 I'm very pleased to be able to participate in this celebration of Sampson's coming of age. I've known Sampson for many years. I was trying to remember when it was that we first met. What I think it was was at one of the spring schools in Trieste many years ago. I think I was having a coffee with Michael Green. We were having a quiet English conversation when Sampson arrives and this transforms and enlivens the conversation with his characteristic combination of stories, anecdotes, jokes, critiques of what's going on, all combined with the critique of what's going on in theoretical physics. I was talking with one of the other speakers about the difficulty of knowing quite what to say about Sampson in these morning sessions since there's so many great stories of Samp without Sampson, most of which are probably not right for talking about this time in the morning, especially not when the lecture is being recorded. Sampson also has his serious side and I think it's very good that we have a scientific meeting here to recognise his many scientific achievements. He's done a great deal of important things. Perhaps it's probably true to say that he hasn't perhaps got the credit he perhaps deserves of many of the things he's done and it's very good that a lot of the talks here are recognising and remembering some of the things that he's done. Unfortunately, yet I haven't had the pleasure, or if that's the right word, of collaborating with Sampson, but we very nearly did. We applied for a research grant together, a joint British Irish one. So we heard yesterday about Khan's criterion that we shouldn't think about whether something is mathematics or physics, but only whether it's true or false, whether it's right or wrong. It's not clear what happened, but it seems that one of the judgments on our grant proposal was that it was too much physics and not enough mathematics. What I want to talk about today is some of the things which we might have got, some of the work I've been doing which might have been part of that collaborative effort involving special halonymy manifolds and related things. Someone asked me whether the word degeneration in the title had anything to do with Sampson. Of course it doesn't, it's referring to degeneration of geometries. I thought there was no connection, but then as many of you know, Sampson has been involved in an art film project over many years. I learned yesterday that two of the films from that project have been nominated for the Berlin Film Festival. One of them, it turns out, is called degeneration. I was wondering if there was any... Not believing that there can't be a coincidence, I looked to see what it was about. It was about a physics research institute. Where the members there were facing what it called an existential crisis. Being the kind of film that it was, that resulted in some degeneration and it all ended very badly as far as I can tell. I was very relieved to see that although some of our esteemed colleagues were involved in this film it seems that Sampson was not part of the degeneration film. So I hope that he will long stay free from such existential crises. So the work I'm going to talk about was some work I've done, I've been doing over the last few years with my student, Nipol Chumjomras. It's taken me a few years to learn how to begin to pronounce his name. I still haven't got it right from Thailand. The general theme is understanding relationships between string theory and geometry. We have a picture of string theory formulated as a perturbation theory and we now know a lot about the ingredients involved in that. But the dualities we've discovered have revealed a surprising non-perturbative structure to the theory. Still very poorly understood and these duality symmetries which help us understand some of these non-perturbative and strong coupling generalisations. What I want to talk about today involves some very recent mathematics which are some very interesting recent geometric results which turn out to shed a great deal of light on some of these dualities. In particular it gives a very interesting relationship between some of the ingredients in perturbative string theory involving brains and orientive folds and related things and certain geometric properties. It does it in such a way that it immediately lends itself to all sorts of generalisations and it looks like it's going to be a very profitable way of trying to understand and unravel things both about non-perturbative string theory and about geometry. The starting point is the duality is involving K3 and it will go on to look at special hillonomy generalisations. The starting point is the duality between two A strings on K3 and the heterotic or tight one string on a four torus. In fact it's almost exactly 25 years since Paul Townsend and I proposed this duality and it's quite intriguing that there's a lot of evidence in favour of that. There's still a lot of mysteries involved in this. One way of understanding some of this is a standard duality argument where if you take type one string on a four torus then via a chain of T and S dualities you can get the type two A string on an orbifold of T4 modded out by a Z2 a four torus modded out by an orbifold of a four torus and this is a special orbifold point of the K3 modulized space so it gives a proof of a derivation of this duality at one particular point in the modulized space of K3. Then one can try and argue how to move away from this point by understanding how to move away from this point in the K3 modulized space and the corresponding moving away from that point in the type one theory. This is quite a fruitful way of understanding this but one of the issues which arises in thinking about this directly is that when you move away from this point in the modulized space there are no longer any isometries and no longer any conventional T dualities to use so that the duality can no longer be understood in this simple way. Some of the new mathematics I'll be talking about involves particular identification of a region of the modulized space of K3 in which we can understand some of this quite explicitly. In particular the chain of dualities taking type one to two A on K3 takes the D-brains in type one theory to the Klutzklein monopoles and it gives a picture in which the D-brains here are represented by Klutzklein monopoles and I'll be saying a lot more about that. But there are also orientafold planes in the type one story and this whole story will lead to some kind of, should lead to some kind of geometric dual for these orientafold planes and in this way will give us a new picture of K3 and a new understanding of orientafold planes at strong coupling and a rather explicit picture of the dualities. As I mentioned I'll be using some new results from geometry but in fact this stringy duality picture in some sense could have predicted that there should be this picture of K3 which involved the modulized space being understood in terms of positions of Klutzklein monopoles and so on. So a starting point will be to think about type one string on a circle so thinking about a circle across nine dimensional Minkowski space if you do a T duality it replaces the circle by a circle moded out by Z2 so corresponds to reflection in one line and so there will be two fixed points and in this picture in the string theory one learns that one needs to insert an orientafold plane at these two points so we're looking at a circle across nine dimensional Minkowski space so we get a nine dimensional plane at each of these two points and because there's this reflection symmetry we can think about just considering a half of the circle with these at either end and we have a picture of an align interval with an orientafold plane at either end so we're taking the product of this with nine dimensional Minkowski space so we think of this as being a slab with nine dimensional orientafold planes at the two ends and we have a picture of this with some nine dimensional D brains which move along this interval and in fact there are 16 of these D8 brains in the perturbative theory and at strong coupling this picture is modified there was arguments originally by Cyberg and Morrison which said that there's more things which can happen at strong coupling and instead of having orientafold planes there's some objects which are referred to as o-star planes instead and their whole structure is rather mysterious in the type one theory but this whole picture should have a dual representation in the dual K3 theory and in particular if we take the case where we have this interval being very long compared to various other scales in the theory we should expect a picture of K3 which has got a long neck and that's so we have some long neck, a four dimensional space with some long neck region a various divided into various segments and these are joined together by various bubbles which involve insertions of colloquial monopoles I'll be more precise about how this works as we go on and these positions of these colloquial monopoles will be dual to the positions of these D8 brains and this being a smooth geometry there must be some kind of smooth hypercaler cap which goes on these ends and one of the things we'll be talking about is what these caps are what these geometries are and it turns out there's an intriguing relationship between these and del petso surfaces and the rather limited classification of these has a lot of implications for the dual string theory so this is the sort of picture we might have expected from string duality and as I'll explain in 2018 a very precise construction of a matrix on K3 in a limit in which the metric degenerates to something which has got a long thin neck of this kind was constructed and has precisely this kind of structure and this whole story leads itself to further directions one thing I'll say a little bit about is the very interesting generalisation to special holonomy where this picture in terms of brains gets generalized to one involving intersecting brains and one of the nice things about this story is that dualising the type 1 prime gives a consistent string configuration for various brains that can't exist in isolation if one slots off dualising D brains one ends up with various as one gets to D7 brains and D8 brains one's already familiar with the fact that these don't exist well on isolation and you need to take into account their back reaction and put them in a consistent string background for example in the type 1 prime as we've discussed if formally one takes further dualities it needs to various things which have been referred to as exotic brains in low co-dimension and again these don't exist in isolation but this chain of dualities starting from the one we've just been concentrating on in this talk gives rise to consistent backgrounds for these to live in and in particular it takes the K3 to certain non-geometric backgrounds T-folds and related things and then the picture of and then we have a picture of instead of K3, instead of K3 being understood in terms of Coulout's Climb monopoles living in this background geometry we understand it now in terms of exotic brains in terms of T-fold background so let me now start talking in a little more detail about some of the ingredients which go into this so the first thing I want to talk about is a tourist bundle with a duality twist so having a two tourists for example fibered over a circle if we could start a geometric bundle we could look at a bundle where the monodromy around this circle involved a large difumorphism of this two tourists but in string theory we can generalise this and we can think about having a two tourist conformal field theory here and we can look at having a monodromy around here which is an automorphism of this conformal field theory in other words a T duality transformation which for a D tourist would be in this group and for a two tourists you get O22Z which has got an SL2Z cross SL2Z subgroup and this subgroup acts on two moduli of this two tourists one is the complex structure modulus for the two tourists the other is the complex modulus given by the value of the B field on this two tourists the constant value of the B field plus I times the area of this two tourists so this these are both symmetries of this of the conformal field theory if we had a monodromy which is in the SL2Z acting on Tau we'd construct a geometric bundle if we had one which involved the full group then we'd construct something more general a tourist bundle with a duality twist so is it not that T dual group is S special optimal? so if in the bosonic string it's ODDZ in the super string it would be more properly SO no it's a bosonic so I'm not being very precise here so there would be a slightly different story depending on whether I was looking at the bosonic or the super string and then there's in either case there's an SL2Z cross SL2Z subgroup but there's a question about whether I need to divide this by an Z2 and I'm not going to be very precise about those sorts of features in this talk so for the purposes of the next couple of slides let's think in terms of the bosonic group where I can put everything in O22Z and everything fits for the super string I would have to be slightly more restrictive so a simple case is where we take a case where the product is topologically just a three-torus but we have a B field which depends which is proportional to the volume form dywedd by wedge DZ on the two-torus and depends linearly on the coordinate X around this circle so the curvature of this two-form the exterior derivative will just be the three-volume of the three-torus and there's a monodromy here for the modulus rho as you go around this circle because B shifts by M times the volume form if we take a T duality we turn off the B field and get a two-torus bundle over a circle and we get a manifold which is referred to as a nilfold which is a two-torus bundle over a circle but it could also be viewed as a circle bundle over T2 with this metric and there's a monodromy, a geometric monodromy in which the complex structure modulus shifts by an integer M here and if we take one further T duality we end up with something we get a metric and a B field which look fine until you remember that X is meant to be periodic and these are not periodic metrics or B fields but it turns out that there's a monodromy which involves a T duality transformation as you go around this circle the metric and the B field come back to each other related by the action of an O2 to Z transformation which acts in a complicated non-linear way on this geometry but because this is a symmetry or an automorphism of the two-torus conformal field theory this is still a good string theory construction so this gives a nice duality orbit of geometries which is often discussed but there's an issue in that none of these are solutions of the string theory for example three-torus with flux doesn't satisfy the equations of motion doesn't define a conformal field theory but nonetheless one can embed look at these in string theory by looking for bundle solutions in which these geometries arise as fibres and then the duality transformations act as fibre-wise duality on this construction and the simplest case is where I take these three geometries fibred over a line and in particular the case where we have the nil fold fibred over a line gives a construction which admits a hypercaler metric so this metric can be understood as a kind of Gibbons-Hawking metric so as we'll be familiar to most of you the Gibbons-Hawking metric is a four-dimensional calerici flat or hypercaler metric which is given as a circle bundle over an R3 which is where you have R3 with these coordinates and we have a circle fibre with coordinate y and we have V being a harmonic function on R3 and omega satisfying this equation and if we allow the harmonic function to have delta function sources at some points we have a circle bundle R3 with some points removed and remarkably this is regular at the sources of these the sources R i of these things if M equals one and this gives a multi-tub nut space and there are simple orbital singularities for M greater than one so in this way we construct a metric over the whole of R3 not just on R3 but the points removed so one way of part of the ingredients will be looking at trying to put this on a torus so we can think about so-called smeared solutions where we take this harmonic function on R3 and choose the potential to be independent of one or more coordinates and the coordinates which it's independent of we can then choose to be periodic and it can be periodically identified so for example if we used to smear on X and Y we take V to be a harmonic function just of tau not with X and Z independent of X and Z so that Y should be a Z then we have a harmonic function in one variable which is just a linear function or a piecewise linear function where we can allow there to be a kink at say tau equals zero where there's a jump in the gradient and we can think of this as giving a domain wall with the two plane dividing the transverse three space three dimensional space into two parts for tau bigger and less than zero and the difference between M and M primed can be thought of as being associated with the energy density or the tension of this domain wall so this is a very nice construction but one of the advantages of the metric one constructs this way is typically singular and that's something we're going to have to deal with so let's look at the smeared Givens Hawking metric where we take this linear potential then we get this metric and if we notice that if we take and look at it for a fixed tau the metric becomes precisely the kind of nilfold metric we were talking about before which is a circle bundle over T2 with representative of the first churn class given by this two form and M is required to be an integer and this can be thought of as quotient of the group manifold of the Heisenberg group by a discrete subgroup so here we have a picture of a nilfold fibered over a line, the line parameterised by tau and there are domain walls at the places where the gradient jumps here and so we have a set of domain walls separating these nilfold bundles over a line and to get a string solution we take a product of this with six dimensional Minkowski space getting a ten dimensional metric and this can be thought of as a metric for a smeared coluysiglymonopole with the original tau nut metric it's precisely what's referred to as a coluysiglymonopole metric and now we can look at the T dualities if we take the nilfold fibered over a line we've got three circular directions the x, y and z directions and we can look at T dualising in any of those so if we T dualise in this coordinate the one of the circle fibre the y coordinate we end up with a metric which is conformally flat it undoes the topological twist but turns on a b field and we end up with a three form h given in this way so we get a conformally flat space but with the conformal factor given by this piecewise linear potential the ten dimensional solution is that we take the product with Minkowski space and this is the metric for a Nervyschwartz fibrain smeared over the three directions x, y and z which are then identified to give a transverse space instead of being a transverse space being r4 it's r cross T3 and we can do a further duality and get a T-fold fibered over a line but I won't say a lot about that today this generalises straightforwardly to having a multi-domain wall solution we're having a general piecewise linear function with kinks at a set of points tau i and with a domain wall with energy density given by the jumps at each of these points and it's also interesting to know that you can also think about single sided domain walls if we have this linear potential this is invariant under reflection tau to minus tau and if we quotient by this reflection this gives a solution just for positive tau and you could think of there being a single sided domain wall at the point at the boundary tau equals zero so these are going to be some of the ingredients we're going to look at but as it stands these are not consistent string backgrounds away from the domain walls we get to have a hyperscalar space which is a good solution and the duals also give conformal field theories away from those but the domain walls are singular and the linear delaton and the potential v will quickly blow up unless we end it with single sided domain walls and when we do that we will need some negative brain charges to give total net zero charge to get a consistent solution so to understand this what to do about this we can use the chain of dualities I mentioned in the introduction so we started off with a smeared collucicline monopole taking the potential function to be independent of what is a number of object a number of directions we could then compactify we could take those directions to be periodic and then we could T dualise and we end up with a number of Schwartz 5 brain smeared on T3 we could then dualise this to S dualise this to a D5 brain and then T dualise to get a D8 brain wrapped on this same T3 and this same chain of dualities will take the nil fold to a T3 with flux fibered over a line and then we end up here with a D8 brain which is this domain wall in 9 plus 1 dimensions the D8 brains to exist in a string background we need to introduce orientafold planes in particular we need the consistent string background is the type 1 prime string I've mentioned which has an interval with orientafold planes at the two ends and 16 D8 brains moving between that and in this case the singularities which occur at the domain walls reflect the presence of physical objects which we know occur in string theory the D8 brains and when we dualise to get consistent backgrounds for the nil fold the T fold and the T3 with H flux there should be we should be getting duals of the D brains and the orientafold planes and there's questions about how these singularities are resolved in particular by the time we get to the Klutz-Klein monopole there shouldn't be physical physical objects represented by the singularities in the way that the D brains were but instead we should be getting smooth geometries so understanding how to resolve all this will tell us something about the kind of smooth geometries which should be dual to these various objects so a bit more precision about the type 1 prime theory we have 16 D8 brains each of charge plus one in my units they can be arranged so there's ni a set of points tau i so the total number is 16 and we have two orientafold planes of each of charge minus 8 at the two endpoints tau equals 0 and pi and then the metric away the supergravity metric corresponding to this configuration is the kind we've been talking about it's this form so it has a nine dimensional Minkowski space and then five it over this line with parameter tau and if we have a number of if we have a number of the D brains coincident with the orientafold plane if we have n minus coincident with the orientafold plane here we get charge minus 8 plus n minus and minus 8 plus n plus here and the total number of brains in the middle will then be given by the sum of b minus and b plus which will be less than or equal to 16 so now we look at the chain of dualities we've talked about we can first of all think about dualising the supergravity solution wrapping on T3 to get smeared Gibbons Hawking or Nova Schwartz V brain that we've talked about and the 16 sources will then become Glut's Clive Monopole or Nova Schwartz V brain sources smeared over this T3 and the issue is understanding how these singularities should be resolved and at the ends of the interval there should be duals of the 08 planes which again should be somehow geometric and so the chain of dualities we've talked about is starting off with the D8 brain going T dualising to a D5 brain S dualising to a Nova Schwartz V brain T dualising to a Glut's Clive Monopole and then maybe dualising further to one of these exotic brains which I mentioned and so the chain of dualities on the orienta folds has been understood for the first few steps the 08 plane goes to a 05 plane the S dual of this has been referred to as an ON plane in the literature but then one of the questions is what happens if we try and look at the further dualities what do we get here what are the in particular there should be something geometric arising here and the analog of the fact that we have a geometry which is dual to these brains arising here so looking at the full string theory we start off with the type 1 prime string theory on which the type 1 on a circle cross 9 dimensional minkowski space which is T dual to the type 1 primed on this orbit fold S1 mod Z2 cross 9 dimensional minkowski space and then we can compactify both sides on the three torres and do the chain of dualities we've talked about so the type 1 prime can be thought of as the 2B string theory identified under the operation omega which reflect which is orientation reversing on the worldsheet and then we get the 2A theory the type 1 prime is the 2A theory modded out by reflection in the 9th coordinate together with which is the circle coordinate together with this orient orienta fold T dualising we get the 2B modded out by the Z2 which involves reflection in four directions and an orienta fold the astualising we place this by minus 1 to the left handed firmy number and then T dualising again we get a geometric orbit fold of the 2A on T4 modded out by this geometric Z2 which is an orbit fold limit of the K3 and so we understand as so this is the understanding of the duality between the type 1 on T4 and 2A on K3 at this point in modulised space and so we have the corresponding transformations of the various brains that we've talked about it the brains in the various series starting off with the 9 brains in the type 1 theory and the brains become gravitational solitons and one of the issues is what happens to the orienta fold planes in this limit so dualising the super gravity solution with D8 brains to one with Klutz-Klein monopoles we end up with a space which is the nil fold fibre over a line with smeared Klutz-Klein monopoles and at the ends of the line interval we get some sort of geometric dual of the orienta fold planes and this is the same set of dualities which we get from dualising the type 1 prime string which takes it down to 2A on K3 so this is an argument which predicts there should be a region of the K3 modulised space where the K3 looks like this picture I've drew here where we have a long neck regions which are the nil fold and we have it over a line and there are jumps in the degree of the nil fold the vatntige M characterising the bundle and at these jumps there will be regions which have Klutz-Klein monopoles and at the ends there will be geometries at the ends of the interval which look like the duals of the 08 planes and this prediction was very safe to make because just such a limit of K3 which looks just like this was constructed in 2018 by Heinz, Sun, Wieikowski and Zhang they constructed precisely a family of K3 metrics dependent on the limit gt such that in the limit t equals 0 it collapses to a line interval so it's constructing a geometry which has got a long neck region for small tiles so we're looking at a particular region of the moduli space of K3 near a boundary and the neck is split into segments each of which is a nil fold fibered over a line and the nil fold as I mentioned is the circle bundle over tile 2 characterised by its degree or churn number and different values of this integer will arise at different segments with insertion and the gravitational instantons or occludsclined monopoles at the plate of the junctions and the ends are cut by spaces which were constructed by Tian and Yao which I'll be saying more about in a moment but these are complete non-compact hypercaler manifolds which precisely have the right asymptotics to glue on to here in other words the asymptotic region is a nil fold fibered over a line precisely this kind of geometry so here's a picture of what this looks like from their paper here there's a case where we just have one bubble in the middle where we have all the occludsclined monopoles at these points here we have smooth caps at the end and the metric is in fact constructed using each of the ingredients is a smooth hypercaler manifold I've talked about the nil fold fibered over a line the Tian Yao spaces are precise hypercaler manifolds and we'll be talking a bit more about the metrics the occludsclined monopole metrics here and the hard part of the mathematical analysis was showing that you could glue these together in such a way as to construct a smooth compact hypercaler metric in other words a K3 metric so the first approximation to their metric is precisely the one which I've been constructing from these supergravity solutions away from these domain walls who have this kind of geometry and the way they resolve these the singularities is that the at the end point the single sided domain walls at the end we resolve with the Tian Yao spaces and at the the jumps points the tau i it's resolved using aguri buffer construction for hypercaler for occludsclined monopoles so the idea here is if you look at the Gibbons Hawking metric replaced by r cross t2 the first approximation would be to smear over t2 but better you could take start on r3 take a periodic array of sources in a 2 plane and construct a sum of potentials which is singular but it could be regularised to give a good harmonic function and then when you've got a periodic array you can periodically identify to get a single source solution on r cross t2 and near the source it's non-singular and looks like a tau of nut construction and this all works fine except one of the tricky points is that because of the regularised sum the solutions are only regular on a finite interval in r if you try and extend it beyond that then the potential v which we talked about becomes negative and the signature flips so resolve the Gibbons Hawking metric by an aguri buffer metric on r cross t2 with a monopole charge n and so these bumps in the middle will be precisely these aguri buffer type solutions near the sources it looks like an n-center multi tau of nut solution and if we have coincident ones we get a normal fold singularity and in this way we get a consistent hypercalometric on some interval in this region here and far enough and so each of these kinds of metrics only works for a finite domain and the hard part is showing that these can be glued together so let me say a little bit about the Tianiel spaces that arise at the endpoints these are complete non-singular, non-compact hypercalous spaces, asymptotic to a nil fold fibre over a line and these are constructed from so-called del petso surfaces by subtracting, by cutting out what's essentially a two-tor it's more precisely a smooth anti-canonical divisor and on this space here one can Tianiel had an existence proof of the existence of metrics of this kind so just like the Calabiol story it's an existence proof rather than a construction the del petso surfaces are complex algebraic surfaces characterised by their degree which can run from one to nine for degree nine it's Cp2 for degree B it's given by blowing up nine minus B points in Cp2 and there's a second one of degree 8 which is Cp1 cross Cp1 and this gives a complete classification of the del petso surfaces and hence of these Tianiel spaces so the Tianiel space is constructed from the del petso surface of degree B and it's asymptotic to precisely to a nil fold where the degree of the del petso becomes the degree of the nil fold and we can extend this to degree zero by taking the case where M is what's referred to as a rational elliptic surface and then in that case the nil fold just becomes a three torus it becomes untwisted and we get the cylindrical region the cylindrical region will just be R cross T3 and we get asymptotically we get a cylinder T3 cross R in that way so we take so we resolve the singularities here with a grwy buffer matrix at these domain walls and at the endpoints we introduce Tianiel spaces the degree of the Tianiel spaces goes up to nine because of the classification of the of the del petso surfaces so the charges of the insertions of the Clutes Clive monopoles have charges up to 18 this almost agrees entirely with the type 1 prime picture but it said but whereas the type 1 primed we got there were 16 D8 brains here we get 18 instead of 16 so we're finding this almost precise agreement from what we'd have expected from duality apart from the fact that we get 18 instead of 16 here and the resolution is as I mentioned in the introduction is that the idea that you have 16 D8 brains is correct at weak coupling for the perturbative type 1 prime theory but at strong coupling it turns out that an 08 plane can emit 1 D8 brain to leave a new kind of plane of charge minus 9 the so-called 08 star plane so then at strong coupling this would give a picture of the type 1 prime where we had 08 star planes at either end and 18 D8 brains on the interval or we could have some of the brains coinciding with the 08 star planes and then we get exactly the same equations as we get for the degenerate K3 and both cases have 18 sources and this allows for these 18 sources are needed to allow for some of the gauge symmetries which we know for example from the heterotic dual should arise here for example the possibility of SU-18 gauge symmetry so if we look at matching the moduli spaces now we see that the type 1 prime moduli space is given by this coset which has got 16 D8 brain positions the value of the dilaton the length of the circle or the interval and after dualising this gives an embedding of this type 1 prime moduli space in the moduli space of K3 which is of course a much bigger metric moduli space and in particular we can understand the moduli spaces in this way so for example if we look at the oriental fold of 2B T4 mod Z2 which arises as one of the duals we to understand how T4 mod Z2 looks like a long interval or in other words what this K3 looks like in the as a dual of how this moduli space is embedded in the moduli space of the K3 we can parameterise T4 mod Z2 we can regard this as T3 cross an interval where at the ends of the interval we identify the three torus at the ends by to pin a Z2 so we have a long neck which looks like if we take the interval to be long we have a long neck which looks like T3 cross I cross the interval but the ends pinch off to get this restriction and then the moduli comes from the positions of brains on this picture and if we then dualise one further we get the picture of the K3 with a long neck cross the dual of this T3 which becomes the nil fold and we get the moduli from the positions of the Cluth's Clive monopoles so we see that the duality between the heterotic or type 1 on T4 and 2A on K3 are understood as TNS dualities at the orbital point but not at general points in the K3 moduli space it doesn't extend there because there are no isometries however matching the moduli spaces of the two theories means that we can understand how what the corresponding duals are and the duality at one point in the moduli space formally leads to a duality at all points and in particular we can translate moving in the type 1 prime moduli space into moving in the 2A moduli space in this way how much time do I have 3 minutes ok so I'll say a little bit about the non geometric generalisations there's a generalisation where we replace the three dimensional nil fold with a higher dimensional nil manifold which instead of being a circle bundle over a two torus becomes a TN bundle over a T, a torus bundle over a torus and rather remarkably it was shown from supergravity arguments by Gibbons et al that if you take certain bundles of this kind fibre them over a line just like we fibre the nil fold over a line you can get not a not a hypercalometric but a special halonomy metric on this space and one of the work and with my student we looked at this structure and looked at what happens under duels of this and we found that the special halonomy metrics dualised instead of two nürbur schwarz five brains or Klutz-Klein monopoles instead to intersecting nürbur schwarz five brains or the corresponding intersections of Klutz-Klein monopoles solutions so the kinds of bundles we get which arise in this way the simplest ones are these which give rise to these kinds of halonomies including G2 and spin seven as well as Calabi Al so looking at the chains of dualities we took what we've been talking about was based on the duality between the Tau of Nutt or the Klutz-Klein monopole and the nürbur schwarz five brain and the corresponding multi source generalisations of these so we showed that with my student we showed that the special halonomy metrics of Gibbons et al T dualised to intersecting five brain solutions with one function and it was then very straight forward to generalise this in such a way that there was a different harmonic function for each of the intersecting five brains and then this in turn gave rise to a special generalisation of these special halonomy metrics with several functions all of which were highly smeared over many directions but we also know from the intersecting five brain intersecting brain story that there's we can do a little better and go explicitly to semi local solutions and this gives rise using that same generalisation here gives some semi local construction of special halonomy metrics and the full story will be understanding the relationship between fully localised intersections here which would then give rise to special halonomy metrics here and one of the and one of the and what we're still working on is trying to understand this duality further and in particular there's a possibility that just like we constructed a complete compact hypercalometric for a general limit of K3 that there could be there's a prospect of compact special halonomy spaces arising in this way from these intersecting brain solutions in a way which again would be again it's something which is formally seems to be predicted by these dualities and it would be very interesting to understand in detail whether this works and whether this could be used explicitly and then one of the nice features about this picture is that we have a very explicit model geometries for these different regions of this space and because the individual regions apart from the endpoints we understand they have isometries we can explicitly do T dualities and other dualities for the model geometries which are glued together in this full metric and so we can go some way to understanding the effects of string dualities at least to this in some detail at least at this region of the modular space and in each case it gives a picture which is consistent with what we'd expect from our understanding of string theory it's also intriguing that dolpetsos are arising here in an interesting way and it's also intriguing that dolpetsos have been invoked as having structures which are dual to many which seem to reflect many other structures which arise in string theory in particular relations to u duality but the chairman's standing up so I'll just quickly come to the conclusions I've talked about the nil fold in its dual giving local solutions of string theory fibreing over type 1 fibreing over an interval dualising gives full strength, dualising in type 1 prime shows how to incorporate these into full string theory solutions and it gives an interesting realisation of K3 in a particular degenerate limit as a nil fold fibreed over an interval with Knut's Klein monopole insertions and Tianyau end caps and it's a very interesting approximate geometry for K3 that allows explicit duality transformations in an interesting way and we see that some of the non-perturbative effects of oriental folds like the O8 star planes now after these S dualities become perturbative aspects of the dual geometry in type 2a and we can understand that the O8 in the type 2a picture and the O8 star and the normal oriental fold planes are very much on the same footing and it's and it's led to a lot of some interesting ongoing work involving generalisation to special helonomy manifolds and to non-geometric duals to all of this story but that's for the future and I'll stop here and once again wish so the this new folds T4-1-D2 for special helonomy is basically replaced by joises or before construction or it's rather this is similar to covagia so these don't seem to be related to the joise construction but it seems to be giving other to other constructions which are looking much more like duals of intersecting brains and because there are two types of constructions I think now one is joises and another like eight years later by covagia I'm sorry if I'm not pronouncing no you're probably pronouncing it better than I am but I would but I think it would be very interesting to understand that kind of question more fully so the short answer is I don't know but I think that's that kind so I walk to be I mean looking for the adelogs of the orbifold point T4-1-D2 in this story is a very natural place to try and understand these dualities and then Z2 square was for joise right and it was almost like T4-1-D2, Fibre, Dolmer and other T3 right singularities were always cummer so my impression although I'm not an expert on this I haven't looked at this in detail is that these seem to be corresponding to different orbifold points which is one of the things which makes it quite interesting Any other questions? One of the early slides you had this convention this blowing up of the metric and the linear dilaton so what does it mean that the linear dilaton blows up? So for example if you looked at the type 1 prime theory without putting on these if you just looked at a single a couple of D8 brains without the orientafold planes then you'd find that the dilaton would be increasing would be going off to infinity somewhere and that would mean that in those regions you wouldn't have a perturbative string theory and you wouldn't really know how to deal with that So it's no longer linear then? So if you've got a non constant dilaton you can always choose a coordinate so the dependence is linear so it depends which coordinates it's linear in one coordinate system but it means that the perturbative picture is not it's not valid so you might look for a hajava wittent type construction or some other kind of construction to try and understand it So you just want to pursue regarding when instead of having say two, three, five of a having the flags for something or having new poles when you have this generalisation of multiple new poles which is a T-dualised to the input second NS5 which in turn generate the whole KK monopoles sector do you have u-dualities in this case or is it possible to construct a transformation which would lead to what would be u-duual of such construct when you have a collection of n-folds So that's a good question so all of this story has various u-duality generalisation so I talked about the case where we had where we used combinations of S and T-dualities which could be thought of as giving a u-duality but we could certainly there's questions about stories about more general u-duality transformations one of the problems is that a lot of these other duels there are things where we don't understand the perturbative string theory formulation so it's harder to make progress there but I think that's a very good question for the future sorry I missed something there are no more questions