 Okay, so this I would thank you for inviting me and actually it is my I graduated here 90 years and 90 not 90 years ago but in the 90s and it's fun to be back. So I mean a regular tiling is what you also see outside actually on this doors is a relatively stupid design but it's like it's like that. I mean this is there are four there are three of those and this is one of them maybe I make it a little bit I'm going to push it up here and so that has you see the vertices have valence three and then there is of course the even book mobile in tiling where the vertices have a valence four and then there is a tiling which where the vertices have a valence six hat to make it impressive but one thing is of course I mean here's for instance a vertical valence six and then you see but this one is actually dual to the first one so here so this is the hexagon tiling so that is for the x so x quad that triangular and now I wanted to start my talk with a geometric construction m g these are certain graph manifolds and they will be color b o l folds actually so it's a bit similar and what I uh similar to the talk that was given on where for instance we had the series of banana graph but here I have a little bit more more different I mean different graphs and different variety so what I do is I first associate to every vertex a p1 in all of these graphs and I just number them and of course secretly this has a second corner that's a x i and u i and then I take I take a certain contour which is a convex contour and and sort of this should be a generic so for instance I can go like this or I can go like this and then I pick up a couple of these p1s and this I call the graph so so as I said I will not focus too much on this lattice because this lattice is and we locally do to the hexagonal lattice so um so we have this curve which cuts out a graph g and that's why this manifold gets labeled by this graph and then I want to uh I want to uh take certain propagators so I mean graph has propagated and this propagates so we have internal propagators which are just x i minus x j to some weight and these are internal propagators they'll give to the x and then I also um maybe I forgot here that external legs which of course appear by this cutting will be labeled by a i in some labeling so I have also external legs they are x i minus a i w i j so these are my propagators and um it is conveniently negative sign and the weights are also and now I want to have uh in general a property I mean this is not the most general situation but so generally um we want we want um for each vertex vertex i uh we want the property that if I let's say if I take this weights and sum over all the until new for this and maybe multiply this two then should be two d i p and in a sense I take a very symmetric choice so I take the choice that uh w is just equal for all and is equal to itself so that would be um two thirds for a new equal three and one half for a new equal four and one third for new equal six and you see this is the the way so this is uh is a conformality condition as we will see in a second and you see this is sort of a resonant choice so you can say I make the simplest resonant choice and that uh that is this one and the resonant comes from the fact that these are corresponding at the end of the day to certain hyper geometric system and this is a particular symmetric and resonant draw of course you see that in this uh lattice the angles play a role so basically uh as you see I mean if if I have here I can so to say I put the weight simply here and then but I I didn't I mean I can sort of say and then take an orientation but I will put the these uh weight on the propagators as it's appropriate okay so then um is so what is uh what is mg so um so I defined um my mg and as I said I don't have to define it for for all for the hexagonal lattice because of this duality which I explain a little bit more but what I do is I take y to the d maybe over d I take a w and then I take a polynomial in specifically speaking in x u and this a's this outer things and I um take d equal three for new equals three and d equal two for new equal four and and this I do because I want in a sense I want to recover the Feynman amplitude and I define this mg as being uh this d sorry this is now I just have to specify this one so this is the mg but the I haven't said what the p is so the p is basically uh that I take the product of xi minus xj over these internal lines and the product of xi minus ai over the over the internal lines over the the external lines and now you see why I'm doing that what is lattice is because we have I mean technically speaking we have a base which is p1 to the l uh because I associated to all of them uh a base and then of course we have that uh the canonical class of the base is something like the product of two times of the hyperplane class of all this of all this p1s and if I do this now then I have that the branch locus I mean I have then that d minus one over d uh times uh kb um I mean I I have I need the kalabiao condition and that will be that happens to be uh a new times h i and now you think that I see this works because um because um d I mean if if the if the valence is uh is if there is a four valence then every x has to appear four times of every pure and therefore you get four times the hyperplane class and the branch locus is okay and if you have the valence three then every x1 has to appear three times but it does because the valence is three and so I get uh so then therefore um the mg is the kalabiao that is and of course the reason that I'm doing that is that I'm uh sort of uh want to uh I want my my period integrals to be related to uh to the climate integrals for this graph so um so you can um you can uh say I mean let me first give a couple of the examples so um so let's say we have an example would be you look at this and here I have an important mission so these a i they are all in the uh one point compactification of c so they are in c and there is actually a psa to z action is uh p to z action and what I should also have said that I will make a couple of comments when this d is general but here I took actually as you can see from this thing I took actually d so it's a d it's a two-dimensional conformal field three but it's actually uh not um it's not um unitary and it's a very interesting two-dimensional field three but you see because of this sf2z action I could not I can now put my points like that so what I get here is um is uh simply a curve um which is so this is a uh genus I mean it's a it's a genus one belly curve so this is uh it's branched at three points and it it turns out to be at the point of complex multiplication and then this is important for the rest because in a sense the higher dimensional Calabi house for this uh three diagonal lattice will be also at complex multiplication in a sense um and then well the this is for the tree balance lattice and then uh I just want to show you that you know this guys uh so we can take one infinity z now I have of course a four point so I take the cross ratio and that is just in the jump so in this case uh remember that I take this uh covering x minus one and then x minus z so this is the the one of the curve so of course as you see uh from this branching locals this guy becomes relatively singular if I go to higher so these are these are now these are still smooth but mg is singular for l greater than one and but you can quite trivially you can smooth them and you will find actually a pair of reflexive polyadera so we can smooth them and smooth them and then we get uh from the Newton polytope from this smooth sky we get uh find that is actually a reflexive this is a pair of reflexive polyadera and that means that um that this uh becomes now a smooth a smooth um l fold but I don't want to uh I mean I can sort of say give now that the first you will see has of course Euler number 24 it's a k3 and the next one have high Euler numbers I don't want to be fooled but this is easily calculated so that's that's that's this so um so now I come a little bit closer to why I'm doing it and why what is the physical interpretation of all of this so um I mean of course I'm doing it with a very similar intentions than in all calculations of Feynman graphs by Calabiao periods I want to see that the Calabiao period has to do something with the with the Feynman graph and yeah no no that's an important point of course these these model are very restricted and so this is the guy that why this is singular you're basically because your branch points has some I will say more about this so we'll see actually uh how you can live with a subfamily and resolve the thing with blow up so that's possible but there's a very interesting twist in this in this construction so um so anyway let's define for each p1 or this p1i I will define the measure and then the so to say the reason is we want uh Calabiao periods uh to be equivalent to Feynman integral so and and that works because if you let's say take a period over this graph mg then I take the normal cryptus residuum form so I take some cycle in h l m g maybe z we will talk about this later let's just keep it open for the second and then I take um I take uh this measure so I get this product of these measures um and then I take one cycle around omega so this this this goes this is gamma goes it circles with omega equal zero and then I get this one and if I now do the integral I get of course uh still this one and I get here a y d d minus one and then if you solve this equation what y is then you get actually uh then you get uh the product over all the p1s and you get uh just this uh this propagators yeah I don't have to write the numerator we have just this sum of all the propagators the ij the external and also and you can also remind me of this and that I know yeah okay anyway we know it right now uh otherwise actually okay so so these are the internal ones that they and and as I of course this integral precisely makes sense because it is a Calabria because this weights and the fact that this lattice has the right um valence uh is responsible for the fact that this um this is that this one is a nice Calabria form and so this integral makes sense as a Calabria. So let me tell you something which is different than uh then uh uh then in the previous talk on primer graph so these are these are uh uh integrals or these are prime and diagrams in momentum and position space in position space so in other words when I have uh let's say uh this configuration which would be a Calabi R3 fold because it has three points so it's a three-dimensional uh thing then what it really means in position space is that I have to take the dual graph and that means that this dual graph now um has indeed is a train track graph and then you see that it has as many loops as the guy has positioned and in a sense it's just this idea okay I I fix all this external positions AI and I integrate over the internal uh possibilities of momentum of positions and that is my integral in the position so then uh so that's this uh that the loop is indeed the number of p1s I mean the the measure is always the same so the measure is just the product over these these guys for each p1 well I mean here okay yeah this is you're right I mean here I should uh should actually put ui yeah that I can take the residuum when I'm in a circle in w for zero or dw so I'm in circling uh w equals zero in the w thing I don't have to specify no no I mean this is sort of say the standard differential uh residuum differential of Griffith I mean and that uh that one uh yields this one so for the elliptic curve you could do exactly the same thing and you would get uh the differential square root of the the branch of the when you try to merge on this one but it's it's just it's just here you would get from the same thing you would get the x over y and that is y is the square root of these guys and that's how you get this property no no you have to uh this is the pass it goes to generically around the locus y at w to zero okay so here I have no choice but here I have a big choice I can put all kinds of of aspect with you so that's that's also important and now it comes to the second part of the talk so this is a little bit about the physical origin origin of this problem and this started with so this is a suggestion so it starts with n equal four super young notes young notes in d dimensions and people have discovered this has a like this has a young instrument this has a young young individual system individual system and so of course now why I do this so general it has so to say fixed propagator weights it has it has a fixed dimension it's best in four dimensions and so on but but the early people I mean there's a lot of literature on that but the early people were good good good on and kaza cough sorry I wrote this you go down and kaza cough suggested in 2000 I mean in 15 to study these young in symmetries uh young in symmetries more general more general in particular in all dimension but if you so this is a deformation as deformation but it's still true that your Lagrangian I mean they call this the fishnet theory because you have this net section and it was actually invented by and so this so let's say we have some some variables in this n equal four young mill series we have these are matrix variables and then your Lagrangian has to be a trace so it's I should also say I think the gauge is that and color and then you have you have the trace and you have the new x the new x bar and then you have the new z that bar and then you have this in this interaction term x z x bar that and and they uh so to say they discovered that in any dimension this is a corresponding to the twirling graph because you have this interaction but they discovered that in any dimension this has these young in symmetry and and so they proposed it to be an interesting problem to to study it in general and it indeed is quite quite nice x and z sorry x and z you're just too Hamish made Hamish well this is just how the series of build up I mean this is basically um it's basically this part of the of the amplitude of the super young mill series which has this young in symmetry and and you can yeah yeah right so um let me yes yeah if the dimension I mean here here I didn't fix the dimension but the claim is that in all I mean the most interesting dimension in which this problem was studied was six four two and I studied mostly in two dimensions but it turns out that that some properties of these and I come to this now some properties of these things are interesting in all dimensions and the most challenging case is actually the sixth dimension so and beyond sixth dimension there is no the series actually yeah it's still as a young in symmetry and anyway so this is the setup and now I should also say that the there isn't I mean there's the series of course a little bit special but there is an amplitude or at least something which is monodromy invariant invariant and and that is simply so I call this a a graph and this was the period of the graph and that is simply defined by p d mu veg p mu bar p from one to l and then and then again the propagators but now the propagators raised to the absolute power so this is a real quantity and that is is an interesting real real quantity and it's particular is um is single value so it has no monochrome that's what is expected cute and if you do this indeed in in four dimensions so again so I do it in four dimensions then what you get is that this a uh let's say of the cross indeed before is something like so now I introduce certain real parameters don't worry about it too much and then I need some factors and here I have essentially real parameters in the propagator so here sigma i minus ai but these are not the same ai this is a real parameter and this one is famously uh is uh the block we can a deloc so that has some one two and three four and then it's the block we can a deloc of that and this z is is is so the set is essentially of course and is d d is is block deep in a dialogue and so so that's to do to remind you that it's something like the imaginary part of v2 of set plus log of that log of one so so this is this is also the this is actually the volume of tetrahedron with respect to the hyperbolic metric and and so of course in a sense the one of the punch lines of this new developments in in um um amplitudes is that maybe these amplitudes are all volume uh so um now um I want to go to the next session which is is the conch so to say I bring together the physics and the geometry now in a in a way in a way so this third session is a combining geometry and I'm back in talking mainly about the two so um so we have um in this work uh studied this thing uh quite quite intensely and uh we first of all you will worry what is so to say the analog of the uh of the of the quantum of this block we can a dialogue and the answer is um this is my claim one in the fact it is pretty obvious to each to each uh as a new equal three new equal four graph and then by extension to something which I don't make very clear but uh new equal six um we can we find we find that the analog with my dialogue is basically a p dagger sigma p with some factor of i so the amplitude so the amplitude is is that let me write this a little bit okay but though this is the amplitude of the graph so it's actually some uh rational function that is uh that you have to pull out and the z's are generated at cross ratios and then um it is p dagger sigma p and in other words is something that you a rational function times e to the minus k so essentially you see this is like here that also a cross ratio but here this is actually also real quantity so essentially what was the quantum dialogue becomes the caler potential of this e to the caler potential of the WL which of course with some appropriate factor which is i to the um to the l is actually um this is a rational function and a rational function comes when you take this cross ratio so because i don't talk about the ace but rather about cross ratios of the i mean there are many cross okay so then um i mean of course since this is uh related to reflexive polyadron you can also say this is the quantum volume of the mirror so that that's what it is and i mean to make the statement very precise you have to pull out one of these coordinates i mean you have to pull out this is all homogeneous uh in in one of these coordinates you have to pull it out but then up to this it's the peak care is the quantum volume of the mirror so um i have uh almost 20 minutes because of hope yeah yeah yeah almost ah sorry this is not an hour you're right well i don't i don't want to i mean basically the proof of this statement is simply you just follow this construction of this kalabiao and then that's it so uh so there's not not much that to prove and um and then i want to make a remark about the um new equal six case so the new equal six case is that if you will see a y with this weights um then you can sort of replace it maybe i throw it so you can replace it by uh delta and this delta has then the weights one third one third one third and if you do that for instance let's let's do it for for uh for these so this one is a is a k3 um so this uh so i can now go go ahead and put this uh delta y rule here and then um then i erase of course this inner point and then i get a graph which looks like which looks uh like this and so i get go to this graph and now we have again this uh conformality because uh this all this thing adds up to two but you see what's interesting is that is actually now one-dimensional thing it's actually uh elliptic it's not an elliptic it's a space the general elliptic of the general walker but it has and that is my one of the punch lines is actually the motive of the kalabiao when you pick a appropriate real structure on the case so let me um uh go to the second claim and the second claim is is uh right far reaching i mean you know that um that the that if you if you have um the conformal group in two dimension it's basically uh the uh conformal group is a signature in two dimensions higher but this sort of say factorizes and no i mean this is here now still a family of curve because uh because you have you can it's it's basically a one one and and if you would have started with the kalabiao three-fold which uh would be uh this case and did the uh the delta y rule two times now you have two delta then you actually end up with a little different curve namely you have just uh you have just this curve but it's still one-dimensional so this is again a differently generated genus walk but yeah so one parameter well i mean i should uh no no no i should uh this is too quick so here i should set i is one parameter if i if i identify this post to infinity and that is actually what the people call pitch additional so then they they say okay this one was a zero this one was uh one and this one is that and if it's a one parameter model it's a one parameter um okay so now comes a really really beautiful claim namely that um that this uh splits in the yang yin of sl two r times the yang yin of sl two r bar and now you pick uh this piece here and of course we have in the for the kalabiao we have a flat gauss manin connection connection and and we have we have uh also uh this is equivalent to a demodel that degenerate that uh annihilates this period or this element annihilates this period and that one is exactly equivalent to the uh to the differential equation that is generated by this yang yin symmetry so that is um the yang yin symmetry of sl uh two r and this i'm not quite correct because that is different so times and this times a little bit tricky because that doesn't commute with each other times the automorphisms of the graph and you see the only difference between the three valent graph and the four valent graph is in this piece but you can actually I don't know whether you know but uh if you uh this yang yin symmetry is basically just uh is just generated by the momentum but now the momentum is for for each i so it's i the ij mu so this is for each of the external vertices that the yang and symmetry act and then there is of course uh k mu the special Lagrangian transformation the rotations but they are also all specialized to all external points in the differential operator and these are differential operators and these differential operators are equivalent to the demodel that is for this kalabiao so then we have an li mu nu you know what it is and we have also a differential operation uh because the A i mu is down so here's the A i and there's a mu so these things have now in general I mean this is for any space time dimension so here can mu can run from zero to b minus one or if it's a three and one to b so that is uh that is uh quite interesting that to the that is so to say the family of kalabiao's whose picker-fuchs differential idea is exactly generated by these yang yin symmetry so it is a very nice relation between between a kalabiao spaces and integral systems and also a relation between what wortland calls the world of that comes from large endualities like this uh these integral systems for the yang yin theory and and kalabiao geometries but here's is a super concrete just telling you that all these picker-fuchs differential equations are given by that and I can give you examples that you will know that's also nice so so basically if you if you take if you take this fishnet graph and you take the um the kalabiao and you I mean you can take as many things that you want but you just identify all these edges here and this edge is here to zero one infinity z and then you get a picker-fuchs operator which is um d l plus one minus z theta one plus z to the l plus one so this is the picker-fuchs operator where this is the logarithmic derivative uh in this set and you know this picker-fuchs operator is one of the hyper geometric picker-fuchs operators that occurs from the for the for for then I should skip this point but um I want to say a little more I'm I this is the front of volume I already said so but but let me give you a flavor of how to deconstruct kalabiao motives with this implement kalabiao motives so first I think um shak asked this question namely if I have um these graphs which is a k3 um and I have here six parameters but secretly I have only um so I I make a I make now a plot of the branch locals with this guy so here I I say a one a two is picked and here it's uh this is in the uh so remember this where you are now I have this branch locals and of course uh when x is one of those guys then I have a factor and if these two factors meet then I get a note so I get this picture and whenever these factors meet I get a y squared equal x squared locally but I also get it because there is a factor x minus y I also get it from the diagonal and these are all a one singularities and therefore you see that the picard group of this guy is uh two from the two p ones and and then uh nine from here and six from here so it's 15 so it's 17 but on the other end this guy has uh six parameters minus three of the pss fourth solution so you split actually the uh the middle core more I mean the h1 h2 two you split it in 17 and three and that suggests that this is still the generic color in case three that you power drive so first of all you have to resolve it this one parameter family makes sense as a smooth case read secondly these there's no a secret symmetry in this problem but this will change if you take this one so if you take this one then of course here we have we have only two two lines because this can be a one and a two for x one so it can be a one a two and then it can be a three and a four for x two and still you have to diagonal so you have this line and so you get here these points but now these points are cast because you have the three fold covering so you have y three equal x square and therefore if you make the same counting you get two plus eight times two because you have a two singularity so you blow up two for p1 so then the p-card group of this guy is 18 and you see that you have here four parameters you have only you have pss to z you have only one parameter so basically this is still not the generic calabi out case three subfamily but it's rather restricted and what we claim and what we can see more generally so we have here this one and the reason that this one becomes a degenerate elliptical uh and generate um higher genus reman surface is because you have um you have an galore action from the covering group of the of the uh three covering so you have some three action on the covering and and therefore what happens is that you split the hodge chronology but you don't split it over z so in the in the case three you have to first pick a real structure and then you see if you look in the book of boycotts and heckman the famous book on differential equation you see that for these cases they already observed that you have that the monodromy is in q or in z even but you have to extend q by one over three and that's exactly what happened and it happens because you have to sort of pick the invariant things under a set three covering action and so in this way for instance if you ever wondered why uh this case um this case um the analog of this case here with all threes were never appeared in duke's list it is actually a calabiano motive of a seven fold which is a three fold covering and once you pick a the right real complex structure it actually does appear and so in this sense we can solve all this series and in particular we have also always despite the fact that the monodromy is not interval we have the monodromy invariant expression and and maybe with this I should let me go to deny i'm sorry for overtaking thank you very much we have time for questions so and then you will make it if you look at our work and essentially we can now how we can make the figure out I claim that we claim this for one of the little things that we are thinking but it's only one from the pathogenesis I mean all these copyrights now well I mean they are not all most of them are not only one that's well I mean they're not all right so they won't be for this this one but then you know and I can go all the way I mean this the important thing is but of course in this construction you will know how to realize all the another thing of course is it's really the origin is really that the that the galorec room actualizes the fact that and what I can make the monotony I mean you see the and then and if I now and then then I get there will be set up and there will be a cloud, but at the end of the day. And yeah, we have. I mean, I believe we can. Any more questions. Let's thank Albrecht again.