Diandian Wang - Open and closed triangulations of 3d gravity

Thank you for the introduction and also I'd like to thank the organizers for this organizing this wonderful workshop. Uh so today I'll tell you uh a little bit about this uh recent work based on this paper from a few months ago with Daniel Jeffers. Uh there will be uh some other related work that will be mentioned uh the reference will be uh in in the slides. Um okay so uh to give uh some motivation uh to this work uh I guess the big picture here is that we try to understand the CTF ensemble due to uh 3D gravity which as already explained in the previous talk uh 3D gravity uh we expect the deal to have a continuous spectrum so some of results for compact uh group doesn't um doesn't work immediately. Uh so there are some difficulties. Uh but here in in this in this talk um we'll try to introduce the idea of triangulation uh which is a well-known mathematical um um method uh for studying 3D geometry. The the conceptual uh benefit of doing that here is in some sense uh to to make the continuum gravity into a discrete problem. uh in some sense. So we only have to focus on the local building blocks and the hope is that by working with these local building blocks uh certain problems can be uh solved uh more locally in some sense which makes it easier. And another uh reason is to uh focus on um well another benefit of the framework I will that I will introduce is to uh focus on the uh statistical data without worrying about the conformal blocks which are themselves um difficult um or complicated mathematical objects. And um from a slightly more mathematical uh perspective uh there is a a well-known terra visual uh method of theory of uh TQFT that we revisit in the case of gravity. Um it does not um um the TV theory for compact groups does not immediately generalize to uh to a non-compact group. Uh so there are some subtlety that are involved and um then there's also uh some other motivations such as embedding rearchf into a slightly bigger framework and also uh as I'll um explain some examination of a proposal for studying uh the duel of uh uh CTFs with boundary which is known as the adsbft proposal. So these are just a list of um motivations um which I'll try to uh clarify in the talk. So I'll begin by reviewing some uh results in 3D gravity. Um I will I won't have time to introduce the most basics. uh so I'll have to assume uh certain certain things uh but I'll try to clarify and and explain at least stage the most important results uh in the recent years that will uh that we'll need for this talk. So uh one of the uh main uh thing that that's understood in the uh recent years is to um I guess people that people now even in higher dimensions and lower dimensions than 3D we expect that 3D gravity um can be understood as computing average ensemble averaged quantities of uh 2D cffts. So as a uh example uh this is a very simple uh 3D manifold. It looks like a three punctured wormhole three puncture sphere times interval. So that is one of the simplest wormholes. Um you need at least three because uh fewer if you have fewer then it doesn't support such a geometry. So these lines are conical defects and they have to be heavy enough um to for this to exist. And this if you compute a um the you know is literally action it would agree exactly uh as a function of the positions of the three points um with an average quantity um cigar that is obtained as independently using bootstrap methods from CFT. So CJK here is op coefficient. So a uh in a in bootstrap you can work with an ensemble or slightly averaged quantities in a CFT so that this ck squared has a distribution. >> So which >> uh so here uh is assuming only the the conformal symmetry. So if you work with high enough uh energy in in uh CFT you can obtain this as a universal what's known as a universal dynamics. It's similar to like the cardi formula works universally if you only assume modular symmetry. So this is a um this is a um analog >> average of operators in a given CFT over small window or all CF. >> Sorry. Yeah, I was I was being a little uh unclear about that because uh the original work was in a in a single CFT average over some energy window. Uh but uh here uh we are sort of interpret interpretating that as a average over a class of CFTs. uh uh I think there are um I mean it's part of the open question what is exactly d23 gravity so at this point we are just assuming there's not a big difference um so um I guess more generally um when we say theory gravity especially at finite gton we we think of vority cap which uh in practice This it gives you you very precise uh quantization of 3D gravity and in particular at finite uh G Newton or central charge it just gives you a method for computing the ping functions for example these are two wormholes with um four boundaries each of them being a three puncture sphere without var uh you know you have to try to write down the metric on this complicated um manifold and they're like I six uh conical defects in each of them and to compute the action that's very difficult but vis just gives you u almost immediately a way to write it down as a function of you know all the positions and and weights of these conical defects the weight is related to how heavy uh or how big the conical defect angle is um >> for not for nonexpert you say the 3D gravity gravity partition function you assume there's a single 3D gravity part theory what is assumed here >> um I so here I guess um I guess there's some um I would say uh in general I guess there are different formula I would in principle there are different ways to quantize 3D gravity so gravity supposedly gives uh the the correct one in some sense um at least it reproduces the classical answer the one loop etc. and it also has the right symmetry and is heavily constrained. So in certain sense it is the uh decronization of the gravity [clears throat] >> on a fixed hyperbolic >> um uh yes sorry [laughter] on fixed hyperbolic uh manifold uh and that is uh kind of important later on I will try to explain um yes so uh in these two examples because it has four boundaries each being a three punct sphere it would compute some cotic moment of the CJ case. So now we should think of each CJK as like a random variable X and then we don't know what probability of X is but we know you know this average uh X bar X square bar and X to any power bar except that there's more than one X there's X1 2 3 4 each being a CJK um so um there is already some interesting you know mathematical question here which is uh If you are given all the moments you know the the x to the n bars uh can you invert it to obtain the probabil probability on that distribution um the that for for single variable say that is a well well studied math problem there are theorems it will tell you exactly when uh it can be done and when it is unique here um it's a multivariable and Um uh I think the currently there's no uh answer to that. So we don't know yet if there is a um uh probability distribution on these answers. It's related to uh what's known as the positivity constraint uh that are also used in our field uh to constraint for example uh quantum um u quantum matrix models etc. Oh sorry matrix quantum mechanics. So you're basically saying that we don't know if gravity is reflection positive. >> Um I wasn't thinking about reflection positivity but >> positivity you just mentioned. Yeah, >> matrix that's the reflection symmetric reflection symmetric configurations >> I think it's not obvious computation I don't obvious >> um okay so yeah so that was a a side comment but uh is in some sense it is something that can be done. Uh technically it's not that easy because these functions are complicated and there are a lot of terms to to consider. Um okay so in data gravity um which people understood uh um uh this people understand this pass integral quite well uh we usually you know um decompose uh this geometry into smaller pieces u for various reasons for example uh we can study is a geometric recursion uh in particular there is this asmtoic region uh that can be essentially removed so that people can focus on the smaller uh core geometry. Um so uh the reason why we can remove it is essentially because this is a gnome trumpet function that uh essentially after a integral transform uh we can obtain the ping function on the on the core part of the manifold. Um the reason I'm mentioning this is that uh so this is uh from last year by Hartman. Uh 3D gravity is uh very similar. So uh we usually we think about these uh you know 3D manifold relevant for ads the ones with asymmettoic boundaries. Um in fact that is still true. It's just that these asmtotic boundaries have the analog of the trumpets. So why don't we sort of remove them so that we can focus on the the core uh manifold uh where the asmtoic boundaries are replaced by a finite boundary that makes uh the for example the the manifold compact. So you can look at as a volume um it's literally a number whereas if you have asmtotic boundary you have to renormalize the volume you have to consider uh the anomalies u so that is uh makes the things easier and in fact the the technically what's removed is the conformer blocks um so which I was mentioning earlier that uh although the in some cases they're known um they are complicated functions so it's useful to not consider them. >> Will you be focusing on hyperbolic manifolds throughout the talk? >> Um, yes. Yes, I should have said that. Um, I will uh as uh in the in the closing marks I to comment on offsh um so the this core geometry that we will now focus on uh is also known as the fixed um P basis. Uh the reason is that uh also as a result of uh paper from Tom Hartman um um if you look at this geometry look at it volume etc uh is literally given by the visc partitioner function in the fixed p basis where p uh should be thought of as a um um sort of conjugate to the moduli. So usually when we have asmtotic boundaries we work with you know the complex structure or uh some you know um as a function of the the moduli and then we look at its value. Uh but if you write the component blocks you can you can [clears throat] pick a uh particular basis just by convention called the p basis. Uh the the reason why it's called p is that p stands for momentum. So it's like the conjugate to the position basis. Um these are very useful because um they're already considered in in the work of virtual TKFT but previously they were not known to have a geometric meaning. So you know they essentially defined by transforming the uh the Z basis or modular basis to this basis but now uh they they literally have a geometric meaning and you can look at this uh um you can look at the metric you can work out this geometry look at this uh volume it does is indeed given by the pent functions um of course I haven't really explained what what is meant by these finite boundaries uh so precisely More precisely, they are um they're called pleated surfaces. They have this extrinsic curvature ki j equal to zero everywhere except at um these uh marked circles that were labeled uh blue and at these circles uh although it's not visible in this picture they they have a corner. So the 3D geometry has a corner at this at these points. um but the intrinsic uh geometry uh does not see it. So it's a feature it's a distribution um on the extrinsic curvature. Okay. So um these are nice bases and we'll work with them. So one um one thing that that sort of comes out of that uh realization that you can work with this core geometry is that you can now build it uh in the sense of taking a bunch of building blocks and gluing them together. So this is called uh essentially called triangulation as well studied in math. Um the building all the building blocks are given by this list. And so the first one uh is sort of a more uh it's a more well-known one is called the generalized tetrahedra. So it take a you take a tetra tetrahedrin you chop off his uh uh little um you drop off corners at his vertices and then the way you chop it off has the feature that all the um all these triangle faces are orthogonal to the hexagonal faces. And the hexagonal faces um they have a uh they meet at an angle and that angle is uh is is important. It's not uh it's not pi is par two. Um but then there's all these other ones. Uh how are they related? They all essentially they're actually all called generalized hydrohedra. And the the they're related by for example starting with the first one uh you make the top and bottom triangle bigger and bigger. uh at some point they will join and then that's how you get this uh surface and they draw at an angle. So this little uh horizontal line uh is now a corner in the 3D geometry and then you can do this uh by making other uh other uh triangles bigger bigger until they meet another face. So you get all these uh this way. Um so uh so at this point this this is not interesting. is just saying that we can chop it off and then we can build the total the big uh you know the manifold this way. Uh although there is already interesting that it can be done any manifold hyperbolic ones uh can be done by using these building blocks. Um but what is uh more interesting is that uh the volume of each of these building block is all related by a single function. Um that's called the classical 6J symbol. So it is a single function. Um by taking these the six number is six J symbol. It has six numbers. By taking the six numbers to or some of the six numbers to be uh imaginary and the other ones to be real, you obtain the volume of all these uh building blocks. So this is already uh interesting fact. Uh furthermore you can um I was calling this the classical 6J symbol because they are related to the the classical volumes. But then you can take them to be finite. So the six G symbols are quantum uh objects. So you can take the uh sorry finite in B which is related to the central charge C. It's use uh it's called B because the B here is the Louisville B is related to central charge. Um okay so if you take the 6GML to be quantum and just take the same you know way to build this particular manifold which by the way could be very complicated uh because to capture a very complicated topology the way to build it requires a lot of gluing and uh um usually that is otherwise difficult to describe. Um so then um now you have a function that's uh uh that depends on finite B or central charge C and you don't change the way they're glued to each other and then uh you just multiply these um uh these functions together. What you then obtain is something called the uh conformal to visual part function. Um so that is essentially a um uh I think following uh some logic for the in the the way when the tri visual theory was uh was uh first proposed the logic still uh essentially tells you you will get a topological invariant although in the in the case of gravity uh you have to deal with um I guess a convergence issues. I I'll make I'll make a reference to a proof that proved it in restricted cases. Um so um this way of building you know particular uh manifold uh which or is a topological environment um you could ask how is that related to TQFT which is another topological environment. Um the reason why is a topological environment is well it's already called a TQFT. So uh so it it should be and but also uh 3D gravity doesn't have the local degrees of freedom. So uh at the end we we should uh expect a topological variant. So these two topological variance are indeed related and they're related by this um by this by this equation and um so roughly speaking you take cap you square it um um then you do some integral transform where s is a modular crossing kernel and then uh you do it for each internal edges. Um uh the internal edge is uh is the following. When you do the when you glue all these blocks together um some of the edges will be surrounded by by the uh the building blocks and then they they're not visible in the final object. Uh so they the the edge the length of the edge for example is not a it's not a observable it's not a boundary data. So uh is becomes internal and is integrated over in tap sense it's like a state sum uh usually it's a sum but here because the spectrum is continuous it's integral okay so uh that is uh uh that equation is uh proved in also in the paper by Hartman um and it follows a uh I would say slightly u mathematically heavy uh approach where it would involve some construction that is not clear um a priority why we should work with that by the end it does does the job and obtain this formula and it it generalizes some compact uh proof in a compact case but you have to deal with divergence issues as I was saying um so one of the goal I think the main goal of this talk is to prove this again but using a different method method um and this new method will be conceptually simpler and also I think it would be useful uh to connect it's also useful for connecting uh certain things that has been studied in recent literature. So that's another uh advantage of thinking about it. Yes, if I if I consider [clears throat] a closed hyperbolic pend there are no piece so there is no integral to it just uh left hand side is equal to right hand side mod squar uh how uh trivial or obvious that statement is I need to prove it or it just follows from definitions >> yeah let me think um yeah so I I will talk about a case uh that does not involve the inte integral because oh sorry um the there will always be the integral these are the internal edges um so even when we uh look at a closed manifold uh generically we will have internal edges uh to form the the total uh geometry. Uh the difference is that um here in in general there are in addition to the internal edges there are also external edges. So in the close gate there are no >> this this transform and this formal is only over external edges right >> uh yeah I think I said it yeah sorry did I say it wrong yeah sorry you're right I yeah I wrote internal I I wanted to say external okay um yeah I I got confused yes so these are only external edges you're right you're right >> I'm just curious when there are no external edges Um >> is this an obvious formula or one has to actually prove it? >> Um yeah sir okay now I understand your question. So um just to clarify again uh the the integrals over uh external edges there are also integrals over uh internal edges that I'm not we're not seeing in this equation because they are in the definition of this uh the left hand side. Uh but to answer your question um um you are in this way of computing things we actually never get close manifold. The reason is with these blocks um we are allowed to glue along these white faces. Uh but each of them will have some green parts to it. So the green will always be part of the uh final boundary. Um it's just that um when we don't have the white uh boundaries left over, the final boundary will be a totally geodessic boundary. There are no uh these corners. Uh in general, there will be these corners that are important. Yes. Um if we're really looking at a uh like a closed manifold um with no uh not even the green boundaries and then this method uh doesn't actually doesn't quite work. Um yeah except the cusp case I will try to comment on at the end. Yeah. Okay. So um that was the review of u some recent work. Um so we have uh I guess until 40 that's the >> yeah till 45 including questions. >> Okay good thanks. Uh okay so let me give you also a quick review of the uh BCFT which are you know CTF is living on 2D these are 2D BFTs so CTF living on remma surfaces with boundaries um so why do we want to study them first of all you know uh boundaries are important and also uh you know when people studied uh 2D CFTs uh um in the T contact strength theory it's very important to have the open strains so uh These boundaries CFTs are super important for that reason as well. In fact, we we'll use a language of open strain and closed string when we refer to these these states in the uh 2D CFT by the the open and closed uh versions of the state operator correspondence. So when we say open string by that correspondence it also means operator living on the boundary. uh when it's a closed string it just means that usually just the usual operator living on a point in the interior of the 2D manifold. Um another reason you know is that it just uh super interesting because it's just not a it's it's not just an analog of the closed case there are interplay between the the closed string and open string. So that makes the whole thing more interactive in some sense. Um also there you know also recent uh reasons for why it's interesting including for example uh even uh even when you are interested only in the closed sector uh the the open sector can be a useful way to compute quantities such as the entanglement entropy. Um so to uh give some the some of the basics we uh we usually we just have the uh the the spectrum uh for the closed strain and the ck for the OP coefficients and those are the data in the uh into the closed sector. But now we also have a a single weight h for the open uh strength or each uh boundary operator. uh it does not have u a second um second weight essentially because on the boundary uh you have to match the two uh two copies of the virus uh generators. So reduce to one number. Uh another way to say this is that u the open strength does not have uh the spin uh in some sense. Um so uh and then there's also uh the analog of the uh closed strrain OP coefficients. Uh you can think of the OP coefficient as three points three operators on the sphere. But now if you have a disk with three boundary points that's the boundary uh op. And uh it has three additional labels because you can put different boundary conditions on each of the intervals. And there's another uh uh piece of data now that we have both is that if you have a disk uh you have a one open string and one closed string and um that is an independent data you have to specify. So with all these data now you can specify a BCFT and when we now when we talk about ensemble we just mean uh there will be averaging in uh in all these quantities. Um because we are focusing on the hyperbolic case we'll mostly be thinking about averaging in the OP coefficients. Um and then there are work uh uh also on uh considering offshell topologies which will be averaging over the the first two lines. Um just to give you an idea of how complicated this might become. So in addition to the usual two uh bootstrap conditions which are the crossing symmetry and the modular uh symmetry uh you will have six independent uh bootstrap conditions uh for BCFTs. uh for example just to give you a example for example let's look at this one that is uh if you have a two bulk operators and one boundary ones you can first fuse the two bulk ones together and then uh take the that with a boundary fuse that with a boundary operator or you could do the other way by taking each of the bulk operator to the boundary and now you have three boundaries on the three operator on the boundary uh so uh that is becomes a E uh OP coefficient. So these two uh once you sum over all the internal states that are dotted lines represented by the dotted lines they have to be equal. So that is a more general crossing symmetry and we'll we'll see how you know what role it might play although I won't give all the details. Um so um that was a quick review and then we'll uh try to understand what is the uh the bulk theory now that we we have the boundary have boundaries. So the bulk theory will now have additional pieces in in its boundary because otherwise you know otherwise it boundary wouldn't have any boundary. So, so the gray um so let let's look at example uh let's say this one is um is a disk times interval except that now um um the disk has three boundary marked points. So the 3D wormhole is given by this geometry where the three colors represent uh end of war brains uh with different tension in general. Um so that's why there's ABC label and the red lines uh in the classical geometry these are uh kinks like geometrically the same as corners just giving a different name um and the the size of the kink or the angle in that corner is related to the the weight of the boundary operator uh here and the tension is important um as well uh if you change the tension you change the geometry and there's another one where there's a conic ical defect in the bulk. uh there's a kink on the on this uh in the war brain and they will compute uh this case they will it will compute the b uh average quantity the second one will compute a d squared quantity you could say whether this is just a uh you know a bulk definition uh in principle yes because we don't know the boundary ensemble but there's some evidence why it is the right one uh uh because similar to the c squ case you could work with a single for uh CFT use bootstrap conditions to to look at is a heavy or high energy uh spectrum average over some energy window. You will also find that uh the function bootstrapped uh will be exactly the uh the right one uh will be exactly agreeing with the 3D gravity computation as a function of the positions and tensions and weights. other in principle other different topologies contributing to this or >> uh yes uh is uh yes yes so so for example when we compute B I JK square we should allow any uh or for now hyperbolic any hyperbolic manifold um with these two boundaries so it could be higher topology >> but insertion of JK could introduce selection rules so you might not have like normally if you have you should have disconnected topologies. >> Yeah. >> But if you have JK there might be those might not be allowed. >> Yeah. So I'm uh Yeah. So I I should say I think um I'm I'm saying this for JK this OP coefficients uh but just like in the you know in the usual case uh we could frame every all the question as you know genus super functions etc. So these are just it's consistent. So we can work with either you can go between the two. Yeah. So in that case uh another you know relevant comment would be JK's are it shouldn't really uh uh really be a particular JK. We don't know what they are but once we insert these into the uh ping function when you average over it it will look like you know all these coefficients times confform blocks etc. And then this go into that pattern function. Um so the the bulk theory is uh essentially similar to the virt uh except that now is uh we call it open closed vers um it just generalizes the varicity by including the brains um so uh uh so you know what is open close tap is not a madeup name uh it's well studied in 2D uh the the I guess one of the explanation for the name is that usually when we [clears throat] have a TFT we glue along some cut and that cut itself is a um is a lower dimensional manifold without boundary. For example 3D TFT we glue along um you know the boundaries of handle body etc which are themselves closed rima surfaces. uh a 2D closed decap would be the the one you know the usual ones where you glue pairs of pens and each of the cut is a circle. Here you're allowed uh to glue along only a sector of the boundary. So in the 3D case we we could glue along um these gray uh uh 2D surfaces which themselves have boundaries and that's why it's called open closed. Um so so the the the way to construct them um they they are in 3D they are less formulated less well formulated than the the close case for example the modular tensor category precisely implements the more cyber conditions. So you know you have the bootstrap it gives uh gives you a mathematical well- definfined uh tap t these are slightly less uh well formulated but they exist um there are a lot of literature on that um I'll just give you the the building blocks uh in the in the var case uh they look like like these uh they look slightly complicated uh so first of all each color is end of war brain um uh generally with different tension and Each um um each gray uh each gray disc is something you that you're allowed to glue along with another disc and each uh blue line is a bulk wasn't line the usual uh wasn't line and uh each um red line is a is a boundary line which is a thing in the open close cap. Um so let's say if we just to give you some idea if we take all the uh uh the ways to be imaginary and then each blue line becomes a conical defect each red line becomes the kink as I was saying um yeah so yeah so these are finite central charge objects so they define tap t as finite c but you can take the classical limit and they obtain give you some geometry Um okay so uh uh then it's quite natural that uh that since we already have the fixed p basis for the uh for the close sector we we could do something similar for the open sector. So for example here uh we could have a manifold with some endor brain in green and then the red sorry the white boundaries or components of the boundaries they are uh just some u 2D surfaces with boundaries and and they also have the property that they the extrinsic curvature vanishes everywhere as a tensor except at these little um uh intervals. Now uh each is related to the weight of a p. So the why that's why these are fixed p basis uh objects and uh you can comput it in open close tap tical limit and then you can check it agrees with the geometry that satisfy these boundary conditions or more precisely you have a variation principle under these boundary conditions. um you you look for the settle and the settle or sorry the volume of that settle is a classical limit of this uh ping function. So this is very much analogous u to the close case but to make the things even simpler we could um uh I think this is a good well definfined thing to do it just that let's focus only on open strains. So on the CFT side suppose there's like because we're looking at ensemble it doesn't matter uh I think it doesn't matter if we just say there are no closed strings um and then there are no uh there are no H and H bar uh there's no CI K because there's not even an I uh there's no D I I so the only thing that exists is the the open strength so the data will be the HI for each open string and then BIK K uh among the open strings. So um and then something um that simplifies significantly significantly is that uh all those building blocks I was showing earlier uh they they won't exist except this single one. Uh this is the the first one that was in that list but I'm drawing it slightly differently. And the reason why I am drawing it like this is that that single uh building block is P function is the six J symbol. And we saw earlier that the classical limit of the six J symbol gives you the hyperbolic volume um of these uh uh uh geometric uh objects. Um uh furthermore in this purely open case uh you don't uh ever need the imaginary piece. In other words, uh in that list of building blocks with all these, you know, green faces touching each each other, all these green triangles will be disjoint. Uh so, uh this is the only type of building block we need. And then uh yeah, then this is a simpler problem to to think about. Uh the reason essentially is because uh if you only have the open sector there's one bootstrap condition that that remains and the crossing kernel for that u bootstrap condition is the 6J symbol. Uh I think this is probably a profound thing. Uh but at this point we could just understand it at the level of the function. We it just happens to have both uh features. Um so now we are sort of um I'll say almost ready to to come back to prove the the main goal um which is this relation between the uh CTV and the visotic cap. So first of all just let me just give you slightly more detail because we now have this purely open ensemble the relevant sector in the bulk is just the was what's called the purely open varicity. uh is like is at the level the the actual function they're pretty much closely related to vers but conceptually uh is that we have all these open closed uh building blocks uh now we have just one and purely open result uh can therefore be written as a multi uh as a product of the W's each w is six J symbol um integrated over the now these are internal ways that uh that we have when we glue these tetrahedral together. Um uh uh as I was trying to say earlier I we don't have to glue along all these hexagonal faces. Um because in fact when we study uh these uh OP statistic we shouldn't glue along all the u hexagonal faces. each second of phase represents a u bajk op coefficient. So it is important we don't do that. Um and then the the uh the triangular green faces they they form smooth in brains. So the uh the earlier pictures we saw that we you know these wormholes we want to study uh can be computed this way and you can check is uh is exactly you know the volume etc are correct. So um with that uh we could think about uh whether you know this tells us something or gives us some insight about this relation uh between CTV and vot. Um so the relation is I would say is a more mathematical one because I was just saying that we shouldn't glue along all the hexagonal faces uh because then we don't have you know data at the end uh to to interpret but if you do uh at the mathematical level you just have a uh all the green faces green triangles left which they they form join to form totally geodessic boundaries. The reason why they're totally judici totally jodessic just means u um they're yeah so in informal language uh there are no no corners there are just a smooth uh um boundaries and the jodessic just stays on the boundary. Okay. So, uh uh if you do that, um it turns out that uh that gives you a topological invariant at finite central charge. And so, this is actually quite hard to prove and it's proven by mathematician only last year. So, uh so this is uh this is a special case when you do glue along all the hex hexagonal faces. Um furthermore you can try to add more ingredients to this by relaxing the condition a little bit. So earlier we we had all these um um internal edges we were saying that you integrate over w etc. But we also uh implicitly assume that uh all these u variables uh they should around each internal edge should add to uh 2 pi. Um so classically it's obvious why because you the internal edge is not the actual boundary. So there's no singularity there. So if you don't add up to 2 pi there's a conical singularity. But if you do relax that uh you could get some geometry which looks like this. So this is uh in our language earlier this computes a uh so I haven't really specified a topology but I'm just saying let's say the boundaries are these two spheres. uh each of them with three punctures and then there are three conical defects connecting some of them. If you uh now require the these conical defects to be some specified number um you want to specify and then uh you could use the following relation uh to relate this to something else and this relation I I'll say uh you know if you're familiar with this is a relatively like a trivial relation but uh if if you're not uh I would can only explain try to explain what it means uh without giving the the proof or the derivation. Uh so the left hand side is like a it's like cylinder with a conical defect or in a quantum uh um regime. Uh it's a bulk wall line connecting the two discs and the right hand side is um it's a solid disc sorry is a solid cylinder with this middle part removed. So this is a is hollow. You can go from left to right and then in that interior boundary you put a a boundary wall line running in a loop like that. So in the BCFT language is a is essentially a closed string um propagator or closed strrain u fixed p basis um object and is related by modular s is modular crossing kernel uh to the open string um propagating in the loop. Um so uh one easy way to see why it's correct is to at the mathematical level you could do the doubling trick and then it gives you a uh just a modular uh just a modular crossing transformation between the two uh two to so uh if you do that uh although it's it's a it's not totally easy to see but if you like I spent a few minutes try to apply this to this picture uh it's relatively easy to relate that to this object. So you have to uh in this picture you have to remove the neighborhood of each of that uh blue lines and that and replace with right hand side of this equation and then this picture become this picture. So now this is a manifold whose boundary is a uh is a pleated surface. So it's geodessic except at these three circles. And then earlier we saw that this is the um an example of the the call or the um the the part the manifold after removing the trumpet uh which Tom Hartman uh defined. So this uh by by his theorem this is related to the fixed ptar function. So now that we have this relation u it therefore uh relates the two uh a priority unclear um uh probably independent uh tap t invariance to each other. Um okay so um that was the uh the main result by this point maybe it's unclear you know what's interesting about it and u uh why it's useful. Um so I would say um one thing that's interesting about it is um is uh uh is something that I actually didn't say is that it only works for for the scalar sector. scalar sector defined to be when you u have this uh equation there's a square uh which usually is taken to to be uh the visual t with p uh for its first copy and p bar for it second copy where p bar is unrelated to p. Um but uh this uh equation actually only works when p equals p bar. So you could just remove the this modular sign. Um you could ask why is um uh in some sense this derivation explains why it doesn't generalize because uh in this equation the left hand side is only uh non zero when p uh equals p bar essentially by a rotational symmetry or conformal symmetry. So uh uh this derivation works when it is a scalar >> [snorts] >> uh and it's unclear whether there exists some other proof uh that uh generalizes to u non-scaler. Um but but it's open question. It doesn't prove it doesn't work for non-scaler. It just seems to explain why in the scalar sector it works. Um uh another comment is that um I was I briefly mentioned that you know these different parts of the animal brain could have different tension etc. uh that is usually actually hard to deal with because if if you change the tension on each of um um so uh yeah having said what tension is tension is when you have this end brain is classical action has a parameter you could just add [snorts] uh you know plus t in this action uh times the uh the the volume form or the area uh on uh square root of g uh of the andor brain um that parameter uh is important because uh the the whole geometry depends on each t. So if you have you know three five uh tensions uh is a function of these three five different numbers and you could ask you know if the action changes as a function of the tension does it still agree with the boundary um ensemble? uh you could say of course it does because we haven't specified the ensemble but that is not true because um once you specify uh you know these t1s it computes something on its boundary and then um if you consider higher genus higher uh you know uh manifold with more boundaries the g actually carries over as a simple factor so you g sometimes uh will just be g to some power um so that uh on the boundary side even when you when you are computing a ensemble uh the dependence on G is not something we can uh play with it just has to be fixed uh and then the bulk model in fact turns out to be consistent with that so if you do change the the tension of each of the brain the action will change uh at the end as um some something that goes like log of the uh tension so when you look at the part function is e to the uh at least classically e to the minus of the action. So it becomes a polomial uh or in fact not just polomial it's just a single G to some power u if you have different G just G each G to some power and that turns out to be important for this to be consistent to have a consistent ensemble interpretation uh so the reason why I'm sort of explain emphasizing this is that now you could say what if I I write a different action for the end brain uh then I guess it just uh it's unclear whether it's going to work because then it depends on the tension or on these parameters will not be consistent with ensemble interpretation. Um another thing was that conform blocks have to be related to reormalized volumes. Um something interesting uh uh to check. Um I may not have time for describing the cusp geometry but uh let me say that uh in that case um there's a lot of mess results uh and you can try to include the cast geometries to uh to this holographic uh ensemble dictionary but uh it's not totally clear how to do it. Um I think I'm going to uh these are sort of easy to understand and they're less uh concrete. So these are the general questions uh that are are still open. Uh let me just uh uh okay I think you can read all these. So let me thank you uh for uh for COMING [applause] HERE. Uh any questions? I I have a question. So in principle in the bulk you also have some version of crossing moves because you chose some triangulation but you could have chosen a different triangulation. [clears throat] >> Do those play any role in the story? >> Yes. Um they um yes so they play a very important role. So the reason why there were all these building blocks uh is that um you can't uh pick all of them like independently uh because if you do then you can pick different blocks uh to form the particular geometry and then you see you don't agree you don't get the right answer. So the uh uh the fact that it is a tap t requires all these indep uh different blocks to have uh to have particular the right uh uh p functions and then when whatever way you use to build a particular manifold you'll get right answer. So uh so that is uh I mean in some sense it's not uh it's not that non-trivial because uh these building blocks they are built uh or by design using the BCFT uh crossing moves or crossing kernels and then they each of them do have some 3D uh understanding and once you understand how you translate them to 3D uh then uh the fact that the BCFT bootstrap uh u the BCFD bootstrap conditions uh the fact that the B BCFD bootstrap conditions ensure when you have a BCFD correlator no matter how you cut it into smaller pieces you'll get the same cutting function also ensures that when you have a 3D manifold no matter how you cut it you get same answer >> more questions Let's thank again. >> Yeah. Thank you. [applause] [music] >> [music]