 Thank you very much. It is a real pleasure to be back here and I Was here several times in the 1990s helping to organize. I think actually the first time I ever came to Italy was a trip here in around 1986 and three things Really struck out or three. I remember three things from that first trip One was getting to meet Abdul Salam who wanted to know about all the latest developments in string theory The second was Italian pizza and the third was Going into a back room at the Adriatic on being given an envelope with a million lira in it for expenses He's all created a very strong impression It's really nice to be back. So let me start by giving you a plan of the lectures so the first two Lectures I'm going to start by giving you kind of an overview Then I'm going to talk about what might be called Classical moonshine and This involves some elements of what's called monstrous moonshine The subject which was developed starting in the late late 1970s through the 1980s and 1990s so I'll talk about a Particular construction of a conformal field theory due to Franco-Wapowski and Merman that has the monster group as a symmetry I'll talk about what are called Mackay Thompson series and Their connection with all genus zero functions Then I want to talk about something that I think is not Perhaps so well-known in the string theory community, which I've called deconstruction And Called the Grice algebra things called Miyamoto involutions These are topics which appear in this conformal field theory But which I think might have some broader applications to conformal field theory and maybe to ADS through ADS CFT I'll briefly mention Another example classical example Conway moonshine and Briefly perhaps talk about the construction construction of the CFT that has this Conway moonshine So the third and fourth lectures I'm going to talk about a more modern version of moonshine which really started in 2010 and is continuing with the efforts of lots of people and Here I will Tell you a little bit about jacobi forms and Their connection to the n equals to super conformal algebra I'll tell you about the result that revitalized interest in moonshine which Is a result connecting the elliptic genus of a k3 surface with the Matthew group m24 and then I Will tell you a little bit about umbral moonshine, which is an extension of this and I'll talk about the relationship of this to Meromorphic jacobi forms a little bit about something called the discriminant property and and Most of this is going to be fairly mathematical so in the sense that the title of my talk was Moonshine in physics I Will towards the end and sprinkled throughout try to talk about how this might connect to conformal field theory string theory ADS CFT and things like that, but there will be a fair amount which is just Mathematical so let me start by talking about one of the Great theorems of 20th century mathematics Which is the classification? finite simple groups So I assume you all know what a group is and that you all know what a finite group is finite groups don't play as big a role in particle physics as Continuously groups, but they obviously show up in things like the study of crystals and Just to make sure that we're on the same page recall that a normal subgroup And of a group G is a subgroup that's that the left and right cosets are equal and that's written with this funny triangle symbol and Give it a normal subgroup you can put a group structure on the cosets to define the quotient G mod n which allows you to decompose a group G into a smaller group H and finite group theorists often use the notation which I will borrow by writing G is n dot H which means precisely this that G has a normal subgroup n and the quotient of G by that normal subgroup n is the group H so There's a kind of analogy simple groups Are groups that have no normal subgroups? So they can't be decomposed into smaller groups So you can kind of think of a finite simple group as being like a prime number in The sense that you can always decompose numbers into products of primes You can decompose finite simple groups into sort of products not exactly products but using a chain of normal subgroups into smaller and smaller groups and So if you understand the finite simple groups you understand finite groups in the same way that once you understand primes You kind of understand all numbers So the classification theorem is the statement that all finite simple groups are either in one of a set of infinite families of groups and the first family is usually written by physicists like this it's the cyclic group with P prime the alternating groups a n for n greater than or five consisting of permutations even permutations of n objects and Then there are what are called simple groups of lead type and These come in families, which I'm not going to discuss in great detail. They're classical exceptional listed and basically what these involve are taking groups of lead type that you lead groups that you're familiar with and Writing them as groups over finite fields. So for example, we're all familiar with the group SL2 But you can take SL2 two by two matrix with determinants one over the group with seven elements and That defines a finite simple group turns out to have order 168 and Similarly, you can take some plectic groups or thogonal groups and there's even a way of doing this for exceptional groups And there are various twistings of it But essentially what you get are something like six or seven infinite families of finite simple groups where you look at, you know Classical matrix groups over larger and larger matrices over fields with a finite number of elements and then There are 26 sporadic groups, which don't fit into any of these infinite families and these 26 groups are sometimes usefully viewed as a group of 20 Called the happy family finite group theorists like funny names These are all subgroups of M Which is the monster group, which is the largest sporadic group and Then there are things called the pariahs, which are not subgroups of Monster group there are six of them These funny initials don't necessarily mean anything But actually they're associated with names so J stands for Janko, O for Onan, R U for Rudvalis, L Y for Lyons and The groups in the happy family can be further Usefully organized into a set of kind of generations so the first generation Which are the first sporadic groups to be discovered are the Matthew groups M11 M12 M22 M23 4 There's a second generation All related to the leech lattice, which I'll talk about a little bit. These are all groups which either arise from automorphisms of leech lattice or automorphisms of the leech lattice that fix some number of points in the leech lattice So they're related to subgroups of a group called Conway zero, which is the groups of automorphisms of the leech lattice and They include other so simple Conway groups three of them This is a Zuki group Bofflin group the Higman sig sig Higman sigs group. I think and one of the Janko groups and Then there's a third generation consisting of some Fisher groups Thompson group Herata Norton the held group the baby monster and the monster now So far I would have to say this is a really bad introduction because all I've done is put a bunch of letters down Without explaining what any of them are all these funny names This is the monster. This is the baby monster so The reason that this is interesting is that this classification and this kind of structure just comes out of pure mathematics But it seems that the best way of understanding many of these groups and many of their features At least for those in the happy family Involves conformal field theory string theory string compactification on Calabi outspaces and that's very peculiar so in the string theory literature I went through The papers that I could find and I may have missed one or two, but I looked at all the papers Involving various aspects of moonshine both from the old days and more recently and in those papers You will find a set of groups that are Linked to the automorphism groups of particular super conformal field theories. So the monster is The automorphism group of a C equals 24 conformal field theory The baby monster is the automorphism group of a conformal field theory With C equals 23 plus a half and I'll tell you roughly how it's constructed The Conway group is the automorphism group of a super conformal field theory with C equals 12 M23 and M22 Are also automorphism groups of super conformal field theories that preserves an n equals 2 or an n equals 4 super conformal algebra. These are all kind of classical examples And It turns out that in further pursuing moonshine many of these other groups like the Thompson group the Suzuki group and the Clawson group also show up so in some way that I would say we really don't understand the right framework for studying these sporadic groups seems to involve conformal field theory and aspects of string theory and That's kind of what I want to explain I want to use techniques that come from conformal field theory to explain some of the structure where these groups are coming from what kind of things you can learn about these groups and You know the eventual goal I think here is not just to use string theory to learn about finite group theory, but rather to understand Why it is that there's a connection between these sporadic groups, which are rather bizarre exceptional strange objects and string theory and string theory is supposed to be a theory of quantum gravity Why is it also a theory of so many of these sporadic groups and? I Think there are two reasons to think that there might be something to be learned One is that although some of these theories in terms of compactifications are Compactifications down to two dimensions so for example for a sequence 24 theory you could use that to compactify the bosonic string or the heteronic string down to two dimensions Which doesn't seem very physical, but by ADS CFT we might learn something about three-dimensional gravity and some of these other groups like M24 are connected to compactifications on K3 surfaces and Compactifications on K3 Are kind of you know everywhere in string theory Both in string duality and in looking at string Compactification on Calabi outspaces many Calabi outspaces are K3 vibrations And so if you understand something about Special about super conformal field theories related to K3 you understand something about more general kinds of string compactifications so that's sort of my motivation and I Won't say you know, I think it's enough justification to look at something. It's rather mathematical All right, so the examples that I've described here are examples of what I would call classic moonshine classic in that they were first done You know a number of years ago and also in the sense that there is an explicit conformal field theory construction Which realizes these symmetry groups so in the constructions the particular sporadic group of interest commutes with the very sorrow algebra and so if we look at the partition function of the theory and For the moment, I'm just going to be focusing on the holomorphic part of the theory Then this is going to have a Q expansion The function of the parameter tau Labels the one-loop partition function of the theory Such that Z of tau will be a modular form general of some level meaning some subgroup of the full modular group and Such that this coefficient C of n will be sums Dimensions of irreducible representations of sporadic group G That is a symmetry of the conformal field theory commuting with very sorrow That's since you know C to the n is counting the degeneracy of states at a given l not eigenvalue That follows from the fact that the group commutes with the very sorrow algebra It's trivial, but what I'm about to say will put some meat on it so It's sometimes I think said in the physics the physics papers that moonshine is essentially some kind of connection between Finite groups often chosen to be you know interesting large groups and modular forms But it's really more than that because if that's all it was it's trivial to make Examples and the things that really deserve the name moonshine are more special than this so for example Take any lattice L You can make lattices with all sorts of interesting symmetry groups You can just take a hexagonal limit lattice which has a you know hexagonal symmetry But you can take very large lattices that have all sorts of interesting exotic symmetries and you can consider string theory on RD mod L if L has ranked D and This defines a C equals D conformal field theory or in the math literature of VOA with a Symmetry which is the automorphism group of that lattice and a partition function Z of tau which will exhibit the symmetry So in this way you could exhibit moonshine in this sense for almost any group And I think many of the sporadic groups even the ones that are not part of the happy family Can be viewed as automorphism groups or subgroups of automorphism groups of large lattices But moonshine requires more than this It requires a certain rigidity and that rigidity is usually connected with genus zero subgroups of SL2R so you need also a kind of rigidity and Exactly what form this rigidity takes Buries from case to case But in the case of monstrous moonshine and also in the Conway Shine this rigidity Appears in the following way if you have a symmetry Group G and you have an element G in G That acts on the state space of the conformal field theory then you can consider a twined version of the partition function where you take the trace of G In each eigenspace of L naught and These Objects will be modular forms Well in general you would expect them to be subgroups of SL2Z Because only certain elements of SL2Z will leave this boundary condition fixed But in fact they often turn out to be modular forms for certain finite or discrete subgroups That are a little bit larger than SL2Z. So I'll just say subgroups of SL2R and In the case of monstrous and Conway moonshine. These are all genus zero subgroups so Sorry Discrete sorry. Yeah, discrete subgroups. I was just about to say that Yes, they're weight zero the partition function is weight zero and these twine versions of them are also weight zero All right, so what what do I mean by genus zero? Well, let's Recall from string theory That the fundamental domain For the modular group is the following region bounded by So this is the tau plane So bounded by these lines at the real part of tau is equal to a half and minus a half and lying above the semicircle How equals one? So in other words under Elements of SL2Z any point in the upper half plane can be mapped to this fundamental domain F and If you want to be a little more precise you should actually include this part of the boundary Because this is mapped to this by tau goes to tau plus one and this is mapped to this region by tau goes to minus 1 over tau so this is a fundamental domain in its boundary and You can take this fundamental domain and what does it look like it looks kind of like this It has these cusp points here because their elements of order three that leave these point fixed There's kind of a cusp off at infinity and you can map this fundamental domain by a function It's usually called J of tau To the Riemann sphere in other words as tau varies over this fundamental domain There's a complex function J of tau which maps this fundamental domain one to one onto points on the Riemann sphere So whenever you have this situation H being the upper half plane with coordinate tau and you have some Subgroup Which I'll call let's say gamma G Of SL2R which acts on the upper half plane And whenever there's a function TG which maps this to the Riemann sphere Then you say that gamma G is a genus zero subgroup and the function TG of tau is Called the help module for that group these genus zero subgroups are Kind of special There are many many subgroups, which are not genus zero And some examples will come up later So if you consider gamma not n Which consists of elements of the modular group such that C is congruent to zero mod n Then this is genus zero and is equal to two three four up to ten and then twelve thirteen sixteen eighteen and 25 if memory serves me right But for no other values event So going back to Zohar's question It's absolutely true that if you take The coefficients C of n you can always decompose them Into trivial sums of the trivial representation or you can decompose them into different ways But this condition That first of all this be modular and in some cases that it be modular and be a genus zero Subgroup is extremely constraining and untrivial and will fix what the decompositions are in general And is also what makes moonshine More than just a connection between modular forms and finite groups It really Turned it into something that's rather special. Yes. I guess so. Yes. I mean I'm not sure what is the question of terminology or I've seen this terminology used but I'm not a hundred percent sure what it means. So I I'm not sure I should Say anything but I can but it is true That if gamma g is a discrete subgroup of SL2R I can take the quotient of the upper half plane by this group and this will define a two-dimensional surface Which can have genus zero one two three, etc. Depending on the exact form of the group. So I mean, I think that's the correct statement and the particular cases that arise in in moonshine or genus zero So I'm going to I will I will spell out some of the details of this because I know this has been very broad brush But I just wanted to say I mean still it's kind of part of the overview that These examples which I'll talk about in more detail are the examples of classical moonshine where there is a CFT But the more modern versions of moonshine involve Different structures which are less well understood So in particular they involve Mach modular forms which I will define for you Rather than modular forms they involve groups which are related Emeyer lattices which I will Say more about in a moment and There is a connection Which I would say is not perfectly understood To case three surfaces and maybe to more general Calabria manifolds So for this more modern kind of moonshine There is no general explicit CFT Vertex operator algebra string theory background wherever you want to call it construction There are hints of connections black hole Eps state counting which I'll mention when we get there and there are also some hints of relations between Classical and modern moonshine the whole situation is murky because the hints are just that they're not direct connections and Because we don't have an explicit Construction there are a lot of results that that Can be derived without complete understanding So all right, so that's kind of what I would like to tell you about first starting with the classical moonshine and we'll start with a discussion of monstrous moonshine From a conformal field theory and string theory point of view So we can think about this in terms of a compactification of the bosonic string So you know that if we compactify the bosonic string on a d-dimensional torus There is a norene moduli space of compactifications which has the form of a double coset and You move around in this moduli space by varying the constant metric and two form fields on the torus and The general point norene moduli space is you know Not a complicated, but it's not a rational conformal field theory in that you have an infinite number of Terms in the partition function it doesn't holomorphically factorize in terms of some finite number of Characters of some larger algebra, but we can go to special points and one of those special points arises when d is equal to 24 Where we can take the even self-dual lattice of momenta to simply be Direct sum for lambda is an even self-dual rank 24 lattice So this situation generalizes a Construction which shows up often in string theory That is if you want to have an even self-dual lattice then that only occurs when D is a multiple of eight and When D is equal to eight the unique answer is the E8 root lattice when D is equal to 16 You can have the E8 E8 root lattice or you can have a weight lattice of spin 32 mods E2 Those of course are the gauge groups of the heterotic string And then in 24 dimensions, there's the leach lattice and there are 23 other lattices called nemire lattices These nemire lattices are not Well, they play various roles But let me say just a bit about them because they will show up later when we talk about umbil moonshine So they are they are constructed starting from a root lattice of a D or E type Meaning you take sums of root lattices of a and d n or e6 e7 and e8 Yes, could you speak up? No, no, it's no. No, this is purely a spatial compactification So I'm assuming that time and one of the spatial dimensions are non-compact. So if you take an ade Some of ade root lattices with total rank 24 such that each component has equal coccidron number and What is the coccidron number? Well, if you have the root lattice, let's say so here's the root lattice of Su3 there are always vile reflections where you reflect in the hyperplane that is Orthogonal to the root and For any root lattice there is a group an element of the group of such reflections called the coccidron element Which is the product over the simple roots it's the positive simple roots of these vile reflections and the order of that Element is called the coccidron number So I can give you a little table of what these are So for a m minus one The answer is m So in particular for a one, which is su2 It's the order is two. It's just a single reflection for a two, which is su3. The order is three Corresponding to the three-fold symmetry of that lattice for d1 plus m over 2 So for the orthogonal groups the answer is m and for e6 e7 and e8 The answer is 12, 18, and 30 So here's a little exercise using what I've just provided you classify the root systems x that have rank 24 and equal coccidron number and you'll find That there are 23 examples a 1 to the 24th is obviously ranked 24 and all the components have the same coccidron number a 2 to the 12th is another example Then they're turned out to be more complicated things like a 11 d7 e6 Which also has ranked 24 and using that table has equal coccidron number and then This continues and you get things like e6 to the fourth and e8 cubed Alright, so these things will appear later But for right now, I want to consider a compactification where we don't choose any of these But we choose the leach lattice So we'll choose for our compactification the leach lattice Which is the unique Even self dual rank 24 lattice with No roots Meaning that the set of elements in the leach lattice With length squared 2 with the usual normalization Is the empty set there are no points of length squared 2 Instead the shortest vectors have length squared 4. Yes, but which classification? well in the I Haven't really told you why The Neymar lattices only involve ADE, but it's just a statement. I mean for B and C There's certainly a coccidron number. There's a coccidron element, but they're not they have roots of different length so they're not simply laced and For reasons that I think would involve much more detail You can't use them to construct even self dual lattices in this way It possible. Sorry say that again my vial reflection in what I'm really so I'm I'm really considering compactifications Where I have a 24-dimensional torus and then either R11 or if you want to work in Euclidean signature R2 But everything that I'm talking about all the vial reflections are Occurring in this 24 Dimensional spatial part of the compactification and I really haven't assumed or said anything about what's happening in the remaining two non-compact dimensions so You know you can work in In Kowsky or Euclidean signature sort of as you please and the vial reflections don't act here at all You could consider Compactifications down to you know one or zero dimensions where you try to compactify this part as well That gets into all sorts of kind of complicated and subtle issues which are not relevant for what I'm discussing and I wouldn't you know I don't necessarily have to talk about this in the context of string theory, but I could just talk about C equals 24 conformal field theories But I think there are a few useful things to be said when you think about it in terms of string theory Compactifications, so that's why I'm doing that. All right, so for the theory that's related to monstrous moonshine we pick the leach lattice and The partition function of this theory Then completely factorizes Into the theta function of the leach lattice divided by 8th to the 24th Where the theta function of the leach lattice is a sum over elements of leach lattice due to the lambda squared over 2 and From now on we can focus just on the holomorphic degrees of freedom and Once we do that we can then decide what we want to do with the anti-holomorphic degrees of freedom We can do something to them we can consider boundary conditions to define d-brains etc etc But the construction that I want to talk about just involves a holomorphic part And that's a legitimate thing to do since we've chosen a point in the ring moduli space Where the theory factorizes and purely into a holomorphic and an anti-holomorphic part. All right, so let me First of all tell you what the construction of Frankel and Wipowski and Merman Involves in terms of conformal field theory language So if we look at the states and the CFT that we've just described Defined by string theory on the leach lattice and we look at the holomorphic part there are states which are Fox-based ground states I have 24 free bosons, so I have 24 Oscillators which annihilate the ground state The zero mode I'll call p hat I and I build up a Fox base from this ground state in the usual way by acting with the creation operators clearly the partition function of this theory has a contribution from the vacuum It has a contribution from the first excited state of which there are 24 and then this factor of 1 9 6 8 8 4 comes from states alpha minus 2 Alpha minus 1 I alpha minus 1 j and states on the leach lattice that have length squared equal to 4 So there are 24 states here There are 24 times 25 over 2 here and There are 1 9 6 5 6 0 states here And if you add up these three numbers you'll get 1 9 6 8 8 4 so this Partish the partition function of this theory is equal to the j function up to a constant and It is a weight zero modular function Which is the help module or a help module? or a modular group and It's unique given a constant term and the fact that it starts with Q inverse and It was famously Noticed by John Mackay That 1 9 6 8 8 4 Equal to 1 9 6 8 8 3 plus 1 and This is a song of the dimensions of the first new irreducible representations of the monster sporadic group at least where you order them by their size and This was regarded as you know being so shocking and astonishing at the time Because at the time there wasn't wasn't an understanding that you naturally get weight zero modular forms in conformal field theory so 24 however is not the dimension of Anything other than 24 copies of the one dimensional representation, so that looks a little bit odd and What Frank Lopowski and Merman figured out how to do was to take this theory and Modify it to get rid of that term and To give you a theory that has the monster group as its automorphism group, so I want to tell you what the modification is and In the next lecture, I'll tell you some things about How you can see that the monster is the automorphism group, but I'm not going to be able to give you a full Explanation because that's a rather long involved story But I'll try to tell you about some useful things that you can do using the conformal field three point of view So what Frank Lopowski and Merman proposed to do was to consider a asymmetric Z to orbifold of this theory Where you take the holomorphic part of the left moving coordinates and do an orbifold By a minus one Action so the partition function of their theory as usual is going to be Trace in the untwisted Hilbert space that we've just constructed of The projection operator on to G invariant states Maybe I shouldn't call this G. Let me call it theta So I don't confuse it with elements of the monster And then in order to have modular invariance, we know in orbifolds we have to include a twisted Hilbert space and In that Hilbert twisted Hilbert space. We also have to project on to invariant states and this it will turn out is equal to J of tau with no constant term and with the monster group as It's automorphism group So my automorphism group I really mean that there's a symmetry of the theory which would preserve OPEs correlation functions, etc. The full structure of the conformal field theory Yes, yeah so, I mean Well, let me so I'll tell you briefly what this involves So what happens when we do the z to orbifold? Well, first of all we project on to the invariant states So if I look over here These states have eigenvalue minus one under the z to action so they will disappear These states have eigenvalue minus one these have plus one of these states Well, I'll have to construct linear combinations Which have eigenvalue plus or minus one under the Orbifold action So it's not too hard to see That this part of the theory Simply gives a contribution, which is one half data leech over to the 24th, which is just one half of one and Then the action of theta well theta takes states with momentum lambda to those with momentum minus lambda so they don't contribute to the trace so the only things contributing to the trace here are the oscillator states and Because they have eigenvalue minus one Under the action of theta I get a one plus q to the n rather than a one minus q to the n from the trace of theta and Then we need to compute what the trace is in the twisted sector and In the twisted sector We have boundary conditions the Xi has to come back to itself up to a minus sign and Normally this would mean that the ground state is labeled by the fixed points But in an asymmetric orbifold, that's not the right answer Modular invariance requires The a to go from one to the square root of a number of fixed points if you were to think of this as a symmetric transformation and I'll say a bit more in a minute about why that's the case so actually a goes from one up to two to the twelfth rather than two to the twenty-four and Given that you can then compute the contribution to the partition function in the twisted sector So there's a two to the twelfth from this ground state degeneracy there's a q to the half and this a half is minus one Plus three halves this three halves is 24 over 16 And that's using the fact that the twist field for a single boson has dimension 16 and we're twisting 24 of them And then in the twisted sector I have half in a demoted Oscillators and then I have to include the projection on them. So if you Add these up to compute the total partition function You can write it in terms of the usual Jacobi theta theta functions And you can show That it has precisely effect of killing the 24 and leaving everything else as it was before Now this more or less had to work Because as long as the orbifold Satisfies level matching so preserves modular invariance you had to get a weight zero modular form here The tachyon survives so this has to start as q to the minus one So the only thing that's undetermined given that it's modular invariant is the constant term and the constant term is Clearly zero because you kill The states that contributed to it and in the twisted sector The all the states are massive because the dimension of the twist field is three halves So this form follows just from the fact that this twist preserves modular invariance. What time did I start? 1030, okay, I guess that means I'm done with this lecture but I'll just I guess I'm giving another one soon. So I Just want to tell you a little bit about what we're going to do It turns out Well, so one thing we would like to do is to see some evidence that this theory actually has the monsters at symmetry group I'm not going to be able to show you that in detail, but I Do want to show you how a number of aspects of The group structure of the monster are evident in this orbital construction And then I want to tell you about an interesting construction That constructs a kind of an algebra that was connected with the monster before this conformal field theory Understanding and that I think has more general application And I want to also show you how the baby monster can be extracted from this using a technique Which really follows just from the structure of conformal field theory. So we'll do those three things this afternoon