Royal Society meeting on Physics & Mathematics of String Theory

2:30 And what you do is take the assumption in general is uncomplicated and you have to toss in invariance under the change of metric. You have to in volume element of the cohomology is determined which is still interpretation. I might comment before I go on the interpretation of what you would do if you turn the lattice gauge theory and whether you see the combinatorial invariant that might be amusing reproduce. The original Riedemeister-Franz approving invariance within context of the last gaze theory. I just said, in terms of gaze theory, this reminds me very quickly for myself, I think I'm calling them chromodynamics rather than string theory, and I've always felt

5:00 understanding this orbit space, the configuration space, based on three-mount fault for gaze theories. I've always wanted to understand the string or operator on this space. I'm happy to see all these topological invariants associated with three-dimensional gauge theories. In any case, here is the space that we want to deal with, and I want to emphasize several facts. There's a natural vector field on this space, essentially the magnetic field. Here's how it acts on a tangent vector and dual to that one form is that vector field vanishes and that means it's a flat vector potential and therefore it takes pi 1 and so going back to what I said for Riemann services we have that the zeros of this vector field are the same as the characters or representations of pi 1. It's also interesting to remark, although I will not get to it, that if you take the integral curves of this vector field and you think of a zero equal to zero, that that will be actually anti-dual.

7:30 It always depends on which direction you're going. Connection. Talking about that one, it's easy to check. The one form that represents the field is actually closed on the EMIG, just directly computes variations, and you'll find that it's closed in one form, and therefore it's the differential of a function. There are various ways of describing that function. One of them is in terms of the alien variant of the operator, which is really the curl operator, except it's associated with the vector potential. That is, it's the differential, which takes a one-form to a two-form, and it takes the dual epsilon ij to get back to the one-form, except that it's D plus bracketing by the vector field you're at. And then, the Aden variant is simply the sum of the, what's the sign on it? The critical thing is to put the appropriate signs. This function is discontinuous as an eigenvalue passes through zero, but continuous in this way, if you can ignore that jump, which is always two, an eigenvalue passes from zero, from minus to positive, its differential is well defined, and half the differential is equal to this. There's another way of describing subsequently, and that is the Chern-Simons invariant.

10:00 You can define the Chern-Simons action for a vector potential, how it transforms by the winding number, so again the differential is well defined, and it gives you one form that's so natural on a three-dimensional, on Yang-Rills based on a three-dimensional, from the point of view of Kornfield here, reproduced. These words as a result describe quantum field theory. I have to describe the action. The action is omega d omega in one form, so the type I've described, and it's this we choose the usual gauge function, the edge on the edge of omega, too much is the technical point for those of you who are not familiar with it, but it's bread and butter to all field theorists who've had to use Gauss determinants. And if you compute the ratio of determinants, lo and behold, you get the combination I described earlier to the fourth power. I want to emphasize that we have interpreted the determinant of star d as the determinant of the absolute value of star d. I haven't worried about the phase involved in the possibility of negative eigenvalues.

12:30 I hope to say more about this later. Now you can think of this action as really abelian Chern-Simons from what I've described earlier. Non-abelian Chern-Simons, the ordinary Chern-Simons, is what I've described with a constant 1 over 8 pi squared. Then you'll find that if you make the same computation in the weak coupling level, The recoupling limit, and what I mean by that, is you have along here the rest of a-mi-g, you only look at the normal, only use the quadratic approximation of the action in the normal direction, and this involves the determinant function. When you do that carefully, you end up with a measure of moduli space, which is independent of the metric on n. Once you understand the torsion invariant, go back and make a comment about this, at least it was to me, but you see, if we look at this three-dimensional action, it of course, there's no metric in here at all, it is sort of topological in some sense, in the sense that it doesn't depend on any metric on the three-dimensional, but it is a non-connection action, it's not a constant function, it doesn't depend on the metric.

15:00 On the other hand, when you look at this, Professor Allup emphasized in a previous talk, this will depend on the method, but it normally does depend on the method, and so it's kind of surprising that this partition function should be compensated for by the ghosts determined that you need to make this whole thing make sense. So, that's the reason why this gives you a metaphor. There's something independent of the metric. And now I can make the comment that it does make sense to think about probing field theory. John's polynomial was a function of k, I put coefficient k in front of the term Simon-Stern, and as a function of n in the gauge group S-unit. Polynomial invariants generalizing with Jones to three manifolds that are not simply connected. I've said enough of that invariant. Let me tell you about one other invariant that I want to discuss associated with these three manifolds and three-dimensional angles. And that is the Casson invariant. And I want to describe it briefly in context. I want to make an assumption. About pi 1, that when I make it a billion, as I divide out by brackets I get 1, this could mean the first homology group of the manifold is 0, and then it's a homology tree sphere.

17:30 So if we look at it from a homology point of view, it looks very much like S3. Now I should say a word about the purpose of this invariant before I talk about it. It actually occurs as a no-ghost theorem. A no-ghost theorem. A no-ghost theorem. The No-Go Theorem is a no-go theorem for all dimensions, but it's a no-go theorem in that there were various attempts to show that there were other simply connected three manifolds besides S3, and some of them poses boundaries, etc., and there's a no-go theorem that Cassin and Varian says that you can't write one of these homology true spheres as a boundary of an appropriate four-dimensional space, so it's kind of technical. It's present time, so you might have to use it. In low-dimensional topology, a striking example of something that is interesting from the point of view of Yang-Mills and ones that tend to understand three-dimensional Yang-Mills for purposes of quantum physics. So I always consider the modulized space of the flat vector potentials, the scales of this vector field, as so natural on AMRG. And the simplest definition of the Castan invariant is that it's one half the order characteristic of the space a minus g in the sense that it's the sum of the zeros of the spectra field. So to put you in the standard geometry of a manifold, you have the order characteristic of a manifold, which is... There are a number of variables which you can describe in so many different ways and one of them is as the alternating sum of the zeros of the vector field, sorry, the sum of the zeros, singularities of the vector field, namely let's assume the zero is isolated and that you simply count the winding number around that zero. It also has an interpretation as the alternating sum of the very numbers

20:00 Or, and that means that many numbers is interpreted in a number of different ways depending on what kind of cohomology you're talking about. If you want the sum of the singularities of the vector field, there are technical problems. There are, the zeros of this vector field need not be a discrete set, but you can perturb the vector field by gradients to make it discrete, zeros discrete. What you do, if you have, then around that loop, you have, think of that, you have the Wilson loop function, which is parallel transport, parallel transport around, that's kind of singular, what you do is take a little neighborhood of it, and look at the nearby loops, and then take this function of nearby loops, and smooth it out by integrating over the normal coordinates. Take this singular function, so she concentrated on the loop, smoothing it out a little bit, and those functionals perturb your vector field and end up with something with a zero. Your vector field is not too bad. So you can solve the one problem of making these zeros discrete. And then you have the problem of deciding on what the sign should be. And what you do is say it's zero, you decide the sign is going to be zero. The path will assign the sign here depending on the number of eigenvalues of this operator that change sign from here to here.

22:30 Luckily, it turns out that if you took another path, the total sign doesn't change. And thus one gets a way of attaching plus or minus one to the zero. We find the eight invariant in terms of the plus and minus eigenvalues, and then you can see that the spectral flow is the difference between the differential and the values at the ends because of the sums. So you could write a plus or minus sum in terms of this expression. On the other hand, many of you know that if you actually wanted to get determinants as opposed to absolute value determinants, The eight invariant enters, so in fact you get sine of the zero in terms of the ratio of determinants of the phase killed by Chern-Simons factor. I emphasize this for there's a relation between Casson and torsion that isn't so clear to me yet. It's just being clarified. Both by work of Witten and Dennis Johnson, there seems to be a deeper invariant than just Casson, and Witten has taken the torsion invariant and thrown in the appropriate phase that you get an invariant that isn't just a positive number that takes into account the phase of the determinant, and here in Casson you also have that if there's some relation between the two that needs clarification. If one thinks of the Bacchus invariant as a cohomology, then one would like to understand what that is any cohomology, and this cohomology is the four cohomology, and in keeping with talking about canonical formalism, I'm going to quickly describe Atiyah's version, and what you want is a Duran cohomology here, the basic operator is D plus

25:00 We all know in infinite dimensions we've got to have a potential term to dance things to infinity for supersymmetric reasons. The basic operator is d plus multiplication by this one form. But you don't operate on ordinary forms in amyg. But on semi-infinite forms, the tangent space of amyg splits into two spaces, plus and minus, We will essentially go through the Dirac C construction. We will take the volume element as being the dual product of all the positive space and the forms you're dealing with are finite. That volume element is the vacuum. It unfortunately is not well defined because of the zero modes. It's an anomaly, so to speak, because as you move around the closed curve, when you come back again, you might shift by zero modes going. Positive to minus, but the amount of shift is divisible by 8. The closed forms and exact forms are well defined. The degree of the form is an amount away from the basic volume of this. It's not well defined at mod 8, which is why you deal with this interesting analytic problem to actually show that cohomology from the analytic known, which is described by Fleur in terms of...

27:30 Morse theory. You might try and do the following. If you've got the characteristic of this cohomology, then it is the index of that operator I described, plus the adjoint. And then one might try and compute the other number above via the methods of the supersymmetric quantum mechanics. That is, you should be looking at loops on a mod g based on a-manifold. If these were just paths, we'll know that there's an AMOG on M3 cross R, but here we're doing the closed path, and then you have to look at the subset of AMOG based on M3 cross S1, or parallel transport around the basic circle of S1. In fact, if you probe the flat vector potentials in this space, you will, in fact, get back to this, in fact, the zeroth order. Donaldson polynomial for this space. Just formally apply the principles of supersymmetric quantum mechanics and write down the appropriately rounded. What you do is get a part whose quadratic piece is the same as the one Witten used generally for topological Yang-Bills in interpreting the Donaldson polynomials. Well, as I suspected by time, I can't get the four dimensions.

30:00 So, I will stop with making the following comments. I don't emphasize dimension three. I've just indicated how you might go up by taking loops on a three-manifold to get up to four dimensions. In understanding the case of M3 cross S1, as one did classically in electromagnetism, we would then try and write down the Lagrangian more generally to talk about topological Lagrangians for a four-manifold. Or you might go down the two, as Witten does, with the Jones polynomials and talk about associated component field theory, and one then has this marbles relationship between two and four, Professor Donaldson will be talking about. I have dreams about this relationship. Classical level already on survivalism noted the relationship between solutions of certain nonlinear partial differential equations in two sigma models. Instant times before, we're hoping that there's a quantum field theoretic relationship of some kind of non-linear periodicity between two and four. I myself am still enamored by dimension three. If I go back to my description of the proof that a two-sphere is really a two-sphere, it required a lot of analysis and complex variable theory. I've always looked and felt that there was some other structure on a three-man fold, simply connected, that one ought to put on the three-man fold, which would play the role of a complex structure for S2, and then there was some kind of analytic tool that would play the role of Riemann-Roch and Riemann-Eppinger. ...by ordinary geometric things, quaternionic structure, for example, in full space, which doesn't work as far as I can see. And now, on the basis of what I said in terms of the Kastner invariant, if you had applied that instead of to MIG, had applied it to the gravity case, you would have natural space would be met, divided by diffeomorphisms.

32:30 And you would ask about the zeros of that field and the other characteristics of this space. If you could prove that this vector field had a zero, say if you could prove that the index was not zero, then you would be able to prove on this simply connected manifold that that three manifold would have a conformally flat metric and therefore would be a three-scale. Everybody's got his pet way of dreaming about how to prove the Poincare conjecture, and I'm really asking... Whether there's anything in the associated quantum field theory which might lead one to believe that the corresponding index could be non-zero. In any case, I'm dreaming that somehow, maybe the structure that one, I think, ought to exist for three manifolds is not so transparent in ordinary geometry, but it's some kind of infinite dimensional structure related to conformal field theory. Well, all these are dreams. That's what a child does, dreams while they're waiting for the tooth fairy. Thank you very much. Thank you very much for this talk, even though you didn't keep to three. But, uh, I'd like, uh, Mr. Stiegel, you're going to get the microphone, too. To get an example of lens spaces which are three manifolds with, um, the same homotopy, I mean, first homotopy group, which were not, have they got the same homotopy groups for all ends?

35:00 No, it's only the first one. Yes, they are. The covering is by two. The covering is by two. One of the things I found when I come down talking about these things, people that I talk about are not from real training, or they are not trained in this field. Immediately, actually, they could learn, of course. Then, on further reflection, when you go and write a piece about afterwards, you see actually they're not obviously, they're challenging problems. ...meeting...