 And in particular on the, yeah, for the physics side, they were important for formulating the standard model on the math side that led to the formulation of Donaldson invariance, which distinguished the smooth structure. And of course, as we know, this led to a very kind of fruitful interaction between the subjects. So for these lectures, I was planning to review with the historic story, starting with what goes back to the work of Witten and the computation of Donaldson invariance balance goods as a game for rational surfaces. And if you want to reproduce these Donaldson variants for rational surfaces, in fact, it turned out that you have to understand the theory very well. You need to use the full cyberquitten solution. And then using that full solution, you can evaluate the so-called u-plane integral, all which I will explain in more detail. And there are some more recent works which I will touch upon, hopefully, tomorrow in the lecture tomorrow. So please interrupt with any questions. I'm not quite familiar with everybody's background, so if there are any questions, let me know. And I'd like to start the story with the physical theory in flat space and then work towards the theory in a compact form manifold and set up this so-called u-plane integral. And as Professor Nakajima mentioned, the u-plane is essentially the Coulomb branch for SU2 cyberquitten theory. So we are considering pure SU2 cyberquitten theory. And this pure means that we only consider the vector multiple if we don't add any matter. And the field content then consists of a gauge potential, a mu, together with a scalar field, a complex scalar field phi. And two while fermions, which are psi and lambda, and then lambda bar and psi bar. So this is a complex, so we have a phi and we have a phi bar. And let me also mention the representation of these fields under the global symmetry group. Symmetry group, which is SU2L times SU2R. It forms the SO4 rotation group. And then there is an r-symmetry group with inconvenient terminology, maybe. Let me denote it by as u2i. And there's also a u1. So this part of the global symmetry is just because we have cyberquitten theory has n equals to 2 SU2. And therefore, you can rotate the two super symmetries in a sense into each other, which leads to this global symmetry group. And if we add the quantum numbers here for these fields, we get the following representations, 1 half, 1 half, 0, and a charge 0 under the u1 group. And for the phi, we get their trivial representations for the rotation group. So we have 0, 0, 0, and a 2 for the holomorphic field and 0, 0, 0, minus 2 for the antiholomorphic field. And for the fermions, we get 1 half, 0, 1 half for the two chiral fermions plus 0, 1 half, 1 half minus 1 for the antichiral fermions. And the supersymmetry charges of the theory they transform in the same way as the fermions. So they have the same representations as this. OK. So this theory comes with a potential, which we aim to minimize, 1 over the coupling squared. And so all these fields are valued in the Lie algebra of SU2. We get a potential trace of phi, the commutator phi x and phi bar x squared. And we aim to minimize this potential to have the vacuum model space of this theory. And so in particular, this means that phi and phi bar commute. That implies that they are valued in the carton of the SU2 Gates group. And therefore, and then using the Gates transformation, we can set phi equal to 1 half times a times sigma times the third Pauli matrix is 1 half a times 1, 0, 0, minus 1. So a is a complex number. And for this, in a sense, parameter space, the potential will vanish. And we are at least the classical vacuum space of this field. So in order to parametrize the vacuum modelized space, we want a Gates invariant combination. So we take a quantity that is proportional to the trace over phi squared. And so if this parametrization this would go as 1 half times a squared. And this is what we denote by u. And so the space where u lives is the u plane. So this is valued in the. So this a has, in fact, so here it just appears as a parameter and it seems directly related to u. This is in a sense a weak coupling relation between the two. A has a more independent interpretation if we consider BPS states and their central charges. Since the electric magnetic charge is of the theory, sort of electric charges, once we have turned on a non-venancing A, we have broken the SU2 Gates group down to u1. And therefore, we have one set of integers gives us electric charge. And one set of integers gives us magnetic charge. And so this is under the unbroken u1 in SU2 on the Coulomb branch, where u is non-venus. So these are two integers which in principle can take any value. And we have the notion of a central charge or a complexified mass, which is then a times ne plus nm times ad. Sorry, I'm looking to put the n on the left side also, n e times a. And at weak coupling, where g is very small, we have that ad goes approximately like the complexified coupling tau times a. So tau is theta divided by 2 pi plus i over g squared. There are maybe some non-trivial factors here that go down from the top of my head. But this is a tau is a complexified version of this coupling constant. And we see it is always valued in the upper half plane because g is positive. And in fact, this tau and a, due to the rg flow of the theory, are also determined by u. So we have that ad, in fact, reads i over pi times the square root of 2 times u times the logarithm of u. And that a basically corresponds to the square root of u divided by 2, which then reproduces that relation. Any questions? So these are the, you should view these as the leading terms at the weak coupling. Both terms get an infinite number of corrections. It's called inverse powers in u. So the cyberquit and solution, one could say, is the solution to know all these infinite corrections to these parameters. Or you could say differently. So the cyberquit and solution gives a of u and ad of u. And we have the relation that tau, in fact, is by definition the derivative of ad to a. So we know tau as function of u, or at least the series expansion. And we can also invert this and express u as function of tau relation, where we had u tau is 1 over 8 times q to the minus 1 over 4 plus a series in q, where q is e to the 2 pi i times tau. I won't go into much detail here, but so I can explain that both a and ad, you should view them as a period of a one form over an elliptic curve. But for our purposes of evaluating the u plane integral, since this is for now enough, I will give some more details I guess by the way. We can include, if you want to work with the dimension full parameters, then u has a dimension of the scale squared. So we can involve the scale lambda. So we can make a drawing of the u plane given the solution, and it looks roughly like this. We have one, the weak coupling, singularity which I already mentioned, i infinity. Now you might think that there will be a point in the interior where u is equal to 0. But it turns out that if you include all the non-poderbative effects, there is no special point with u is equal to 0. But there are two special points where u is minus lambda squared. And u is lambda squared. Two singular points where either a monopole becomes massless, or a dial becomes massless. The weak coupling, the monopoles and dials are these very heavy objects. But if you go to strong coupling, it turns out that they can be massless. And these two points, they correspond in terms of the coupling constant to tau is equal to 2, I think is this one, and this is tau is equal to 0. There's a so-called wall of marginal stability, which is across which BPS stays decay. It's not so important for us. But for us, these three points, u is plus or minus lambda squared. And u is infinity. I infinity are the special points of the u plane, or the column branch of this theory. Any questions? OK, so this is the theory in flat space. Maybe one thing I should, before going to topological twisting, I should add. So I defined u as a trace of phi squared. If you go to the quantum theory, I should really define this as a correlation function of the quantum theory. And since this was the theory in flat space, this was a correlation function in R4. So we would like to formulate this theory on a compact manifold. And in order to have the power to evaluate, in fact, partition functions and correlation functions, we want to preserve some of the supersymmetries. And for this, we will need to apply a technique called topological twisting. And so basically why we need the topological twist is because these supersymmetry generators, they transformed us to the fermions. They all transformed in representations, which had more than one dimension. And we want to have a scalar, at least some of the supersymmetry transforming in a scalar representation or a one-dimensional representation. So to this end, we need to identify an isomorphism between the as you to R symmetry and the as you to inside the rotation group, which practically then basically says that you need to take the representations of a diagonal group of, say, as you to R group times, say, one of the components of the as you to components of the rotation group. So the new representations we get for the fields, as you to L times as you to R prime. R as follows, the gates potential, a mu, remains a vector. So it still transforms with the four-dimensional representation. Phi and phi bar, they remain scalars. So they still transform with the one-dimensional representation quantum numbers, 0. And then lambda psi, lambda bar, psi bar, they do change. And they transform as follows. One is the trivial representation, three-dimensional representation, and the four-dimensional representation. So basically, you take the diagonal of the previous as you to R group and the as you to I. And then if you look at the representations, you find this new representation for this as you to R prime, which is the diagonal of these previous two groups. So we give these three terms. They'll represent a different fermionic field. The trivial one is a Grossman valued zero form. On this side, eta zero form. Then the middle term, 0, 1, is actually chi. It's a self-dual. It's also a Grossman valued because it comes from a fermion. And then it's a self-dual two form. And the last term is a vector. So I could maybe write in this explicitly. Chi mu nu dx mu dx mu. And then we have a vector psi mu dx mu is a Grossman valued. And so since we have this one, this trivial representation, we also have a scalar supercharts q with squares to 0, which is nil potent. So let me just maybe list how it acts on A mu and phi. q acting on A mu gives us psi mu. q acting on the holomorphic scalar field gives us 0. And q acting on the holomorphic scalar field is proportional to eta to the zero form. So since it squares to 0, this is a BRT. q is a BRT type symmetry. And for that reason, basically all the observables in the theory will fall into three classes of observables. Namely, those which are not annihilated by q. I represent a generic observable by Curly O. So the first class is those for which q acting on O is not equal to 0. They don't pay so much of an overall for us. Then we have the so-called exact observables O. And those are the observables which can write as the q acting on some different observable w, or some w. And then the third class are the closed ones, namely those which are annihilated by q but are not exact. So these are the O observables which are in the kernel of q, but they are not in the image of q. Now this q is a global symmetry. And there is a famous statement from the field theory, the Ward-Tacassi identity, that the correlation functions of a q-exact observable, they should in fact be 0. So if you determine the correlation function of qw, q-exact observable, let me write it on the left-hand side, although this may be not well defined. But on the right-hand side, we get a path integral dx physically with the insertion of this operator e to the minus s and x, where x represents all the fields of the theory, very general statement in point of field theory. This is equal to 0. And it in particular says that q-exact observables decouple from the q-closed ones. If you would add a number of q-closed ones here, then it would still be 0 because you could bring them into the bracket because q is acting on those fantasies. So maybe tomorrow I will explain that in the topologically twisted Cyberg-Witten theory, we found observables for which this did not appear to be true. We found q-exact observables which seem to diverge, in fact, or be infinity rather than 0. And we needed to develop a new regularization and renormalization to demonstrate that it actually does evaluate to 0. And therefore, in a sense, this confirms again this general statement in quantum field theory. So that was the result of a paper from January this year. But before being able to explain that, I need to introduce a few more things. So this is quite a general statement for topological theories that you have a q, a BRST type fermionic symmetry. Your observables then fall into these three classes. And now in our topological gates theory, the Lagrangian basically takes the following form. This i over 8 pi sum numerical factor times the complex five-coupling constant tau times the trace of fx, where it's f. And then the rest of the action is q-exact. So this is the q acting on sum w for some specific w. In the particular, since this is a topological term, there is no metric dependence in this term. All the metric dependence in the q-exact, so the metric dependence is in part. And for this reason, if you take the variation of a physical, if you take the variation of a correlation function to the metric, then this is equal to zero. And this will infect a zero if q acting on o is equal to zero. If you do this variation, you basically bring down this q-exact term. And we see with the vorticasse identity that the q-exact observables decouple from the q-closed ones. So if o is q-closed, then it is independent of the metric. Topological observables, q-cormology. So in principle, the theory has many other observables which one can very well evaluate. But they are just not topological. Question so far? OK, so for our topological twisted cyber-witten theory, also known as Donaldson-witten theory, there are two observables in the q-cormology which are particularly useful. One is one observable. First one is always equal to 2 times u for some to get the orientation right, actually for the matching with math. We multiply this u by 2. We discussed that if you act with q on phi, it gives zero. And since u is built of phi, if you act on this, you see it is annihilated by q. And the more complicated one is o is minus of x, which is 1 over 4 pi squared integral over x, which is an integer 2-cycle in the manifold x in h2 times z. And then on the right-hand side, we get the following. We get a trace over psi, which psi, minus 1 over the square root of 2 of phi and f. So this is a bit more work to demonstrate that it vanishes if you act with q. But it is, and there is a so-called descent mechanism which constructs q close observable starting from u, basically. I will discuss these. These are the two relevant funds for this talk, which will be relevant to if you rate these observables or these correlation functions with these observables, you will produce the invariance defined by Donaldson. So I already mentioned before we are considering a compact four-manifold m, let m be a compact four-manifold smooth. And so let me say a little bit more about it. There is the lettuce, or maybe let me write it the lettuce, or the two cycles, h2 and z. Gives naturally a rise to a lettuce, l gives. So we allow for the possibility of torsion. So this l is basically this group, but modding out the torsion. You could say you embed this group into h2m r, and then l is the embedding of h2m z into h2m r, because in r there will not be any torsion. And so the signature of l is b2 plus, b2 minus. And so on this lettuce we have a quadratic form. l is denoted by b. So this is l times l to z. And we can extend it to l times r to get a form over the reels. So we are interested to evaluate the full path integral. Let me note it by girly z over this four-manifold integral of dx, which are these fields of the cyberquipment theory. And then we integrate, or we multiply it by the exponentiated action integral over m of the Lagrangian l. And once we put it on the compact manifold, it turns out there are some curvature couplings, which I denote by mu of u. So these are some curvature couplings, which you don't see in the Lagrangian of the cyberquipment theory on the flat space. But once you go to a compact manifold, they become important. And it turns out that this, so we do a path integral on a compact manifold. And what is a bit different from doing at the integral over a compact manifold than doing the path integral over flat space is that we now also integrate over u, because it is no longer a boundary condition in the non-compact space. m is compact, so we can integrate also over u. There is an integral over the u-plane if you work over a compact manifold. And so let me just maybe sketch again the u-plane. The u-plane, since I have these three special points, a weak coupling limit, and then two strong coupling singularities, u is minus lambda squared and u is lambda squared. And now this path integral turns out has delta function support on these singularities, which is known as the cyber-griton contribution to this integral support, plus or minus lambda squared. And this can be completely specified in terms of the cyber-griton invariance, which was discussed by Lothar this afternoon. And then there is a second contribution from the remaining non-compact part of the u-plane. Let me just denote it by a set of the u-plane. And this contribution from the u-plane is an effective u1 theory, as I already discussed for the BPS states, because once you have a web for u, then you have broken the gate symmetry. So in a sense, it's now at least the gate scope. You don't have to work with the non-a-billion gate scope anymore. You can work with the u1 connections. My interest lately has been in this contribution from the u-plane. And in fact, it does not always contribute. So I was discussing the signature of the four-manifold, B2 plus, B2 minus. If B2 plus is larger than 1, you can show that the contribution from the u-plane is 0 set u-plane is 0 due to too many chi 0 modes. So this field chi was a fermionic self-dual 2 form. And if B2 plus is larger than 1, you get too many of those. And just the rules of the grassman integration will put it to 0. And so it does contribute for B2 set u-plane is not equal to 0 for B2 plus smaller equal 1. Although actually the case of B2 plus is equal to 0 is not very well studied. So we will focus on the case that B2 plus is what will continue with B2 plus this one. OK, so even for this class of spaces, you would figure it happens that, in fact, the cyber written contribution venesis, then you can put a metric of constant curvature on your manifold. This is, for example, the case for the rational surfaces. And then the full answer comes from the u-plane integral. So if you want to reproduce, for example, the Donaldson invariance for the projective plane, or P1 times P1, then physically you need to evaluate this u-plane integral to reproduce those two discuss. And we will also set, in fact, B2 B1 equal to 0. This is more for simplicity to avoid later the zero modes coming from the psi field. In principle, you can include them. There's paper by Mourinho and more. But to keep things simple, we set this one equal to 0. But I'm not imposing that it is simply connected. So there might still be torsion in B1, which would lead to torsion in H2 as well. So we are aiming to determine the full path integral of this theory on the u-plane. And there is, in fact, a very, quite a beautiful analysis of what is actually contributing to this integral. And it turns out, let me give the reference for this. This is the paper by Mourinho and Witton from 97, that the u-plane integral reduces to an integral over the zero modes of the fields, the zero modes. So they argue this by taking a limit of the metric, g sending the metric g to d squared times g, and using that the theory is topological. And then it turns out that only the zero modes survive this limit. So this, now we come pretty far in setting up this u-plane integral. It takes the following form. It turns out that this, most naturally, or one unnatural way to formulate it is in terms of these parameters a we saw before. And then we get the following zero modes. We get the zero mode of the fermionic zero form, the self-dual two form. And there is an auxiliary field, d, which I haven't really mentioned. But there is one field to make the supersymmetry algebra close. And then we have these, the curvature couplings I mentioned before. I call them nu of, and they are more specifically given by as follows, au times pu power sigma e to the minus the integral m over the manifold m. And now we just have the zero mode Lagrangian. So you can put all the derivatives in the fields, all the kinetic terms you can put into zero, and just keep the constant terms in the Lagrangian. So this was what I earlier called nu of u. And now I'll notice the zero mode. So let me see what we want to get. So maybe let me just give this Lagrangian. Actually, there's not so much left after you produce the zero modes. So it reads as follows, 1 over 16 pi. And then we get tau bar f plus the self-dual part of the u1 field strength, which the self-dual part of the u1 field strength plus the holomorphic tau, which couples to the anti-symmetric parts of the u1 field strength. Then there is the part of the auxiliary term, y, maybe I should just call this imaginary part of tau, over 8 pi d, where it's d. It's a self-dual real two form. And then there is a term with the fermions, i over square of 2, 16 pi d tau bar dA bar eta chi wets f plus d. So that's the zero mode Lagrangian. And then these factors, a u and b u, I can give them a little bit more detail. They are proportional to dU dA to the power 1 over 8. And bU is u squared minus 1 to the power 1 over 8. Chi and sigma are the Euler number and the signature of the manifold m. So these are just topological data of the form manifold. Yes? Yeah, we don't really see it. It would just be multiplied by the order of the torsion group. And it comes out of the integral. For some choices of signature, you're forced to have torsion by the Röklund's theorem. So in that sense, implicitly, it will be dependent on sitting in chi and sigma. But you don't really see it. Yeah, that's right. There are other questions. And there's one thing missing from this expression, which is the sum over the u1 fluxes, these f plus and f minus. So it's a sum over h2, so the u1 fluxes. So we have normalized the field strength. So it's at 1 over 4 pi times the class of f is contained in h2 mz. This is for SU2. If you want to consider gaseous group SO3, which cannot be lifted to SU2, then there will be a shift by a half integer. OK, so let me kind of write things a little bit more explicitly. If I want to consider correlation functions, so in general, I will note them by this phi. And the field we put in the correlation functions I put between the straight brackets, I mentioned that for as a free gaseous group, you can turn on a hoved flux. So this results in this mu, which is then contained in 1 half h2 mz. Our lattice elements would lie at 1 half spacing in the lattice. And we saw in the zero mode Lagrangian, it depended on f plus and f minus. So there is a dependence on the metric there. And in fact, we find that these expressions are only locally constant as function of the metric. And to indicate this dependence, I put this j here, which is the period point of the four-manifold. So then on the right-hand side, what we get is the following. I will write it as an integral over d tau, where it's d tau bar. We saw, maybe I shouldn't have erased the zero mode Lagrangian, but if you have it on your notes, you see that there was a term there, which was d tau bar dA bar. And if you integrate over the fermions, the term comes down and suggests a change of variables to tau, which I have implemented here. Then I multiply it by the curvature couplings, but now it's function of tau, and the sum over fluxes, which I denote by psi mu j tau bar. So let me just write down the expressions for these two terms and then I will stop. Newtale, the tau, is this change of variables with now for the holomorphic side dA d tau times nu of u, and then u viewed as a function of tau. And the fluxes, maybe, it's better to psi mu j is one divided by the square root of the imaginary part of tau times the sum over k in the lattice plus mu. So this L is really the lattice. Then there is this inner product between the flux k and the period point j, which is a self-dual 2 form, q minus k minus squared divided by 2, and q bar k plus squared divided by 2. So that's the sum over fluxes integrating over d eta and d chi brings down this linear term in the flux. And then we can also insert observables in this integral O to complete the expression. So then we'll continue tomorrow by explicitly evaluating this integral partly using the mock model of forms, which also came back in the title. Thank you for your attention.