Twistor String Theory & 2T-Physics (contd.) / Two Faces of String Theory in Twistor Space

0:00 I have to play with the film, I'm going to introduce with the system, I'm going to introduce precisely what could be super reality, AES 5 plus S5, that could be the full extent. So, all I need to show you is that I do have the correctness of the film. So, let's come to your opinion. Z is, you know, 8 by 4, so it has 32 x. Half of them are bosons, 60 of them are bosons, 60 of them are fermions. All the x's are complex, so therefore that's how many bosons, 32 bosons are 32 fermions. I'm starting with the Z. Now, then I have these constraints. The constraints are the constraints of SU2. SU2 already has three parameters. Another three parameters, that's it. And then another parameter that is seven. Okay, three plus three plus one. This is a gauge transformation. It allows it to remove the visual freedom. I also have the strength, and that's additional. So I need to multiply the number of digits that I can remove is 3 plus 2 plus 1 multiplied by 2. Okay, so that's 7 times 2, 14, so the original bosons, 32 minus 14, that's 18, so that's correct. and you go through the Agnus source of work. So this is the correct space to give you, in frozen twisters, to give you ABS-5 process 5. It will give you the same representation of SU-2-2, SU-2-2 slash 4, that is, you will get from super-garity, factory super-garity, compacted by ABS-5 process 5. And then, if you go over and do the string version of that, then you would expand the S-5, as you were going to do the string version. Now, king-fine physics gives you the same thing also in a slightly different way, as a cosine. So it's also worth noting what this cosine is. It gives the Lagrangian,

2:30 the clear-eyed, like the original Lagrangian remember there was a Cartan connection, and then there was, it was being dotted with something, with some kind of matrices. In the gauge that I, that I need, in that gauge, that kind of matrix becomes just a constant. It just becomes a diagonal matrix with pluses and minuses, in such a way that it defines for you that it's an invariant of the PSUQ2 times the other PSUQ2. So, over here you have an action constructed out of the Cartan connection. The single power of the Cartan connection, not twice, not like a signal model. A signal model is constructed out of the Cartan connection. So this is a single power of the Cartan connection, but dotted with some time matrix, like this, which tells you that there are symmetries in such a way that G is here as a concept that sits in this particular process. If you have the degrees of freedom, again, you will find, if you learn it out of time, you will find again 18 volumes in system terminals. And this way of writing things is exactly equivalent to the previous transparency. The previous transparency looks more linear. The previous transparency looks like oscillators. oscillators and then you apply the oscillators on the states and you find the physical states, whereas over here the constraints are already completely solved and instead we have a non-linear type of Lagrangian based on this. So this process again has all different or in degrees of freedom, to give the same information, the same thing that will engage in dynamic information. Okay, so no time to tell you in more detail about twisters in six dimensions. But let me say what the spectrum is going to be associated with that. If you do the particle, if you do the counting and then do a little bit of algebra and find the spectra, you find out that it is, the spectra is the one with the following fields. A self-dual form on circuit of a gauge convention in 60 inches, a self-dual is 60 inches. A fermion, classified on the list P4, is a 4, and on the list of 5,1 also is a 4, and then a scalar, which is the anti-symatic representation of this P4, which is a 5.

5:00 So if you, so you have five bosons over here, from here if you go through you have another three bosons, so eight bosons all together in the physical space, and then also four times four, four times is actually two because you have derived by physical fermions, so eight fermions. So this is this section, and we can discuss that for this content, one should be able to find an interactive conformal field theory based on these degrees and fields. But it is, the content of the indices is such that it is impossible to write down a polarion filthily based on such forms. However, one expects, and we know it was given in various pictures, and in papers and others, also, on the side of the head, discuss various things about electric theory. It should exist, but one knows little about it. Now, the twisted version physics, is based on OSP-8-4. So I have Z, where capital A is on the left of the presentation. It's a quack. It is eight bosons and four angles. But that's bound. Okay, closer. And then the other two transforms on this. So it's, again, a triangular matrix arranged like this. But then there are some pseudo-reality conditions because it's coming from a group So, basically it's equivalent to two posons classified with a 4 and a 4 bar under a C4 and then two canyons classified with a 2 and another is a 200 under the C4. The Lagrangian is very similar to the previous case. And again, it's a system of constant oscillators and one knows how to deal with that. And the time we use a theorem is exactly like that, to correspond to the spectrum of the dimension here. Now, the four-dimensional case also goes exactly like that.

7:30 And now let me finally describe the full two-time covariant version of all of this. Okay, so up to now I did not give you the two-time covariant version, I gave you the result. But here are the two-hand provided versions. It goes just like I was writing the particle to start with, containing the next square and the P square and the next up key, but now it is for strings. And then the carton connection coupled to the appropriate component So there are all kinds of local signatures I have no time to discuss, and a global signature which classifies all of the states and governments over facilitation, the physical aspects of the property. So anyway, so it is this theory when it is gauge fixed. In four dimensions, it gives you exactly the vectorless version of the viscous to this time. And when you do it in the other dimensions, when you do it in six dimensions, it gives you what I was mentioning earlier, the distinguishing. So it gives you the workiest version except, as I said, with the shape in something which you can't have time, it should be of interest to most people, but I can describe it in there. So let me also tell you what is different when it comes to the 10-dimensional super stream. It is the same structure except that over here I had before, for, say, four dimensions, I stand up for six, gamma and m for s of s of six. But now I extend x with additional six x's, p with additional six p's, and then I couple them with the angular momentum for s of six over here. So that's the only difference in going from, say, the four dimensions to the five dimensions. Still, it would be a CPP2-structure. So, and then that reduces, as I said, the spectrum of the hyper-disciplinarity anti-GIS assessment. Okay, I was going to speak a thing about crystals in more general spaces,

10:00 arbitrary dimensions, and how do you define crystals in arbitrary dimensions from this point of view. So in the article, what is a twisted description of it coming from this S.O.D. comment you said after that, this is a twisted, the equivalent of that, and it is highly twisted and called a twisted. So anyway, let me summarize. So there are, by now, all sorts of astonished facts about two-time physics, but that works. That does describe ordinary physics in various circumstances. And all that is based on this local institute, which they produce, and from there one-time physics emerges, the dynamics of one country, the space times, various space times emerge, so they are all emerging concepts, and they come from this holography of coming from deeper sub-dimension to deeper sub-dimension, so it's a stronger type of holography compared to other holography that we've been talking about here. So from there we can come down to crystal space as being one of the holographs, And then we can get to do, in relation to familiar things, we can get to do some new things, such as . Now, let me ask you this also to do some speculations. But since it seems to be working so well, the final aim should be to try to find out, to use the method, to try to find out whether or not you can learn something about what end theory is all about. And so this matter of two-time physics is making any similarities which are very difficult to see in other ways. So systems that have the similarities that we all knew throughout our lives, we were not aware of what possibilities were there. And the two-time people's formalism just makes it completely clear that they are there. And then you can go back and check that they are there. So, in particular, we are looking for the fundamental theory, which is similar things we don't know yet. But, we know that that theory has eleven dimensions, and two-type physics suggests it's not two more dimensions,

12:30 in order to see this energy as fully as possible. Alright, so then, I would like to state it, that I would like. But this formalization is useful to construct anti-array in its fullest form, in the most difficult form of the A2, in 11-kara-2 dimensions. And 11-kara-2, when we get supersymmetry, we don't know what supersymmetry can exist, but involves the spin of 11-kara-2, that we have to go to spinners that are 64-dimensional, and therefore the only unique supersymmetry that will do that is always 31-60 degrees. that in one way or another this entity is going to come in and teach us things. As I have already in several cases in the past pointed out in several ways that it connects in one way or another to various types of human beings. So I'll thank you very much. I'm so grateful that you allow me to talk to you. Thank you. I don't think we have time to question this time, because we only have until 2 o'clock tonight, and it's not sooner today, and we're going to get caught back. So can I... Various people that have buses and trains to catch, so they're going to be tripping off during the course of the afternoon. So I'd just like to say that various people have come to me and said that this has been a successful confluence, that's my impression. Also, I just wanted to say that you can't have no confluence without the speakers, with a conscious of the fact that many people have come along the way, taking a lot of trouble at a very busy time of the year. Thank you very much. I would like to thank the organizers for such a great conference

15:00 and for giving that opportunity to take care. The talk will consist of two parts. The first will be based on the work of the Lubritch-Mothel and the Neisting. The second part is based on different work of Robert Deidreff, beginning with the Neisting and Kummer-Datang. Although the two topics I'm going to cover are quite different, there is a common theme. In both cases, we shall find some interesting irregularities which suggest new structures not explained by current formulation of the theory. So I also should give you a warning that this is actually the first time I'm going to attempt such an experiment, so I hope I'll be able to finish on time, but I don't have any prior experience. Alright, so, let's start with the first topic. We're going to study the relation which has been discussed in a self-study-all previous response between primitive n equals 4 super-align units, and a psychological stream, the B-model, on the twister space, U3 slash 4. As usual, we'll parameterize the twister space by bosonic coordinates lambda nu, nu being the Fourier transform of lambda-tildes, and for parameterized coordinates, which altogether parameterized C4 slash 4. So, the precise relation, well, the precise check of this relation would be in connection with the commutation of the scattering amplitudes in the whole theory. And as we've heard again many times, the amplitude, which is a function of the mental linear variables than the mu can be, literally, can be obtained by some integral of the moduli spaces of polymorphic curves in CP3-4, where a degree of the curve is related to the number of minus-holicitive luons participating in the scattering amplitude. So, and the integral is of the modulized spaces of holomorphic curves with n-marked points, where, roughly speaking, each marked point corresponds to an external neuron participating in the scatronomages. In the topological B-model, such holomorphic curves are interpreted as the one-brain-instinct

17:30 pause. So, let's consider a simple example. If you want to study scattering amplitudes, scattering amplitudes which has only two minuses, only two minus, at least, at least, we'd have to consider degree one curves, according to the internal formula, therefore, Well, these are simply lines, Cp1s and Cp2s. Well, in the case of, if we go to the next case, which is next, then, we should bring out these degrees with three minuses. The degree should be two, and there are two possibilities. One can consider either Connick's degree two curves, like this one, which I draw in a real version, And for two lines, Christophel did we result in two, which are connected by the proper gate rather than the polymorphic or sinusoid. So in principle, there are these two contributions, which we hope to contribute to the substitute, and we're going to analyze both. So, in order to get into the hand itself, connecting amplitudes in Young-Mills theory to curves in this first case, let's consider yet another example of degree-free. This corresponds to an amplitude which has 4 minus 1, and there are several, again, and contributions in this case. They could be either a single pubic or a combination of a colony kind of line or three lines connected in various ways by propagators. Again, marked points are distributed in various ways and there are various kinds of graphs corresponding to such data. So, let's first look at the contribution of the maximally connected instant positive, which corresponds to the curve of the maximal positive green.

20:00 Well, in this case, we are talking about modular cases of geno-zero curves, I could say this, I could have said this at the beginning, of terms of degree d, with n-marked forms in Cp3 slash 4. Dimension, well, superdimension of this modular space is 4d plus n, 4d plus n, and 4d plus 4, even n plus n, as you mentioned, respectively. And there are various ways to represent such curves in the FISTER space. And following the prescription which works in the simplest situation, we tend to realize such curves as automorphism classes are mapped from CP1 into the FISTER space. This modular space is non-compact due to the possibility of curve to degenerate, like in this cartoon, where a conic could go into intersecting lines. So we have to compactify the modular space if you want to study such integrals, and one way to do it, which is commonly used in the Gromov-Britain theory, is to simply add such degenerate curves which correspond to an intersecting line of curves of lower degree, and this This is going to be the compactified modular space, which is the modular space of stable maps. So, to proceed with the formulation of the problem, we have a map from C2-1 to C2-3-4, which are one-hand, right-hand. With a positive choice of coordinates, the main disease with this voltage index A represents all eight bosonic as well as fermionic coordinates of CP2 slash 4, homogeneous coordinates. And sigma is an homogeneous coordinate of CP1, so we confirm the trisection of a bunch of coefficients, beta k, which parameterize with modulus.

22:30 There is an actual measure, an invariant measure, as was explained in previous talks, on this modulus case which simply is the product of the betas for all possible values of k and n. So, the expression for that integral, or for the staring amplitude that you want to compute, is simply given by integrating over all these moduli, as well as positions of the marked points, which have two extra pieces that I have to explain. First of all, there is a term which comes from the termian parallel actors, as well as all to explain in some of the previous talks. And we also have external wave functions, which are represented by Del Barcloth's 0, 1 forms, CP3, 5, 4, and integrating them over a modular space, we use that pullback of that information map to the modular space. There is a usual GLTC acting on sigma, which leaves the integrand invariant, and we have to quotient by the symmetry, so to be more precise, the amplitude, which is over here, should look like this. It should be in fact integral over betas and sigmas divided by GLTC, modded out by GLTC symmetry, and later on, I'll try to condense all these notations into three, into the vector which involves three factors, the vector on the moduli space, which is which contains the moduli of the curve, the factor which comes from the fermion parallelizers, and the factor which corresponds to the wedge products of the pool facts of the external wavelength. So this is to do over the modulized case of Gino 0, endpoints at degree d curves in CP3 slash 4. So this integral was evaluated in a number of cases by Roebb and Sparklin and Golovic, and they found that it perfectly

25:00 reproduces the correct Daniel's scattering amplitude. Well, we're not quite done yet, because there In particular, they are completely disconnected instantons, which correspond to a bunch of lines in the fissor space connected by fissor space propagator. In the physical space, each line would correspond to a point, and these are precisely the emissions that differ in a number of times. In this case, we have several maps, which I'll call QI, which maps CQ1 and CQ3 and CQ4, which correspond to each of these lines in the sister space. And each of these maps can be similarly parameterized by moduli beta. Now, there are several groups of such moduli corresponding to each of the curves. In the final, they characterize that space for given dimension AD and for dimension AD. There are several LTC acting on elliotropic curves in time B of them, because we're talking about degree one curves, of many degree one-thirds in this case, which lead to super dimension 4D8. So the moduli space looks like a product of degree one moduli spaces for each other term. Again, in order to write the formula for that amplitude, as an integral over such modulized cases, we have to specify several ingredients that are quite similar to the ones we had in the case of maximally connected instantons. There is, again, a measure on the modulized cases of such maps, which permetrides a modular I'll call it mu for lines to emphasize that we're talking about 3-1 curves. There is a measure coming from permanent correlators. There are, again, external wave functions pulled back to the moduli space. And finally, there are propagators, which are similar to external wave functions

27:30 in the sense that they're represented by 0-2 forms on C2-3 slash 4. So, then putting all this together, you get a similar expression for contribution of disconnected instantons to the Daniel's amplitude, scattering amplitude, which is given by some of all such graphs, which represent different lines, degree one curves and pisterous space, with different distribution of marked points. And for each of such graphs, we would have to do an integral of the moduli space of such maps. And again, it has several factors. One corresponds to the moduli. Peramon correlators, wave functions, and properties, which are probably combined into the same entity. Again, everything was divided by action of L to C acting on each other line. So, this integral was evaluated for essentially, when we talked earlier, this is essentially the MHD prescription for computing and mill substitutes. So, it was studied in a very nice work by Schroeder Storker from Lytton, which was presented and reviewed several times. And, again, they found that this computation essentially reproduces the complete coaxial So, when I actually come to the puzzle or question, how come two contradictions, or two possible types of instantons in the B model happen to deal as the correct answer to the annual Now, some things could, here it's illustrated in the case of degree two-thirds, which could be either connected one of the degree two-thirds, or two lines, whose total degree is also two, connected by a fixed-first plane. Well, the integrals in both cases, as we've seen, are performed on different moduli spaces.

30:00 almost deeper-deepers, whereas here it's a monolized space of lines of 3 plus 4. And these monolized spaces are very different. They have different dimensions, in fact, and so on. But they do share a common boundary divisor which corresponds to the degenerate curve, which in this case would be represented by two lines intersecting each other. So, it's clear from this picture, you just collapse the propagator and bring the two lines together, and similarly, the planet can degenerate into . So, one might hope that if these two integrals here happen to have a simple fold along this boundary divisor such that the residue is the same, And then, properly defined integrals over these two modular spaces should indeed be related. So, I'm not going to address this problem in complete detail, in particular because, well, we don't understand the point of the contour which goes into the definition of these integrals. Now, going to analyze the residues, first of all, we will see that there are indeed holes in each of these integrals over this boundary divisor, which corresponds to the 2% curve, and secondly, we find that the residue is indeed the same in both cases. So, in fact, the first of these cases, which corresponds to disconnected instantons, has already been analyzed in the same paper by the shortest version of Mitten. And I'll just briefly give you the details, the sketch of the details in both cases. In the case of disconnected instantons, we have two lines, and the relevant part, the corresponds to the relative position of the two lines. Remember that, according to the incidence relation, each point in the Minkowski space is represented by a line in the distance space, and this is precisely the equation for each of these guys. And here, the moduli, I call them x,

32:30 in analogy with the fermetrization of points in the physical space, because that's So, that relative position, relative modulus of these two lines, x prime minus x, is denoted by y, and the integral over the moduli space of such disconnected lines contains a piece which looks like the integral over this relative position, which does have a simple hole, as was already pointed out in this paper line, which I just started to make. And then, localizing the integral to this pole, the form of the integral, one finds a resulting integral over the molecular spaces of intersecting curves, in fact, whole sum over such integrals, which correspond to different ways of distributing the marked points over such intersecting lines. And, again, the integral has a very similar form. It contains the integral of the micro-intertaxing curves, as well as fermions, correlators, and interrelations. Here I am, for simplicity, presenting a picture of how it goes for . So... The same story, or similar story, happens in the case of maximally connected curves. In the moduli space of connected degeneration curves, there is two-dimension one locus, which is the boundary divisor with a dimensioned area, which corresponds to degenerate curves. We can characterize this extra complex parameter, which takes us away from this degeneration marked as I'm called epsilon, such that epsilon equals zero would correspond precisely to the location of the boundary divisor where it occurred generally. With some specific choice of modulo and geodeficiency, we heard and essentially represented, well, in case it was funny, by

35:00 this equation for the hyperbola, which does indeed break into a union of intersecting lines one epsilon is equal to zero. So, the measure of that integral that we had earlier has several factors which correspond to the moduli on the curve that permanent correlators have internal wave functions, and we are going to ignore the factors which correspond to internal wave functions, it doesn't seem to play a significant role, although I'll comment on it in a moment. But the first two factors in this measure do the interesting epsilon is kind of, that measure corresponding to the moduli of the net has a factor of epsilon over epsilon cubed, and the case of the measure, which comes from permanent trial, which goes like epsilon-square, so we do find a single hole in this. However, I have to point out that this fermion, macro-cumming, parlor-permionic correlators, knows about the sanctified ordering of the coins, and this leading epsilon-square behavior occurs only if we distribute the points on these two intersecting lines in a way in which risk accidents type of order. This is actually a very nice feature, which is necessary if we are in contact, and we should be diagrams where the points have to be distributed over different components of intersecting lines without type of order. So, as I mentioned, you also find that the residue in the integral is precisely the same as the one we find in the k-cell. One can go ahead and repeat this argument iteratively once we have the basic step, which allows us one curve of degree d to curves whose total degree is alpha d, and each such degeneration one step at a time, localizing the integral from the moduli space of degree d curves to

37:30 boundary advisors which correspond to such intersecting configurations of curves of lower degree, and eventually getting us to intersecting lines, which correspond to MHD diagrams in the physical space. So, this essentially includes the first term, and brings us to my favorite part of this topic, which is the list of all confessions, which is, in fact, very long. and that was the whole purpose of this part. First of all, showing that the integrals have a pole that the boundary divisors and the residue of time size that doesn't prove that the integrals are related. One would have to know the precise definition of the point and the integral that goes into this description. In fact, in the case of the connected instanton, in order to reproduce that same residue as the one which comes from MHP diagrams, we had to go around all possible holes. So in that sense, the contour should be like over infinity in the modulus. As I mentioned, we also ignore the external wave functions, which might play some role, and there are many other questions, such as loops and similar functions for high genus, choice of restrictions, because we seem to find that there are several ways of permalizing the rules for computing the mean sumptitude of such integrals. And the natural question is all such integrals are proportional to each other, based on the constant, one might wonder if we should add we are contributing together, or there is indeed a relation between such integrals, which perhaps might be used to deduce some precursor relations among the mean sumptitudes, once we know how to break more complicated substitutes into more basic constituents. And I hope that all of these questions,

40:00 which are a little bit more specific, could eventually lead to proper physical interpretation of this phenomenon. So showing that some integrals happen to find science with other integrals, to me, suggests that there might be some good physical reason or at least to happen, as usually it happens in the case of string theory, qualities, and so on. And perhaps it might also shed some light on what the dual string theory in the future space should look like. In fact, yesterday, we can give one possible definition or idea for what the suitable integral might look like. Alright, so, I think we're doing a lot of time so far, so this brings us to the second topic, which is, um, which emerged in, uh, studying many, uh, seemingly different phenomena in string theory, including, uh, geometry of chemicals on swing back, uh, various theories of gravity, we shall explain in a moment. Black holes, their entropy and fracture phenomena, as well as topological string theories and their goddities. In fact, the last topic was presented the subject of Andy Knight's talk this morning. He also mentioned connection to black holes and entropy. And there are, as I say, many other subjects that we investigated And which, all of them, although they're seemingly different, led to measure Gb2, some structure in seven dimensions, which seems to involve many folds of G2 polandemarks. All right, so let me start with a topic which is somewhat connected to geometry of n equals one-screen vector. If one believes that all supersymmetric polymorphic quantities in screen theory are computed

42:30 by topological string theory in one way or the other, then we definitely have to expand the space of classical solutions, classical configurations, and topological strings, because by now we definitely know that space to join the trio, and it goes on string data, goes far beyond Calabiao's spaces, which includes spaces such as non-Kellar and non-Calabiao-Manical. But let me come back for a second to the ones we are most familiar with, namely the good old manifolds of special polonomi. So, as you know, in general, the polonomi group is a subgroup of SOS, and the special polonomi manifolds are precisely the ones whose polonomi group is pretty smaller than SOS. So, particularly interesting examples, these plots include LBL spaces in dimension 6, whose holomony group is S2-3, or potential labels whose holomony group is S2-2 in dimension 7. such manifolds have many nice properties. In particular, they preserve supersymmetry, and this is precisely the reason they play an important role in the screen theory, which which geometrically means that they have a constant spinner, and using such a spinner of psi, one can construct a form by sandwiching a gamma matrix with sine, say, dagger, and therefore components of the differential form omega are simply given by this expression, which guarantees that the form is convergently constant and in various other phallony groups, unless it's zero-cost. And such forms play a very important role in geometry of special phallony mayfolds as well as in physics. In particular, they can characterize minimal or supersymmetric cycles of dimension P, where P is the same as the true of the form. For example, if you want to compute the volume cycle, all we have to do is to integrate such form over the cycle. And although this is

45:00 probably very familiar, I want to stress it because here something interesting happens that using differential forms rather than the metric, we happen to compute the volume of the cycle, which normally would be written as integral of square root of g. And So, let me illustrate in a little more detail how, using forms, one can construct a matrix and bias a version. So, it turns out that a condition that manifold has P2 halonomy is equivalent to existence of an invariant tree form, associative tree form, which is denoted by, which is poles and co-close, and given five, one can reconstruct the emetric cloud with polonomy and vice versa. In particular, if you have a metric written in terms of wild vines like this, then three-forms can be constructed by taking the products of the wild vines of the one-forms with positions, which are simply the structure constants of measuring octonions, which are very, uh, the, uh, multiplication relations, multiplication, and so forth. And vice versa, uh, well, we can use the same formula to locally bring, uh, three-form five in the same form and then extract the bow binds. If that's not very satisfying, if you want to know the explicit components of the metric given, uh, components of the three-form, which does it, namely, one would have to take three such, three forms, altogether they have nine spatial indices, and we can use the epsilon symbol to contract seven out of these nine indices, so we're left with an object which has only two indices, and it turns out that it has all the right properties to be the matrix. In fact, the matrix of the two hologramy is nothing but this logic Vij, rescaled by the experiment of V. And once we have the metric constructed, we can use it to define the fourth star,

47:30 as well as any other operations one might wish to consider. So, this phenomenon is a precursor of, a more general idea that can be used in various theories of gravities, where we describe the metric that geometry of space reforms, differential under forms, rather than engaged fields, rather than the metric itself. So, these are some notable examples which include, in dimension 2, 2D topological gravity. In dimension 3, famous trans sinus gravity. There is a four-dimensional theory of gravity in force, which is described by two forms. That's why it's called two-form gravity, and it appears as self-dual sector of loop quantum gravity. In dimension 6, there are several different theories which describe either career geometries or, hello, the other geometries and the theories of either career gravity or career spencer gravity, were mentioned in earlier part by Einstein. The cousins of these two theories proposed by Fitchin are based on four forms and three forms, and equations, plausible equations of motion in these theories are, in fact, exactly the same as equations of motion, Filler-Greis or Federer-Spencer theory, so one might indeed expect that these two theories are related pairwise. All of these theories are, again, based on either differential tensor forms or gauge fields, and don't involve metric in their construction. In particular, partition function in all of these theories computes some interesting typologically diametric invariance of the underlying geometric structure. So, let me explain some relevant examples. In particular, let me say how it works for case of four-dimensional two-form structure. Let M be a four-dimensional manifold, and

50:00 an A and SU2H connection of this manifold, and sigma, a triplet of SU2, a joint, a triplet of the two forms of the manifold, transforming in the joint of SU2. So, well, here I made a joint in this case manifest, and the action is written as sigma of HF minus lambda sigma This is a pretty simple action, and equations of motion that follow from the action are theory that the triplet of two-form sigma is currently closed due to respect to the connection, and the field strength curvature of the connection A is equal to option multiple to the two-form sigma. Well, it turns out that given solutions to these equations, one can reconstruct a metric on a four-manifold M, which has some nice properties. In fact, given the triplet of the two-form sigma, we can combine them in a similar way as we've had in the case of G2-manifold, namely we take three two-forms, have six indices, using the epsilon symbol in four dimensions, we can contract four of them, A1, A2, A3, and A4, and we are left with an object which has only two indices, and using epsilon in the SV2, we can also make it SV2 symbolized, so this object indeed has all the right symmetries and properties of the metric, and it turns out that if sigma Then, this metric is, in fact, cell-dual-lines time, which means that power is equal to g, and w plus 0. Let me discuss another theory, which is not on the sleeves. In fact, I should mention that all of these theories are of a special, and such a possibility of describing gravity theories, tensor forms, is special to certain dimensions.

52:30 And the largest dimension in which this is possible, at least as far as I know, is dimension 7. And the corresponding theory is based on a functional, which was proposed by Hitchin, and it uses a three-form in seven dimensions. So that's why we call it three-form theory. And the action using the three-form in seven dimensions can be constructed in the following We already saw how to use three-form in dimension seven to construct the Hodge-star operator, which I didn't know the stars apply to emphasize that this is constructed using the metric produced from the three-form form. So, therefore, we can write this nice, seemingly simple action, phi range star phi, which in fact is nothing but square root of g, where g is the metric you can construct it out of that three-point. And if you want explicit expressions, a determinant of this two-index object p, which we had earlier, raised to the power of 1.9. So, in fact, this function is pretty complicated. It's very nonlinear. In fact, it's homogeneous of degree 7 over 3 in terms of phi, but it has very, very nice properties. As in the case of the other gravity theories, in particular this cell-building version of the loop on from gravity. Classical solutions in this theory have very nice geometric interpretation in terms of certain geometric structures on seven manifolds. Specifically, this operates nicely due to the structures. Let's see how it comes to that. Let's view phi as a gauge field, as a curvature of the gauge field, and let's assume that it's closed and has as a fixed homology class, such that we vary the field to form B, whose curvature is the 3 form phi in this action. So then, critical points, and using homogeneity of this functional, it's easy to see that critical points of this functional

55:00 are precisely such that D star phi is equal to zero, and since we already assume that phi is closed, points are characterized by 5, 3, 4, 5, which is closed and co-closed, and as we saw earlier, these are precisely G2 structures, so that we can construct the metric out of this 3-form, which has G2 along on it. I want to stress that this formula for the metric in terms of 3-form 5 is extremely similar to this formula on the other slide for the metric in coordinate It has three basic objects in the theory, three other forms, contracted in a suitable And in fact, the contraction is extremely similar because it looks as if this epsilon symbol in seven dimensions is broken into epsilon, which has three indices, and epsilon, which has four indices, and this is not the maximum. It turns out that this seven-dimensional theory, based on this action, has very nice properties. It refuses to all other theories in lower dimensions, all other theories of supremacy in lower dimensions, as long as the manifold that we choose contains some calibrated or supersymmetric cycle of a suitable dimension. So differently, if we find an isolated cycle, which I call M, inside seven-dimensional manifold X, then we can try to restrict this seven-dimensional gravity and see what kind of theory is in use on the cycle M, people on the associative three cycles, we can find for Simons' gravity, on the associative four cycles, we can find that two-form gravity, which we had on the previous slide. So, I'm going to explain this little bit more detail. So, of course, in order to find such a reduction which encodes

57:30 only geometry of that cycle M, we have to assume that everything depends only on data which we have on this cycle, and therefore it means that we have to consider the following where this cycle has some normal bundle, which is essentially the bundle dimension M vector space over our cycle, such that the metric, seven-dimensional metric, restricts the flat metric on the normal direction. In particular, it means that we have to consider S1M comparing tons-outs. If you choose local coordinates, like on the normal direction, in normal direction, we can write one-forms where, in general, it can include a gauge connection A, a gauge connection on the manifold M, which keeps track of the way this vector space fibers over the side hull. So, in particular, in the example we're interested in where that normal bundle has dimension 3 and the cycle is 4-dimensional, we can naturally split the three-form into components along the normal bundle alpha, and components along the four-cycle, which I'll call sigma. These are two forms, and alphas are one forms, so this is a very natural, in fact, the most generic split of such field form, and as you can see now, everything depends only on fields on the force cycle. Mainly, there is a gauge field A, which tells us how normal bundles, which is connection on the normal bundles, and there is a two-form signal. In fact, it's very easy to check that equations for the GQ structure, the equations of motion in this seven-dimensional theory that phi's, three-form phi's, close, and co-close are precisely, in this case, precisely reduced to the equations of motion in this two-form gravity. And expression for the metric in seven dimensions, as we talked earlier,

1:00:00 does reduce to the right expression for the metric on the four manifolds on the solid-luenstein four manifolds So, in a way where three of the normal directions to the manifold now become gauge, play the role of the gauge indices, gauge group indices in the four-dimensional theory. In fact, the geometry of the resulting bundle is nothing but a bundle of self-dual two-forms on the four main folds, and this is non-compact space, which is our three-bundle over four, at that boundary of such non-compact space, and this is obviously the same as bundle of whole two forms of unit norm of a core, and this is the connection to twister space, plus this is the twister space. This point of view was explained in the first talk by LeBrun. There are several more ingredients we can add. In fact, the metric, the G2 metric, on this non-compact space, total space of the R3 bundle over R4, is asymptotic to a cone over the cluster space. And two-source space has integral complex structure in case of four-manifold is self-dual, as was explained in the talk by the group. And furthermore, if four-manifold is Einstein, the two-source space has a greater structure. So this is very attractive for constructing topological A model in the two-source space. To conclude with a wild speculation, it seems to suggest that there might be some holographic kind of duality between topological gauge theory on four-dimensional cycle M and topological gravity, that's why it's gauge theory gravity duality, on complement of such cycle in this non-contact space,

1:02:30 which is essentially the same as unbundled or just the filter space. Well, if we had that precise microscopic definition of this topological M-theory in seven dimensions, one would perhaps derive this holographic duality by starting with some D-brains wrapped on the four-dimensional cycle. And as usual in the ABS-CFT correspondence, one can describe such system in two different ways. on T-brains, which should be the gauge theory on the four-dimensional manifold, or as gravity, which in this case is the topological answer imposed on the complement of the space, and this is precisely what we have here. Well, one can go further and try to speculate what precisely should be the gauge theory on this side and the gravity on this side. And since we're talking about pathological and self-dual structures, it's natural to think that it might be pathological, self-dual, and milled here, and perhaps a model on Cp-free, which is the first type of a four-sphere. Both theories are, in fact, trivial, so if that's really the complexure, then the duality It does fall apart in a rather trivial way, because self-dual and Niels-on-4-square as well as A model on CP3 are trivial. Well, partition functions are extremely simple, so in order to get something interesting, we would have to deform both theories. It has to deform self-dual and Niels-on-3-basire, and Niels-on-OS-4, and this A model on CP3, by incorporating certain observables. I want to emphasize that here I'm really about just trying to confer partition functions of such theories, so it has nothing to do with scattering amplitudes in certain functions about partition function of each theory on a contact state. Well, but this is very well speculation, and the lesson is that there are these different theories of gravity which seem to naturally emerge in a very unified framework from this seven-dimensional theory. And it has a few more aspects, such as, it nicely explains various relations between topological spring theories, which Andy Narsky talked about in the morning, as well as a certain phenomena in black hole physics,

1:05:00 which I didn't have time to go through, and hopefully we'll hear about them for some other time. So, thanks a lot. Any other questions? So I have one about the first part of your talk. If you compare the case where you've got the higher degree curve with the case where you've got disconnected lines, you have to make a choice about which fields end up on which line in the case of disconnected lines, and it seems that you don't have such a choice in the case of the curve, is that right? You have to add over all possible distributions or something. Does that manifest itself in the case of, and how does that, it seems that the two things are different in that respect. No, no, not really, in fact. Apparently, in the case of high degree curves, one would have more choices, in fact, because there are many, many different ways the curves can break, and there are many possibilities for external points, for more points, to see from different components of the degenerate curve. But suppose it's not degenerate, that's my point. Yeah, if it's not degenerate, then they're fairly democratic. That's right. But it made it degeneration of some components that degeneration do correspond to specifying which point goes into which component of the degenerate curve. And in particular, on one of the slides, I had to make sure that the distribution of such points amongst various components does restrain the site of ordering, which was in the case of the distinct act. Are there any other questions? If not, let's thank our speaker again.