 Hello? Is it amplified? Good. OK. OK, so I guess I just fire up, right? So thanks for coming back. So last time, I talked very generally, pretty generally, about this notion of BPS states. And I talked about the example of n equals 2 supersymmetry in one dimension. And the example there was, if you have a Riemannian manifold m, you can get, as a space of BPS states, the Dirac homology of m. And so the philosophy is that m is very complicated. It produces, for you, some very complicated Hilbert space. But then the inside of there, you find this BPS subspace, which is the harmonic forms, is more perfect to the Dirac homology. In the notes, I mentioned a few other examples of this phenomenon. And maybe I should just say them out loud because they'll be more interesting for some of the people in this school. So particularly when I mentioned one example, the example is let's fix an A, D, E type Lie algebra, a link in R3, an R, a representation of G, irreducible representation, finite dimensional irreducible representation of G. Then there's some physical construction, which I'm not telling you, but you can look in the notes to see broadly how it's supposed to go. And I should say, this comes to these ideas of, it starts really with Gukov, Schwartz, and Wafa. And then it was kind of reinterpreted by Witten. And the version I described is the version described in Witten here that you can construct a physical system which has exactly the same formal structure. It has n equals 2 supersymmetry in one dimension. It starts out from something six-dimensional, but it winds up having n equals 2 supersymmetry in one dimension. And in the end, what you're supposed to get is some version of the, well, hopefully exactly, the Kovanov-Rozansky homology of L. I guess maybe to be exactly the Kovanov-Rozansky, I better say that G is like s l n, so a n minus 1, and R should be the fundamental representation. Okay, see the notes for something more precise. So I just want to say that here it's kind of the same philosophy. So you would construct from this data some very complicated Hilbert space, which depends on particularly every little detail of how this link is embedded in R3, just like the spectrum of the Romanian manifold depends on every little detail of the metric. But then when you study the BPS states of that system, you're supposed to get something more tractable, which is this, not homology. Okay, that's just for inspiration. That's not anything we're going to do, most likely. So the other thing that we talked about, so this was kind of quantum mechanics, and the other thing we talked about was quantum field theory, n equals 2 comma 2 supersymmetry in two dimensions. And so there, let me just say a little more about that. So there we had a supersymmetry algebra, which I called script D, A, it had Casimir operators, M, which acted in unitary representations as a real number, and Z, which in unitary representations was a complex number. And the idea was when you study one particle states, you think of this as being the mass of the particle, that's a very familiar kind of thing. And then they have this other invariant, which is not so familiar, because it doesn't come from the Poincare group, so it's not so familiar, a complex number, which is called the central charge. Yes. Let me just, yes. So the two is the fact that there were two Q's, Q and Q bar, that's the two. The two two is the fact that there were four Q's, Q plus bar, Q plus, Q plus bar, Q minus, and Q minus bar. The fact that it's divided into two two has to do with how these guys transform under the SO11, the Poincare symmetry. But I think, let me not say, let's not say more about that now. So what we said was that the representations of this algebra came in two flavors. There were the short ones for which the mass was exactly equal to the absolute value of the central charge, and there were the long ones for which the mass was bigger than the absolute value of the central charge, so it can't be less. It has to be one of these two. The lowest the mass can be is equal to the absolute value of z, and if it is, then the representation is small and it's kind of protected. That's what we talked about last time. And particularly we defined an index, the index mu of h, which was a kind of signed count of the short representations. So the short representations came in sort of a plus flavor and a minus flavor, and what you do is you count the number of pluses minus the number of minuses. Now, as it came up in the questions, there was a slight refinement of that, where the Hilbert space is broken up as a direct sum of pieces labeled by a charge. So this a, I'll call the charge, and we'll see what it is in an example in a minute. And the central charge, the central charge operator just acts as a scalar on each of these graded pieces. So when you look at one graded piece, you just think of the central charge as just some absolute fixed number. There's an additive structure on the charges. This I didn't mention last time. Again, we'll see it in the example in a minute. And the central charge is additive in the sense that if I have two particles, one with charge a, one with charge b, the combination is a state of charge a plus b, and the central charge of that state is just the central charge a plus the central charge b. Now, here's the key question. So as we vary the quantum field theory, so we're gonna be interested in varying the parameters of this quantum field theory in the same way as here, we vary the Riemannian metric. And we ask the question, is this index constant? Constant. Is it really a deformation invariant of the structure? And the answer, what's believed to be the answer, is yes, and it follows from the kind of formal arguments I showed you in deformation theory last time, except or let's say as long as the continuum of multi-particle states stays away. What do I mean by that? Well, let me draw a picture of what the situation could be. So in my Hilbert space, HA, what could the spectrum be like? Well, so I'm drawing the spectrum of this Casimir operator, the mass. What are the masses of the states? Well, we know they're bounded below by the central charge, the absolute value of z. And so the good situation would be that there's some BPS states down here, and then maybe there's some other discreet states that occur here, and then at some point the multi-particle continuum sets in. But it sets in up here someplace safely above absolute value of z. In that case, all of our formal deformation theory stuff will work fine, because you just apply it to the part that's below the continuum, and you don't worry about the fact that there's this continuum. So this would be okay. And the situation that you don't want, the dangerous situation would be if the continuum comes right down to the absolute value of z, the BPS bound. In that case, you can at least imagine a problem, and it'll turn out in fact that there is a problem. So, okay, this talk about multi-particle states maybe seems a little abstract, so let me try to make it concrete. Is it kind of diagnostic for what we're gonna be looking for when we do the example? Let's think for a second about two-particle states. So let's imagine we have two particles, one of them carrying a charge, call it A, and one of them carrying a charge, call it B. And let's look at this particular state, so they have maybe some momentum. There's gonna be some two-particle state here, and let's just calculate its value of M. Well, what can we say about it? So first of all, I guess this is a little exercise, but hopefully you'll believe it, that the mass defined as the eigenvalue of this Casimir operator in this state is at least the sum of the masses of the two particles. The least it could be is if the two particles are just sitting at rest, then you just have their two rest masses individually. On the other hand, each of these particles individually obeys the BPS spout, so mass one is at least absolute value of ZA, mass two is at least absolute value of ZB. Now that on the other hand, okay, so the sum of the absolute values, that's greater than or equal to the absolute value of the sum, which is equal to, okay, it's A plus B, let's call that C. So A plus B is C. So we've got this combined state that's carrying the charge A plus B, which is C, and the mass of that state, we've just shown, okay, no surprise, it obeys the BPS spout, right? The mass of this two-particle state is greater than or equal to the absolute value of ZC, as it has to be for every state in the theory. Now our question is, when is this bound, when is this inequality exactly saturated? If it's saturated, that's gonna be the situation where this continuum state comes right down to the BPS bound. What you're worried about is that this bound might be saturated. When is it gonna be saturated? Well, just if, okay, two things have to happen. First of all, well, I guess three things have to happen. So first of all, the particles have to be at rest relative to each other, then this will be in equality. The particles have to be separately BPS, then this will be in equality. So they separately saturate the bound. And then the key thing is, so when does it happen that the absolute value of ZA plus the absolute value of ZB is equal to the absolute value of ZA plus B? These are two complex numbers. That happens just if the two complex numbers have the same phase, right? So the key condition is that the phase, the argument of ZA has to be equal to the argument of ZB. So just in that situation, this two particle state could exactly saturate the BPS bound. So, okay, so let me sum up. So what's the conclusion? What's our expectation for this BPS index? So we ask the question, either is not this BPS index mu of Hc constant when we vary the parameters? The conclusion is, is constant or should be constant except if when we vary parameters, we reach a situation where there exist two BPS states, two BPS states with charges A and B, say, adding up to C and the argument of ZA is equal to the argument of ZB. Just in that situation, it's possible for, and the way you sort of say it informally is that it's possible for a particle of charge C to decay into two particles, one of charge A and one of charge B. But generically, it can't happen. Generically, it's forbidden by this calculation here. The only time it can happen, the only time when that two particle state can be BPS is when the phases of the central charges are lined up. And so the situation is gonna be that generically, that won't happen. Generically, all the phases will be just random numbers. And then as you vary parameters, sometimes it's a kind of co-dimension one thing that two of these complex numbers get lined up and at that moment something can happen. Okay, so now we'll go to an example. Yeah, yeah, so they're valued in, so we're gonna, in fullness of time, we'll discuss two different examples of this kind of business. In one case, they'll be valued in how is it, how is it called? In one case, they live in some lattice, that's the easy case. In the other case, they live in, it's like a set with a partially defined sum. So they're not numbers, they live in some set. You'll see in a second what it is. Other questions? Yes. Yeah, yeah, short representation, yeah, thank you. The representations that appear in the states that are in the short representations are the BPS states, absolutely. So I could equally well call this a BPS representation. Absolutely, yeah, thank you, good, other questions. Very good, so in this case, in this short representation, so this is in the notes, but in the short representation, so we have four supercharges, two of them, two linear combinations of these will act as zero. But it's actually an important thing, oh, okay, look at the notes to see the exact linear combination. But I will say one important thing about it is it depends on the phase of z. So there's sort of a one parameter family of subalgebras parametrized by an angle theta. And so particles with different phase of z preserve different subsets of the supercharges. But that's another way of seeing why this condition is relevant. This is the situation where both of those particles separately are annihilated by the same supercharges, the same two out of the four supercharges, great. Okay, so let's talk about an example. So now I want to do something that's mathematically sharp. So we'll do what's called the polynomial in the Ginsburg models. So let's fix a polynomial called W, complex polynomial, so in K variables. Or, in general, we interested in a family of polynomials. So here I've just got one polynomial, but in general I might have a family of polynomials, so I'll have another parameter, call it z, z living in some parameter space, I don't call that parameter space c. And we'll mostly take K equals one, so then it's just W from c to c, just polynomial. So for example, we can take our parameter space to be c, the complex numbers, so I'll call the parameter z, and then W of x at parameter z will be the cubic polynomial, one third x cubed minus zx. So okay, for every fixed z, I have a cubic polynomial, and then as I vary z, the cubic polynomial changes just by changing the linear term. So it's a family of them parameterized by c, the complex plane. Or another example would be to take a two parameter family, so now c will be c2, and I'll just take quartic polynomials, W of z of x is one quarter x to the four minus one half z1 x squared minus z2 x. Okay, so here I took a quartic polynomial, and I'm varying the linear and quadratic terms. Okay, so now what do I do? So here's what we'll do. For a given W, I'll define a classical BPS soliton, oh sorry, maybe I should first say. So one kind of physics fact, for given W, there exists a two comma two supersymmetric quantum field theory called the Landau-Ginsberg model, LG model. Okay, that's just to motivate what I'm about to do. So now the construction is definition. So suppose we're given some particular W, I'll define a classical BPS soliton, classical BPS soliton with phase theta. Yeah, so I guess for given W and theta, I'll define a classical BPS soliton with phase theta. There's just gonna be a solution of some differential equation. It'll be a map from C to the K, sorry, what am I saying? From R to C to the K. It'll be an equation, so we're just looking at solutions of some differential equation. The equation is dxj by ds, so this R I'll think of is parameterized by a coordinate S. So think of it as being space, we have a one, this is a theory in one space in one time dimension. This is gonna be just a function of the space. dxj by ds is equal to E to the i theta times the jth derivative of W and valued at X of S. There's our equation. So it's some first-order differential equation, right? So it just determines the evolution of this thing as a function of S. So you're just describing a point in C to the K evolving as a function of S according to this equation. So this is, I think you can also think of this equation as like a gradient flow. So one way to think of it is gradient flow for a function which is minus the real part of E to the minus i theta W. Anyway, this is the equation. And what we're gonna be interested in is some particular solutions of this equation. So suppose I take two critical points, X i and X j. If X i and X j are critical points of W, then we'll define an ij soliton is a soliton, meaning a solution of this equation, a classical soliton, which obeys X of S goes to X i as S goes to minus infinity and X of S goes to X j as S goes to plus infinity. So the picture of what these things look like in space. So of course, the thing is actually complex value. It's valued in C to the K. So this is kind of a toy picture. But the toy picture of what it looks like is, way over here, it's at the critical point X i. Way out here, it's at the critical point X j. And then somewhere in the middle, should make it go a little faster. Somewhere in the middle, it kind of goes like this. It switches over from vacuum i to vacuum j. Of course, one thing it could do, one solution of this equation is just to be stationary, just to sit at the critical point for all time. We want things that don't do that. We want things that approach the critical points at the two ends, but it interpolates between them. Okay, because, okay, were I to write down the equations of motion of the Landau-Gindberg model, then this would be a field configuration which would be classically annihilated by two of the four supercharges. I'm trying to avoid introducing that language, but yeah, that's why it's called BK. Okay, so what can we say about these things? So, I mean that, so this particular quantum field theory is defined by a Lagrangian. The Lagrangian has a space of fields. The supercharges act as kind of odd vector fields on the space of fields. And so it's meaningful to ask whether that odd vector field is or is not zero at any particular field configuration. So here, you make a field configuration which is this thing at every time, just as a function of time, it's trivial, and as a function of s, it's this. That one is actually annihilated by some of the supercharges. In that sense, it's kind of classically BPS. It automatically obeys the classical equations of motion by virtue of being BPS, actually. Yeah, that actually happens even here. So, there's this subtle thing where the question was, does that ever happen that the classical one is not actually a quantum one? That's how I interpret the question. Even here, so what we're gonna define is a kind of index that counts these things with signs. But the more correct thing to do would be to construct, to actually try to construct the space of BPS states. And the way that it's constructed in this case, and I refer you to the 400-page paper of Guy Atom Moore and Witten, the way that it's constructed in this case is roughly speaking, you make a complex where for every classical solution, you have a state in this complex, but then there can be some differentials and those differentials can actually be not zero. So, in the quantum theory, it really does not have to happen that this thing lifts to an honest state in the quantum theory. Yeah, that's right, you have actual examples of this. So, in the notes, I give a page reference to where you should look. Okay, yeah. But I'm not trying to construct the space, I'm not so ambitious today. All I'm trying to do is construct the count and for that, it'll be enough to just count these things with signs. Okay, so let's just say a few words about what these things are like. So, one thing you quickly derive from the equation is that W projects X of R to a straight line in C. In other words, so you have this trajectory in C to the K, you look at the image of that trajectory under W, it just maps to a straight line. I'll draw a picture, yeah, let's draw a picture now. So, let's draw what it looks like in this cubic example here. What would the picture be? So, in the cubic example, it's actually very simple, there's only two critical points, right? So, there's two critical points and they're lying in the X space, which I actually wanna draw up here. So, here's gonna be the X space, here's gonna be the W space. So, W is this cubic polynomial, it's a three to one map. So, I guess it looks something like this, you know, cartoon picture of it, three to one map. Here are the critical points. Here are the images of the critical points, the critical values. So, call this X1, call this X2. Here's W of X1, here's W of X2. And if you have a soliton, it's a straight line from here to here. So, the soliton is gonna be some pre-image of this straight line. Okay, and what we're gonna do, so we'll define mu of i, j to be the number of i, j, classical i, j solitons counted with signs and see the notes for some discussion of how you sign them with signs. I'm just saying, for each soliton, we're gonna give it somehow a plus one or a minus one. We're gonna count them that way. So, in this example, let's just see what could happen in this example. So, it's actually not so easy to solve this equation sort of analytically, but it's easy to understand what goes on. So, for example, here, we could start at X1 and just look at what are all the possible gradient flows out of X1. Well, in this example, it's completely trivial because we're just in one dimension. All we can have is we can have a flow that goes this way, or a flow that goes this way. Because we know that they're gonna lie over, sorry, we already know that they lie over this interval. And so, just lift that interval up to here. Coming out of X1, there's just this, or this. Coming out of X2, there's just this or this. Well, okay, okay, now I take a poll. Is there gonna be a soliton? Yeah, how come? Yeah, right? I mean, look, just look. Here's a path going from here to here. This one comes out of here. This one comes out of there. They have no choice but to meet, right? So, there's the soliton, right? So, in the cubic example, right? So, luckily, I didn't have to solve the equation to find out that there's a soliton. So, yeah, let's call this, since I'll use it a lot, I'll call this just the cubic example, and I'll call this the quartic example. So, in the cubic example, what we just said is that the soliton count from, say, vacuum one to vacuum two, so I'll call it mu12 equals, well, I'm not careful about the sign, but it equals plus or minus one, and that's no matter what the parameter z was. So, we had a parameter z, right, which changes our cubic, but no matter what the parameter z is, that's gonna move these critical values around, but no matter what it is, that the prodigy is the same. I'm always gonna get one soliton. Ah, right. So, all this is for the case when z is not zero. So, in order to have all these soliton numbers well-defined, I'm gonna always be in the situation where I make everything, in this case, I'm just in one variable, I just wanna make all the critical points non-degenerate. So, for z equals zero, we don't ask the question. Then, there's not even the two vacuum to talk about, right? Likewise, mu of two and one and z is plus or minus one. Actually, if you work out the signs, this is plus or minus, this is minus plus. So, let me draw a picture in the z plane, very stupid looking picture. So, now, I think this is the first time I draw a picture in the z plane. Now, we're drawing a picture in the parameter space of these polynomials. That's ultimately gonna be the most interesting space. And the picture is just, okay, there's a special point at zero, where the vacua, where these two critical points come together. Everywhere else, nothing happens, just mu of i, j equals plus or minus one, nothing interesting. Okay, fine. Now, how about this chordic example? So, let's look at the chordic example. Well, now it's already a little bit less clear, right? So, let's draw the picture again. So, I've got, now there are three critical values. So, let's imagine that they're like here, here and here. So, I'll call this, let's see, we'll call this w of x one, call this one w of x two, and call this one w of x three. And suppose I wanna understand the salatons going from vacuum one to vacuum three. So, I draw a straight line like this. So, now, this is some, now we have a chordic polynomial, so it's a four-fold covering that we look at. I don't mind if you're confused. No, no, no, sorry. I do mind if you're confused. Yeah, what can I, yeah, yeah, yeah, yeah, yeah, yeah. Yeah, you always pick on the hardest things. So, I was deliberately, you know, a little bit vague about how I fixed these minus signs, right?