 Before starting on the board, there are some info for people that sign up for the dinner. I just write down the latest bus you can take, but indeed you can go anytime you like. So that's the latest bus, and the stop is just in front of the road here. And here there are some instructions. It should be pretty simple. You can either take two buses or just one bus to go to the center essentially, and then you walk there. And the dinner is at 8 pm. OK, so let's start with Upamanium ojtra. We talk about finite entanglement and drop-in string theory. Please. Am I audible? So good afternoon, everyone. I would like to begin by thanking the organizers for inviting me to give this talk here. I'm really honored that I'm giving the talk in this hallowed hall in front of such a distinguished audience. And this occasion is all the more special because I have my teachers, including my advisor in the audience, as well as some of my students from the ICTP diploma course. And I especially encourage the younger people to ask questions during the talk. It's really important. I'm sorry, probably it should come back. I would encourage them to ask as many questions as they can, because I really want to make it understandable to everyone. So I'll be talking about finite entanglement entropi in string theory. And just to emphasize the title, I will parse it backwards. So it's really string theory or super string theory that I'll be talking about in its full glory. And it's the entanglement entropy, which I'll explore. Actually, not even the entanglement entropy, but as I'll argue something that has more information than entanglement entropy itself. And the crucial part of the title is, of course, the first word, its finiteness. So, this work was done in collaboration with my wonderful collaborators. It has really been an amazing experience for me to work alongside him and to learn from him. And as I'll explain the key idea he had developed about three decades back, on which we built this work. So this is a paper that appeared on the archive last week. And I'll also discuss briefly some unpublished results. So, let me start with the basics of entanglement. Entanglement is, of course, quintessentially quantum phenomenon. And it lies at the heart of quantum mechanics. Spoken simply, it qualifies the failure to factorize a pure state for a generic bipartite system in two states, of its constituents. The simplest example is, of course, the maximally entangled belt state, which in this picture corresponds to just two spin-half particles, given as to a spin-zero state, which we cannot really factorize. For a bipartite system, whose Hilbert space can be factored into two parts, in the HA tensor HB, there's an easy way to define a density matrix, which, obviously, the pure state density matrix is psi k, psi bra. And we do the partial trace over the B Hilbert space to obtain a reduced density matrix, rho A. And this entanglement entropy is nothing but the von Neumann entropy of this reduced density matrix, rho A, which is a quantifier of the entanglement present in such a pure state. And in entanglement and its discussions, the density matrix is of utmost importance. And so, here, I'll quickly review some nice aspects of density matrix, which might be obvious to many, but which is nonetheless important to emphasize. So, in order to calculate the von Neumann entropy corresponding to some density matrix rho, we usually define something called this quantity z hat, which is trace of rho to the script n. And please, you know, notice the distinction between script n and something called n that we'll discuss later, because that's going to be important in what I say. For typical systems, we have information only about some positive integral script n. And it's using this information, we would like to find an analytic extension of z hat in this variable script n. And this point was very nicely emphasized in the introductory section of a paper that Edward had written in 2018. The density matrix rho is, of course, a Hermitian and positive semi-definite. And it has, of course, unit trace. So, if we want to make an analytic extension on the complex plane of script n, this z hat n is bounded on the right half plane, where real part of script n is greater than or equal to 1. And this is really an important input that, as we take more and more powers of rho and we take the trace, it has to be very nicely behaved on this right half plane. In fact, as real script n goes to infinity, this z hat should go to zero because of these aforementioned properties. That said, the behavior on the entire complex script n plane may not be so nice. In particular, to the left half plane, to the left of real n equals 1, it can have possibly sick behavior, the sort of which we will encounter in due course. The von Neumann entropy is simply given by a derivative of this analytic function at the value script n equals 1. But this script n equals 1, and this derivative is just one number, and that doesn't contain as much information as we would like. So, we would like to know more, we would like the behavior of this quantity as an analytic function in script n. So, from quantum mechanics, it's just a quick, or actually not so quick jump to quantum field theory. In quantum field theory, we can roughly imagine that there are degrees of freedom associated with each point in space. Put this way, we can think of trying to partition space into two halves. For example, we can try to think of partitioning this room into two halves and calculating the entanglement entropy between the degrees of freedom between these two halves. However, as we also learned from Edward this morning, in this speaking, such an operation is not valid in continuum quantum field theory, because the Hilbert space itself doesn't factorize in continuum quantum field theory, and more than that, it cannot be even written as a direct sum over direct products of Hilbert spaces associated with these two parts. However, we are physicists, and we would like to calculate something, and even this naive physical picture, this can be given some nice meaning with sophisticated machinery. This sophisticated machinery goes by the name of Tomita Takesaki theory, which was nicely summarized by Edward in his 2018 APS Medals RMP article. And again, as we heard this morning, there's universal divergence in the entanglement entropy on account of short distance correlations across the entangling surface. Why is it universal? It's universal because it doesn't depend on the specific state being considered, it's just a property of the algebra of observables. And the algebraic point of view in question is quite useful in describing entanglement in QFT. For example, in quantum field theory, the algebra of observables in a region corresponds to type III von Neumann algebra, which is the worst kind, as we already heard today. But obviously, in quantum field theory, we still proceed and there is a host of, actually not a host, there are a few methods available, by which we can calculate entanglement entropy, or some analogous quantity. Foremost among these is replica method, in which to calculate trace rho to the n, or some analogous quantities, we simply take an n-fold cover of the spacetime manifold and just glue it together along the entanglement surface and in the process we introduce a branch cut with a branch point on the entangling surface. So this is a nice formal manipulation and it does give very sensible answers. In particular, for example, the replica method is a key ingredient in the proof of the celebrated holographic entanglement entropy formula by Levkovic and Maldasena, which appeared roughly ten years ago. But so far we have been discussing QFT. There is obviously one force, which we all know about. It is just this force, namely, when we drop something, it falls, the force of gravity. Again we heard this morning that the situation improves considerably when one includes gravity in the picture. And it was shown by Edward and several other collaborators, as we heard, that sometimes the Type 3 algebra actually gets elevated or promoted to a Type 2 algebra in the presence of gravity. And in this context, I would also like to mention the work of my friend Sam, who identified a Type 3 algebra in Super Young Mills pretty recently and which also is an important part of the story. Heuristic arguments of the sort that were mentioned also during this morning's talk, they have indicated that the divergence of entanglement entropy, this can be absorbed into the Newton constant, and which roughly suggests that the total entropy should be finite. And this is what was sort of the driving, the main point made by Saskind and Uglum in early 90s and also by Kabat and several other people subsequently. However, there is a key problem, which is that we do not yet know how to rigorously define an algebra of observables in quantum gravity or that they would even make sense in a theory of quantum gravity. However, this question, I hope all of us will agree, is an incredibly important question and we should be able to find at least an operationally meaningful way to deal with entanglement in a quantum theory of gravity. I mean, this world we know that gravity exists, quantum mechanics exists, and this quantum entanglement, so we shouldn't just throw up our hands and say that we should be able to calculate something. And as far as quantum gravity is concerned, we should be looking at perhaps the theory of quantum gravity and to my knowledge, there is only one string theory. However, even in string theory, this is a serious problem because we don't even know how to formulate an algebra of observables in string theory. Then, that's the next part of the problem that strings are necessarily extended objects and whenever we introduce some sharp boundary in a theory with extended objects, it's not clear whether we can even do that and this is also related to the very soft UV behavior of string theory. And again, yesterday from Edward's colloquium, we learned that in theoretical physics, the quite counterintuitively, the more the restrictions, the better the theory is and string theory is the most restrictive of them all and the structure of string theory is so rigid that things we take for granted in QFT often fell to work in string theory and there are many such examples. For the case at hand though, such an example is the following that in string theory, we do not know of a string background in which it corresponds to some enfold cover which is branched at the entangling surface. In string theory, the most well understood aspects of string theory are of course on-shell, so we must at least start from some background which is a consistent background in string theory otherwise it would not make any sense. So, that's another serious problem. Therefore, what do we do? Any questions at this point? If not, let me proceed. So, the answer was actually given by my wonderful collaborator about 29 years back. No, actually, I am serious and it's a remarkably precious paper and in my personal opinion, this paper is at least two decades ahead of its time and it's no wonder that we are only realizing it's important relatively recently. So, this appropriate method considers propagation of strings on a ZN orbifold where for certain technical reasons N is obviously positive and N is also an odd integer. So, in replikatrik, obviously we multiply it, we put many things together, so there's a conical excess and in this case, there's a conical deficit. This doesn't work really. This, quite amazingly, describes an exact string background. There are tachyons, but these tachyons do not propagate in the bulk really and these are localized at the tip of the cone. Nevertheless, I'll argue and we'll see that in spite of being tethered to the tip of the cone, these tachyons are quite dangerous so we shouldn't ignore them. We'll be considering type 2 string theory in 10 dimensions. The spectrum obviously includes closed strings with massless spin-2 excitations. In the early days of string theory, this massless spin-2 particle, it was regarded as a nuisance until it was recognized that it is really the force of nature that was known to us since the dawn of time. So, gravity is an inseparable part of the story. So, what is our goal? We want to calculate the contribution of the one loop partition function to the quantum entanglement entropy using the Orbefold method. And although I do not go into the details right now, because I mentioned one loop, you might be wondering about the classical piece. As it was shown by Ateesh again about more than 20 years back, this classical piece actually gives the Beckenstein hocking term. It just comes in a very natural way from the string partition function. Sorry, yes. Expected to give us the entanglement entropy of the half plane, right? Yes. Okay, so, just to make a couple of comments. In this situation, you could probably define entanglement entropy in gravity even in other situations. But in this situation, we don't expect even in the theory of gravity that there would be any problem in defining entropy, right? Because we can always think of dressing operators to the side of the half plane, which includes asymptotic infinity. I'm just saying, it's a good case. In the presence of dynamical gravity, I think even defining areas, surfaces might be an issue. So, you might say that, okay, we are starting with the simplest possible non-trivial scenario, which makes sense in a quantum theory of gravity. So, yeah. The scenarios also might make sense. But in ADS, for instance, we consider these kinds of regions all the time. Any subredator you consider the RT surface goes off to infinity and there's no problem in defining the entropy. Yeah, but I think just calculating this quantity in a theory of gravity is a subtle question. Okay, probably we are absolutely in agreement. Yeah, of course, yeah. So, thank you. Thanks for that comment. So, here's a brief geometric picture of the model. This cone is sort of the cone on which the strings propagate and there's a delta function type which is singularity at the tip of the cone, which really doesn't matter. There are closed strings propagating in the bulk and then there are closed strings which are localized at the tip of the cone. But this, as I said, is tethered to the tip of the cone and it cannot really move very far. It just, you know, there's some oscillation degrees of freedom at it has and it remains confined there all the time. So, what do you want to do? So, this is some particular orbifold and we want to perform an analytic continuation in this orbifold index n to, you know, make it look like one picture that I had shown earlier in which one essentially has a replicated manifold of half space. You know, this, as Subrat was mentioning, this is one half space and this is the other entangle half space. And as we take an analytic continuation, so this is a very different analytic continuation. This involves the temporal coordinate. We find that, you know, this is just our favorite Rindler geometry which is often a good approximation to the near horizon geometry of non-extremal black holes. So, what we are computing is the entropy associated with this horizon, with the Rindler horizon. So, in order to do that we have to calculate the string partition function in the orbifold geometry and the one loop string partition function and this one here stands for the one loop part which I am calling z1n. This corresponds to a toroidal world sheet. But, and in which obviously we, you know, sum over all string degrees of freedom. However, this is not the space-time partition function and that's really worth emphasizing. There are two partition functions here in play. The torus diagram, when we look at it from this point of a space-time, you know, at long enough distances, this simply looks like a vacuum bubble and the space-time partition function as students from the quantum field theory 2 course will remember is simply an exponentiation of the vacuum bubble. Therefore, what I was calling this z hat is simply the exponentiation of the world sheet partition function. However, there is an important distinction that I would like you to remember and which is central to our story is that for positive odd integral n this z hat, this space-time partition function does not correspond to an integral script n. What this integral script n corresponds to is 1 over n, which is some rational number less than 1, which is definitely not an integer. But, we would like to go to the scenario or the exam in the cases where this script n is an integer and that information is not available to us from this orbit point of view and what we are calling the physical region is the region where this n, the orbifold n is between 0 and 1 and we would like to make an analytic continuation and study its properties. Here, I must mention a very interesting story that Edward had explored in 2018 in that paper that I had mentioned. This picture was implemented in a really fascinating manner in the case of open strings attached to d-brains which crossed the windler horizon. And one loop open string diagram so let me draw a picture here. So, this is the open string attached between two d-brains and the ends of open strings are really fixed to the d-brains and it is the open string now one loop diagram in this picture but from another point of view which is really a beautiful thing about string theory when I was mentioning in the beginning that by this open closed duality this corresponds to a tree level closed string exchange between these two d-brains and just by studying the open string partition function we could learn something about the closed string in the spectrum in the analytic continuation and it turns out that the open string it had no tachyon in the spectrum and there was correspondingly to the UVIR duality between open and closed strings there was a very nice behaviour of the closed strings in the UV. However, for n greater than 1 which is for generic orbe folds there are open string UV divergences which correspond to tachyons in the closed string channel. However, and this is the most interesting thing that Edward found, these tachyons disappear in the physical region after analytic continuation and in this physical region as far as the open string partition function could tell us about closed strings but in deodorit made very good sense and there were no divergences of this kind and there are some other nice aspects which I will not mention here. In fact, it was mainly this result which made us hope that something could be done along these lines for the closed string všetnje veliko izmeni, zelo, da ga jo smo tudi taho neki nekaj zelo užitečno načine. Nama sva viša ničo novinna delarstva, na še dobro dar ležimo do in n materials začala ima sva svih. To je vgladnje, da je nekaj tega rizma, da nič nader gustno začala, nič nekaj tega rizma. But, let me not bore you with that story and let's jump straight ahead. Any questions or comments at this point? Three level contribution that give rise to the piconsten hocking term of the area. It was done by a paper of Athish из 2002 on this tekin condensation, Pravamo, da pa vsuru. Prejšljenje. In, prejšljenje. Prejšljenje. Poj Komen. Reaper. Zazve. Reaper. Reaper. Reaper. Reaper. Reaper. function, the one loop or before partition function, and this is a complicated formula, but we will get somewhere. So, this is the area, which I will explain in a moment, and these are various functions, which you can just read. Here, this quantity tau is the torus modular parameter, and the integral region is the fundamental domain of the SL2Z group, and in this integral, there appears Jacobi theta function and the Dedekind eta function, and only yesterday we heard that it was really 19th century mathematicians, whose work we have been using in string theory all the time, and here are two shining examples of these functions, one named after Jacobi, one named after Dedekind, whose functions they continue to be useful to us deep in the 21st century. But, as I understand very well that this slide is very unilluminating, because this is just a bunch of symbols with no meaning. So, let me unpack this formula a little bit. This AH is the regularized horizon area of the transverse directions. So, we essentially have R2, which is the orbifold plane, times R8, which is the remaining transverse A directions, and this AH is the volume of these directions, which is suitably regularized in string units. And, let me say a few words about the spectrum. It is best to analyze this story in the Green Schwarz Formalism in the Light cone gauge. And, in each sector, there is one twisted complex boson, which corresponds to twist on the orbifold plane. And, this one twisted complex boson, as we will remember, this goes to the denominator, and this gives rise to a denominator, this theta function in the denominator. There are also three untwisted complex bosons, which are left unaffected by this orbifold, and they contribute to this theta to the sixth year. And, because we are starting with type II supersymmetric theory, there are superpartners of the bosons, which are fermions. And, this, all the four fermions are charged under rotation on the orbifold plane. And, correspondingly, these fermions go to the numerator. And, here, this, there are two integers, k and l, you know, each of them range over n values. And, they label the different twisted sectors of the orbifold. And, ls correspond to the twines. If this word is unfamiliar to you, don't worry, I'll also explain it in the next slide. So, it's actually most intuitive to, you know, also study it in the Hamiltonian picture. In this integrand, this can be written as a trace, as a particular trace involving q and q bar where q is our favorite, you know, function of the modular parameter e to the 2 pi i tau. So, what does this say? This says that this function g mod square is nothing but the trace over the twisted Hilbert space. So, there are, in orbifold, there are, you know, k different Hilbert, in different Hilbert spaces. And, this particular term in the sum is this trace over this Hilbert space. And, because it's an orbifold, on orbifold, we also have to act on with this rotation operator. And, this g to the l is nothing but this generator of the orbifold group. And, over here, obviously, it's, you know, string theory, we have some oscillator vacuum, and we act on this oscillator vacuum with raising operators. So, nl and nr are the oscillator energies, and epsilon lr are the vacuum, or the oscillator vacuum, or the ground state of the harmonic oscillator. The sum 1 over n, you know, and this l sum, this affects projection operation. You know, again, this is another beauty of string theory that, you know, even in orbifolds only this zn invariant states will contribute. And, there are many states, which you would have nively thought would be present, get projected out by this operation. And, there's a tau 1 integral, which forces a constraint named after the second director of this institute, which simply says that nl equals lr, nr. And, physically, this simply means that no point on the closed string is more special than the other. So, we also want a spacetime interpretation of this picture. So, in the spacetime, if the particle has mass m, the mass square of the particle is given simply by the sum of this, you know, the ground state energy plus the excitation energies. And, here, however, is a trouble, because there are states where m square is negative, namely, there are tachyonic states in the spectrum. Again, we are led to, you know, the beauty of string theory. We have to do this integral over the fundamental domain, which is defined like this. And, we split this, you know, integral in a nice form, in which the way Peterson measure appears in the integral, and this f is whatever was left behind from that. And, this really is sort of one of the most beautiful things one can imagine in string theory, which, you know, connects mathematics and physics in a very deep manner, namely, the fundamental domain. As the Edward had explained to us yesterday, string theory naturally provides a cut-off, and this integral doesn't extend over this entire upper half plane, but there's an overcounting, which we must not do, and that, in fact, is obtained just by cutting this off. So, string theory naturally takes care of, could have been, uv divergence, which appears beneath this section. However, you know, from our childhood, we are taught to be very scared of uv divergences for good reasons. In, as we grew up, of course, we realized that uv divergences, we can deal with, you know, more easily they are sort of more innocuous in certain sense. What is not so nice is higher divergences, which appears from the far end, you know, the tau 2 going to. So, this is the tau plane, and this tau 1 is the real direction, tau 2 is the imaginary direction. From very large values of tau 2, we could get infrared divergences that should make us worried. So, I mean, again, infrared divergences we probably also shouldn't be scared of. It simply means that we are doing something wrong or asking a wrong question, and for example, we are probably doing perturbation theory about some unstable vacuum. So, the, what is some thing in our picture is that there are severe infrared divergences, and this is infrared divergences, this is severe infrared divergences arise from tachyons, and of these tachyons, there are many. So, first I will discuss the leading tachyons. The unquisted sector is given by the k equal to 0 term of the orbifold sum, and that is tachyon free as we would expect, and each of these remaining sectors has a leading tachyon. So, let's just analyze it, you know, from a simple oscillator calculation, and just confine our attention to, you know, something, this values of k, which is less than n minus 1 over 2. Remember, n is an odd integer, so n minus 1 over 2 is an integer. So, there are three complex untwisted bosons with the ground state energy of minus one twelfth. So, for the younger members in the audience, let me tell you that this minus one twelfth is actually the sum of all positive integers, and this is also related to the fact that the bosonic string lives in 26 dimensions, and also related to super stream living in 10 dimensions. So, when the fields are twisted, obviously, there is a shift of the ground state energy commensurate with the twist. So, we get some additional terms, 1 minus 2 k over n times k over n. Then there are four complex twisted fermions, and for the fermions, obviously, these ground state energy is positive, and this is actually half twisted, because the fermions are half twisted as compared to bosons. So, when we add them up, we get minus k over n, the constant negative term disappears, the quadratic term disappears, leaving us with only a negative term, and there is a factor of 2, which comes from the two sectors. So, in total, in every sektor, there is a leading tachyon of energy minus 2 k over n. This is also what we find from an analysis of the Jacobi theta function and the Dedekind eta function, that when we do an asymptotic expansion, it is slightly involved and not too involved, fairly straightforward, but we want us to be careful. We find that this gives precisely this structure, when k is below half of n roughly, and when k is above half of n, we get this. And in fact, this is a very nice feature, we find that in the tachyon spectrum, it is nicely symmetric about the midpoint. Namely, there is a k going to n minus k symmetry in the spectrum. So, from now on, I will stop discussing this second line, when k lies on this upper part, and look at only this part, when k ranges between 1 and n minus 1 over 2, and just multiply it by 2 to get the answer we need. Any questions at this point? However, so, if you thought the tachyons were bad story, let me add to the bad news even further. There are subleading tachyons. I mean, even if we have the tachyons, we can add more and more oscillators on it, and the spectrum obviously goes from negative and it will keep rising, but there is no guarantee that it will, the leading tachyon is all you will get, and there can be subleading tachyons. Quite remarkably though, in every twisted k-sektor, there is a very nice structure of the subleading tachyons. And from an analysis of the theta and eta functions, we find this. So, this is what we had seen in this previous slide, but now we have to understand the corrections to this formula. This is the leading exponential divergence, and then there are subleading exponential divergences, which I can write as 1 plus something something. And this something something is just this. It's, and we see that it follows a very nice structure. It's a nice exponential series with regularly spaced excitation energies. And it appears, every, each one of this appears with degeneracy 1. And this is actually one of the series that's been known to humans for the longest time. It's a finite geometric series, about whose importance we'll say more later. Okay, yes, thank you. So, here RK is the largest non-negative integer, such that this condition is satisfied. Roughly, this corresponds to the number of oscillators by which you have to act on this tachyonic vacuum before you read something massless or massive. So, RK by design includes only the tachyonic terms in each k-sektor. If you have RK plus, so in this sector, if you multiply this with that, and if you have RK plus 1, that RK plus 1 state will either be a massless state or a massive string state. And it's also interesting, as I said, this all sublating tachyons appear with unit degeneracy. And furthermore, although it's not obvious from this formula, that if you simply remove this d tau 1 integral, this does not change. Namely, the ZN invariance, which comes from the L sum and level matching, which comes from this d tau 1 integral, one seems to imply the other, which was very surprising to us in the beginning when we saw this. So, in order to substantiate what we saw from analysis of the theta functions and eta functions, we had to find a way to justify it using an oscillator calculation using operators. So, this is the twisted boson, oscillator expansions. And there's no zero mode here, because it's twisted, and that's the nice part of the story. That's the reason it's really localized at the tip of the cone. And then these alphas are the oscillators in the right moving sector, and alpha tilde is the oscillator in the left moving sector. And then there are the oscillators, which are complex conjugates of this oscillator x. And obviously, there are also untwisted boson oscillators and fermionic oscillators, by acting with which we can obtain higher string modes, tachyonic massless or massive states. So, this subliting tachyons, which really are excited states on top of this harmonic oscillator vacuum, this can be created only, it turns out, you can try to find it, but you wouldn't be able to, that this subliting tachyons, in which the energy still remains negative, they can be created only by the powers of alpha and alpha bar tilde of this quantity of this specific oscillator mode. And notice that alpha and alpha bar appear together, namely the complex and the complex conjugate quantities appear together, which implies Zedian invariance. And there's also alpha and alpha tilde, alpha is the left moving sector, alpha tilde is the right moving sector, and we have an equal number of oscillators. So, this is a way to see that Zedian invariance and level matching actually go hand in hand, as far as the tachyon is considered, this is obviously not true for higher string modes or massless or massive modes. And as we saw from the slide before, the excitation energy, as also can be seen from this picture, are just twice of this number. And if it's just one oscillator and one vacuum, all these states of tachyons, they manifestly have unit degeneracy. So, which nicely explains the observations that we made earlier. And just in case you are wondering, the fermions and the untwisted bosonic oscillators do not make any contribution to the tachyonic spectrum, but they do make contributions to the massless and massive spectra. So, here's a pictorial representation for a large enough integer n equals 13. So, remember I told you that we'll be only considering states between k equals 1 and n minus 1 over 2, which is 1 over 6. And the full spectrum obviously, you can just reflect it about a horizontal axis. So, it turns out that most of these subleding tachyons in number, they are sharply picked around the halfway point. For k equals 6, there are five subleding tachyons. And as we go down just one number, there is just one subleding tachyon. And for this below, there are no subleding tachyons at all. But as we increase n, we can take n to be 57 or 73, it will always be sharply picked, but more and more tachyons would permeate into lower values of k. So, what do we want to do? We want to calculate the contribution of all these tachyons. So, in order to do that, let us do a very crude approximation at first. We do something called a leading tachyonic sum, in which we take a sum over these odd values at every k. And essentially, we'll approximate them by 1, for a reason you will see why. And then just take a sum of these leading tachyons. So, just so that I don't have to go to a previous slide again, here's a tachyonic contribution to the integrand. This factor of 2 comes because of the symmetry in the spectrum. This is the leading tachyon and this is the subleding tachyonic contribution. And fk, I have already told you, is a finite geometric series, which we learnt in high school, how to sum a finite geometric series. For this finite geometric series, it's simply this quantity. And here notice something interesting, that it's 1 minus exponential of some negative quantity always. And the denominator is also 1 minus exponential of some negative quantity. So, as we go to the infrared region, namely we take tau 2 to infinity, we will simply approximate this fk by 1. And therefore, we simply neglect this and sum this series. But again, this is another geometric series, which unit coefficients and this is where we land. And here, as seen members of the audience is one of the most important features of the sum, that once we have summed it, on this right hand side, we have lost any information about whether n is an integer or not. This here is an analytic formula in n. And what is more, as we go to the physical region, 0 less than n less than 1, we find that this exponent actually is negative in the physical region. And the exponent below was already negative in the first place. So, again, in this physical region, after analytic continuation, we obtain something 1 minus e to the minus of positive quantity over 1 minus e to the exponential and other positive negative quantity. Therefore, as we take the limit tau 2 going to infinity, we, this gives a finite answer for large tau 2. This tells us that as we sum the leading tachyon, we actually seem to be getting closer to the kind of answer that we want, namely finite behavior in the region n less than 1. However, I have not really solved anything yet, because we have not yet dealt with all the tachionic divergences, namely, in approximating that quantity fk by 1, we have simply forgotten about the subleading tachyons, but this, although they are subleading tachyons, they are still tachyons. And an integral over the modular space for n greater than 1 would still give divergent answers. So, what do we do? So, we do something that we need to do, namely, we need to add all these subleading tachyons into the story. So, let us remember that in this subleading tachyonic sum, this fk has this property, where this integer r runs from 0 to rk. And here is what is sort of the unappealing part of the picture, this appearance of this integer rk, which is an obstruction to doing some nice sum. Therefore, what do we do? And obviously, as it might have been obvious to you, this presence of rk does not help with the analytic continuation. So, the simplest possibility, like any pragmatic physicist, is to take rk to infinity. And what does this mean? This means that in this subleading tachyonic sum, we are actually adding an infinite number of states, some of which are massless and almost all of which are massive. And this is an infinite tower of massive states. And these are all fictitious. So, let us see in pictorial terms, what this means. So, these are the fictitious states. These are the leading tachyons. These are the subleading tachyons that are physically present in for n equals 7. And now, in order to do the calculation, we simply add a host of fictitious states here, which can be massless or massive. And we just populate this whole lattice in the rk plane. And now, I spoke about doing a certain sum one way. Now, this is really the modified sum. We take k from 1 to n minus 2, and r goes from 0 to infinity. And this modified sum, I am calling by the name f tilde t. This t here stands for tachyon. And yet, oh, yes. Yeah, there can be some states that actually exist, but we will eventually add and subtract this. So, at this point, that's immaterial. So, these two states could exist. So, from the point of view of what I was doing so far, namely just looking at the tachyons, these are fictitious, but indeed. I mean, what it will end up doing is, it might change the degeneracy of some states later. Indeed. Thank you for that question. So, now, in this anticontenuation, we will just do, for each values of r, the whole k sum. The k sum, it turns out, is still a finite geometric sum. And this, there's an infinite r sum, which we will, of course, retain. And quite amazingly, this sum still has the same structure. It's a geometric sum, and for each values of r, the sum has an r-dependent factor, which appears naturally. And the sum still has this factor of n minus 1 upstairs. So, when n is greater than 1, of course, this is barely divergent, as we take tau to infinity. But when n becomes less than 1, this quantity goes to 0 at last tau 2, as does this. And this is a very nice convergent series. And we end up with precisely a finite answer in the physical region, 0 less than n less than or equal to 1. And in fact, quite pleasingly, this answer, if you put n equals 1 here, this is 1 minus 1, which is 0. And that's quite pleasing because we know that the answer n equals 1, the z1 or b fold is trivial, that's just the 10-dimensional super string partition function, which vanishes on account of spacetime supersymmetry. So, this is nicely consistent with that picture as well. And therefore, we find that although in this unphysical domain that tachyons have menacing appearance, they are not actually physically threatening in this actual domain. And there are several features, which go behind this result. And these are just the features that super string theory happened to be born with. So, what are these features? First, there are exactly n minus 1 leading tachyons with unit degeneracy in each twisted sector. This is an important fact for the nice behavior in the physical regime, 0 less than n less than or equal to 1. The structure would be lost otherwise. If we had n plus 1 leading tachyons, we wouldn't have this, we would have some different analytic property of this function. The leading tachyon mass-squared spectrum is also linear in k, and which really is the reason that we could do this geometric sum nicely. And this arises from some non-trivial cancellations because of the supersymmetry of the parent type 2 string theory. This is a non-trivial feature. This does not happen, for example, in the bosonic string orbifold, where the mass-squared spectrum is quadratic in k, and we don't know, at least I don't know how to do that sum in a nice way. And remarkably, the degeneracy of the subleading tachyons in each sector is also unity, and even there the mass-squared spectrum is linear in k. So, it seems that the amazing features of super string theory are conspiring to give us a very nice result. So, what have we done here? This is the f tilde t function that we had seen a few slides ago, and this f tilde r is a remainder, which this is nothing, this just says that from the actual function we have subtracted of something. But, quite remarkably, because this second sum that I described, that takes into account the effects of all subleading tachyons. And therefore, this f tilde r that's manifestly tachyon-free, even in the unphysical region. Otherwise, this full modular integral is quite divergent. So, we have split the integral into two parts. Now, this part of the integral is obviously divergent. This is the tachyonic part. But this, as I have added here and I will show you more evidence, this does not diverge in the physical domain, which is the region 0, the region between 0 and 1. And so, in order to say more about this, we would have to deal with this whole function. And how do we deal with that? So, here's something, you know, perhaps one of the most pedestrian approaches that one can take, that this integral involving f tilde r, the remainder integral, we would presumably find a smooth function in this region by some numerical interpolation method. So, we, you know, after having subtracted of this integral, this integral is just one number, that function is just one number we'll get as a function of n. And for n equals no, we know that its value is 0 for n equals 3, 5, 7. We would presumably be able to fit some nice function, so that we are able to determine its behavior in this region. And we could try to add more and more points and check for its numerical stability of a smooth curve in this region, which would tell us about the full function in this region, because we already know that the second part of the function is finite. And presumably, something like Newton's method, you know, which is one of the oldest known interpolation formula, that might be able to help us, because all these points are actually quiz based, as it must happen in the Newton series. If we add these two integrals in the physical domain, we would then get a finite answer. So, here is the tachyonic integral in the physical region, which is the second integral. And this is, you know, just evaluated numerically. So, this first part, you know, it is a crude approximation in which instead of the fundamental domain, really this rectangle is taken, leaving out this part. And this gives a very nice result, you know, which saturates near 0 and actually approaches 0 as n goes to 1. And doing this integral over the fundamental domain is also straightforward, but, you know, takes a bit more computational time. And we see more or less the same structure here, as in, you know, doing the very crude version with this truncated fundamental domain. So, which shows that this modular integral in the fundamental domain in the physical region is actually finite number for tachyon, and it can be calculated. So, that is that. So, that is an analogy that I would, we made in the paper that I would also like to emphasize here. So, one of the classics, classic analogs is finding an analytic extension of the factorial, which is defined only for non-negative integers. And this classic problem was addressed by Euler, among others. And the function with the right properties is nothing but Euler's gamma function. So, here we see the values of the factorial at these integer values. And, you know, having known it, known these values, we can analytically extend it on the entire complex plane. And this, you know, just show, obviously, is showing the real axis. And the right property is obviously gamma n plus 1, which is the factorial function for non-negative integers. Our situation here is, of course, the opposite. We know for a fact that for certain integers, our function diverges, like the gamma function diverges at non-positive values of its argument. And we want to find something that's nicely smooth and analytic on other values on the other half plane. So, this is an analogy that's probably worth bearing in mind as we set about exploring this in more detail. So, obviously, this is not the end of the story. On quite the contrary, this is just a beginning. There are several ways we can go ahead and deal with this problem. Some of the immediate questions that we can address are as follows. So, I spoke about some special, you know, conspiratorial behavior of the super string. So, please. Yes. Yes. In the n going to one limit. So, the quantity was going to zero, but not necessarily. So, this is not the full answer. This is just part of the answer. The full answer, we don't yet know. We have to do the remainder part and obtain something smooth. But this is the tachyonic part, which is going to zero, indeed. But as we can see from the graph, its derivative is not going to zero near n equals one. And precisely at n equals one, it's exactly zero. But even if we add n equals one plus 10 to the minus 50, that would be badly divergent, because, you know, this is really a tachyonic integral. Yes. This full answer. So, this is not even the full answer of what we said about to explore. This is part of the answer, which I believe is quite important. But, obviously, the full answer for entanglement entropy would be, you know, some G string expansion. And, you know, this is, obviously, the genus one term. And there would obviously be other contributions, which can be obtained as higher genus contributions to the partition function, you know, from, you know, the genus two demand surfaces and so on. So, it is obvious, at least, that for this answer at one loop, I'm sorry, maybe obvious question, that the cutoff is somehow set by l string rather in l plank, that the whatever term you get looks like area over l string. Yes, yes. That is obvious that this happens? Yeah, yeah, yeah. Sorry, why is it obvious? No, I mean, this comes from the calculation, right? I mean, I'm sure it's obvious to you, but you could explain why it's not obvious to me. So, one particular case, no, this... Oh, there is no G string, so you can only get area over... No, no, no. So, there's an expansion area over G. And let's talk about four dimensions. So, A over 4G. And 4G would be area over l string squared G squared. So, this is the classical three-level term in the action. What we are calculating is the leading calculation. Let's call this A1G squared plus dot, dot, dot. We are... This is in the limit of large area, you know, where other terms are suppressed, you know, log A and 1 over area. So, let's just look at this term. So, this is what we are calculating. So, it's area over 4L string squared, where the dependence on G string has disappeared. And the fact that you get the area term, that's also just obvious somehow in this... Yeah, it just emerges. Yeah, so, A is it possible to quickly explain why you get the... Maybe it's completely obvious. I see. I mean, essentially because of delta zero, which, you know, is the volume of spacetime. And here we don't get the complete volume of the spacetime because of the orbit fold. Yes, Daniel. Yeah, it's just one term in the expansion that we are computing. Yes? It's X. Yeah. Yeah, I mean, so, alpha prime is L string squared. Yeah, yeah, yeah. We have to know, yeah, I mean, as physicists, we care about the number, so we have to... I mean, just to show that it was... And it was not obvious that it would be a finite number and, you know... Yeah, yeah, because to begin with, you know, it's obviously 10 dimensional type 2 string theory, which, you know, maximum... I mean, or before it breaks it, but if you look at it from the point of view of replica trick, it just, you know, is a sort of replicated manifold of type 2 string theory, in which there's no renormalization of the Newton constant. Can I just ask one more obvious question? The obvious question is, is it surprising that the tree-level answer gave you A over 4g Newton in perturbative string theory? I mean, one might have thought that somehow that requires degrees of freedom that go beyond perturbative string theory. Is it... But it somehow happens that you get a finite A over 4g Newton finite, so everything is finite in perturbative string theory, somehow. Okay, okay, so, let me just... Yeah, please, please, please. This is a very... I'm sorry, didn't... Could you just replace it for a second? Yeah, yeah, sure. Yes, yes, yes, yes, yes. Yeah, yeah, yeah. No, no, no. So, indeed, we do... No, no, just to emphasize this point, please. Yeah, yeah, yeah, not yet. Yes, so that's this part, that's the remainder. And this is the divergent part for integer n. And thank you, Edward, for this answer. Thank you very much. Yes, exactly, exactly. Yes, yes, Marko. Please, yes. So, that's a great question. So, this is... Which is the sector k equals 0, l equals 0, which goes like volume. And that is 0 because of the fermion zero modes, or in other words, space-time supersymmetry. So, one obvious question is... You know, this is type 2B string theory, in which there are many conspiracies. So, how generic are the features that we have studied? And how can we generalize it to other important contexts, including possible string compactifications? Would it still hold? We don't know. And obviously, what we would like to do is make a more mathematically rigorous treatment of this whole integral. Are we able to rule out some pathological possibilities? Because, okay, you might ask that, the tachions analytically continued, gave a nice answer, but can we rule out the possibility that the massless and massive states won't behave like tachions in less than one region? We don't know yet. That's a possibility. We think it's physically unlikely that, at least from the point of view of effective field theory, that massive and massless states would do something so strange. But we are in the process of examining this from a more mathematical point of view, and hope to report it on soon. And obviously, we want to garner more numerical data of this whole modular integral. And this is some really precise numerics that we have to do sometimes probably with a numerical precision of 1000 or 2000 digits, which would be important. And we are working on some of these questions already, and you will get to know more about it, hopefully in the near future. Please. Yeah, no, I'm nearly at the end. Yes, yes. We hope that. We don't yet know if that's true, but indeed we hope that something like Carlson's theorem would be useful in this case. That's why I'll also mention Newton series, because Carlson's theorem has often been used to justify the uniqueness of the Newton series expansion. So that's the hope, and we have been thinking about it, you know, even before we put it out, we have had a lot of brainstorming over these ideas. So, this is almost my last slide, and I thought it might not be so inappro... I apologize in advance, but I'll put forward some of the implications of our results and maybe some vague speculations, and I apologize for having this vague slide. So, perhaps the most important question pertaining is the black hole information paradox. And quantum entanglement, or more specifically, it's some theorems related to quantum entanglement, namely the strong sub-adjectivity theorem, is the key player in Mathur's formulation of the information paradox, which was also subsequently used by Ulmeri, Marov, Polchinski, and Sully. So, that's something that we might like to address. And we could learn something about black hole entropy in string theory, perhaps, by this question, because the geometry that we have considered, it approximates the near horizon region of non-extremal black holes really well, and also in the context of ADS CFT, that would be an important question to address. The orbit fold breaks the supersymmetries. Is there any hope that we can study non-super symmetric black holes with this method, because the black holes in the real world are obviously not supersymmetric, and some of the most rigorous studies of black holes in string theory actually come from supersymmetric contexts. And in this sense, it might also be useful to look at an off-shell formulation, which is, obviously, much less developed than on-shell string theory. So, as this technology gets more and more developed, it might be worthwhile to use it to understand some of these non-classical geometries. And finally, as a hope as we were discussing also with Edward, the day he came here in Trieste, some of the most well-defined quantum field theories are obviously defined on the lattice, and on the lattice we typically have some sort of a type-1 von Neumann algebra. And string theory does something very close to what a lattice does. Namely, it gives us a natural ultraviolet cutoff. So, would it be too much to hope for that there can also be some type-1 von Neumann algebra in string theory? Obviously, at this point we don't know, because we don't even know how to define an algebra in string theory, but that is that. And for the last part, I would actually like to make a golden jubilee tribute. So, this year we had a golden jubilee of QCD, quantum chromodynamics, so one of the most successful theories in recent memory, and we had a very nice colloquium by one of the people who discovered QCD in a certain sense, and we had a very interesting colloquium. But in the same year, and at the same time, there was another paper that came out, which happens to be directly responsible for this hurdle that we are organizing, and we are still trying to grapple with its implications. I must say that this I didn't remember at first, but during a family video call yesterday, my mother pointed it out to me. So, it's this paper, which was published almost exactly 50 years back, and I urge you to read the abstract, and it's really full of fascinating ideas. Even if you look at the abstract, you would realize that this is really what we are talking about. And the time that this paper was published, obviously the author probably didn't anticipate that his ideas would be verified sooner or later. After the publication of this paper, it took two years before Hawking supplied the factor one quarter, and actually 23 years before Strominger and Wafa gave substantiator that this is truly statistical entropy. Coming back to our paper, obviously we are very excited about our work. At least I definitely don't know what impact it will have, or whether it will have any impact at all. That's really for posterity to decide, but okay, I can't really wait that long because in a sense, I am much more fortunate because I don't have to wait for posterity. I am privileged enough to have this distinguished audience before me. We will probably give me more insightful comments that we can build on. Thank you very much. This question is very physically important, but wouldn't there be an infrared divergence from massless states? Great question. Actually no, there could have been an infrared divergence from massless states, but let me go back and show you the formula. Because of the momentum zero modes, there are always sufficient powers of tau2 in the denominator. So even if you just look at the way Peterson measured, this is 1 over tau2 square, and inside that f, there's a 1 over tau2 cube. Well, I just meant, in the spacetime, the massless modes in the spacetime, you're talking about entanglement entropy in Rindler, in Rindler, wouldn't you expect of some long distance logarithmic, divergent contribution to the entanglement entropy, so how do you end up with a finite answer? That's a great question. I think that's different than the massless state. No, no, no, that's indeed a great question, and this simply is in an expan... Such sort of terms are responsible, for example, logarithmic terms in the corrections to the Baconstein hocking formula, but we are working in the limit of large a, in which we have sufficiently large, so that term is neglected, and this is the leading term in that expression, but yes. Okay, of course, that term is infinity, red, so it's bigger than a for all a, but I understand. I mean, a is much bigger than no. Right, sure, but the term is log a divided by zero, right? A in what, log of a in what units? I think a priori, this has nothing to do with black hole entropy. I mean, also the fact that you get area upon four g from a tree-level calculation, it's quite similar to Gibbonson hocking in spirit, but I think it's conceptually quite different. I mean, the way you can argue about it. So, first of all, you know that space-time partition function is logarithm of the space-time partition function, but of course, a tree-level partition function vanishes, so what happens to the space-time partition function? But I think this equivalence ignores boundary terms. So if you include the Gibbons hocking boundary term, you can argue, so it has to be there, and then you can argue that the bulk action has to be zero, as long as the deleton equations of motion are satisfied because it's the exact conformal field theory. And therefore, you only receive the boundary contribution and that has a nice analytic continuation in n automatically. It's like n minus one or something like that, and if you differentiate it to get area upon four g not. So it's actually, I would say it's a bit surprising because there is no black hole. It just supplied to you by gravity, the string theory. Yeah, e to the minus two phi r has to be zero because otherwise, there would be deleton table. But there has to be Gibbons hocking term, which is key divided by eight pi g, and that contributes. No, no, so it's a space-time calculation. It would be nice if there is a worksheet method of doing it, but I think one can argue that effective action has to have that term. Yeah, there is no, at present, no worksheet methods of computing. Even for a Schwarzschild black hole, you could ask, how do you compute the Gibbons hocking term from a worksheet calculation? And at the moment, we don't know, but we don't doubt that that term exists. Just in terms of type one versus other type algebras, if you take the zero brain theory, but maybe more generally, I want to ask it in the context of holography, but if you just take the zero brain theory as the boundary theory for the supergravity dual, that theory clearly defines the type one algebra, isn't it, because it's just quantum mechanics, and at finite n, there's no scope for any divergences. And so, isn't that an example where the non-perturbative definition of a theory of gravity gives the type one algebra? And one could do higher dimensional analogs, but then you get field theories, and I'm not sure whether that creates some issue, but here is a clear, isn't there a clear example here in matrix theory or zero brain theory that gives the type one algebra? This is tied to my talk, but some people won't be there, so I just want to ask. I think the answer is yes, but in those descriptions, we don't know why, what spacetime locality means in so on. That's true. I can also add that if you take a two-sided black hole, whose near horizon looks like a render, then we do know that there is a type one algebra, because it's a direct product of left conformal field theory and right conformal field theory, where there is an irreducible representation of the algebra of the right. So, that also is a motivation why you expect a well-defined answer in the bulk, corresponding to this thermofield double state. More questions or comments to upo? Okay, if not, let's thank you again.