 Thank you for coming out early in the morning. I'm going to be talking today about a recent revival of ideas related to moonshine, which have been developed starting in 2010 to the present, going on. And this new kind of moonshine involves, at least in one way of formulating it, Jacobi forms N equals 2 and 4, super conformal algebras, objects called mock-modular forms, the nemire lattices that we already encountered, and a number of other things. But I think it will suffice to try to explain some of these things. Now, I know that the first two lectures might not have been crystal clear, but the situation regarding moonshine in those kind of classic examples where there's an explicit conformal field theory is much clearer than the situation involving this Matthew and Umbral moonshine, where there's a mathematical framework, which is perfectly consistent and makes sense and satisfies all sorts of non-trivial properties, but the physical understanding of what's going on in terms of symmetries of something that a physicist or string theorist would recognize that as a conformal field theory, sigma model, et cetera, is confused and a lot has to be done to try to really understand what's going on. But I'll try to at least be as clear as I can regarding the mathematics. And then towards the end of the last lecture, I'll try to talk a little bit about attempts to fit it into some kind of a physical framework and what ideas people are exploring, although nothing has been completely successful so far. So let me start out by talking about Jacobi forms and how they show up in computations and string theory. So when you study string theory, you encounter what mathematicians would call an elliptic curve. And in string theory, this arises because you have one loop string amplitudes on the torus. And any torus, you can choose the metric to be conformally flat and then up to conformal transformations that are uniquely specified by a parameter tau, which lives in the upper half plane. And on this elliptic curve, you can choose a point z. And there are two classes of transformations which are very natural to consider. The first are elliptic transformations, bending z to z plus mu tau plus lambda, which just corresponds to taking this point and shifting it up this way or over this way by one of these two lattice vectors that specifies this elliptic curve. And the other thing that's natural to do is to look at modular transformations, which correspond to global diffeomorphism of this torus. And those correspond to transformations of z that look like this and tau, which have the familiar form from one loop amplitudes in string theory, where a, b, c, d are integers and the matrix has determinant 1. Now, if you ask for functions that transform nicely under the elliptic transformations, you get elliptic functions, Bierstrass function. But those functions secretly depend on a parameter tau, which is used to label this torus. And if you then kind of ask just to look at this modular transformation and ask for functions that transform nicely, you get modular functions. But in a way, the most natural thing is to consider functions that transform nicely under both of these, and those are Jacobi forms. They're functions of z and tau that have nice transformation properties under both the elliptic and the modular transformations. Now, what do I mean by nice? Well, it's useful to look at a kind of motivating example, which is the Jacobi theta function, which gives Jacobi forms their name. So the Jacobi theta function of z and tau defined by following sum, it's convenient, as usual, to let q be e to the 2 pi i tau, and to call y pi i z. And it's easy to work out what the transformation properties of this Jacobi theta function are under both the elliptic and modular transformations. So in particular, you will find that under elliptic transformations, it transforms like this. And there's nothing very complicated about this. You just stick in z is equal to z plus mu plus lambda tau. You rewrite the sum by changing the summation variable around a little bit. And this pops out after a couple lines of algebra. And you can then look at how it behaves under modular transformations. And I'm only going to write down special cases here. So first of all, it's clear that if I take tau to tau plus 2, then this term doesn't change in some of the theta functions invariant. And a slightly less trivial computation using Poisson summation shows that if you take tau to minus 1 over tau, you get the square root of tau over i. e to the pi i z squared over tau, theta of z tau. And so this follows from using the Poisson summation formula. Now, these transformations are clearly, well, sl2z is generated by tau goes to tau plus 1 and tau goes to minus 1 over tau. So we don't really have here transformations under all the generators because we're missing tau goes to tau plus 1. And if I do tau goes to tau plus 1, you can see that this term is going to change by a minus 1 to the n. And so I get some other function. This factor of the square root of tau is what you would expect for a weight 1 half modular form. We could try to make precise what the modular transformations are here. But if we just take this as a motivating example, we could try to extend this to a more general framework. And that's what Eichler and Zagier did in their book on Jacobi forms, which is quite readable, very explicit. So they said, let's take this combination of transformations and try to make it make sense for the full modular group and rather than for half integral weight for integral weight. So they defined a Jacobi form of weight k and index m, or k and m are integers, be a holomorphic function by from the, I guess maybe I should write it as the complex plane and the upper half plane to the complex numbers, thinking of this as the variable z and this is the variable tau, which obeys an elliptic transformation property. And if we look at this elliptic transformation property, and although we're not allowed to in this definition, we put m equals 1 half, you would see that it's precisely the elliptic transformation property of the theta function, which is why this is generalizing it from a half integer to general integers. And then under modular transformations, I'll use a shorthand notation where gamma z of tau is z over c tau plus d, a tau plus b over c tau plus d. So under a modular transformation, you get c tau plus d to the k, which is the usual factor for a modular form of weight k. And then you get a slightly more complicated formula here. But again, for m equals a half and for tau goes to minus 1 over tau, you can check that this factor agrees with the factor that you have here. And these somewhat strange looking factors are needed in order that the combination of these two transformations, which generate the Jacobi group, acts nicely on this set of functions. So that's the definition of the Jacobi form. And there are a couple of properties which follow quite easily. I won't prove, but they involve fairly straightforward manipulations. And they're useful in analyzing them. So the first property, kind of trivial, the transformations tau goes to tau plus 1, and z goes to z plus 1, d phi invariant, as you can see from these transformation laws over here. So that means that you can write a Fourier expansion in terms of q, which is e to the 2 pi i tau, and y, which is e to the 2 pi i z within an R integer. And then you can show that these coefficients using the elliptic property, you can show that these depend only on quantity, which I'll call the discriminant, or which they call the discriminant, d, which is 4mn minus R squared, and on R tilde, which is the value of R mod 2m. So in other words, I can view these c of n and R as really being some other coefficient I'll call capital C. That's a function of d and R tilde. And in the sum, any c and R, which have the same value of the discriminant 4mn minus R squared and the same value of R mod 2m, will be equal. So that equality allows you to pick out the coefficients that are all equal and reorganize the sum. And when you do that reorganization using this property, you find that you can write pi. So I guess I should say here, I'm looking at something that has weight k and index m. That's where the m comes in. So the second property is that you can, by rewriting the sum as I described, you can do the sum over R first over all R's that are equal to the same R tilde mod 2m, and then over the R's from 1 to 2m minus 1, or whatever. And when you reorganize the sum that way, you can pull out all of the z dependence into something called a level m theta function, which is given by a sum on integers n to the 2mn plus R squared over 4m times y to the 2mn plus R. And where hR of tau is equal to a sum on these discriminants d, these new coefficients c of d and R due to the d over 4m. All right, so that's a lot of somewhat complicated formulas. What is this guy? Well, these theta functions, since they have a nice explicit form in terms of the sum of exponentials, you can use Poisson's summation, which is sort of the general tool for analyzing modular transformations, to deduce the modular transformation properties of thesestatas. I'm not going to write them down, but you can find them in all sorts of places. And you can use this and the modular transformation properties of phi to deduce that these guys are vector-valued modular forms. And what that means is that under modular transformations, there is some matrix, depending on a, b, and c, d, and then some hs of tau. And I guess also a weight, c tau plus d to the k minus 1 half. So there are the sort of things that are often encountered in conformable field theories, where you have different sectors of a rational conformable field theory and the different sectors mix into each other under modular transformations. All right, so after that little barrage of math, I want to pause for a minute and ask if there are any questions. And then I'm going to tell you how these things arise in spring theory or in super conformable field theories. The elliptic transformations are, they're not global dwarf morphisms. Essentially, what they are is just translations by a lattice vector. And if you had a function which was invariant under these transformations, you would just have a function which was well-defined on the torus. But you can't have a holomorphic function which is well-defined on the torus because it's not constant. So you either need to look at meromorphic functions or you look at functions that are not really defined on the torus but are rather sections of a line bundle over a torus. And these phases in the transformation laws under elliptic transformations are really telling you that you should think of these Jacobi forms as sections of a line bundle over the two torus. So that's their geometrical interpretation. It's not connected to any kind of global dwarf morphism of the torus. Could you just speak up a little bit? I don't quite understand. Oh, it could have been written in product form? Some Jacobi forms can be written in product form. But I don't know if there's any general result that every Jacobi form can be written in a product form. I'll write down some examples a little bit later. I think they can maybe be written at least as sums of products because many times they can be written in terms of theta functions. And then you can use the Jacobi triple product identity to rewrite those products. But I don't think there's necessarily just a single product that you can write them as. OK, so I now want to explain how these show up. One way of understanding where they show up and where these properties fit into physics. So when we consider compactification of string theory on a Calabi-Aus space, x, we know that that leads to an n equals 2 super conformal field theory. And provided there's a quantization condition on the u1 charges of this n equals 2 super conformal field theory, you can always construct spacetime supercharges. Now, this theory has a stress tensor. It has two dimension 3 halves super currents. And it has a dimension 1 u1 current. I'll call the modes of these ln, gr plus and minus, and jn. And as is familiar, I think you either have half integer r or you have integer r, depending on whether you're in the Niver-Schwarz or Ramon sector of the super conformal algebra. And I'm not going to write down the commutation relations exactly, but they have the rough kind of form. And they tell you what the charges are of the g plus minus under the u1 current. The commutator of a g plus with a g minus gives you verisoral plus a u1. And j is primary under verisoral, et cetera. You can just, there's a Wikipedia page on n equals 2 super conformal algebras, and you can find the commutation relations in detail there. But there's an interesting fact about this algebra, which was pointed out by Schwimmer and Seiberg. And that is that there is a spectral flow isomorphism between the Niver-Schwarz and Ramon versions of this algebra, something that is not true for n equals 1. So what this involves is taking ln and redefining it to be ln plus mu jn plus c over 6 mu squared delta n0, redefining jn to be jn plus c over 3 mu delta n0, and taking gr plus or minus to gr plus mu plus or minus with mu equals 1 half. So you just take the algebra, write it down explicitly, make this substitution, and you'll find that you get exactly the Ramon algebra starting with the Niver-Schwarz algebra or vice versa. So they're really isomorphic under this. And you can extend this to integer mu. For integer mu, this takes the Niver-Schwarz sector back to the Niver-Schwarz sector, the Ramon sector back to the Ramon sector. But it's still an untrivial isomorphism because it shifts ln by something proportional to jn and shifts jn by a constant for j0 and leaves the sector fixed. But it's still an isomorphism of the algebra. And you can check that there's a quantity 2c over 3l0 minus j0 squared, which is invariant under this shift. So it's going to turn out that this spectral flow isomorphism implies essentially the elliptic transformation property of a certain quantity, which is purely topological in this theory. So in an N equals 2 superconformable field theory, we can define the elliptic genus. And we'll think of this if we have some collabial space x, which gives rise to this N equals 2 superconformal field theory. As a sigma model of x, we'll call this the elliptic genus of x. It depends on the collabial manifold x and then on two variables z and tau. And it's given by a trace in the Ramon sector, so Ramon for left and right movers, u to the l0 minus c over 24, bar to the l0 twiddle minus c twiddle over 24, minus 1 j0 plus j0 tilde to 2 pi i z j0. This quantity is also written as minus 1 to the f left minus f right. So what is going on here is that we have weighted things on the right. That is the q bar dependence. By minus 1 to the f, we're basically computing minus 1 to the f right times q to the q bar to the l0 minus c over 24 on the right. And this quantity is the written index and is non-zero only for ground states with l0 tilde equal to c tilde over 24. Otherwise, if you're in an excited state on the right movers, the supersymmetry, the G-naughts in the Ramon sector, pair states that have opposite values of minus 1 to the f just like in supersymmetric quantum mechanics. And so all those excited states on the right cancel when you take this trace. So in fact, this is a function only of q and of y, which is e to the 2 pi i z. There is no dependence on q bar because this q bar dependence completely goes away when you take the trace and include this. On the left, this cancellation doesn't happen because you're weighting the states by their J-naught eigenvalue. And this differs for fermions and bosons. So if I were to set z equals 0, then only the left moving ground state would contribute. So the elliptic genus at z equals 0 and tau will be a constant because then you'll have cancellation both on the left and the right except from the ground state. And this is essentially trace minus 1 to the f just in the supersymmetric quantum mechanics on this space. And that gives us the Euler number of x, a famous fact from long, long ago. So this elliptic genus is something that is counting right-moving ground states, arbitrary left-moving excitations. And those are well known to be BPS states. So this is really capturing some kind of topological information about x. And that intuition has a space-time interpretation in terms of the kind of infinite set of indices of Dirac operators coupled to various tensor products of the tangent bundle, which I won't try to go into. But there's a space-time computation of this in terms of index theory. I'm just talking about the world sheet description. So this is a very interesting gizmo to compute. And you can use spectral flow, which turns out to be equivalent to the elliptic property, and the usual modular properties, one loop string amplitudes, and the fact that the Ramon-Ramon sector is left invariant under modular transformations to show that the elliptic genus of x, z, and tau is a Jacobi form of weight 0 and index m, which is 1 half of the complex dimension of the Calabi-Alspace x. This is essentially following from the formula for what the central charge of the conformal field theory is in terms of the complex dimension. I should add one small caveat here, which is that technically speaking, this is what's called a weak Jacobi form. And a weak Jacobi form is a Jacobi form that satisfies the condition that these coefficients are 0 whenever n is less than 0. So in other words, there are no negative powers of q in the expansion. There's a slightly stronger condition, which applies to other classes of Jacobi forms. But this is the Jacobi form that this is the condition that's suitable for the elliptic genus. I guess one could make a connection. Yeah, I mean, well, one way of making a connection would be if you look at string theory compactified on a K3 manifold, then you get a theory which can be chiral in six dimensions, and that theory has various anomalies. And those anomalies are related to the index of various operators on K3. So yeah, one could connect it to anomalies, but I'd have to think a little bit about what the most complete statement is. But yeah, there is a relation, but it's not going to be essential for what we're doing. Yeah, yeah, it's right. It's essentially, well, I won't try to write it down, but it's a kind of, there's a really complicated formula for it, but essentially you take a Dirac operator, but instead of the usual coupling to the tangent bundle, you take it coupled to some tensor product of the tangent bundle, various anti-symmetric and symmetric products. And then you arrange these things, you kind of grade it, and then the sum of all of the index of all these graded Dirac-like operators are what the elliptic genus is. And you can find explicit formulas, or I can tell you where to find explicit formulas for what that infinite set of Dirac operators is. There's no left or right. The operators I'm talking about are operators in spacetime. So operators on, Dirac operator on the Klavier manifold x. The left and right is purely a world sheet description in string theory. The spacetime interpretation involves index theory on the Klavier space x. So it's interesting to compute this. And one of the first non-trivial examples that was commuted, computed, was the elliptic genus of K3. And well, it turns out that there is a unique weak Jacobi form of weight 0 and index 1. And that unique weight 0 index 1 Jacobi form can be written in various ways. But one formula for it is to take theta 2 of z and tau over theta 2 of tau to the fourth power, plus the same thing with theta 3, same thing with theta 4, where theta 2, theta 3, and theta 4 are the usual theta functions. Sometimes I guess they're called, well anyway, these are the three non-odd Jacobi theta functions. And we can figure out the proportionality constant because the elliptic genus of K3 at z equals 0 should be the Euler number of K3, which is equal to 24. But if I look at this thing evaluated at z equals 0, it's clearly equal to 12. So we learned that this is equal to 2 times 501. Now, this was the starting point for the following computation. For K3, because it's hypercaler rather than just caler, there is actually an n equals 4 superconformal algebra. And therefore, the elliptic genus must have a decomposition into characters of the n equals 4 superconformal algebra. And Oguchi Oguri and Tachikawa, 2010, worked out what the decomposition into characters of n equals 4 is equal to. I understand it involves a formula, which was actually in Oguchi's PhD thesis. But they interpreted it as well. And so I'm going to tell you what they did. So characters of the n equals 4 algebra are defined by taking a trace in a highest weight representation where h and l are the l0 and j0 eigenvalues of the highest weight state. And then as usual, you act on that highest weight state with the creation operators to fill out a multiplet. And I'll define the characters to be the trace of minus 1 to the f, q to the l0 minus c over 24, y to the j0, and a highest weight representation labeled by h and l. So there are formulas for these characters, which were worked out by Oguchi and Taormina. They're somewhat complicated. And I'm not going to write them down, but it's a mechanical thing to decompose one function in terms of the sum of other functions. And so you do that. And what you find is that you can write the elliptic genus as the odd theta function squared over the dedicated eta function cubed times two terms. This is a kind of complicated object, but it comes from what is called a massless n equals 4 character, which has h equals a fourth and l equals 0. It's called a massless n equals 4 character because a multiplet with this h and l value gives rise to massless states in spacetime. And I'm not going to write down what that function is, because I don't need it right now. And then there's a term h2 of tau. And this is a function that has a form, which I'll write down explicitly. It encodes the degeneracies of what are called massive characters where you compute the characters with h equals a quarter plus n and l equals 1 half, which in the simple example are the only other characters that appear. So in other words, this guy has an expansion q to the minus 1 eighth, which comes from various things. And then there's an integer expansion in positive integer powers of q that label the characters with h value a quarter plus n. And that sum looks like minus 1 plus 45 q plus 231 q squared plus 770 q cubed plus dot, dot, dot. And for further reference, I'll write this is the sum equals 0 to infinity c of n minus 1 eighth q to the n minus 1 eighth. Now what Aguchi Aguri and Tachikawa noticed is that 45, 231, and 770, and the next coefficient as well, are all dimensions of irreducible representations of a sporadic group, the Matthew Group M24. So that observation is the starting point of what's called Matthew Moonshine. And you'll notice that it's a bit different from the starting point of monstrous moonshine. So the starting point of monstrous moonshine was the fact that J of tau up to a constant term, which doesn't matter, looks like this. This had a nice decomposition into dimensions of irreducible representations of the monster. But this guy was a weight 0 modular form. And weight 0 modular forms can appear as the partition functions of conformal field theories. This guy is something else. It is a weight 1 half Mach modular form, which I'll have to define for you. But it has a more complicated transformation property under modular transformations. It's not weight 0. And it's not interpretable in any way that I know as the partition function of a conformal field theory. So the other thing is that when you interpret this as a trace of q to the l not in a conformal field theory, all the states contribute with a positive sign. So these numbers are really counting positive dimensions of vector spaces. But this arose in a computation of the elliptic genus. And this elliptic genus involves trace minus 1 to the f and other stuff. So states can contribute to this with a plus sign or a minus sign. So these coefficients are not manifestly positive. As a matter of fact, this first one's negative. It turns out that all these coefficients are positive. But it's not completely manifest from the definition that that should be the case. So people have been trying to understand. I said, yeah, why do you say that? Somehow I'm going to swore it was 4. Let's see. It should be index 1. He's saying these guys transform like index a half. And so when you take the fourth power, you get index 2. But I said it should be index 1. I think that's what you're saying, right? Let me, I'll check. I think this is right somehow. But all right, I'll. What? Yeah, so I think it must be that these theta functions, if you look at the index, if you look at their elliptic transformation properties, it must be essentially like, it should be, no? Well, OK. Yeah. Look it up. Somebody look it up. I thought this is right. But I'll have to reconcile these things. OK. So what are we supposed to do with this? Well, I don't know. You can just forget about it. It's a weird curiosity. But it cries out for some explanation. In monstrous moonshine, the answer was to find a conformal field there with the monster acting as a symmetry group. So it's the first thing you would think is, let's find, a K3 surface that has M24 as a symmetry. But that doesn't work. It should be squared. My apologies. At least I got the factor of 4 out in front, right? So there is no K3 surface with M24 symmetry. Well, you have to be a little more specific by what M24 symmetry means. What this really means is an automorphism that preserves the symplectic form, which is needed in order to preserve spacetime supersymmetry and to preserve the n equals 4 superconformal algebra that you've used to do this decomposition. So it's known that all the symmetry groups that can act this way in K3 are subgroups of M23, which is another Matthew group. But they're smallish subgroups. And you can't get M24. So you might say, OK, but we know that in string theory, there are things that look like K3 compactifications that might not be so easily described geometrically. And really, what we should be looking at is an n equals 4 superconformal field theory with a discrete spectrum of states corresponding to a compact space with c equals 6. So that's there. And with such that the Euler number is 24. So that is sort of the conformal field theory definition of what a K3 surface is. It's just a superconformal field theory that has n equals 4, c equals 6. But you can analyze this question. This was done by Gabberdeel Olpato. And again, there's no such beast with M24 symmetry. So the most obvious attempts at an explanation of what's going on fail. And well, if you can't do physics, you can always do math. So essentially what people did in this problem was to try to investigate mathematically whether this structure was consistent in the same way that the structure of monstrous moonshine was consistent. There, people computed these Mackay-Thompson series, Conway Norton did, before the conformal field theory construction by Frank Lopowski and Merman was known. And you can do the analog here. So that's what was done in papers by Chang, Gabberdeel, and collaborators, Gucci, and Hikami. So essentially what you do is you play the same kind of game that we had for the monster. So remember for the monster, said, well, we could assume that there's some infinite dimensional module that was graded such that the dimension of Vn was the coefficient An in the expansion of J of tau. So we can play the same kind of game here and say, let's suppose that I should really interpret these numbers as being dimensions of vector spaces on which M24x. So that would be the statement that there is an infinite dimensional graded module for M24 such that the dimension of this bit labeled by N is equal to these coefficients N minus 1 8. And then if we assume that there's a structure like that, we can certainly compute the analog of the Mackay-Thompson series. As we can look at twisted versions of these things by computing the trace of an element g in each one of these representations weighted by u to the N minus 1 8 and try to choose a decomposition of each one of these spaces into one of the 26 irreducible representations of M24 with positive integer multiplicities. And in the case of the monster, we demanded that these things be modular forms on a congruent subgroup and actually weight genus 0. What are we supposed to choose here? So what kind of function should be these b? Well, the answer is they should be Mach modular forms for essentially gammonaut of N, where N is related to the order of g. That's not quite correct, but it's close. Now, the problem with this formulation is that I haven't told you what a Mach modular form is and how that's supposed to replace the modular forms that we had in monstrous moonshine. So I think I'm out of time for today. So that will be part of the subject of the next lecture I give. I'll tell you what Mach modular forms are. And then I'll tell you of an extension of this structure to umbral moonshine, which is also not understood. That is, there's no explicit string theory, conformal field theory, black hole, whatever background that explains what's going on. But I'll talk kind of qualitatively about some of the ideas for trying to explain what's going on. And maybe one of you can implement that or find some better idea to try to explain the mystery of why these group structures are showing up here.