 dA. Yes, dA of FA is zero. So if dA, if you apply dA to this, then you get that dA star of FA equals zero. But the solutions to this are the BPS states. So this is a second order PD. This is a first order PD. And furthermore, it minimizes the energy. So these solutions minimize the integral of FA squared. So basically, anti-self-dual connections are an example of BPS states. But a lot of other things that you've seen in mathematics are BPS states. So examples include, okay, so AST, but also cyber-witten solutions. And also, if you like symplectic geometry, then J-holomorphic curves. These are maybe the most common, but in some of the lecture series, you'll see other BPS states. They depend on the theory. Okay, I think these are, oh yes, one more thing. In physics, you also hear about, I don't know, something 4Dn equals 4 or 3Dn equals 2 theories. And I wanted to say what that means. So if you have something like 4Dn equals 2 theory, well, this is just a dimension in which the theory is. So well, it's d plus 1. So this would be like a 3 plus 1 dimension. And this relates, well, it's related to the number of superchargers. So theories can have maybe just one supersymmetry or more. And n, well, n is kind of a constant depending on the dimension times the, no, sorry, the number of supersymmetries is a constant depending on the dimension times n. Well, the constant is kind of the minimal number of supersymmetries in that dimension. Let me just say we have 4n superchargers in dimension 4 and 2n in dimension 3, because these are the dimensions we care about mostly. Okay, so now I'm going to give four examples that we will encounter in this lecture series of written type dqfts. So next examples. So first is a, well, it's, let's see, what's called the pure super young Mills theory. This is a 4Dn equals 2 theory. And it's topological twist. It has a topological twist. It gives some theory that includes basically two things. It includes two things that mathematicians like Donaldson theory based on the young Mills equations and cyberwitten theory. These are two limits of the same theory. So Donaldson theory is kind of the high energy for small scale. So this is kind of the theory that you see at small scales. And this is the one that you see at large scales or at low energy. All right. So basically this gives you some PDEs that are interesting. And this gives you two 3 plus 1 dimensional dqfts. And they are related. So in dimension 4, for closed 4 manifolds, we have the cyberwitten invariance and the Donaldson invariance. The cyberwitten invariance were the object of lecture one by Hadish. The Donaldson invariance, I guess we haven't seen, but they're based on the young Mills equations. And these two are related by something called Witten's conjecture. There's a physical explanation for this, but there's also been mathematical progress in showing that kind of, if you know one of them, then you know the other. And they're related in a slightly complicated way. Now in dimension 3, you have cyberwitten, or also called monopole fluorophomology. So now you associate a vector space to a 3 manifold, right? And the cyberwitten theory, this is called monopole fluorophomology. It was, yes, there's a book by Kronheimer and Rovka, which does this in great detail. And then here it's young Mills or instanton fluorophomology. And this will be the object of lecture series number 8 by Tamrovka. By the way, I should say that these are, in principle, there should be a relation between them, but this is unknown. This is one of the big conjectures in the field. I mean, the big questions like what is the relation between the two fluorophomologies. Okay, so, oh yes. So cyberwitten fluorophomology is actually the same as something else called Heggart fluorophomology. This was introduced by Osvat and Sabo, and they're the same. So, nowadays, it's known that they are equivalent. So, but this is based on gauge theory on PDEs. And this is constructed using symplectic geometry and pseudo-holomorphic curves, basically J-holomorphic curves. And this one is actually more computable. So, that's why people like it. I like to do computations with it. It's somehow easier to work in symplectic geometry. And there is a similar, there's also a theory for knots, knot fluorophomology. It's kind of Heggart fluorophomology, but if you have a knot in the three manifold, and this kind of stuff is gonna be covered by Jen Hum in lecture series number, number what? Number four. So, this is really cyber-witten theory, but in disguise. Okay, so that was one example of written type TQFT. There are some more TQFTs of these types. So, also, so, yeah, so there are some theories of class S. So, these are also 4DN equals 2. And they're determined by some data consisting of a Lie algebra, a Riemann surface with punctures, and some data, the punctures. Let me not say what it is. Right, but an example of this is actually super young Mills. When you take G equals SU2 and C is P1 minus, well, with punctures at zero and infinity. And yes, you give it some data. So, this is a generalization of this theory. Yes, let me say, okay, I'll say this comes from something called 6D20 super conformal field theory, also known as theory X or the five brain theory. This is not a topological field theory, but it's some quantum field theory that physicists like. And it actually gives many of the other QFTs that we've seen. So, basically, this is in six dimension and you write it on a four manifold times this curve, times this Riemann surface. And then it gives you these theories of class S. Okay, so the reason we care about these is when M4 is a circle times a three manifold, these class S theories are described in terms of, they basically have something to do with maps from Y to some space M, which we will encounter over and over again, this is the space of Higgs bundles on C. Yes, this is the modular space of Higgs bundles. It's something associated to the Riemann surface. Well, there's lots of interesting things about M. M is a hypercaler manifold. Let me say a few things about them. It's hypercaler, so it has a metric and it has three complex structures and three, well, it has a whole family of complex structures and caler forms. And basically, if you know and basically these theories of class S, give you information about the metric, about the hypercaler metric. And this will be the focus of the lectures by Andy Nightski next week, so that's number seven. Yeah, so he will tell you about basically counting BPS states can be done in these theories in some way, and they give you some information about the hypercaler metric on this space. Now, the space M appeared in mathematics long time ago by Hitchin, so Hitchin is the one who it's also called the Hitchin modular space. Maybe I should say one thing. So if you with the complex structure J, this is not something complicated. It's just the space of, well, it is complicated, but it's the space of representations to SL to C from the fundamental group mod conjugation. So it's the character variety of the surface, of the every month surface. But okay, from this description, you cannot see the metric and the other and the hypercaler structure. So this space will be, so some, it's some of its mathematical properties will be studied in lecture series number six by Laura Schaposnik. Okay, so, okay, so what's another example? This one we're not going to see too much in these lectures, but I thought I would mention it. So there is also 4DN equals 4 supersymmetry. Yeah, super Yang-Mills. So this is Yang-Mills, but with more, more symmetries, more supersymmetries. You can think of it, I mean, one aspect is Yang-Mills with complex league group, like SL to C instead of SU2. This comes, so all these theories came from the 6D20 theory. This also comes from 6D20 theory in some way. Kind of everything comes from that. And it has some topological twists, which give you some other equations that I think we won't, won't be discussed in this lecture series, but you might see in other talks. So these are the Kapustin-Witton and Bafa-Witton equations, among others. And for low-dimensional topology, this is relevant because of a conjecture by Witton in 2011. So based on physics, he says, he says that the coefficients of the Jones polynomial found solutions to Kapustin-Witton equations on something. Right, so this is somewhat related to, I mean, we saw the Jones polynomial coming up in Schwartz type theory as some path integral, but it also comes in a Witton type theory as counting some solutions to some partial differential equations. And mathematicians are trying to make sense of this conjecture and prove it in some cases. Right, so this is somewhat related to lecture number two, but okay, because, well, it involves the same thing. Actually, Witton said a bit more. So in lecture number two, you've also seen the categorification, which was Havana homology. And Witton also says that Havana homology should come from counting solutions to something called the Hadish-Witton equations, which are a five-dimensional extension of, well, they're kind of like Kapustin-Witton, but one-dimension higher. And yes, one more thing. So these Baffa-Witton and Kapustin-Witton theories, if you reduce them to two dimensions, you get back the Hitchin equations. So the moduli space M also can be, yeah, you can also get to it like in this way. It's related to this kind of equations. Okay, so the final example is something in three dimensions. So there is a 3D N equals 2 theory. This comes from the 6D theory again. 6D to 0 theory. By the way, the 2-0, it still counts supersymmetries, but in dimension six, there are two kinds of supersymmetries and there are two kinds of the first and zero. Well, yes, anyway, it counts both types. Yeah, so basically, you write this theory on a three-manifold times S1 times D2, and you get some three-dimensional theory. So to a three-manifold why? It should give some numbers and these are, well, there are some invariants that were introduced by Gukov, Putrov, and Vafa and also another paper by these three people and Dupé. And yes, I guess we can call them invariants of three-manifolds, but they have no mathematical definition yet, except in some cases, like for plummings. But they are related to something we've seen before, namely the WRT invariants. So the WRT invariants of three-manifolds make sense for Q equals e to pi over k over k plus 2. So basically, at roots of unity. You can extend them to roots of unity. I mean, these are the primitive roots, but you can also extend them to other roots of unity. But yes, how should I say? So the GPV invariants are power series with integer coefficients and they converge for absolute value of Q less than 1. So they converge in the unit disk. And yes, and when you take the limits at certain roots of unity, you should get back the WRT invariants. So in principle, they study similar things to the WRT invariants, but in a different sector. And what's important is they have Z-coefficients. So by the way, they count, I mean, they count BPS states in this theory. And this will be the focus of lecture number five by Pavel Putrov. So even though there's no mathematical definition, you can say what they are based from physics in some class of examples and you can study them. The reason why, for example, I find them interesting is because also from physics, you can expect them to be able to categorify them. So there should be, oh yeah, by the way, they're called usually Z hat A of Q and should be categorified. So this should give something like Kovanov homology, some version of Kovanov homology for three manifolds. Well, let me put it this way. So the Jones polynomial is a case of the WRT invariants for knots in R3 and, well, in that case, it is a polynomial with integer coefficients and it has a categorification, namely Kovanov homology. But for three manifolds, people don't know how to extend Kovanov homology to three manifolds. That's an open problem mathematically. And the problem is that this is just defined at the roots of unity. But this invariant should have Z coefficients and therefore there's some hope of them being the Euler characteristic of some theory. Okay, so I think that's what I had to say about this theory. Questions? Yes. Could you say something more about this procedure going from? No, that's a topological twist. Maybe Max Zimert will talk about it on Friday. I mean, I would have to tell you what the QFT is and yes, and I'm not gonna do that. Okay, other questions? Yes. Well, quantum field theory is all kind of conjectural. Yes. I mean, yes. I mean, the whole theory, I mean, mathematicians don't usually try to, I mean, some mathematicians try to make it rigorous and they have some success. But in general, so people study the field theories, the physicists, and then they get some equations and then mathematicians just study those differential equations. All right, so let me stop here. Let me remind you there's no problem session after this, but the