Matilda Delgado - Anomalies and the structure of String Theory

Thank you, Reuben. And thank you very much uh to the organizers. Um it's very much a pleasure to be here. So, and thanks to all of you for being here for the last talk of the day. I know there's the dinner afterwards, so some of you have been lured here, but uh um yeah, I guess maybe the last thing you want to do is hear me fluffing on about anomalies for the next 45 minutes. But okay, it's 45 minutes and I expect you guys to ask a bunch of questions and I will try to go slowly and uh avoid as many technicalities as I can. Good. So, let's start with some uh motivation. As you know, uh, efforts to connect string theory to low energy physics and the physics of our universe has mostly relied on super symmetric setups. But this is restrictive because ultimately string theory in of itself does not need super symmetry. What do I mean by that? I mean firstly that you know you know the the classic picture we usually consider super strings. So these are 2D SCFTs. So super symmetric on the world sheet. But once you take the massless modes of the string compute amplitudes and derive the effective action that describes the massless modes you can end up with something that is super symmetric or not. So in particular people like to discuss the five 10-dimensional super symmetric strings but people forget about for instance these three string theories which are all perfectly fine it seems from the world sheets modular invariance and everything checks out. um they're free of tachons in spaceime. So if you don't like tachons then you know that's a point in their favor. um uh but they're just non-super symmetric in spacetime and so there's one heterodic one and there's uh this one called the sugimoto string which is sort of non-supmmetric counterpart of type one and then there's a sani string which is also an orient but not of 2b uh an orient of zero b that projects out the tachon here and these are all the gauge groups of these theories okay so you have these things to worry about already in 10d now even if you start with one of the super symmetric ones you know they're super symmetric on 10 dimensional manowski on flat space you if you pick any background you want without being careful you will most likely break all super symmetry so by that I mean okay don't take the Tori the K3 the calabia which are very special manifolds that preserve some amount of super symmetry instead just take pretty much anything else and you can take something much less exotic than this AI generated image take like just reman surface for genus 2 that has no coariantly constant spinners all of susie is broken good so string theory does not need super symmetry because there are already strings that are non-sup symmetric in tend so all the way up to the string scale and then beyond that you can always break it at some sort of intermediate scale good now of course we use super symmetry and we need it to have computational control over all of the quantum corrections GS corrections and alpha prime corrections which become abundant when super symmetry is broken. Uh so in other words yeah we need super symmetry because it makes our lives easier but you know it doesn't have a reason to be there in the first place. So you know what with having learned at CERN the hard way that our universe is presumably not super symmetric at least at energy scales that we can probe it is still interesting to try to understand non-supymmetric string theory. Good. So let's set out to do this. What can we do with all of these control issues I just mentioned, right? It's famously a hard problem. Otherwise, everybody would be doing it. So, um what can we do? What are the tools at our disposal to try to understand string theory without super symmetry? Well, one thing you can do is look at fundamental consistency conditions. And by that I mean things as basic and tractable as anomaly cancellation. And beyond that, you can also try to apply these like consistency of black hole evaporation through no global symmetries type of arguments. These tools are neat because first of all they apply to quantum gravity EFTs in general. You don't even need to talk about string theory and in a sense you know there at no point do you need super symmetry to implement them. They're inher inherently non-super symmetric. Beyond that they make use of some fancy topological tools like bortism groups which I will get to in a second which allow you to really sort of give yes or no answers to some specific questions on non-superymetric strings. So the types of questions you can ask are shed light on things are u of course the fundamental consistency through anomaly cancellation which is basic requirement but then with the second type of arguments you can try to understand what kind of particles and extended objects might be in these theories uh which are of course both of these are very basic things that you should ask about any theory but it's already hard enough when you don't have super symmetry good so what I'm going to do today is tell you how you can use gauge age and gravitational anomaly cancellations to firstly uh argue for the fundamental consistency of the three non-super symmetric 10-dimensional uh string theories the nontionic ones that I briefly mentioned earlier and then I'll tell you like the main part of the talk will be about uh studying anomalies on the world sheet and uh using these anomaly cancellation to put constraints on allowed space-time backgrounds without relying on super symmetry at all okay so I said anomalies uh you know on both sides so I don't I'm just have this slide so that nobody's confused there's the world sheet there's the 10-dimensional EFT for the massless modes both of these theories have chyal matter both of these theories can have anomalies and in the first part I'll be talking about the anomalies in the EFT and in the second part about anomalies on the world sheet so that's the gist plan for today is to first give you a very brief crash course on what uh anomalies or the anomalies that I'm going to be discussing today have to do with bordism groups. Then I'll tell you about the spacetime anomalies for the three 10 dimensional non-suzie strings that are not taken. And uh finally I'll go to the world sheet anomalies afterwards. Any questions on motivation? Okay, perfect. Then let's get straight to it. uh in a theory with uh dynamical gravity and gauge fields, you can of course have gauge and gravitational anomalies and anomalies in a symmetry that is supposed to be a redundancy of your theories. So gauge symmetry is of course a huge problem unlike anomalies in global symmetries where you can do a bunch of to anomaly matching and all that neat stuff. So you really have to get rid of these things. The basic way to describe them is to say that it they correspond to a lack of invariance of the path integral under a gauge transformation or a diffmorphism. And people like to talk about two different kinds of anomalies. First people talk about local anomalies which are those anomalies that are associated to a gauge transformation or a diffmorphism that you can make arbitrarily small that is part of the gauge group that is connected to the identity that you can make literally infinite decimal. No, if you have an anomaly with something a gauge transformation that is infinite you could see how that would be a huge problem which is why you discuss them first. So you compute them if you have PTSD from triangle diagrams in the masters or undergrad or hexagon diagrams in 10D. Um these are all local anomalies and for all that I'm going to say today I'm going to assume they all cancel which they do for the cases I consider. Then there's leftover anomalies which are the ones associated to the global part of the gauge groups to those transformations that are that you cannot make infinite. So of course the historical example of this is Whiten SU2 anomaly [sighs] and these are the anomalies that I'm going to be discussing today. Good. So now let me tell you how global anomalies are computed uh in the modern way. So in the absence of local anomalies which again is what I'm assuming throughout this talk um you can compute global anomalies in the following way. What you do is that you start with your theory on XDU with its partition function and all of the you know problematic anomalous content it may have and you construct a whole auxiliary theory an anomaly theory in one dimension more that lives on this yg+1 whose boundary is the manifold that you start with and you should not think of this anomaly theory as being some sort of higher dimensional theory on some higher dimensional spaceime it's literally just engineered to give you exactly the opposite anomaly from the theory that you start with such that when you couple the two of them together, the anomalies cancel each other out and the whole thing is anomaly free. So if you want to think of it this way, you can think of it as giving you some extra space in which to trace out some sort of non-colapsible path in the configuration of your gauge field or of your metric. You're sort of making this transformation local u u using this extra dimension. Okay. In particular, you have to be careful that you know if you have a spin structure on this, the spin structure extends here and all other um yeah possible structures you may need. Good. So the you just create this theory to each anomalous degree of freedom in your theory. You associate a contribution to the anomaly theory and this is what you obtain. And so the reason that we've constructed this whole auxiliary theory in one dimension more that has the exact opposite anomaly from the theory that we start with is because it's much easier to identify the anomaly with the anomaly theory. And the reason is the following. You have to ask yourself how do I pick this extension this extended manifold here this yd +1. Surely the anomaly is something that should not depend on this choice of yd plus1. Or in other words, I should be able to deform any choice of yd plus1 into another choice of yd plus1 and still have the same anomaly. So maybe you get the point. The idea is that the anomaly itself is a bortism invariant. So bortism groups very briefly are the mathematical objects that categorize which manifolds can or not be deformed into one another. You say that y plus1 and y+1 tilled are in the same bordism class. if their disjoint union is the boundary of some D plus2 dimensional manifold WD plus2 and if you draw it like this you can really think of it as some sort of movie where you're deforming YD+1 into YD+1 tilt and so there are quantities that are conserved under these deformations which are called bortism invariance and again I want my anomaly to be the to be invariant under deformations of this higher dimensional manifold YD+1 which is why you can the the bortism invariance are going to be detecting anomalies for us. Good. So now what do you have to do if you want to determine whether or not the theory you have under consideration has a global anomaly? Well, you just have to compute the relevant D+1 dimensional borism groups because again this is you're always talking about the extension manifold which is in one dimension more and then okay there's two cases either the borism group itself is not trivial in which case you may have an anomaly okay you don't you can't say right away that you have one what you have to do is evaluate the anomaly theory that you've constructed on the relevant generators of this borism group and if that evaluates to something non-trivial then you have an anomaly. You know, it makes sense because ultimately whether or not you have an anomaly will depend on what kind of chyal content you have. You can just add perhaps add chyal content and cancel anomalies. So, uh this step is very crucial in ultimately determining if there's an anomaly or not. Uh then of course if the bordism group itself is trivial then okay whatever theory you may have whatever anomaly theory you may have with that amount of structure won't have anomalies. So you're good to go. Good. questions. That was the very brief uh crash course on anomalies. Fabulous. Okay. Well, then let's quickly then talk about the non-super symmetric strings in 10D. So again, let me show you the duality star where I've added the three 10-dimensional non-supers symmetric and nonachionic strings here with each of their different gauge groups. Now again these world sheets are modular invariant are perfectly fine uh but they're just non-supery symmetric in space time and so people I mean tend to set them aside because it's hard to do computations with them but okay um all three of these theories have a gauge have gauge symmetries and they have firmians that are charged and uncharged under these symmetries caral firmians which means that of course they can have anomalies so local anomalies were checked back in the day when these theories were made which is more than 20 years ago now and they were always shown to cancel through what is known as the green shorts mechanism. The same one that you have in type one or in heterodic strings where you have this diagram compensated from that one such that anomalies cancel. Okay, it's just a fact of life. So local anomalies were checked fine but global anomalies were never checked. So that's what we're going to do now. So we have three 10 dimensional theories. We have to compute 11dimensional borders groups. uh and the point is always figuring out what is the correct uh bordism group to compute. So what are what is the correct structure for the bordism groups we need uh to to be representative of these theories. So because of local anomaly cancellation or as a sort of side effect of local anomaly cancellation you have this non-trivial bi identity for the H3 uh field strength which is field strength of the B2 uh field the calberand B2 so it's just this statement that okay you need this whole thing to be zero where this trace f is a gauge field instance on so for each of the different gauge fields in the three theories and trace r is just the first pontriagon class of the spacetime background. So gravitational thing good. So to mathematicians this is known as a twisted string structure that this should vanish. Okay. If it were just P1 so trace RG or equals zero it would be a string structure. But because you're allowing for P1 to be non zero if you compensate for it with a little bit of gauge uh instance on is called twisted string. What can I tell you? Good. So then we have to compute three 11dimensional twisted string borism groups which are all twisted by the different gauge fields that we consider. Um good. Uh so again you have the gauge groups here. It's 11 dimensional and the string structure is is what is written down here. And these things were just not known. So you had to compute them from scratch using atom spectral sequence and all the heavy machinery that goes on there. But I won't go into these details. I'll just skip to the good part which is the results. So what did we find? We found that the first two uh borders groups. So for the Sugioto string and SO6 are just trivial. Okay. So there's nothing more to do for sure. These theories are completely free of global anomalies which of course puts them on much on better theoretical standing and is a great consistency check I think of string theory without Suzie. Now, you're a little bit concerned about the last one that I left out if you're listening. Um, and the point is that we're not sure if it's zero or Z2, which of course is pretty drastic difference because it could mean that there's an anomaly or not. The reason why we're not sure is because there's a differential in the spectral sequence. So, we couldn't figure out what it is. But okay, even if that's the case, we could still try to evaluate the anomaly theory on this would be non-trivial on the generator of this would be non-trivial class, [laughter] but we weren't even able to identify the generator. Okay, so we we're just not sure. Ultimately, we think the theory is fine, of course, because it's related to the others through dualities and we're not properly worried about it, but it's it would be important to check. Um I can tell you what kind of topological invariant has to vanish on this 11dimensional manifold. uh and maybe you can cook it up. >> So is it are these calculations purely mathematical at some level? I mean the very apart from what you just wrote about the DH being there's no other and against groups there's no other >> that's correct. Yeah the the physics come into like what specific structure you pick. So the fact that it's there's this DH this non-trivial bi identity tells you what kind of borism group you have to compute. But then this is purely mathematical. Yeah. >> It does depend on the global structure of the game whether it's >> it does. Yeah. >> Yeah. So we were careful to use spin 16 because it has spin uh files and spinner rep not system. >> Sorry could you elaborate what global anomaly with respect to the gauge group that you're focusing on for these examples? Uh so the these would detect um gauge so anomalies that may have with the gauge group but also with gravity. So it can be a mixed sort of gauge and gravitational anomaly and um yeah if >> 10 10 DOS 10D large DOS in 10D >> exactly large gauge transformations in 10D. Yeah. Great. Thank you for the questions. Um, but that's all I was going to say about this. Now I'm going to go to the newer things, uh, which have to do with world sheets anomalies. So this is your last chance. Well, okay. Except for after the talk, of course. Okay, good. So, >> so [laughter] in the context of super theory, we usually tend to think that the fact that this world sheet is well defined was the underlying reason for the anomaly of the space. >> Yeah. Do you believe the same? >> That's a really good question and I think a lot of times that's sort of assumed, but I know of specific instances where there was like a K theoretic anomaly that you couldn't see from the world sheet. So checking spacetime anomalies is I think still relevant. I can point you to the reference afterwards. Cool. Great. What about here is that have takons in >> uh yeah so okay I consider the ones without tachons but um that was just because I don't know people don't like tachons but everything that I did has nothing to do with the dynamical aspects so in fact there's a bunch of other 10 dimensional strings as you know very well that are tachionic and I think it would still be relevant so local anomalies uh council for these ones as well you can check or it has been checked But global anomalies are still a mystery I think. Yeah. And and this all of its machinery doesn't care about the presence of the tachon. >> Can you say mystery? People have not computed. >> Yeah. Yeah. Well, [snorts] okay. To my knowledge. But yeah. >> And if you were to compactify these uh then there would be additional potential anomalies that would arise. Now you you're not looking at symmetry. So yeah. >> Yeah. Uh so if you were to put these Yeah. So I guess if you have an anomaly free theory and you start putting an on circle presumably it will be fine. But I mean I can't I guess it could be possible that after compactifying you somehow have some um effect uh that that makes it such that the structure you have to consider is maybe different and then you would have to recmp compute it only from scratch. Um >> I mean some non-trivial yeah circle is probably okay but something non-trivial like some K3 I don't know. >> Yeah. So for instance um yeah in some ongoing work with Ethan and Damian we're looking at uh 4D theories just calabial compactifications and then already just um from the duality groups that you have there you can have some anomalies that are linked to and so then you would have to check specifically the five dimensional boards and groups for these theories which is what we're trying to work out. Yeah. And just a small uh comment uh is that um you know because Matilda is assuming like particular structure to spacetime like in this case like twisted string structure basically the assumption or or the underlying assumption behind that calculation is that the string theory makes sense on every 10 manifold with that structure with that gauge background and so on. Yeah. >> So including product manifolds >> including in particular product. Yeah. >> Yeah. That's correct. Good. Okay, perfect. Thank you for the questions. Uh, so again, you know, you know the drill. Um, there's a world sheet and it dictates everything in the low energy effective action. That's just string theory. Uh, but what we what I want to ask today is sort of the inverse question of, okay, we're not really clear on what this world is. Um, are all spacetimes allowed? And what I'll tell you is that no. And in fact, some pretty simple spacetimes you can cook up will lead to anomalies on the world sheets. And this is not something that's brand new. Uh for instance, for the heterodic string, this has been beaten to death by various people written more uh and more recently Kazuya who showed that local and global anomaly cancellation on the world sheets tells you exactly that you need to have this twisted string structure in spaceime. So what we want to do is do something analogous to that but for type two strings. Good. So let me review the type two world sheets. So it has to be at very least one comma one. Okay for 10D flat space it'll be 2 comma 2. And if you have 2 comma 2 then you know there's super symmetry in space time. So you know by definition that there'll be firmians massless fmian. So there'll be a spin structure and everything. [laughter] For one comma one it's not so clear. But okay, let's first review the standard thing you've seen the first time you studied super string theory or type two super string theory have uh your eight free chyro bzons your eight free chyal firmians of left and right kalty for each of them and the S so SOA global symmetry that rotates all of them together and that's not enough famously you need what is called a GSO projection what is the GSO projection well uh it's the fact that you want your one loop diagram for your closed string to be modular invariance so you want the Taurus amplitude to be invariant under reparameterizations of the Taurus. This is the reason why you need the gso projection because if you have ky firmians the left and right moving ones on your taurus on your world sheets it means that your taurus has to admit a spin structure and most of the spin structures are not invariant under modular transformations which means that you have to sum over them in a modular invariant way. This is what the GSO projection is. And you can also think of it as gauging this left-handed firm parody symmetry, the Z2 symmetry on the world sheet. That's a fact of the that it's just a more modern way of saying the same statement >> in your notation. L is the world number >> world number. That's correct. Yeah. Yeah. Okay. That's important. Um good. So have my car boss and fmuls so s8 and I have my gso projection. So I've consistently gauged this minus1 to the fl. This just leads to 10-dimensional manowski with massless particles transforming in the little group of soa 1 which is so s8 and that's how you see the symmetries match on each side on each side and everything's good. That's the end of the story for a 10 dimensional flat space. And what I want to ask now is what about sort of less trivial backgrounds than just tendency. So the first thing we consider are flat space or ffolds. What are flat space orbits? Well, okay, you take flat space. It has some uh isometries and you quotient by a discrete subgroup of this symmetry group of this isometry group. So for the circle for instance, you can quotient by reflections and you obtain an interval. We want to consider 10 dimensional backgrounds. So we're going to leave time and one spatial coordinate alone for technical reasons that have to do with light cone gauge more than anything else. And then we're going to take the remaining eight directions which have the full S so SO8 and quotient by a discrete subgroup of SO8 just completely generally. And okay for at first for now I'm going to assume that um that there's no fixed point. So I not only do a rotation by some angle but I shift along some other directions such that I remove all singularities for now. Good. So let's consider this to be our choice of spacetime. What would be the corresponding world sheet if this were a consistent background? Well, now I wouldn't have the full S so SO8 global symmetry anymore because I would have gauged the corresponding G subgroup of SO8. That's what an orbitold is. You sum over all the twisted sectors which are just summing over all these different discrete gauge sectors. So I'm left with an S8 mod G global symmetry and I've gauged the G subgroup. And so maybe now it's clearer what I'm uh what I mean when I say there can be anomalies on the world sheets because I'm in order to get a type two string I have to gauge both of these symmetries at the same time. Not just firmian parody symmetry but also this G subgroup. And so it can happen that there is a mixed anomaly and that these two symmetries cannot be gauged at the same time. So then you would say this specific or fold is inconsistent for type two strings. Of course I can try to add in other ingredients like an orientant fold or whatever but that's why I'm being careful really about saying orifold. Okay. So I'm really operating at without orients for now. Okay. So then there's two questions for what subgroup of SO is there an anomaly and what does that mean for the space-time backgrounds that so the orifolds that I have in spaceime okay so the first question is a math question I have my two dimensional theory it has firmian so there needs to be a spin structure so I'm considering spin bordism groups and I want there to be a Z2 gauge bundle for the minus one to the FL a line bundle Z2 line bundle and a G gauge bundle and So what that means is that I want there to be a map into the classifying space of these guys which is exactly what this these things in the parenthesis are saying. And now the mixed here is saying that I want to consider only the elements which are trivial if I turn off either two of these uh of these uh gauge backgrounds. So I'm really looking for mixed anomalies. Okay, good. Um so you can compute this thing. We used a sort of generalization of the usual Smith's isomeorphism which allows you to relate this threedimensional spin bordism group to a 2D pin bordism group and then you can use a tahertz of Brook in 2D. It's a lot easier to solve than in 3D. So those are the the amount of technical details I'm going to give you. Um good. And so the result is the following. So okay, I'm not being careful. Technically this is it would have to be the pontriagon duel of the borism group but bear with me. Um the idea is it's it's a non-trivial extension of uh the first and second mu komology groups of G. Uh so in fact the extension is completely solved because it's just corresponding to this group law here. That's a mathematical fact. Okay. Now what we prove is that the vanishing of these mixed terms maps exactly to the vanishing of the Whitney classes in the target space background of the of this orbyfold here. So the way that you so if you don't know what stifle whitney classes are you don't really need to because I'm going to tell you it means if the first one vanishes it means that m admits an orientation and if the second one vanishes it means that this thing admits a spin structure. Okay, there can be flat space orifolds that just don't have a spin structure and they don't need to be exotic in any kind of way. Um, and the way that you can see that there's a relation here between the borism invariance and the firmian on the on the world sheets is of course because they themselves are sections of this bundle which has to do with the world sheet spin structure the minus one to the FL line bundle for the left moving firmians and the pullback of target space onto the world sheet. So skipping over the technical details, this is how you can prove this. >> Okay. >> Oh, just a quick question. So um how should I think about these C4 classes for the non-compact M uh on the right? Uh are you measuring it on the S7 mod G? Uh is that or maybe embedding in a compact manifold? uh or considering T8 mod G. Yeah. Uh >> yeah, correct. >> Yeah, I think that I'd say that's what we're doing. Yeah, we're just somehow Yeah, taking the T8 instead of R8 from the world sheet. Anyways, it's not Yeah. Um did I want to say anything else? Good. So, so you might say, okay, Matilda, great. uh we we know type two strings and we know in spaceime we have like massless for >> and material they insist on not using wall sheet >> sorry >> yes yeah I'm really restricting to orifold backgrounds um you know otherwise I would be saying okay so I I'll get to the orient folds afterwards um okay so in particular this means that uh any flat space or fold regardless of if it's super symmetric or not will be consistent it's a onetoone correspondence if it's orientable and emits a spin structure which means in particular uh if I take G to be some ZN with N odd and whatever huge odd number I can think of then you know this these will be consistent backgrounds uh no matter what um so I think that's still kind of neat uh you might argue that okay Matilda we know that type two strings have massless firmians but that's only true you you don't know if that's true if you have some very weird orifold that breaks super symmetry completely. You could have imagined that you'd have some sort of orifold that projects out all the firmians somehow and we're saying that that cannot be the case. There'll always be a spin structure. >> So this is only >> yeah [clears throat] good. Um perfect. So let's generalize this a little bit. So we talked about uh flat space or folds that are on top of that freely acting. So no singularity. So it's a smooth target space. Let's generalize this to any smooth target space. So uh with a world sheet that is a one comma one super symmetric sigma model in some 8d target space m which again is ad because I'm leaving the ly cone coordinates alone. Good. So the world sheet theory is this great but what you need to realize is that you can think of the world sheet as just some two dimensional partition function with a spin structure for the firmians on the world sheet and a map from the world sheet into space time and a background for minus one to the fl okay so the way you would encode that into a bortism group is as following I have again a two-dimensional theory so a threedimensional borism group it has firmians it has a spin structure so it's a spin borism group and now I want to map in from the world sheet into m which is exactly the statement the this appears here and I want a Z2 align bundle which is the statement that I want to map into the classifying space of Z2. Okay. And you can see again that this is by the same exact arguments related to the anomalies of the world sheet firmians because they all see the different ingredients that we need the spin structure the pullback of the spacetime onto the world sheets and the Z2 line bundle and you can be we were of course explicit on the various global anomalies of these things computing the ADA invariance when not and matching them to the bordism invariance I'll stop there for the details the point though okay is that again we have some sort of extension but this time of the first and second mod 2 kmi groups of spacetime. And the uh group law is exactly the same one. And so once again, you can just show that these anomalies vanish exactly if my smooth target space admits an orientation and a spin structure. Okay. So this is of course a generalization of what we did before. Um in fact you can show that this reduces to the former one once uh when once you take an orifold background a freely acting orifold background so suppose I had one of these theories where the anomaly doesn't vanish uh how practically where would I if I computed a one to partition function would I trouble modul >> ah good yeah yeah exactly so you can take um just simple example you take this some of these they're called hyperlytic surfaces So they're just orbeolds of toi but some of them don't admit a spin structure. So that's how we ran into this problem because we were trying to write down the the the one loop partition function for type two string on that thing and we found no way of making a modular invariant. So that's exactly what you said. Yeah. >> Are you assuming that the target is just a rem? >> Yeah, we're not okay. So this is purely protributive and even without orient. But we we don't know how to treat for mon stuff in this background. >> But [snorts] you would allow somewhat flux on flux. >> Um yes. Yeah. Yeah. >> If you are perturbative and allow for wall sheet parity, you can you can get away with V structure. You can go to DC. >> Yes. If I want, >> but you need wall sheet parity. >> Yeah. Exactly. Okay. Yeah. I I so world sheet parity would not only change the amount of symmetries I'm considering in the problem but they would change the structure that I have on the world sheets right it wouldn't be spin it would be pin so it's a whole different computation which is interesting and a follow-up so stay [laughter] tuned okay um great so um now let's go to the last part okay this is the last level of generalization that we did so so Far we've only considered smooth target spaces. So we could ask you know what about non smooth target spaces singular target spaces. Famously string theory makes sense on singularities. One of the reasons why it's so cool um in particular orold singularities. So you want to ask the question of what we can say for a general orfold m by g. Okay. So now you have a lot more data to keep track of because you have to keep into account the fact that the world sheet can sort of be twisted by G um around these non-contractable loops and you can have all these twisted sectors localized near the singularity. So in particular, you know, if even if you have a loop in the quotient, the quotient forgets about the fact that this loop is perhaps just disconnected in the sort of covering space manifold m. Uh in fact, the quotient will never know if um well, so the only way that these two are exactly the same thing is if your string is at a fixed point. So you have all of this data to keep track of which mathematically is more the following. So you want to associate to each closed loop on your world sheet an element of the group G. So this specifies a twisted sector if you will and then you want a map of your world sheet into spaceime as usual. But you can't just map sigma into M because this map uh need not be single valued. You can see this directly in this example. No if I have consider you know sigma to sigma goes to sigma plus 2 pi. So I take the spatial coordinate on the world sheet and I want to go a loop around it and implement a g transformation there. If I do that I if I consider that in the parent manifold m it can be that this is in fact not a closed string in the parent manifold which means that sigma and sigma plus 2 pi would be at different places and so this map would not be single valued. So what I can do instead is unroll the string and consider sort of the covering space of the string with a periodic boundary conditions and that's the right way to have like a well- definfined map between these two spaces. Technicalities perhaps um and finally of course you impose twisted boundary conditions which is just a statement that if I take this thing once I quotient by G I like I obtain a closed string despite having started with an open string. Why did this structure not just some GH field on the worksheet? Or maybe it's equivalent because this thing is P1 of sigma going to G. Maybe it's the same as just ordinary mass between sigma and BG. >> Yeah. So here you're for sure considering one twisted sector. So putting some sort of G gauge bundle on your world sheet. But I still think then if you want to characterize how the world sheet is mapped into space time, you have to be a little bit careful with singularities. It's not like this data is not enough. Um >> but on the board sheet nothing is singular. >> Sure. Sure. Um but you want to consider you know um yeah so um let me come back to this once I've explained how this case reduces to the two previous ones and maybe it'll clarify. Good. Um so just one more technicality and I leave everyone alone. This data is exactly the same data as um considering a map from the world sheet itself into this borell construction. So here I'm taking the sort of um covering space of my orifold. So this m that I've before I quotiented by g and I multiply it so it's I do a product with this eg which is just a collapsible space with a g action. So it's something with completely trivial topology and is sort of just allowing me to act with G and I'm modding out by an action of G on both sides. So the action of G that I've dictated by the orif fold on M and the G action that is just inherited through this. Okay, this is the kind of construction that you need to consider equavarantology which is the right kind of coalology if you're considering orifolds. Okay, it's the kind of gmology that is not only sensitive to what you see in the quotients but also what uh all of the twistings and all of the stabilizers that arise at fixed points. So if you're familiar with equancomology, you've seen this before, but otherwise it's a little bit of a slap in the face, but at this point it's trust me. Good. So the the crucial point is this though that I can characterize all of this via a map from sigma into this borell construction. So then I'm again in the case where I have a two-dimensional sigma model with a spin structure and a map into this boreal construction as well as my Z2 line bundle uh which is for my GS4 projection which means I can consider this bordism group and um again let me emphasize that this is truly a generalization okay because if I consider flat space orifolds which means that this M here has completely trivial topology so it's like a point then By definition, this EG mod G is the classifying space of G. So I obtain exactly the case that I had when I had freely acting or folds in the first part. Furthermore, if I have a smooth target space, then with no G action, then BG is just a point and I obtain this BZ2 * M that I had before. And one extra consistency check is that if the the orbe fold happens to be freely acting, so there is no singularities. So um I have this this bell construction which you know uh tracks all of the all of the twistings and all of the stabilizers. All of that now is completely trivial which means that I should be able to just consider this to be a sigma model in the quotient. And in fact this is true because these two things between become homotopy equivalent if there are no fixed points. Good. So you can then again compute this thing and show that anomalies cancel. They're determined by not the first and second Z2 commology but the first and second G equivarian Z2 which is just to say that you're again taking this burell construction instead of the space itself and this maps to the cancellation of these anomalies maps to the first and second G equivarian G4 Whitney classes vanishing. Now this is just not just it's not spin structure and orientation for M. It has more data than that that is sensitive to the singularities. It tells you that if you zoom in to a singularity and you have sort of the space transverse to singularity and the transfer space then it tells you that this transfer space has to have even dimension. Okay? So I'm never going to have a setup where something just ends like this. One dimension just ends. There's no end of the world orfold like this. uh unless you have orient folds. Um and then it tells you the second condition tells you that once I zoom into these singularities the orb fold stratum so that the the singularity is spin so it contains more information than what we started with and good um so that's all I wanted to say there's quick questions before I start concluding and doing the outlook and all that stuff no guys are hungry want that sweet sweet conference dinner. Uh cool. So in the first part I showed you that you know the global anomalies for these three non-supers metric and nontionic strings vanish up to this admittedly important subtlety. Um but as I was briefly discussing with Mariana um I mean we consider the nonachionic ones for no other reason than people like them better because they don't like tachons. you can do exactly the same thing for the tachionic ones and this has not been done should be done eventually. Um furthermore now that we have sort of this more traction on the anomaly cancellation in these theories you can try to play anomaly inflow games and try to figure out whether anomaly inflow in these theories can give you some traction on the various degrees of freedom that could be present on the brains in these theories in particular the NS5 brain. Uh so this is something that we sort of sketched uh in the paper but um I I think there's certainly more to be done in that direction. Then um I told you how cancellation of global anomalies on the world sheet can push constraints on some space times. So from the types of space spacetimes that we considered so orfolds possibly singular or uh smooth target spaces the condition always sort of ramped up to some version of orientability and spin structure. Good. Here there's uh obvious follow-ups. Okay. So, what about orient folds? Actually, these people jumped the gun and did type one already. Um but maybe there's some other more subtle orients that I don't think they really took into account. So, that's uh something we're going to work on in the future. And uh more than that, we so far considered F1 global anomalies, but there are other brains in the theory. And you may consider if the anomalies of these other probes can put other constraints on the theory and that's is something that should appear soonish with Max Rubiner Vinnie Navoa and Sanjay Ramen. Good. Finally, you know I told you about uh anomalies but there's other of these topological tools which tell you cool things about non-supery symmetric string theory like these kind of coortism conjecture type of arguments. So bortism groups in lower degrees detect global anomalies global symmetries sorry that was a crucial point global symmetries in um in spacetime backgrounds and so through this no global symmetries or coarism conjecture argument you can argue for the existence of a whole bunch of objects and processes that are there to gauge and break these symmetries. So this is something again that is purely topological so you can just go ahead and do it for some of the non-suzie strings. don't care about the tachon for now. Um and so for instance, you could do it for the spin 16 spin 16 and obtain all of these non-trivial classes which signal global symmetries that have to you have to get rid of one way or another. Of course, um I hope I showed you that you know through these anomaly cancellation and no global symmetries type of arguments you can use these topological tools which tell you very precise yes or no uh things about the structure the fundamental structure of non-supers metric quantum gravity but of course you know it's not the full story and the ultimate goal is to go beyond topology and extract dynamical implications for which the tachons of course will be very important um but there's still some things you can try to do along these lines like study the dynamics involved with borism transitions study non-super symmetric objects and theories in general and um you can even talk about trying to identify uh strong weling dualities in non-super symmetric strings uh so I'm more than happy to talk about all of these things um at dinner or tomorrow preferably tomorrow and thank YOU [laughter] [applause] QUESTIONS. >> Just uh so going back to the non-supermmetric strings. >> Uh presumably at one loop you generate cosmological constant maybe a potential for the correct has that been looked at you to recoupling. >> Yeah, it drives you to recoupling for all three uh of those ones that I mentioned. In fact, I think generally for the 10 dimensional ones, but once you consider um I think Angelantoni and collaborators constructed some examples uh where you could in fact so all of these had positive cosmological constants that drive you to weak coupling, but they found examples where you could have negative cosmological constants uh that also were runaway type of potentials. Um so so yeah um it seems that there's not one clear rule that these things have to satisfy but all the ones in Tund are are like you said yeah and so the for the ones that drive you to weak coupling the theory is really well defined at zero coupling >> the theory is very well defined as zero coupling well not strictly zero then because you would have the fundamental string becoming very light but like weak yeah um and for the tachionic ones you can also just try to do perturbation theory around the small bit of the tachon like people have tried to do in the past which is of course limited but I think makes sense. So, >> so yeah enough related but also um going back to the earlier question. So you might think uh could there be non super symmetric say ADS5 * S5 solutions with fluxes um >> people have definitely written papers about of such examples. I'm thinking in particular of um um some Milan based people or yeah anyways um I I can point you to the references but the problem is always control issues >> but let's say you took ads 33 * S3 with never schwartz plugs then that's more under control. >> Uh yeah presumably >> that I know of there's no case that even is remotely under control. they always have like a classical solution but then there's no way to suppress uh all of the corrections that they're ignoring. So yeah >> there are solutions that we coupling and where the stud >> so it's efficient flags on the3 and so you have flex on the if you do3 >> okay then I'm sorry uh what yeah okay we can we can discuss afterwards with ns plus >> with ns yeah3 on the3s3 So what breaks us? >> This is in the non-s super in simply non super >> essentially just in the bonic sector and everything works. You don't need you just need >> then and you can argue that corrections are completely suppressed. >> You can stud with >> every and that's the only scaler in the game. >> No, you have the to >> Yeah. Then you have the tool. That's another story about the >> okay no I could sorry possibly uh in the case where you were looking at the non the orifolds that didn't have a fixed point yeah >> you found some of them were were the well she symmetries were anomalous >> could you have predicted from space time that you would have trouble >> well presumably if you had like explicitly tried to could you have said immediately from space time that you would have had trouble So from spacetime the only I think signal that you would have had is that this background doesn't have a spin structure. Um and then if you wanted to define your type two string on that you yeah you would have had issues. Yeah thank you. >> Are there possibly other types of anomalies that we have to worry about or these are the only ones? >> Oh no for sure. So as uh Ruben uh was pointing out and I mentioned at the end uh there's other types of symmetries that you can consider on the world sheet like the orientation reversal and then you have access to all the or orientifolds which is another whole anomaly computation. Um, beyond that, you know, you of course you'd like to sort of introduce Roman Roman fields. I don't know how, but you'd like to. Um, non-perturbative effects. I don't know how, but you'd like to. Um, but you know, D brains and sort of classify like the same thing for like open strings and figure out if you can identify all the non-supery symmetric brains in these theories. But yeah, ultimately you're sort of tied to the world sheet and to perturbation as string perturbation theory. Um, that's I guess the big limiting thing. Right. >> Thanks again, Martina. >> Thanks. [applause] [music]