 For those of you that want to get into moonshine and want to look at some of the group theory that's involved, these sporadic groups are much too big to do anything by hand, at least for mere mortals. A lot of information is contained in a huge red book called The Atlas of Finite Group, some of which is online. There's a software package called GAP, which is used for doing many computations. It knows, for example, the character tables of all the sporadic groups, and you can work out many other things. And the material that I talked about in the first lecture is contained in a paper I wrote with Lance Dickson and Paul Ginsberg called Beauty and the Beast, Superconformal Symmetry in a Monster Module, which is in communications and mathematical physics sometime in the late 1980s. So to summarize the lecture this morning, Frank Lepowski and Merman construct a C-24 holomorphic formal field theory. You can think of it as the bosonic string on a special 24-dimensional torus, modded out by an asymmetric Z2 action, and if you look at a decomposition into the vector spaces in which the Virisorrow operator L0 takes the value n plus 1 and defined v of n to be that state space, then the statement that this has monster symmetry, well, not clear exactly what that should mean, but let me discuss some aspects of it. And then try to work out some of the implications. So first of all, if the monster commutes with Virisorrow, that should mean that each one of these vector spaces can be decomposed in terms of vector spaces Ri, which carry an irreducible representation of the monster. What is an irreducible representation? It's a set of matrices that obey the same algebra as the group elements, the same multiplication rule, and a vector space on which a representation can act is called a module. So in other words, you should be able to find a set of positive integers such that you can decompose the states at each level into a sum of representations of the monster or a sum of vector spaces carrying representations of the monster. And the partition function of the theory is a sum on n, the more the better, I think, the more the better. That's good. No, no, no. Oh, no. Oh, OK. The recording. OK. All right. OK. So the partition function, which as I said is this function J of tau with zero vanishing constant term, should be the dimension of the vector space Vn times q to the n. And for the monster, conformal field theory, this partition function takes the form q inverse plus 196884q plus dot, dot, dot. And 196884 is 196883 plus 1. And this is the dimension of a vector space which is one dimensional and carries a trivial representation of the monster. And this, sorry, I should have called it the dimension of V1, dimension of r0, r0 is the trivial representation. This is the dimension of r1, r1 is the 196883 dimensional irreducible representation. All right. So let me ask if there are questions, because I think A went a little fast this morning, and B was a little bit jet lagged. I'm not sure I'm probably still jet lagged, but I think I was slightly incoherent. And so before I proceed, I'd like to know if there are any questions that I could answer that would make it clear of what's going on, because this is kind of the starting point. To do this decomposition, you only need to commutes with l0, but in fact it commutes with l0 added. That will probably be clear from something I'm going to say in just a moment. So, huh? 2-0. Yes. I'll talk about that in just a moment. Well, maybe in 20 minutes. They're not ferrosauros. They're not ferrosauros, because the OPE, they're primary, I mean the stress energy tensor is not primary, it's quasi-primary. They are conserved currents, but they're primary rather than quasi-primary. So their OPEs are not that of an energy momentum tensor, but they do allow you to do some things involving new energy momentum tensors, which is one of the things I'm going to talk about today. But they're distinct from ferrosauros in that they're primary rather than quasi-primary. Okay. That's right. And I'm going to discuss something based on that in just a moment, and I'll write down some OPEs that make that clear. Yes. Right? The leach lattice is a rank 24 even self-dual lattice. Now, I don't know, it's sort of like if I asked you, what is the E8 root lattice? Well, you could probably give me some descriptions of its symmetries, maybe a construction in terms of basis vectors. I could give you a set of basis vectors for the leach lattice, 24 vectors that generate the leach lattice. I don't think it would be very illuminating to just write them down. It has a huge symmetry group, it's a very special object, and its construction and proving its properties is somewhat non-trivial, and I am not going into it just because it would be sort of, you could give a whole lecture on the leach lattice, but I don't think that would really be what I want to do. But part of the problem with this subject is that it involves many highly intricate technical things. Each one of which would take some time to explain in full detail, and so that's just the nature of the beast. The other property of this conformal field theory is that if you look at function Tg of tau, which is the trace in this conformal field theory of g, q to the l0 minus c over 24, where g is an element of the monster. First of all, this depends only on the conjugacy class of g, because if I replace g by h, g, h inverse, the trace doesn't change. So there are as many of these trace functions as there are conjugacy classes in the monster, I think there's something like 192, I forget. And if you have this kind of decomposition where at each l0 eigenvalue you can decompose into representations or modules for the monster group, then it follows that you can write these as a sum n equals minus 1 to infinity times these multiplicities with positive integers times the trace in the representation ri of the element g times q to the n. And these you can extract from the character table of the monster, that is they are, that's what a character is, a character of a group element is the trace of that group element in a representation. So the non-trivial fact which was discovered by Conway and Norton is that these are all help modules for what I described this morning as genus zero subgroups of SL2R. And the amazing thing about this discovery by Conway and Norton is they found this out before this construction of an explicit conformal field theory. As a matter of fact, they figured out before the monster group was even shown to exist. It was just a conjectural character table and using that they were able to essentially guess this. So this was a conjecture and then Borchard's about 20 years later gave a proof and his proof very directly involved elements of string theory, particularly he had to use the no-ghost theorem. So what I want to talk about this afternoon is to what degree you can see some of the group structure in the monster. And how you can see some evidence that it's really a symmetry of the conformal field theory. So a conformal field theory is more than just a bunch of dimensions of vector spaces. In a conformal field theory you also have operator product expansion of correlation functions and saying it's a symmetry really means it should preserve that structure which is more than just giving a set of vector spaces. So I want to discuss how some of the group structure are visible. That's the first thing I'll discuss. And then I want to talk about an algebra that is directly related to the questions starting the 1, 9, 6, 8, 8, 3 plus 1 mentioned two zero fields that exist in this theory because that allows you to get some very non-trivial evidence for the monster symmetry and it also involves a construction which I think might has not really been used to my knowledge very much in the physics literature and might have some applications more broadly. So the monster has two conjugacy classes of order two elements and if you want to sound smart in the math literature you would say these are involutions rather than order two. And no, it means the class consists of elements of order two. Order two means the G squared is equal to one. So there are two classes that contain order two elements. They're called 2A and 2B, you have to call them something. And these have centralizers, so the centralizer of some conjugacy class C is equal to the set of all elements in the monster such that GH equals HG that is the commute for all B and M and all H in the particular conjugacy class. So the centralizer just means you act by group conjugation and the element is left invariant. So the centralizer of the 2A element in the monster has the form 2.B and the centralizer of the 2B element has this form. Now, what in the world does this bizarre notation mean? Well, this is a notation which I introduced in the first lecture. It means that the centralizer of 2A has a B2 normal subgroup. If you take the group which is the centralizer and you mod out by the Z2 normal subgroup, you get a group B and this group is called the baby monster. And I believe it's the second largest sporadic group. This notation means that the centralizer of the 2B has a normal subgroup and the quotient of that is the Conway group. But what in the world is 2, 1 plus 24 mean? Well, this is something that is actually not all that unfamiliar in physics. So it's called an extra special group in the finite group theory literature. But it arises in a way that's not, I hope, totally unfamiliar. So I'd like to explain it. So in conformal field theory or in the lattice, theory of lattice for a text operator, algebras, one of the good things about this talk, maybe it's a bad thing about this talk, is that it all involves old fashioned string theory. You don't need ADS, CFT, string duality, anything like that. It's just all conformal field theory. So in conformal field theory, you know that there is a state operator correspondence. So a state in this theory that has some momentum lambda corresponds to the operator, which is the normal ordered exponential, although momentum times x. But as you may remember from Poltinski or Green-Schwartz-Witton or wherever, there is an additional factor that you need to put in here, which is needed for associativity of the operator product expansion of the operators in this theory. And this so-called co-cycle factor obeys composition rule. What this object does is it defines a projective representation of the lattice that these momentum live in. So I really want you to think of a lattice as being a group of finite translations. So here is a two-dimensional lattice with two basis vectors e1 and e2. And a projective representation means that when you compose two translations by lattice vectors, you come back up to a phase to the translation. So te1 times te2 would be some phase times te1 plus e2. And we know in physics how you obtain such a situation. You look at a two-dimensional lattice and you put a magnetic field transverse to the lattice. And then you get a projective representation essentially because of the Aharonoff-Bohm phase when you get in translating in the presence of a magnetic field. Now exactly that situation occurs in string theory. To get to the point in Norene-Moduli space, the 24 comma 24 lattice is two copies of a leach lattice on the left and right. You need to turn on a constant metric, but you also need to turn on a constant anti-symmetric tensor field Bij. And the field Bij is exactly the analog of a magnetic field. And as a result, translations on the lattice are represented projectively. And this co-cycle factor in the vertex operators is exactly telling you that there's that projective representation of the translation group. Now that's a standard fact that occurs in the construction of affine-Lee algebra in string theory and occurs here for the leach lattice. And when we do the orbital fold, we have the leach lattice. But now I identify points. I identify a state lambda with a state with minus lambda. And if I think about doing that over here in a two-dimensional example, if I have this vector and I identify it up to a reflection, minus E1, these two points are equivalent mod twice a lattice vector. So when I take the Z2 orbital fold, you really want to look at a projective representation of this, but that group is simply Z2 to the 24th. Well, this group is Z2 to the 24th. And the projective representation of this structure above can be viewed not as a projective representation of this group, but as a real representation, an honest representation, of this extra special group. And there is a dimension 2 to the 12th irreducible representation. And when I said that the twisted sector had these twisted sector states with a multiplicity of 2 to the 12, that's exactly where they came from. So it's a construction finite group theory, but it has a perfectly sensible geometric interpretation as starting out with a projective representation of translations, which is caused by the fact that you have a constant B field in the compactification. And then when you do the Z2 orbital fold, that sort of carries over to a projective representation of this group, and that projective representation is what this extra special group is. Yeah, I mean, I'm not sure off the top of my head with the right mathematical languages for talking about this constant 2 form, but a magnetic field is a constant value of the 2-form electromagnetic field strength. Here we have a constant value of a 2-form field, which is one of the massless modes in string theory. And it has a similar effect in terms of the action of the translation group. It's not a U1 bundle, but probably a gerb. It's a gerb. OK, so I said I wanted to explain something about the group structure. So we're almost there for part of it. So the sporadic Conway group is the automorphism group of the leach lattice, mod Z2, which is simply this reflection Z2 that we did the orbital fold by. And the 2B element of the monster in terms of this conformal field theory is simply an element that acts as plus 1 for vectors in the untwisted Hilbert space and minus 1 for vectors in the twisted Hilbert space. And in the string theory literature, this is often called the quantum Z2. Any Zn orbital fold has a quantum Zn symmetry where you twist the untwisted Hilbert space as eigenvalue 1 and the nth twisted Hilbert space as eigenvalue e to the 2 pi i, mth twisted Hilbert space as eigenvalue e to the 2 pi i m over n. And this is an example of it. So there's a symmetry which is plus 1 on the untwisted Hilbert space, minus 1 on the twisted Hilbert space. The centralizer of this group is 2 to the 1 plus 24 coming from this projective representation, co-cycle factors, dot the Conway group, which is coming from the automorphism group of the Liege lattice, mod the Z2 that you're doing the orbital fold by. So this structure of the centralizer of an involution in the monster is sort of completely evident in the conformal field theory because you can see this structure just in the structure of the untwisted Hilbert space. Now there's a result of Bob Grice, who's one of the inventors or discoverers of the monster, that the monster is actually generated by the centralizer of this 2B element and another involution, sigma. And the sigma is a little bit harder to describe. It's a Z2 symmetry that takes you from the untwisted to the twisted Hilbert space. It's in a way like a discrete version of spacetime supersymmetry in the RNS formalism, where spacetime supersymmetry takes you from the Niver-Schwarz to the Riemann sector. Here there's a Z2 symmetry, which takes you from the untwisted to the twisted Hilbert space. And it's a little less non-trivial to see this, but it's explained more or less in our paper and in much more detail in the work of Frank Lopowski and Merman. So that's one way of seeing at least some aspects of the monster symmetry in this conformal field theory. But I think there's a more satisfying way and a construction which has broader applicability, and that's what I would like to discuss now. Who needs a signal model? It's an explicit conformal field theory. It's not that trivial because you have to have things that take you from the untwisted to the twisted Hilbert space. So you have to say how it acts on twist fields. The arguments I know are somewhat indirect. They use triality and SU2. They're not quite completely explicit. They're indirect arguments that I know of. I'm not sure anybody has done anything quite at the level that you would like. All right, so I now want to talk about something that can be applied here, but also could be applied elsewhere. So in this theory, there is the stress energy tensor, which is a quasi-primary 2,0 operator. That is, it has conformal dimensions to under the whole Warfrey stress tensor and under scaling dimensions on the left and 0 on the right. And we have some set phi i of z, which are 1, 9, 6, 8, 8, 3, primary 2,0 operators. And we know this is true because the partition function has that 1, 9, 6, 8, 8, 4 in it. The Virasaur descendant of the vacuum gives us 1. Everybody else has to be primary because there's nothing else for it to be the descendant of. In the monster, we have c equals 24, but let me write down formulas that are more general. The OPE of the stress tensor with itself takes the usual form. There's the OPE that you would expect for a primary 2,0 field, and then this anomaly, which makes it not primary, but only quasi-primary. And the OPE with the phi i's is that of a primary field of dimension 2. So those two are automatic, just from the statement that phi i is primary and that you have a theory with central charge c. Now, what about the OPE of the phi's with themselves? Well, that's more complicated. But you can write down the form it must take. So we'll normalize these fields so that the most singular term has a delta ij. Now, there could be dimension 1 operators that contribute to a 1 over z minus w cubed whole, but there are none in the monster CFT. So in the monster CFT, we don't have to worry about this term. But I'll put it there just for completeness. Then you get the stress tensor plus some general coefficients times phi k for z minus w squared, where I'll say more about this in just a minute. And then you can have cij rho chi rho plus 2 delta ij over bwt, 1 half bijk, w phi k, all over minus w, where these guys are dimension 3 operators. OK, now this looks like a huge mess. I should make a few comments about it. So this term follow from looking at the T phi phi associativity. That is, you have to be able to do the operator product expansion on these guys and then with this or these guys and then with this. And they have to give you the same answer. So that's how you determine these. Everything else I've just written down using conformal symmetry. And we also know by Bose symmetry that the coefficients bijk, which are in the phi phi phi ope, are symmetric. And we know that aij beta is minus aji beta. And cij rho is minus cji rho. So these are Bose fields. They are by Bose symmetry in exchange z and w. So this term picks up a minus sign, which requires the anti-symmetry of aij. Similarly for the cij, this term has to be symmetric on the exchange of i and j. So there's an interesting fact about this structure, which is that in general, we would not expect to get a closed subalgebra of the general algebra of operator product expansions that we have here. Because when we look at this ope, yes, we get the phi's back. Yes, we get the stress tensor back. We don't have any of these in the monster, but you get these dimension three guys. And then if you looked at dimension three and dimension two, you'd get higher and higher. So it would be surprising to have a closed subalgebra. But in fact, there is one. So if you let phi n i be the usual modes of a dimension two-zero operator, and you define a quantity that I'll call the cross bracket, phi m i cross phi n j, is the following combination. So you look at the commutator of phi i m plus 1 with phi j n minus 1, and you add to that the commutator of phi j n plus 1 with phi i m minus 1. You can compute this commutator, of course, from this ope, since ope's are equivalent to commutators by the usual contour deformation argument. And so you can simply compute this. And what you find is the modes of phi, the viro-sauro generators in a central term. And you can extend this to the cross bracket of viro-sauro with these guys. And the cross bracket of lm with ln is 2lm plus n plus 6m squared delta m plus n zero. Exactly. So this particular combination is hooked up so that you get cijro plus cjiro, and the dimension three operators cancel out. So this is, to me, a rather interesting and non-trivial fact that you can define a closed subalgebra of the full algebra of vertex operators among the dimension two zero fields. I would love to know whether this has some analog in ADS-3, but I don't know if it does. The other amazing fact is that the zero-mode algebra, consisting of l0 and the zero-modes of the phi i, is an algebra that was invented by Bob Grice before flm, before any connection to conformal field theory. And the automorphism group of this Grice algebra, he showed, was the monster. As a matter of fact, this was how he defined the monster. It was the automorphism of this 1, 9, 6, 8, 8, 3-dimensional algebra. How he thought of this algebra? I have no idea, but it probably had something to do with this centralizer of an evolution and trying to associate some kind of algebra with the leach lattice. But this conformal field theory construction shows you, which is really due to Frank Lopowski and Merman, although it was explained in this more CFT language in my paper with Lance and Paul, really shows you why such an object occurs and also shows you that it's a more general structure that will occur whenever you have dimension to primary fields in addition to the stress tensor. It also starts to show you that the monster is a symmetry of more than just, I don't know what it means to be a symmetry of the spectrum, but that there's really an algebraic structure involving the OPEs in the conformal field theory that has the monster as its symmetry. Here it's a subalgebra of the full algebra of OPEs, but still it's very non-trivial and involves an algebraic structure on the dimension two operators. Do you want to look at the tensor product of the 196883 with itself? Yes. Okay. Yeah, I think it is. Yeah. How do you say for the CIA? That I don't know because, well, I guess it probably is, but I think there are more than one irreducible representation that shows up by the time you get to dimension three. But I presume you can compute these in terms of Klebsch-Gordan coefficients in the monster. An algebra just says you have a vector space and you have a way of mapping pairs of vectors to another vector. So that's what an algebra is, that the rule for taking pairs of elements doesn't have to be the standard commutator. It could be any rule you want and this is the rule that works. It's a good question. I doubt it, but I'm not sure. But you don't require that? You don't require that. I mean, not every algebra is a Lie algebra. I don't think the Grice algebra is a Lie algebra, but sure, I mean, every statement about commutators is a statement about OPEs, so you can go back and forth. I guess this is telling you that you should take a particular linear combination of phi i, phi j, and phi j, phi i. Because if you evaluated these, you would start with the OPE of phi i with phi j and evaluating this with phi j with phi i, so you just weight them with different powers of z and then work out the residues. It's the monster, it's the monster of chiral algebra. I mean, there is an infinite dimensional algebra generated by all these higher spin currents, but in general, that's infinite dimensional. The interesting thing about this is you can get a finite dimensional closed subalgebra and the symmetry group is evident as a symmetry of that closed subalgebra. So yeah, there's a higher spin algebra, which is a bigger thing, but then there's this closed subalgebra of that. Well, I don't know how many extremal CFTs there are, do you? I don't think that's even required. I think, I don't quite remember, but I believe that term involving a dimension one also cancels out when you compute the cross-backet. I'd have to double check that, but certainly if there are no dimension one, this works, and I think it works even if there are dimension one because I think that term cancels out. So yeah, you could certainly, it would certainly be interesting to apply it. If you have some explicit examples of C equals 24K extremal conformal field theories, it would be very interesting to see what kind of an algebra you get from the dimension two zero primaries and that. It depends on the details. No, I mean, the fact that it's closed doesn't depend on the details. I never told you, I never used any properties of these BIJK that didn't fall from both symmetry. So it's completely general. I mean, showing that the automorphism group is the monster will depend on the details of what the BIJKs are, but the fact that you're gonna close some algebra is completely general because I didn't use anything other than conformal symmetry and both symmetry. Is it possible to what? Presumably, but I don't know. I've never looked at the four point functions in this theory, but presumably there's some information on that. All right, so there's another topic which is related to this, which again has a specific application in the monster CFT, but I think is also interesting more generally and I'd like to spend the last 10 minutes and maybe a few more if I can borrow five minutes or something telling you about that. So let's consider a situation as in the monster where we have a stress tensor and fly eyes, which are also two zero primary and let's ask ourselves the following question. Let me take linear combinations and try to demand that the OPE of T prime with itself has the OPE of a stress tensor with some central charge C prime. Can we do this? Well, as far as I know, this was first looked at in the physics literature in a short paper of Zomologikov's which probably goes back to the 1970s. Lance Dixon and I looked at it a little bit in some work that we never wrote up and mathematicians like Dong, Mason, Miyamoto have looked at it in the context of the monster. So it's not too hard to work this out because if you know the explicit OPEs of the fly eyes with themself or the stress tensor, this just gives you some essentially quadratic equations that you have to solve. And because this is so straightforward and just to illustrate what's going on, I want to give you two exercises. So here's an exercise. Consider the conformal field theory, the C equals one of a free boson with stress tensor dzx, dzx, and left and right moving momenta m over two r plus nr, m over two r minus nr. So go to radius r equals one and show that the operator one over square root of two e to the two i x left plus e to the minus two i x left is a primary dimension two zero operator. So in other words, we can go to a special radius in the S one compactification where there's a two zero operator. And then B, find two solutions to the equations for alpha and beta that you get by demanding that this is a stress energy tensor, both with C equals a half and conclude that you can take the stress tensor at this radius and decompose it into a sum of stress tensors of C equals a half. That is, this is the famous bosonization of fermions, the fact that is well known that the theory of a free boson of special radius is equivalent to the product of two isic models. So that's a standard exercise which you can basically just look up and work out easily. And here's a more fun exercise. Take a theory with C equals two and two free bosons go to a point in the ring moduli space of two free bosons. Momenta P here are going to live in the root lattice of SU three rescaled by a factor of square root of two and PI in this formula are equal to square root of two alpha i or alpha one and alpha three are the simple positive roots of SU three and alpha two is the sum. So by applying this procedure show that you can construct, you can deconstruct if you want the stress tensor into a C equals six fifths plus T equals four fifths. And this theory, when you do that and decompose the primary fields in the original theory into primaries in the sub theory has another dimension two zero field and you can use this to decompose it into a sum of the ising model and the tricritical ising model. So there's a point in the moduli space of C equals two bosonic and formal field theories where you can find three primary dimension two operators and you can use that to decompose or deconstruct the theory into a tensor product of the first three minimal models. So those are fun exercises. A much more difficult exercise was carried out in a paper of Hohn, a mathematician in a paper which unfortunately is only in German. I forget the exact title but it's something like Cilp's dual vertex operator and thus baby monster. So he shows that in the monster CFT well now we really have a lot of fun because we have one nine six eight eight three primary dimension two zero operators. So you can really go to town and he shows that you can find a linear combination of the stress tensor and these one nine six eight eight three dimension fields such that the stress tensor decomposes into a sum of C equals 23 plus a half equals a half the ising model and that the state space of the conformal field theory this monster module decomposes into the following sum. So in this formula L a half of H is the representation space of the C equals a half theory at conformal weight H and it's well known that the ising model has representations with H equals zero a half and one sixteenth. These VB zero VB one and VB two are the vector spaces that are left over so to speak when you decompose this theory under these two stress tensors and you can work out he works out for you the characters of these guys starts with Q to the minus 47 over 48 since it's equals 23 plus a half I won't write these all down but anyway I won't write down the details but the point is that these numbers here are dimensions of irreducible representations of the baby monster sporadic group and so this gives you a conformal field theory construction of a theory that has the baby monster as it's automorphism group rather than the monster and it also allows you to understand something about the group structure because for C equals a half there is a Z2 symmetry of the OPE or the yes sorry that's the word I was looking for symmetry of the fusion rules which is plus one on H equals zero and a half and minus one on H equals a sixteenth and you can show that this symmetry lifts to the two-way element in the monster so these are all examples of how various aspects of the monster symmetry the baby monster symmetry the action of various involutions the existence of a closed subalgebra can all be captured by elements of the conformal field theory by constructions that are very natural and conformal field theory when you have these two zero primary operators deconstructing into smaller conformal field theories looking at closed subalgebras of the OPE and this is really just the starting point you can continue this process and I think there are actually constructions of the monster that do a decomposition all the way to 48 copies of the icing model so there's a very rich structure what is the monster? I mean nobody really knows I think John Conway famously said that he really the one thing he would like to understand before he dies is what the monster really is and why it's there and probably he will not be satisfied and probably we won't either but to the degree it can be understood many aspects are best explained using the language of conformal field theory and string theory well they're both CFTs but you mean this is a free field theory whereas this is it's a sum of direct I mean you have to look at the it'll be the I mean the partition function of this theory will have the same form it will be a sum of representations of these minimal models the sum of tensor products of representations it's not just going to be a it's a sum it's a sum, yeah and I forget I mean Siegel's Fortress has I forget, it has three or four representations so yeah it will be a sum of products so it's not just a direct product I don't know the answer, it's a good question what is the monster? I don't know where the one came from okay I didn't get quite as far as I'd hope but I think there's always time for questions now or in the discussion section so I think I'll end here