Seung-Joo Lee - Towards Explicit Bounds in Supergravity

Thank you. So, um, thanks very much to all the organizers for kindly inviting me to speak here and for putting together such a wonderful workshop. It's been great so far. Um so the title of my talk is towards explicit bounds in super gravity. So indeed I'll be speaking about my recent journey towards explicit bounds on some physical quantities uh which are supposed to apply desiraably to all consistent super gravity EFTs. So the talk will base uh mainly on these uh three collaborations. The first is uh a recent one with uh Koter Berkar at Chinua University in China. Kotzer is a pure mathematician uh with expertise in algebraic geometry specifically in bational geometry. Uh the the second is from three years ago with Paul Oman and Santa Barbara which is very much related to this earlier work with Tim Vagan at Hanukk and the last one is some work in progress with Andre Lucas at Oxford, Paul Oman again and to Shimanek at Utre. So let me begin with this uh onepage motivation. So at the abstract level, our aim is to pursue a well a fundamental understanding and a consistent description of quantum gravity via effective quantum filter models or uh EFTs for short. Um then in order to achieve this goal um we analyze well those EFTs arising from string theory but as you will see um we will pursue general model independent results. So we'd like to make the analysis as global as possible in the set of string EFTs. Now, interestingly, a lot of mysteries of quantum gravity physics may naturally be encoded in geometry as broadly defined. Um, as we know, exciting dialogues between physics and geometry arise all the time via string theory. And we'd like to benefit from them uh in better understanding quantum gravity. And to illustrate such dialogues um I brought four examples here uh from my own work actually. So number one concerns the weakness of gravity. The law is that gravity has to be the weakest force. Um well this happens to be the case in nature but the statement is that it must be the case in any consistent gravity theory. And um interestingly this amounts to well also very surprisingly finding a certain quantity jacobi structure uh as encoded in sections to elliptic vibrations. Um number two concerns characteristic asytoic physics. Uh as we just heard from Mariana's talk um the relevant conjecture here is that any physical theory when its EFT parameters take values in the asytoic regime of the modulized space must exhibit one of only two very universal characteristic features. And this amounts to finding certain nice geometric representatives of modular compactifications. Um number three and number four concern explicit bounds respectively on the particle spectra and on the uh discrete symmetries. Uh here are the precise physical statements along with the corresponding geometrical tasks. In fact, this talk will focus on addressing these explicit bounds. So rather than reading them out here, let me start from scratch now. Okay. So um our general motivation is to characterize the effective field theories of quantum gravity. As you know, uh quantum gravity theory seems to be very much constrained at very high energies. In particular, string theory is sort of unique in 10 dimensions. But at low energies depending on the choices uh we make of the internal geometry uh there arise many different string EFTs in dimensions less than 10 uh each of which serves as a consistent uh EFT of UV quantum gravity theory. uh and interestingly at least 10 to the hundreds of allowed internal geometries are available leading to as many uh string EFTs uh uh as low energy effective field theories. um and they are part of the vast landscape of string theory which of course also includes those not having a geometric origin but today we will focus on the geometric part of it which as you can see is already extremely vast. Um nevertheless just to get the picture right uh let me point this out. So the entire set of EFTs uh splits into the landscape and the swamp plant as we've just seen. Um the landscape is the set of consistent EFTs of quantum gravity theory and therefore uh contains the string landscape as a vast subset uh which may or may not uh fill up the entire uh landscape. Uh and now we aim to distinguish the good theories from the bad i.e. those EFTs that lie in the landscape from those in the swampland plant by revealing general constraints of quantum gravity theory. Um well it's rather difficult to see even where to start because we don't have much control over the quantum gravity then um the string landscape serves as an excellent guiding principle. So what we do here is to try to establish universal geometric properties of the so many internal geometries thereby extracting any common physical properties of again so many string EFTs and also vice versa and the idea is to reccast those universal or common properties as general constraints of quantum gravity. Okay. So having this in mind today we will focus on uh sixdimensional uh super gravity EFTs uh subject to minimal super symmetry. So today's talk will be I mean this talk will be really about super symmetric uh uh EFTs. Um their particle species are organized into super multiplatess of these four kinds gravity vector hyper and tensor. And as for their spectrum uh we always have a unique gravity multiplat and the remaining uh uh multiplane numbers will be denoted in so by vh and t respectively. Um the EFT is of course defined via the action of a special form subject to the uh this aial supercharges but uh we won't go through this except pointing out that uh one comma t vectors here a and b i's there they are some effective filter parameters known as the anomaly equations and we'll come back to them later in this talk. Now given this arena um one of the most basic questions uh we can naturally ask concerns the finalness of the arena and relatedly uh um the main question of this talk uh is if the discrete EFT data and in particular the particle spectrum data VHT are going to be bounded from above then um dialogues arise between the finess of our quantum gravity arena and the boundedness of algebra geometric arena. So ages for algebraic geometry. Specifically our consistent EFTs will correspond to the elliptic fibered calabia tree force and the discrete EFT data to uh a certain topological data of such geometries such as the model by group and the uh uh picard latice of the base and so on and so forth which you will try to bound. And as for the next level data uh after the spectrum we will also consider uh bounding the discrete symmetry order which will correspond uh in this algebra geometric arena to the minimal multisection index in the extended arena of genus one vibrations and uh at this point let me present the uh spirit of our program and also get you some of the main results organized is into these three parts. So as for the spirit um we analyze the various consistence constraints on the geometry of string theory in particular of f theory and hydrotic theory to derive the spectrum bounds and the symmetry bounds in a top- down fashion. Notably uh we will present these explicit bounds on the tensor spectrum and the U1 vector spectrum uh which should apply to each of the 10 to the hundreds of sixdimensional F3 EFTs. Moreover, we will also propose an explicit bound on the discrete symmetry order uh based on the geometry of hetro theory. The idea is then to uh um um gain some bottom-up insights by interpreting the top down derivation in a purely effective filterative language. Um well and some of these bounds can also be proposed as a bottomup bound some others cannot but uh today we will focus mostly on this top down derivation um to emphasize the uh exciting interplay between geometry and physics. So having given this um introduction uh here is the outline. So we will start with some rudiments of uh sixdimensional f theory building upon which uh universal and explicit bounds will be addressed in three parts. The first two parts will concern the spectrum bounds and the last part uh the symmetry bounds. In fact, uh I'll spend most of my time uh for part one to show you how spectrum bounds could possibly arise in the first place for the tensor multiplets. So here um I'll first spell out a couple of key key ingredients um of this bound uh both in the geometrical and in the physical languages and I'll also um um briefly sketch the bounding strategy clarifying the boundedness and then also manifesting the uh resulting explicit bounds and next in part two I will give a geometric derivation of an explicit bound on the U1 vector spectrum and I'll also give a physical argument via anomalies and conjectures for somewhat weaker bounds um just to get you some uh general flavor of how the bottom of argument tends to go and finally in part three switching gears I'll also describe the geometric origins of discrete symmetries in f theory and in hydrotic theory and uh I'll discuss what we may possibly learn by comparing uh these two settings and after all of this I'll conclude with a quick summary and some outlook. So that's the plan. So let's start with some rudiments. Um six dimensional F theory is a type 2B string theory on a compact complex two-fold Z2 but with seven brains on some complex curves therein. So the exodilatone varies internally and here the exodon profile or the internal sevenbrain configuration is encoded in an elliptic vibration over Z2 whose total space Y3 is clabo. The vibration admits a vious description of this form where fng here are uh polomials of an appropriate degree in the coordinates of the base z2. Um and the discriminant delta is 4 fq + 27g square um whose vanishing low size support singular fibers in turn indicating that the seven brain stacks are located there. In fact these singular fibers um encode not only the brain location but also the associated multiplier types which you now turn to. Firstly, the singular fibers in the complex two cool dimension one loa in the base Z uh result in the vector multiplatess living on the corresponding seven brains. The so-called minimal coda fibers are classified via the vanishing orders of these visas uh triplet data fgm delta and these are immediately translated into the um uh the the gauge algebbras of the resulting vectors. Um these algebbras are of a semi-imple type at finite distance in the moduli space and here the divisor classes of the loss IBIS um they serve as the anomaly coefficient of the simple gauge factors but as you can see what's missing in this table are these high vanishing orders that go beyond the 4 6 and 12 and the uh corresponding singular fibers are called nom minimal Uh while these guys uh uh cannot be crepently resolved in a Colombia way, um they were properly analyzed for example in this work where their fate was identified as the ethanization of the gauge algebbras at infinite distance in the modulate space and in turn as decompactification of the spacetime. And for a quick side remark here, um, in a simpler eight-dimensional setting, this decompactification interpretation agrees very nicely with the ninth dimensional hetro classification in this work. Also, Mariana is here of um maximum nonabilian gauge enhancements which I think supports our decompactification interpretation of this codamian one minimality. So all of this is about nonabelian vectors. On the other hand, the aelion u vectors result instead from the sections to the elliptic vibration which I denote by sas. Um so uh via some stringity to m theory uh we can also work out the corresponding u1 anomaly coefficients bas uh as the so-called high pairing classes. this but the details won't matter in this talk at all. Uh all I want to remind you of is that uh the sections to electric vibration form a group very famous group called the model vi group which is known to be a finitely generated aelion group and this is all about the uans. So the rank of this model Y group counts the independent sections and in turn the number of U1 vector multipllets uh which I denote by uh VU1. Okay. So next quime two complex codimenation two singular fibers at points of the base Z are responsible for hyper multiplex. the singularities are further enhanced at such a point of Z and uh meta multiplatess arise charged under the gauge algebbras. However, if the singularities are too severely enhanced going beyond the affformentioned minimality threshold um then uh what we have is the so-cal confformometer instead and here I'd like to emphasize that this nom minimalities at is is at code dimension two which has to be contrasted with the nom minimality in cod dimension one which we discussed in the previous slide which we identified as uh you know triggering the space decompactification. So here it's just codamation to nominality which is fine some meta of an exotic type. Earlier it was really towards the boundary of the modulated space. Uh finally uh unlike the vectors and hypers the tensors are encoded in the base topology alone. So they are simply arising from the dimensional reduction of the fullform potential over the internal harmonic two forms of Z. And therefore the tensor count T is identified with a picard number row of the base Z2 modulo distribial gravity count. And interestingly the tensor sector connects in intimately to the other sectors as can be seen for instance via the uh confformometer. So over the point where the conform meta lies one can perform a base blow up increasing the pad number thereby generating an extra tensor multiplat. Furthermore, the used to be co-omination two no minimality here uh upon blowing up uh may lead to uh some remnant singularity so to speak which in turn leads to uh uh vectors and ordinary meta hypers as well. So these three sectors are mixed together for instance through this conformal meta. Okay. So enough of particles. Just like the um seven brains on a device lead to vectors, an effective string arises from the three brains on a curve C. And as it turns out, the resulting string type is determined by the precise way the curve C complex curve C is embedded in uh this complex surface Z. Now the viable base two folds are obtained from the projective plane P2 or the Hertz surfaces FN um by suitable blowups essentially all of which is in fact P1 fibered as it turns out and here the way the P1 fiber is embedded in Z as a fiber is very universal and therefore our EFTs are genetically equipped with a certain distin distinguished string corresponding to this distinguished curve class which turns out to be the heterotrotic string. In the rest of this talk we will focus on such generic bases or generic vector of F theory with the picard number of the base strictly bigger than one i.e with the non-trivial tensor sector and of course we may assume this for example for the purpose of bounding uh the tensor multiplan number because the only exception here is t equals z. So we are trying to find the upper bound on T. So it's fine to assume this P1 vibration structure because only exception is very trivial. Okay. So having set this up uh let's first address the tensor bounds. Okay. So we first spell out two key structures in geometry. One is the vibration structure that the internal space Z is P1 fibered. So as already reviewed this is genetically the case and also for the purpose of our bounding task we may assume this without loss of generality and um the other key is the singularity structure that the pair Z comma B is 1 / 6 low canonical or 1 over 6 LC for short. So the pair consists of the internal space Z and the divisor B of Z and the divisor B of which precise definition here I won't detail in this talk is roughly the discrement locus delta of the elliptic vibration I the seven brain lowi modulo the scaling by 12 so B is essentially the seven brain lowi um in case some of you are new to this um notion of epsilon low canonical singularity. Uh for our purposes, this epsilon LC structure of the pair ZB essentially means that each coefficient of this divisor B which is the second part of this pair must be bounded from above by 1 minus epsilon. So with epsilon= 1 / 6 which applies to our case here. Um the this um second key structure tells us that all the coefficients of the divisor B must be bounded from above by 5 / 6 or equivalently uh we we must have all the coefficients of 12b which is roughly delta uh must be less equal 10 and this can easily be confirmed uh from all the minimal coda fiber types uh as we pointed out in the paper and um I I must say that um this coefficient bound or or maybe more familiar is this one. Um well had been known to the three interior community for decades. in fact well more specifically in the F3 literature um but maybe in a slightly different language and what I did not appreciate uh at least until 25 is that this question bound connects to the tensor bound which we have worked out eventually um so this is the thing I'm going to discuss in this talk but it's not just about geometry uh in fact very interestingly each of these two structures which are very geometrical has a very clear physical interpretation which I'll discuss now firstly uh the P1 vibration um is precisely what gives rise to this distinguished heterroy string in the EFT which a finite distance is tension full um so this has been already discussed several times and next the um 1 / 6 LC structure of the pair ZB E um turns out to keep the space-time dimension as six. And now let me explain uh a little bit more about what I mean by this. So recall that the singularity structure demands that all the coefficients of this divisor 12b which is the seven brain lossi must be less equal 10. Then we gain this intuition that this criterion essentially prevents the co-omination of one fibers from becoming non-minimal. So if you remember this kodata table this was at the threshold. Um now given that the co- dimension one no minimality uh triggers the spacetime to decompactify um our second uh uh key principle here is indeed that the space-time dimension has to be kept fixed to six and obviously we are very much entitled to impose this because we are trying to constraint the spectra of the EFTs in six dimensions not in higher dimensions. This is not actually importing some content. This is just there from the beginning from the EFT point of view. Um these two key physical principles are particularly interesting because they are directly connected to uh these two characteristic asmtoic physics. So the uh relevant claim here uh is the emergence string conjecture which asserts as we heard that at infinite distance either a tensorless weekly coupled critical string emerges or the spacetime decompactifies. So this singularity structure serves as the decompactification threshold as in here and uh the P1 vibration guarantees the presence of the uh um critical hydrotic string. So these are not just some technical geometric conditions on just a geometry. Uh in fact each of them has a uh direct uh physical meaning. uh well interestingly uh concerning the asytoic physics which sets the uh boundary behavior. So it's not probably totally unexpected that these are related to certain bounds beyond which maybe the theory is illdefined because we're pushing it towards the boundary. >> So you say 12 has coefficients less than 10 coefficients of what basis? >> So cos so I'm looking at the uh decomposition of this divisor into the components I I read >> what components >> reducible components of this divisor. So I'm looking at the uh essentially the brain seven brain loi. I look at the irusible components of the seven brain lowi. I read out the coefficients of each of these components and I check that each coefficient is less equal 10. I cannot put on too many seven brains on each step. That that's essentially yeah. [snorts] Okay. Okay. Then as for the bounding strategy in a nutshell what we do is the following. A we blow Z down to um a Herzburg surface Z node and then we go through a sequence of blowups fiis i.e B, a sequence of tensor transitions, each generating some extra tensor multiplat. Uh, and B, we keep track of the induced divisor BIS along the way by zooming in onto the conforma. And C, we impose effectiveness and one over six elimin so that the seven brains may wrap something physical and the space-time dimension is kept fixed to t= 6. Um and the idea is that in A if we blow up Z0ero as many times as allowed by the constraints in C uh then the picod number of our internal space Z is bounded by that of ZR the end point of the sort of maximal uh blowup sequence. Then as it turns out the number of blowups R cannot be arbitrarily large and um we will discuss explicit upper bounds on R which will apply also to the tensor count T because we have this inequality. So T is essentially R minus one uh sorry R R + one. So if we bound R then we bound T. So that is the upshot and uh now I'll try to present some of the details unless you have any other questions about the general things we have discussed so far. Okay. So uh to start with it proves to be very useful to define the notion of horizontal multiplicity and the idea here is to try to fully uh exploit the uh the fact that all of these zis are p1 fibered over the common base p1b like this. Um and recall that the the divisor bis um they are essentially keeping track of the seven brain lossi along the blowup sequence. So if we split each bi into the horizontal part and the vertical part with respect to the p1 vibration then the horizontal part bi will capture those seven brains that hit the heterotrotic 7 heterotic 3 brain i.e. the P1 fiber as in this picture. So the blue is DBI. Then in this setting we define the horizontal multiplicity small hi as the multiplicity of this horizontal divisor bih at small zi the blowup point for this i blow up. Then as it turns out these horizontal multiplicities they play the role of a very useful order parameter for this blowup sequence. And a very useful observation here we make is that 12 * hi is always integral uh well this number in fact we can physically interpret as the local counting of these seven brains the internal seven brains hitting the hydroic brains. So we are counting something physically. So it's very natural also physically that this number is integral. Then we learn from here that uh all the non-zero his must be big or equal 1 over 12 because 12 hi is always integral. Um so in a sense they are gapped. Now without loss of generality we may assume that the blowups with higher hs are performed earlier. So we can just arrange the sequence in that way. So the his these horizontal multiplicities form a non-increasing sequence and we will denote by R prime where they first drop to zero from something bigger equal 1 / 12. So that is our R prime. So we are ready at last to address some explicit bounds on R prime and then R in turn. So I'll be quickly flushing through the relevant claims. So firstly to bound R prime um I'll make this claim. So the total horizontal multiplicity which is a sum of all the h is bounded from above by a certain number which turns out to be 28. And rather than going through the details let me just point out that this comes from some couple of global constraints uh coming from the compactness of the internal geometry. Then it follows that RP prime is bounded from above because as we discussed uh all of these his up until RP prime minus one they are positive but they are bigger equal 1 / 12. So with this finite lower bound and given this uh upper bound on this total sum we have the bound on R prime. So that's clear and next to bound R we make two claims. The first one is this. So if hi is zero then the corresponding blower point small zi must lie in two vertical curves and this is as opposed to just one vertical curve. Okay. So why? Well, otherwise I if the point lied just uh on one vertical component of some fiber then upon performing the blow up the coefficient in B of the exceptional divisor that will come out uh would become uh uh strictly negative due to the 1 / 6 LNS which we must impose. So in other words uh B cannot describe the seven brain loai anymore. So that's not allowed. And claim number two is that for every point inside ZR prime the total number of the subsequent blowups there cannot exceed 11. And this can easily be seen to arise from the maximum case where the uh two vertical component involved here of this BR prime both take the maximum possible coition value which is 5 over 6 set by the 1 over 6 L again. And this uh uh turns out to correspond to the confirm matter of type E8 * E8. Then it follows that RUS R prime is also bounded I the number of the subsequent blowups after the first RP prime blowups. So this number is also bounded from above because only R prime points inside this intermediate ZR prime can be further blown up thanks to this first claim and each 11 times at most thanks to the second claim. So after the R prime blowups we have only R prime points of the vertical curve intersections and only at those points we can blow up further and well at each of those only 11 times at most from this second claim and that's how uh some explicit boundary r could arise. So what about the expression numbers? Firstly for R prime we get uh 336 from 28 * 12 and then for R uh because from here R minus R prime cannot exceed 11 R prime R is bounded from above by 12 R prime from here which in turn is bounded by 4032 where I'm using the bound on R prime which I just computed here above. Okay. So what's nice about this derivation is that the um the abstract boundedness which we used to have in geometry is now promoted to some explicit bound which we must care for for physics. And furthermore it's also nice that the bound arises as you've just seen from just a few additions and multiplication just from some simple algebraas the bound is just given. Um but we must discuss the quality of the bounds next. Um so in driving this bound uh we were exploiting several inequalities. So if I go back so we we exploited this one this one and this one. And to be on the really safe side we were assuming the worst case scenario in all of this. So we we're assuming that all of these three inequality are all saturated but at the cost of of course weakening the resulting bound. So in this way the initial conservative bound we obtained was our less equal 4032 and in the math version of the work uh we actually improved it down to 566 via some final control of the compactness constraints. On the other hand the conjectured optimal bound is 192 which you are trying to confirm at this at this moment uh using this top down derivation. Um in fact this last bound the conjecture bound had already been realized by the maximal picard base in the uh famous toric data set as constructed by cruiser and skake long ago and was later on conjecture to be the truly optimal bound by Morris and Taylor. And now we are much better convinced that uh this conjecture is most likely true given some bottomup arguments uh uh provided in this work which actually appeared around the same time when our uh top down results came out. So uh for your reference um uh I brought the explicit construction of the wouldbe maximal base here. um the uh so here is the initial uh zero the initial hutchbook's office which I chose to be f12 for instance then uh the um required 192 blowups had already been in fact suggested uh by Morrison Taylor. So we get this z 192 with picard um 194 r equ= 192. But the question is what about at the level of pairs not just at the level of bases. So after all the pairs encode the internal brain configurations not only the internal spaces. So we are very much motivated to keep track of uh the pairs for the full physics in order to understand the full phys what's going on in the full internal uh geometry and the internal brain configuration. Moreover um uh what can we say about the uniqueness? So this um uh bottom of physics bound of 192 is indeed saturated by this uh would be maximal base but are there any other bases that also saturate this bottom of bound. So in trying to optimize our uh top down derivation of a bound um not only do we establish the sharp bound on a solid ground but also uh we can try to establish these points uh properly. So I brought uh some details uh uh of this blob sequence at the level of bases first and also at the level of pairs eventually. Um and I can maybe share some technical details um later if time permits only if time permits. But uh for now in view of time let me just say in words this kind of analysis has um almost uh convinced me by now that uh the sharp bound is in fact provable via the top down geometry of string theory and moreover that the maximal model is in fact unique. But we are not just talking of uh arguments here. We are really working on a proof. So this is still work in progress. But I'm I'm I'm I'm strongly convinced that this actually is the case. So um in the remaining two parts I'll be very quickly flushing through some main ideas only. Um so what do you know about the manifold? >> About the >> the manifold in that case >> you mean about the total space or the base? >> No the base >> the base. So the base is given here. So this is the base >> as but you know it completely. >> We know it completely. We know how to construct it by blowing up say f12 you could also get there starting from f0 but anyway we know the final configuration completely [snorts] >> and does it have some property like the highest uh number of complex structure >> high number of complex structure. So the picard is 194 that we know. So this is the biggest we can achieve if this bound is true. So that's that's one thing we can say at least and also maybe what's interesting is that we have this EA* EA conform bunch of EA time EA conform. So 1212 pair is really this EA time EAform meta. So this you see that all it's all about EA* EA conform and I've given you the flavor of the maximality for EA time meta. So this indeed appears all the time in this maximal base. So we believe that this is really the I mean this is how Morrison Taylor originally believed that this is the maximal base but no proof. Okay. So um in the remaining two parts really I'll be quick. So firstly uh let me uh quickly discuss the uh U1 vector bound. So the claim is that the number of U1 vectors VU1 is bounded from above by 18 for any EFTs of F theory as long as the tensor sector is non-trivial i.e if the apicard of the internal space Z is strictly bigger than one. Um the derivation is actually rather simple. So we start by noticing the uh nested vibration structure >> you mean strictly U1 right not the rank of the gauge group >> strictly U1 so this is just about the abilion sector can we bound at all this abilion sector yeah so the claim is that it's it cannot exceed 18 okay so uh so it's starting really from the nested vibration structure which is I would say generic in the F3 vacua so um as depicted in this picture so due to The P1 fiber nature, the genetic P1 fiber nature of Z2 which is the base of Y3. The total space Y3 uh exhibits a nested structure of elliptic and K3 vibrations. So this is generic in the sense that I mean to the extent that P1 fiber nature of Z2 is also generic. Okay. Then it follows immediately that oops sorry uh each section 2 Y3 uh induces um a section 2 Y2 the generic K3 fiber just via restriction to the P1 fiber of Z2 and therefore we get this inclusion relation that embeds the model V group of the Y3 uh into that of Y2 and this inclusion relation holds in particular for their free parts And therefore the rank of the model vial y3 is bounded by the y2 counterpart which in turn is bounded by 18. And this last bound comes from uh just from subtracting from this uh h11 of the k3 the generic k3 fiber the base p1 and the elliptic fiber. So this is the maximum we can achieve for the Y2 model V and this propagates to the Y3 model V and well as we recalled from the rudiment what's been bounded here is indeed V1 that's the rank of the model V of a Y3 and this works for every Y3 as long as Z2 is P15 but so it's simple as this so this is the geometric derivation of our U1 uh vector multiplier sector size but now well you may say this is just from F theory uh but We can also argue for slightly weaker bounds uh for any super gravity EFTs without assuming any stringity or geometric origin. So um the uh claimed bounds are 22 if the tensor count is bigger than eight and uh 20 if t is from between one and eight. So just to get you the flavor of how the argument goes uh let me very briefly read out the key ideas and the assumptions used here. So the first is to uh impose the gravitational anomaly uh cancellation uh on this uh anomaly equation a and choose um an appropriate charge basis such that the a coefficient and the inner product omega in the tensor branch take these forms. Okay, we just choose those bases. And the next is to assume that strings with an an arbitrary integral charge in the in the charge basis just chosen above do exist in light of the completeness conjecture. Uh in fact all we need is the strings with a particular charge Q0 of this form and this this particular charge form uh well if you remember this F3 context uh corresponds precisely in the geometric acting to the P1 fiber and for instance you can check very easily that Q0. Q0 is zero given this inner product. So Q0 must represent some some fiber curve and indeed you can check that this is in fact P1 fiber class. So now uh as long as you can assume the existence of strings with this particular charge Q0 then uh from the unitarity of the warie theory of this q0 string uh you can you can argue that uh you can bound the the the count of the gauge degrees of freedom by by 20. So okay you may say that this is it but in fact here I'm assuming something important. So each U1A contributes uh one to this uh uh gau count only if the level computed by the inner product between this Q0 charge and the anomal equation BA is strictly positive. So the last task we have to go through is check this positivity of the product and and as it turns out um uh this desired positivity follows immediately from uh the anomalies of these various kinds and well I'm not going through the details here but let me just point out that we also make use of some of these uh koshi show type tricks which work uh uh given the signature one comma t of the tensor branch which we always F and also similar tricks work even when t is bigger than 8 uh leading to slightly weaker bound but it's still pretty good. So in this case we say it's less equal 22 and this is the flavor of how purely physical arguments go uh without assuming anything string theory or geometric but you have to assume things okay so in the remaining uh few minutes I'll um briefly dis discuss discrete symmetry bounds um so okay and this is how we uh define six dimensional f theory Okay. So um um well the idea was to identify the axial value at each point of the internal space Z2 as the complex structure of an elliptic curve there and and this is how uh the elliptic vibration could be used as a bookkeeping device for the axum profile or the seven brain configuration. But in this regard um genus one vibrations without sections but with just a multi-section uh should also work as good in define a six f theory and indeed it was shown that an nsection clabia treefold uh when we use F3 on this geometry leads to sixdimensional super gravity EFT with Zen mode NZ discrete gauge symmetry and this is precisely the next level data we wanted to bound in arguing for the finitness of our quantum gravity arena. So the relevant geometry question here is if we can ever come up with um an explicit universal upper bound on the minimum multisection index for genus one vibration setting. uh so just to say the request construction by now has a five section but the question is can we make it bigger and if so how much bigger so this is an open question so I do not yet have a clear geometrical understanding of this multi-section geometry however via string duality I'd like to guess the possible answer from a different angle and to this end let's consider the sixdimensional hetroic theory instead for which the geometric data is a K3 surface X endowed with a slop stable bundle V. So this geometric data K3 surface and a bundle V also leads to six dimensional super gravity with minimal super symmetry. And notably if the geometric data is symmetric enough then a discrete symmetry arises in this setting as well. More precisely, if the K3 surface X uh is equipped with a finite ailion group G of simple automorphisms and the bundle there on V with a G equivalent structure, then the group G serves as a discrete symmetry of the resulting sixdimensional super gravity. Now uh recently I learned that uh well there had been this famous classification of simple automorphisms of K3 and also that uh all of these but for these last two I= 3 and four are realizable in elliptically fibered K3 services. So why do I care for elliptic K3s? Well, that's because we know that six dimensional f theory is dual to six dimensional hydroic theory and where the ladder the six dimension hydraulic theory is defined on an elliptic surface. So uh in this setting our first guess is that perhaps the heterodic bounds on discrete symmetry orders uh coming from maybe this automorphism classification must also serve as fitic bounds and if true this uh would indicate an explicit uh upper bound of uh if you look at it eight is the biggest for the multisection index uh for the genus one vibration. So >> quick question. So, so you're equating uh how do you know that every discrete gauge symmetry comes from multissection because what if I'm in a separate point in the modul space or let's say I have you know some >> good very good so um so I guess um so here you can say that maybe we are restricting to this particular origin of the discrete symmetries theory coming from this gaug gauge sector and I would think that there must also be some discrete symmetries possibly arising from the uh gravitational part of the I mean from the base but here we are looking at this uh gauge origin for the this f theory but uh even for those coming from the sort of so to speak gravitational sector I think the hodic sort of dual uh and the bound on the automorphism uh orders uh would tell us something if the duality persist so to the extent that this duality works I think from the hydraodic geometry could say something about even that but many ifs is here but yeah it's a good good point yes >> how many tensors do you think arbit number of >> that's also a very good question so here of course the heterotic jewel is well defined with uh um one tensor >> just one tensor however then we have to argue for how we generalize to those with multiple tensors of course and we have to argue for that but it's most clear at least in the t= 1 case um then you have to consider blowing up you know putting in some five brains and so on whether this argument goes through. So all of these details have to be added carefully. So in this work in progress we are trying to make this um this dialogue now between two geometric settings but uh well precise this this dialog is precise but with this common physics of discrete um uh gauges symmetries but there are many things to be to be argued for carefully. Okay. So um just to summarize okay so in this talk um we have discussed how we launched a program of systematically bounding the spectra and the symmetries of supergravity EFTs and in particular in light of the string landscape uh we have specialized at this initial stage to sixdimensional F theory and we have gained the following lessons firstly the U1 vector spectrum is bounded via the P1 vibration structure which indicates the presence of a header string in the effective field theory and secondly uh in bounding the tensor sector uh on top of this vibration structure the 1 /6 low canonical singularity structure also played a crucial role uh which sets the decompactification threshold and lastly uh by noticing how these symmetries arise in F theory and in heterotrotic theory and by comparing these two geometric settings we are proposing some explicit bound on the um um um minimal mortization index in the setting of genus one fiber calabia freeolds and as for a couple of uh interesting future directions firstly um while we have focused in this talk more on this top down derivation of these various bounds u uh I've also given you the flavor of how one could argue for their bottomup analoges um by assuming certain effective theory quantum gravity conjectures. So uh we are planning to make this bottom up reformulation also precise again modulo certain conjectures whether you like it or not that's one possible direction of future research and another uh direction of research is to reduce of course the super symmetries imposed so having constrained the EFTs when they are subject to eight real supercharges very natural next step is of course to impose only four which is the minimal amount of super symmetry that a genuine quantum gramm theory must have if super symmetric at all. And then to see if uh expressive bounds could also be derived in this much more general setting. And in particular um the tensor spectrum bound uh in the sixdimensional equal supergravity setting we discussed in the first part of this talk is naturally generalized to the um um axon spectrum bound in the fourdimensional equals f3 vacua and we are trying to come up with some expressive number there. Uh so we are uh uh pushing it out. Um so that's what uh one way of pushing it further. So all in all uh I'd like to emphasize once again that uh there are many exciting dialogues between uh string geometry and physics notably between geometric bounds for calabial vibrations and physical bounds for string EFTs or more generally for quantum gravity theories. So that's where I wanted to. So thank you very much for your attention. [applause] >> So it is bound for the the rank of the model like the number of U1. So you are essentially assuming that the three had had the K3 vibration. So we should construct it out of the P1 electric vibration. We know that that's not true in general. >> The only exception is actually there are two exceptions. One is when the base is P2 and the other is when the base is NK. Okay. >> N surface NK surface. So so they are the only two exceptions. But now I've actually um I've I've given the reference. So this was generalized to um uh really general setting. First of all, okay, so this uh is really for the genetic basis at least, but as you say there are exceptions. So actually this paper which appeared maybe a month ago uh they discussed the fully general picture of how you can bound model V and in the end it's about the topology of this Y2 guy. So an analog of this Y2. So in this genetic case we have Y2 of K3. But for example in the P2 case we have um something that has uh that leads to bound of something like I think if I'm not mistaken 28 so it's like u uh three halves of K3 oiler something like that uh it's not K3 it's but we can control this also generally but >> that whenever there was a K vibration there would be a network dual and therefore the number of cancers would be one So >> no no no so so this is genetic fiber. So I I I must emphasize this is genetic fiber. So there there can be special fibers of course in that case it's not just P1 but it's a it's a it's a bouquet of P1s of course that can appear non- genetically but at a genetic point we have a K3 fiber. So that's all we need in arguing for this because we restrict this model value to that of genetic K3 fiber and that's a fiber K3 for sure. Okay. Yes. Mhm. All right, let's thanks speaker again. >> Thank [applause] you. [music]