15:00 I don't know how to do that, but it's plausible to try. You start out with something rather simple. A space-time is really a continuum. It's finally generated. You have a quantum mechanics. Quantum field theory comes back in some classical environment. So that's one way we get quantum theory. The other way you might have to take quantum theory is you take L And then you take the moon, sorry, and then you... Now this is a great big sequential complex, but the simplicies are labeled by homomorphisms. Now, the algebra is a free algebra. So to specify homomorphism, you just have to specify values of the images of the generators. So you're integrating over all possible ways to choose forms corresponding to the things that generated the algebra which originally were coming from here and therefore were homotopy classes, and I want to think of them this time being black holes. So you're integrating over all possible appearances of black holes, over all... You see, if you see a black hole in the distance, what you see is some solution to Maxwell's equations. Solution to Maxwell's equations is a closed two-form. So that's how I think we need to interpret, and then d of these things is zero, so the equation that these things would have to satisfy is that they would be closed two-forms. that should be interpreted as a sort of an internal

1:17:30 relation to geometry. That where things seem to be, where the different black holes seem to be to you if you're inside this region, tells you where you are. And that's all the information there is, is the appearances of these sources. Handles are candles. That's my slogan. Handles are candles. And then The higher things that would come from those guys would tell you some forms that would tell you how these things glued together. And then they would have to write down some pathological and they would have to regularize it. Haven't studied it carefully yet. But I think it's going to work because this space has sort of a huge gauge degrees of freedom. If you take a simplex and let me find it, and you give me the values of the forms on the boundary, well, there are many, many, many possible values for the forms on the interior. But you can choose one particular set just as a result of the ones on the boundary, and then the things you can add on the interior is that. anything in a certain ideal. And that ideal doesn't depend on which things you took on the boundary. So, if you take the ansatz that the measure is sort of translation invariant, then if you find a suitable way of dividing by infinity, you will find that as you refine the triangulation, you're just multiplying by the same constant, regardless what values you had. thing to the coarsest possible triangulation. In other words, what I believe is that in this space it's a topological field theory. So what I want to suggest is that relativity is a topological field theory because it only has a finite amount of information. And the reason it's okay is because space-time is really a gas of black holes. So it's got just as much topology as it's supposed to have information. There have

1:20:00 been a number of papers in literature by physicists suggesting that you should model the boundary of a black hole as a gas of little mini black holes. It's a good way for getting the Ekenstein bound on that. And if you're studying any quantum region, it would be dominated by states near a black hole. So that's a description of the whole quantum region. So if you believe all that, then general relativity really is a topological field theory. It's just that the topology on the average is much more complicated than you thought, because down at the Fung scale, there's lots and lots of violent fluctuations, and they're producing a gas of little black holes. So all I have to do is write down a topological state sum model on this thing. And it has, as I say, huge symmetries, which makes me think I can do it. And this isn't the first topological state sum I've ever written down. So I've had some success doing that kind of thing. So that's the other way. And so my big conjecture is that these two things will both work. The factors will duality. They will actually be dual to one another. So they are complementary descriptions of the same quantum theory, and in the suitable limit they will give you back general relativity. So that's a very, very big conjecture, but at least it's a conjecture. I mean, we've been trying to quantize gravity for 80 years And, so this, and now let me go back and say that what I told you before, I think this is a refinement of my older work. Sorry, a refinement? My older work. Now, before, I was picking a supplicial decomposition of the space, and I was getting a very plausible model for quantum gravity. But then I was saying, in order to get over depending on the simplicial complex, I have the sum of it. That was the group field theory picture, and then it's not finite anymore. So I said, wouldn't it be nice if I had a unique composition in the space, one best composition?

1:22:30 Well, in the abstract, if you're just thinking of space-time and space-time, of course, there isn't one. But now, if you think of it as a region in space-time, and you think about only describing it in terms of the information you see in the post-Eckenstein's Bound, then I want to say that if I can take the information passes through Eckenstein's Bound, and I can turn it into one of these bottles. It has a unique minimal. So, taking the information into account, if the lensing hypothesis is right, so that I can pass to a rational homotidity type. But I can solve the problem of finding the one best decomposition. It's not quite a simplicial complex, but I can recover a simplicial complex from it. Like this piece of black magic right here. So that's the picture as I see it at this point. there's still a lot to do I'm rather optimistic actually I was a few months ago even I was much less optimistic what I feel from my point of view happens is that every time I hit on something in the physics which is puzzling I then go back read the next paper in the math, or the next chapter in the book in the math, and it's there. I have the feeling that just globally, this very large, rich branch of mathematics fits all the sort of intellectual aspects of the physical problem. That's the way it seems to me. Of course, they broadcast a recording of the mother of the pirate saying that it was perfectly innocent. People always think that their own children are wonderful and perfect, and it's my child, so perhaps I'm prejudiced. But that's the way it seems to may. So, okay, I think maybe I'll stop here and answer questions, if there are any questions.

1:25:00 You have to digest it. You have to digest it, yes. Okay, so, oh, speaking of digesting, does anybody, so should I stop then? This was enough for one day. Do you want me to stop now? Thank you.