 Thank you very much. So I was giving a task to lecture on introduction to instantons, so I'll try to start with something pretty elementary. So we will be looking at path integrals in Euclidean quantum field theory, which means that the spacetime will be Riemannian manifold, but the space of fields may be complexified. So the reasoning for this step is the finite dimensional analogy. So in finite dimensional integrals, it is convenient sometimes to deform the contour of integration. And so even if your original integral was, let's say, of something real, over a real manifold, you may end up calculating it by something like residues, and that you do by deforming the contour away into the complex domain. So a typical example, so we'll be interested in integrals of this form. So you start with, so this is a finite dimensional analogy. So you start with some, let's say, manifold x with the volume of 4 omega, and s is a real function, and so that's the original integral. But then, so you want to study this exponential integral as a function, let's say, of h bar, the small parameter. But then you view this original manifold x as sitting inside its complexification, whatever it means. And so this becomes, so we would like this volume form to extend to a holomorphic volume form on the complexification. So this complexification contains the original space x, this function extends to a holomorphic function. And then you realize that, well, this original quantity is a particular case of a family of actually, of a variety of possible integrals where you choose, so gamma, you choose a middle dimensional contour in this complexified space with the only restriction that this integral should converge. So typically what you do, you look, so you define, let's say, a subspace where, so you take the real part of s divided by h bar, and so you expect this integral to converge when this real part is very large. So you take the preimage of the set of very large values of the real part. So this is sitting inside this xc. So typically some kind of, we have some sectors. We don't really care about what happens in the interior, but far away towards infinity in the space xc, we want the real part of s over h bar to be very large. And so your contours are anything which goes to infinity along these directions. And so gamma has a homology class, which is in the middle dimensional relative homology group. So it means that the gamma is a chain whose boundary should be somewhere in this region. And so you have a choice of this contrast gamma, and in this way you get a vector value function of h bar. So the original integral is just one component of this function. But the reason we want to study all of these functions is that this function s may have some parameters, and as we vary these parameters homomorphically, the components of this vector will get premuted eventually. So this is some kind of fundamental object. And it's typically a fundamental solution to a system of linear equations called Picard-Fuchs equations. These equations are in the space of parameters of the function s. Is it independent of gamma or your? It's a vector. So it's a vector. Gamma labels the components of the vector. What do you mean it's independent of gamma? It depends on gamma. So gamma is a discrete label. So this is a group. And so you can view z of h bar as valued in the vector space, which is a complexification of that space. Why action is homomorphic function? Well, that's your assumption, that you extend the real function on the real manifold to a homomorphic function on its complexification. It's not always possible, but we'll assume it is possible. And it is always possible in cases of interest, where functions are typically polynomials. So polynomials are homomorphic functions. All right? So the next step in this analysis is to find different bases in the same vector space. So here, to define the integrals, you are concerned basically with infinities in the space xc. So you want the integral to converge. But there is another viewpoint which is actually dictated by the quasi-classical limit. So if you take the limit h bar going to 0, then the integrals are dominated by the critical, by the saddle points, by the critical points of s. And so there is a different basis. Well, it's different. I didn't tell you how we didn't discuss any specific choice of the basis in the space. So there is a basis in this place of allowed contours, which is, which comes from the critical points. And so this is the basis of left-shut's symbols. So these are special contours tailored to L sub p to the critical points. And the idea is the following. I'm deforming the gamma within the same whole of the plus or it's a different, since you're choosing different. So it's a basis of gammas. So it's a basis of gammas. Any specific gamma you can expand with some coefficients. So to define this basis, what you do, you pick a generic, let's say, Hermitian metric on this complexified space. So x up to c. And then you study the gradient flow, the real part of s over h bar. So you look for the solutions of the equation x dot is equal to the gradient with respect to the submission metric h. So the critical points are where this flow stops or starts. And then at each point, so let's look at what happens near a generic critical point. So near p, you can approximate s by quadratic function. So there's a value at this point. And so z i local coordinates. So I'm assuming that the function is a non-degenerate so-called Morse function. Then you can actually find, you can make a formal change of variables so that this function will actually have this form in the neighborhood. And now the real part, so suppose now h is real, then the real part of s over h bar is s of p over h bar plus 1 over h bar. So what you see, you see that this function has half of the positive squares and half of negative squares in the expansion near the critical point. And so if you look at the gradient flow equation, I'm assuming that the metric is flat. You can generalize it, it doesn't really matter much. So what you see, you see that for half of the variables, this point is repulsive. So this is point p. And so for the variables x, this gradient flow repels you from this point. And for the variable, so this is x variables. And the variables y are attracted. And now if you follow the gradient flow along the x direction, then so x increases, local exponentially, but then you don't know, and you see that the real part of the function increases. And so that's the direction you want to follow in order to make the integral convergent. So we take the union of outgoing trajectories, like x lines. And so this is the left shift symbol associated with the contour p. So locally, it looks like the product of a sphere. Well, it looks like a copy of Euclidean space, but globally it may be very complicated. So as an exercise, I propose to study the area function. So it's a good exercise is to analyze the integral, let's say of this form. I put some i's to make things simple. So this is the function a of x h bar. So you can define it as a real integral of something which is oscillating, but you can now use this technology, discover that there are two possible contours, discover that there are two critical points, and it's fun to play with the left shift symbols for this. Say it again. You don't need the metric to be clever. No, I just only require permission. So one nice feature of this flow is that in general, you can't say much about it, but what you can say is that the imaginary part of S over h bar is constant along the flow. And so that you can use to prove, for example, that for generic h bar this gradient trajectories, so the trajectories which emanate from one critical point will not enter the other critical point. But then also using the same property, you can actually find when sometimes they do hit each other and then there is an interesting story of transitions and kind of phase transitions and complex, complex, complex. This was just for the example. The fun is to study for complex h bar. What do you mean by generic h bar? Well, generic up to measure zero. Because this quantity is constant along the flow, if you map the so on each left shift symbol, you have a real function which is constant, which is the imaginary part. So this is constant and the real part grows. So if you map all this into the S over h bar plane, then your left shift symbols, they have this structure. They map to half lines, right? Because the imaginary part is constant and the real part grows. So you start with the, this is the value of, at the critical point and then you're, along the flow, the real part will only increase. So generic h bar means that these lines do not intersect. But if, so when you start changing h bar, basically changing the slope of these lines, at some point you may start hitting each different point. So that, this, when things become complicated. Things are already complicated enough. So let's, let's, in the beginning, let's not discuss all possible complications. Sorry, when you say that Z is a fundamental solution to a system of, in which variable? So the parameters of S. So for example here, the parameter of S is letter x. And so this, this function solves the very famous equation, which is roughly d by d x squared minus x. So that's, that's the Picard-Fox equation. It's a second-order equation. It has two solutions. That's, that's, that's because this integral has two possible contours. That's because this function has two critical points. So it's all, all matches nicely. All right. So now let's go to the, the infinite dimensional case, which is what we want to study. So the simplest dimensional case, path integral in quantum mechanics. So let's study the particle. Let's take non-relativistic particle in the double well potential. So I'm assuming this, my potential has a, has symmetry. That's for historical reasons. That's one, that's one of the kind of dramatic applications of these methods. But it doesn't have to. So let's take the celebrated example, the real section of, of the Higgs potential. So this is, it's a quartic potential has two degenerate minimum and one local maximum. So classically you have two, for low enough energy you have two possible allowed motions, motion around left minimum and the motion around the right minimum is going to end badly. And well, now we want to study the Schrodinger operator. We will, we will reintroduce H, well let me put it, and while we're interested, let's say we're interested in the spectrum of, of this Newtonian and we will be looking at the low lining eigenvalues of this operator. So small low energy states. So to, to, so it means that E is small. Well, we'll start with real and then we'll complexify. So we want to, we will study this by analyzing the trace. So this is H, curly H is the space of states, the space of essentially these are L2 normalizable functions of X, so in real line. But in order to, to have a better control over the, over this trace we will be studying the Euclidean trace. So it will be not the trace of a unity operator, but the trace of the contracting operator, just usually Euclidean partition function, I could put H bar down here. So T, and so what I want, I want the real part of T over H bar to be sufficiently large. So that means that in this trace only the small eigenvalues will contribute, everything else will, will be suppressed. Now this trace can be computed by the usual, well, can be represented by the usual path integral. And so in any textbook you will, you'll find Euclidean, Euclidean path integral representation for this integral. But today I want to, to, to use a slightly less known, I mean slightly less used phase space path integral representation because that's, that's going to teach us something. So this partition function can be written as a formal path integral. So this is the integral of loops s1 into r2. So r2 is my phase space. And so they are, so these, these are periodic functions of one variable called T. So they're periodic because we're confusing a trace. And so what we put here, we put I over H bar integral p dx minus, I will put T explicitly front. So my little T will go stupidly from 0 to 1. So it's just a parameter on the trajectory. The physical time, the physical time period will be explicitly multiplying the Hamiltonian contribution to the action. So this is my minus s over H bar. So this is my, this is my definition of fs. And what you see is that this s is already explicitly not real. So it's minus i integral p dx plus T integral H of p x dt, where H is a classical Hamiltonian. So if when p, both p and x are real, my action has both imaginary and in real part. So this is the real part, this is the imaginary part. Well, now we have this, we have the integral which we analyzed in fine dimensional case. So let's try to apply this Lefcher-Steinball philosophy to it. So what we eventually, what we will be doing, we will be deforming the, this counter. So we use this now as a a contour inside the space of complexified. So the, inside the complexified space of fields, which is the space of loops into the complexified phase space, this C2. And so we will be interested in all possible contours. Okay, so this is the picture. So this is the, now this is an infinite dimensional space of, so these are loops, p of t x of t, p and x are complex, and they are periodic function of t. You can also, inside, you have the real contour, this original, original, original domain of integration, which was the space of real loops. And now what we will be looking for, we will be looking for the critical points of the action, and then some kind of Lefcher-Steinballs, which will emanate from these points, and then you can try to expand this original real contour in this Lefcher-Steinballs. So critical points. So these are critical points in L of C2. Well, you may at first be surprised that we will be looking at the complex values of p and x. I stress that we don't change the nature of time. So time, here, is one real dimensional. It's, it's around the circle. The fact that it's Euclidean, it just means that there is no i in front of this capital T. That separate from the fact that we want to, we allow the space of fields, we allow the fields to become flexified. And we do that because that's what we do. We just deform the contour of integration. There is nothing deep philosophical about this. So well, you just need to solve the, look at the variations of this functional. And so what you'll get, you get minus x dot plus t p is equal to zero plus p i p dot plus t u prime of x is equal to zero. So you need to solve these equations. And what you usually find in textbooks, you find the expression for, for the same integral where p has been already integrated out. And then you somehow miss the fact that the space of fields were complexified because you can still think that you, you're dealing with the real x. And so usually people say that, well, when you do that, you'll find an action of a particle in the inverted potential. And so there's some, some intuition behind that. But the fact that even when x is real, p is already imaginary, this is usually missed. And so that's what I wanted to stress this point. So anyway, so what we need to find, we need to find solutions of this equation where p and x are periodic. And that is not so difficult to do, actually, especially for the quadratic potential. Prime means derivative with respect to x. So x and p are functions of t. So that's what, what, oh, you mean I missed the factor of t? So what, what, what's the question? You prime of x, yes? Differentiation with respect to t or with respect to x? With respect to x. So to solve these equations, you observe that these are, after all, Hamilton equations may be with, what? I don't assume anything, just so you compute, you prime, so this, this is u, you compute, it's with respect to x, it's elementary exercise, you get what you have. Sometimes it vanishes, sometimes most of the time it doesn't vanish. Okay, so what you find is that the value of the Hamiltonian remains constant along the trajectory. So you, without, so, so you know that this will be, so the whole, the trajectory p of t, x of t sits on a complex curve. So this is the equation of the curve for sum e. So for different trajectories the value of e might be different, but for each trajectory there is some value of curly e. And now the good thing about the quadratic potential is that this equation defines actually an elliptic curve. So this is an elliptic curve. You, I mean it's, it's actually non-compact, you can add a couple of points and so it will be complete elliptic curve. So you can actually visualize the trajectory very explicitly. Say it again. Divination of an elliptic curve. Right, so, so it's, instead of, you know, picture is worth a thousand words, so that's the elliptic curve. So it's actually, it's a real torus. So it's, it's a curve in complex geometry. So p and x are both complex variables. So the, I mean, the space of p and x is four real dimensional and this is a complex equation, so it's two real equations. So it defines a surf real surface inside this four-dimensional space. And well it's the surface given by this equation. Well it's, you can bring it to the following form after some change of variables. Let me call it, so it's a different variable. And so you can, so from p and x you go, you can go to y and x and then from y and x you can go to the so-called uniformizing coordinate, z, which is the integral of dx over y. So you, you look at the set of, so g2 and g3 have some constants. They depend on, depend on the parameters and on the value of the energy, classical energy. And then you study the integral of, of this differential. It happens to be holomorphic differential, has no singularities. The value of this integral is not unique because, well I drew this, from the picture you went, you, you, you see that actually there are two non-contractable cycles on, on this, in this geometry. Well you can actually see them if you look at the, at this equation as the covering of the x-plane. There will be four branch points and so this is, so that over each, so for each value of x away from four special points, you have two values of p possible. So this curve is a two-fold cover of the x-plane without four points. And so if you go around these branch cuts that represents one, one of these cycles, let's say the a-cycle, and if you go in between, that's the, that's the second cycle. And so what it means is that this, this coordinate z is defined not uniquely but up to two periods. So these are the periods of the, of this differential around the a-cycle and b-cycle. And so that's the way to, to see this torus, is you take the complex plane of the variable z and make this identification with two primaries omega one, omega two. And so that gives you the torus. But now our trajectory, our solution, it's a real one-dimensional counter. And so this uniform, uniformizing coordinate is actually, let me call it x tilde. It's actually, this differential is actually the differential dx over p in the, on the original curve. And that's really just the time, the differential of time. So what it means is that this uniformizing coordinate, which is complex, flows linearly with time, which is the, the parameter on the trajectory. So the solution has this form when v zero is some constant, which depends on, on, on the energy. And so what we want, we want the solution to be periodic. So as, as t changes by, by one, we want to be back at the same point on elliptic curve, which means that this v zero has to be n times omega one plus m times omega two with some integers n and m. So the, geometrically, the trajectory is just a rational flow kind of windings on this elliptic curve. And so this v zero is a function of energy and the energy should be such that it's actually quantized in, in units of, in terms of omega one and omega two. Omega one and omega two are also functions of E. So it becomes a kind of retrieval equation. So omega these are the periods of, of this, of this differential over the a cycle and b cycle. These are some elliptic integrals. And so the, the, the upshot is that the space of, so the set of critical points is, so the critical points are labeled by some discrete data, which is the homologic cycle of the trajectory on the elliptic curve. You can locally parameterize it by two integers. I'd say locally because there is no a priori choice of basis. So you can choose for some, you can choose some value of an energy and make your preferred choice of the, of the a cycles and b cycles and then transported to the nearby points. You cannot do it globally. So this n and m will be exchanged by some kind of modular group. And then you, so you need to fix the energy in such a way that, that the, the trajectory closes. So this becomes a condition. Sorry, I forgot to say. So this, so this condition eventually translates to the following condition that the period, so this is equal to t, the original, the original parameter t in our problem. So this velocity, I think it's actually, it turns out to be just t. If you follow the equations, maybe it, yes, dx over p is t over i. Okay. So this is fixed by, by our problem. So we will start, we're starting the partition function as a function of temperature or eglidian time. And then we tune e so that this equation is obeyed. It's a transcendental equation, but that's, that's what it is. So this is what labels the critical points and left-shot symbols. And the fact that you have this infinite number of solutions means that if the partition function of the harmonic oscillator solves some kind of word identity equation, it should be an equation of infinite order. And so the solutions will have quite intricate monogamy, which is worth investigating. Bender and Wu started the, in 1969, they started the, the analytic properties of the ground state energy of a different, of an oscillator with different lambda in which you don't see the, the, which has this single minimum and found that they have, it's become some kind of infinite genus in the surface and so on and so forth. But I think this is more systematic. So when t goes to infinity, one can safely expect that this classical energy, which is constantly along the trajectory, actually approaches zero. And what it means, it means that this elliptic curve is nearly degenerate. So it has a, develops double point. In fact, it develops two double points. It has a symmetry. It's not easy to draw it. So there are two places where this cycle gets pinched. So you can see, it's, it's roughly, it means that this branch points start, they almost coincide. So the, the distance between these two branch points is of the order of square root of this epsilon. And when elliptic curve that generates, it becomes a rational curve. And so elliptic integrals become trigonometric integrals and you can compute them. And this equation, you can actually analyze without, pretty much without any calculations. And it will roughly have the form where e naught is something of, of the order of the energy of the barrier height. So it's like lambda v to the fourth over four. And t naught is the frequency, is the period of the classical motion around the minimum. So it's something like one over square root double prime u at v. So you can compute what it is. And so when t goes to infinity, e goes to zero. So logarithm is large. That's, which you, which is what you need to, to get the large number t here. And so the solutions to this equation tend to indeed concentrate near zero exponentially. But there are interesting features. There are some important factors of two, which I'm putting here. So, so, so what happens is now this trajectory, which used to be a rational winding, now passes between these pinched points several times. And the portions of this trajectory, which, which connect the pinched points, you can approximate by, by the solution of a simpler equation. And so that's where, this is where we'll encounter the instantons. So for, for most of the time, of the t time, the trajectory solves, so the trajectory solves the equation p squared over two plus u of x is essentially zero. So this is something very small. So you can solve. So it's p is either square root of u of two u or, and p remember is essentially x dot. So it's like x dot. And so this is the, this is the behavior of this trajectory when it goes between two, between two pinched cycles. And then as it crosses the singularity, it, the plus sign gets flipped to the minus sign. And then it starts over. So you have, you have m times, the changes of sign occur m times, where m was this, the number of times your cycle contains the b cycle. So the a cycle in, in this picture is the one, is the vanishing cycle. It's one which gets pinched. So this is the a cycle. It vanishes when e goes to zero. And the b cycle becomes non-compact, so to speak. So it's, so, so the period of this differential dx over p over the a cycle remains finite. This is a useful exercise. And the b cycle period, you can actually conclude without any calculations that it will have this logarithmic behavior as a function of the small parameter because, because of the Picard-Lefschitz monogamy. So as you, as e being small makes the full rotation goes around zero, the square root changes sign. So these two points gets exchanged. And the b cycle, which was connecting these two points now acquires two copies of the a cycle. So as e goes to e to the 2 pi i e, b cycle picks up two, twice a cycle. I mean, this two is not universal. It's a feature of the specific model in, in, in, in, for other singularities, it could be one or, or three or something. But from this, it follows that the period of this differential should have analytic structure such that when e goes around zero, this period picks up a finite piece. And so that's this, what this logarithm does. Is it, is it, did you just recover this WKB picture in this way? It's not, it's not WKB. We don't, we don't talk about WKB. I don't know what, what, why you, what's your question is really about. When these large means, when these degenerate pictures emerge, then what do you like this, you kind of remind me of? What, yes, okay, might, might, might remind you, but it's not WKB. WKB is the approach of solving the differential equation directly, solving the eigenvalue equation using certain ansatz for the wave function. We're not doing that. We're not, I never mentioned wave function here. No wave function, no WKB. So we're just trying to analyze the structure of the critical points of the classical action because that's what will determine the possible counters of path integral, path integration. And the main point which I wanted to make was that even though these critical points have this intricate structure for large capital T, small epsilon, small e, they can be analyzed as roughly composed out of two types of ingredients which are the solutions of the simpler equations. And these equations, so this, this is the first example of the instanton and this is called the un-tank instanton. So even though the subtle points were the solutions of the second order equation in terms of the coordinates, the, or first order equations in phase space, these building blocks, instantons, un-tank instantons, the solutions of the algebraic equations in phase space or the first order equations in the, in the coordinate space. Okay, good. So the rest is an exercise. Let me briefly say how this picture with the ellipsic curves generalizes to the systems with many degrees of freedom. There's a class of quantum mechanical models, this structure generalizes. Is there a question? Before going to that more complex thing, maybe I should ask, what's, how do you understand physically this large T limit? Is it, is it because you're trying to probe the ground state? Well, just, so it's, it's, it's where, this is where you can analyze the, I mean, it's where you have better control over the, the subtle points. I mean, for any value of T, you can write the, this particular functions as a sum over the, the subtle points. It's just that the, the, it's kind of a transcendental equation to solve. So was it because of more analytic control that you're going to this asymptotic regime or did you have a physical? Well, historically, this was a good example because this is an example where classical mechanics and quantum mechanics contradict, sort of, not contradict, but it's, where quantum mechanics gives you something which is unexpected from classical points of view. Classically, you have two ground states left and right and in perturbation theory, they remain degenerate. So you have two ground states and perturbation theory, but quantum mechanics tells us that's not the case. There is only one ground state and this method actually allows you to, you can actually see how the classically degenerate and perturbatively degenerate levels split. And splitting is non-perturbative and has this characteristic dependence on H bar, whereas zero is the action of instanton, which is the period of this differential along this trajectory. So Nikita, where do you see here that it's actually the energy splitting which is determined? You have to analyze, it's not, it's, you, you don't see it immediately. You have to do the sum over all the subtle points. It's, it's not so easy actually. It's, I mean, it's, I mean, textbooks give you a kind of a sort of simple way of, of doing the sum, but the textbooks actually miss one of these, one of these quantum numbers which is important. So, so in this, you see, in this formula these subtle points are labeled by two integers m, which is the number of instantons and anti-instantons and n, but there is also n, which is kind of a number of perturbative fluctuations. So this is the classically allowed motion, which you have to glue on top of the instanton and anti-instanton pairs. And this is usually swept under the carpet. And so, so the summation over m actually, well, one shows that it has this feature that it has some m minus m minus, so it's zero for each bar, t to the m. So that's, that gives you exponential, exponential of the, of the energy correction. You, it's, I mean, I didn't, I didn't spell it out. You need to actually do two computations. One computation is, is for the periodic loops, another computation for the anti-periodic loops, so, so that to actually single out the odd, odd way functions. So, so there is some twist you can introduce here, but it's, I mean, it's technically it's the same problem. And then from out of these two computations by, you know, edging and subtracting, you, you, you'll get this exponentially small level splitting. And that of course just reproduces what has been known before using a WKB method. So, so that's why. So, this summation of this instanton numbers is kind of index of some elliptic operator, it would read the, would decorate, it would turn out, I don't know. No, not, not here. But there is no, you see this is, there is no topology. There is no topology in this problem because every instanton is accompanied by anti-instanton. So it's, so this topology is kind of emerging only when you fix the energy, the complex energy level. Then you're, then you have a torus which has non-contractable cycles. In the whole phase space, there is no such, there is no topological variant. And so there's no index, index requires topology. And that's because you were using actually Morse function and in the end of the day it would be some flow problem. No, no, no. These are Morse functions which are real functions, real parts of, of Hall-Morphic functions they all have equal Morse index which is half the dimension of the space. It's not, it's not that useful for that story. Okay. If I just replace a plane by something, if I just replace a plane. Well, I mean, so here it was not, not a plane, there was no metric on, on, not no prior metric on the phase space. The metric was on the real line. Where do you want to put positive curvature on the real line? Yeah. So that, I mean, that's why, so you need high dimensional example. So that's why, that's why, that's what I'm talking about. Okay. So the many body systems of interesting type, they're called algebraic integral systems. So this is where the complexification of the phase space, this is a complexified, complexified phase space has this structure of the vibration of the abelian varieties. So you have, so you have some a priori coordinates p and x, we had only one such pair in that example. And then there is a different set of coordinates, so-called action angle variables. So this is in, in the real context, that's more or less more or less unique, but in a complexified situation, there is no unique choice, not the unique choice of action variables or angle variables. And so these angle variables are defined, you can normalize them that half of the periods will be two pi integers, and another half will be, my indices are all wrong, but will be integral multiples of some function of the values of the angle variables or of the action variables. And now the Hamiltonians, so the integrability means that you have the number of Hamiltonians, which Poisson commutes the number of degrees of freedom, so half of the dimension of phase space in these variables will be functions of the action variables only. And so you can repeat this analysis for the saddle points of the similar partition function, for the path integral of the similar partition function, where now you have not one, but several energies, several temperatures, if you like, or chemical potentials. And so the equations, the, this quantization condition for the, for the values of the energy level actually have the nice form of the, that they correspond to the critical points of some super potential. So it's N, A, A-dual minus where A and A-dual are the periods of the Louisville differential over the cycles on this abelian variety. So A, A. So this is, this is a complexified Louisville differential, also known as the Zabekwitten differential now. And so these periods we are familiar from the studies of BPS states and then it goes to gauge theories, but this correction is not so familiar. So, and so you're, so this is given the integers N and M, this is some analytic function on the base of the, of the, of this integral, integral system. And you look for the critical points of this function. So these critical points will be locations where, where the trajectories, which will close after the finite number of revolutions will be sitting. Say it again. Yeah, it's a function on the base from one to N, A ranges from one to N, and N is the dimension, is half the dimension of the phase space. I don't, I don't think this question is relevant at this point. Okay. I mean, if time permits we'll maybe get, get back to this later, maybe not, probably not today. Okay. I wanted to say a little bit about the supersymmetric quantum mechanics, because that's, see, in the discussion so far, so there was this, there were these subtle points, which were the solutions of the second order equations, and you can approximate them by the combinations of solutions of first order equations, instantons and anti-instantons, but they all key, they were entering on equal footing. And supersymmetry is when you actually kind of disentangle them and make one type of solutions more important than the other. So let's, so let me study again the one-dimensional supersymmetric quantum mechanical model. So this is a kind of physical way of, of doing Hodge theory, and kind of Durham homology for its deformation. So I'll have them, so I'm going to describe this in terms of the first order formalism with Lagrangian, so again I have my variables x and p, which are bosonic, and then I'll have the thermionic variables, psi and psi bar, and this is all in one dimension, so they're all functions of time. And so we'll, we'll have some kind of topological supercharge, which maps x to psi and p to psi bar, the squares to zero, so delta square is equal to zero, it's only half of the supersymmetry, there is another supercharge, which I will ignore, and so now my action will be, I will take it to be, for most part, delta exact, and it's going to be given by the integral of psi bar x dot plus i over 2p minus v prime of x. So v is some function, I will assume to be Morse function, the critical points I'm going to generate. So, well if you compute this, you'll get the familiar, maybe I put i in front of it, so you get px dot term from this term minus p squared over 2 plus thermionic terms, and so p enters without derivatives, if you exclude it, you'll get that p on shell is equal to, well let's try to solve it, so it will be, just need to, should your variational for psi bar x should be, variational for psi bar x should be, variation of, sorry, I'm sorry, I'm sorry, it was absolutely wrong, thank you very much. That's what it meant. So on variable x, delta x has the RAM differential from boson to fermion, and here it acts as the casual differential from fermion to boson, and so when you solve, when you do the variation, I think you'll get that p is i times this thing. So here the choice of the differential delta prefers instantons, which will be the locus of, in the space of fields where this expression vanishes, so delta equals to zero, you get instantons. Instantons in this context means gradient trajectories, the solutions of the equation x dot equals v prime. Now one can actually do what Jan was asking about, namely introduce the metric, and so the metric will, well let me do it in the case of several variables, because once again dx a equals psi a, d psi bar a equals p a, and now my action is i delta psi bar a x dot a. Here is the metric, something which probably won't have much time to explain. So one note of caution, so x is a coordinate on target space, so psi is a fermion which is valued in the tangent bundle, in the pullback of a tangent bundle target space, so it means that if you change coordinates psi transforms homogeneously, just multiply with the Jacobian of the coordinate transformation. Now in order for this action to be invariant, you'd like psi bar to transform as the, as the section of the pullback of the cotangent bundle, so also homogeneously, but then this supersimilar transformation will produce some, will give rise to some, so if I change coordinates from x to x tilde and now transform psi bar in the kind of naive geometric way, the result would be that p will transform inhomogeneously. Delta doesn't depend on anything, so this is roughly like, so p is, even though it has a lower index, it looks like it's a one form, as it is in the Bosonic case, in the supersimilar case it's not a one form, it's, it transforms as a connection. And so in order for the action to be covariant, you need to introduce not only the metric, but also some connection on the tangent bundle, it doesn't have to be the Christoffel, it doesn't have to believe a Chevita, but it could be, so that's the origin of this term, it's to compensate for this inhomogeneity to summation of p. And now if you're only interested in the, in correlators of delta closed observables, which are, which correspond to essentially chromology, drum chromology of target space, then you can play with this action, so because it's delta exact. And so there is an interesting limit in which g goes to zero, the inverse g, so it's a large volume limit, and v goes to infinity, so I make my Morse function to be very large, such that the vector field, the gradient vector field remains constant, remains finite. So in this limit, this term disappears and you end up with a theory which on the one hand looks almost free because the momenta enter the action linearly, but it's not quite free because it actually knows about the topology of the target space, it knows that has many critical points and so on and so forth. And so from this you can recover Morse theory in the way Morse conceived it. And if you don't do this, then you get kind of written representation, but then it's not easy to analyze. Okay, so 15 minutes, all right. So let's go now to infant dimensional case. So we'll be interested in two types of theories, now quantum field theories in 2D and 4D, which are kind of generalizations of this model, namely the Sega models and gauge theories, and sometimes we'll combine them. So I mean actually I spent most of my career thinking about this simplified approaches to those theories with the instantons which solve similar equations than the full theory equations, but lately I've been interested in trying to find the analogs of these smooth instanton and instanton solutions in those theories. And so I'll try to squeeze into this lecture some of these solutions, but maybe not all of them. So let's do the Sega model. So we have a Riemann surface, again, so this is a Riemann surface, which we'll be sending to some target space, also Riemannian, and also with some metric G and with the B field. So the B field is locally, it's a two form, more globally, it's a connection on the GERB, so you define it in patches in such a way that the exponential of the action which I will write is well defined. So X are some coordinates in target space, and so we describe the map by saying how the coordinates depend, and how they become functions on a roll sheet. And then in Euclidean roll sheet I will put an I in front of the B field term, so that's the typical Sega model action without tachyons or dilatons. You can actually map this, so this is a kind of a generalization, so this is a bosonic part of the action, of the semi-quantum mechanical action, if the B field and the metric are related in a certain way. So it's the simplest, let's say, so if X is scalar, so that the metric is scalar and B is the scalar form, that's the simplest way of relating the metric and the two form. More generally you can introduce a columnist complex structure and require that the metric is related to, well, so there is some compatibility condition between the complex structure and the two form, so in this case, if this is the case, then you can actually rewrite this action in the form which will look like the action of supersimetic, is a bosonic part of the action of supersimetic quantum mechanics. This is called the Bogomolnii trick, and so it's again an exercise. So we write the action as the integral of the norm square, so it's a norm squared of the anti-halomorphic derivative projected onto the homomorphic coordinates. So again, so J is the almost complex fractional target space and this operator, so J, since J squared is minus one, has eigenvalues on the complexification of the tangent space, has eigenvalues plus i minus sign, so this operator will project you onto the plus i eigenvalues, so like homomorphic components of a tangent space, and so what I want, I want anti-halomorphic components, anti-halomorphic world-should derivatives of the homomorphic coordinates on target space, even though the coordinates themselves don't have to exist, it doesn't have to be an integral complex structure. And so there is some coefficient here, there is some, okay, sorry, so there is a, so omega is built out of b g dot j, right symbolically, it's an exercise, so what happens there? And so if this form happens to be closed, then this part of the action only knows about the topology, the degree of the map, and so you get, you minimize the action by setting this to zero. And so that way you arrive at the notion of instantons in the sigma model, which are the pseudo-halomorphic maps, right? Wrong, this is of course only when this number has certain positivity properties, so because maybe there's another way of rewriting this with the plus sign here and homomorphic derivative here, so sometimes you get anti-instantons, but one way or another you get the solution of the first sort of equation. And so the instantons equals pseudo-halomorphic maps, so they solve this horrible equation, but it's much better than to solve the sigma model equations, which are the second order. Well, we'll say something about them. Now in gauge theory, in four dimensions, there is a similar trick. Yes? Maybe a trivial question. Is there any condition that the sigma model becomes conformal? Yes, of course there are conditions, but this is classical analysis, classical, it is always conformal. Nikita, why did you assume that j is not integrable? To assume that something is not integrable, I did not assume it was integrable, I didn't say I'd assume it is not integrable. I just did not assume anything about j, just almost complex structure. It's just an algebraic manipulation, that's it. If it is integrable you get more structure. Okay, young mills, they're in 4G. I'm really embarrassed to do this, but well, I was asked. You suffer, and I suffer. Okay, the action, my tools may be wrong. So here we'll be talking about the SUN gauge theories later on. And so the trace here means the trace in the n-dimensional representation. So we work on the four-dimensional Riemannian manifold, and so the star here is an operator which squares to plus 1 on two forms. So it has eigenvalues plus 1 and minus 1. And so again, there is a way to rewrite this as the norm squared of f plus tau plus 2 pi i tau, where tau is something like, so nf plus is the self-dual part of the curvature. So again, it's the same Bogomolini-type manipulation, and well, you use it. So if this quantity is, I guess it's, let's see, so what do we want to say? If this quantity is negative, then the minimum of the action is achieved when this is zero. Because the action, the real part of the action, the way I defined it, well, actually it may be negative definite because, so if you represent the connection by, you know, anti-Hermitian matrices, which is what the algebra of SUN is, then this is negative definite. But in physical normalization, we usually write A is the Hermitian matrix, and so this is positive definite. Anyway, so if you trace the positivity of the action, then of course, if this is zero, then this is negative the density of the pentagon number, and so if this is positive, then this is the way to make it minimized. If it's positive, if it's negative, then you have to do this different writing, tau bar here. So here these are just energy considerations that it's sort of favorable to look at instantons or anti-instantons depending on the topological charge. But once you embed this into a supersonic theory, then again the supercharge which you single out will tell you that the correlation functions of operators which are annihilated by the supercharge will be saturated by instantons or by anti-instantons. And so that's what we will be analyzing, I guess, tomorrow. I have four minutes. So there is one more thing which I wanted to mention, just kind of educational thing, that you can combine sigma model and gauge theory. If there is an, let's say, H, it's X on X preserves G plus I B. So it's both, it's both isometry and let's say X by Hamiltonian transformations on B. Then you can couple the sigma model to the gauge theory, gauge fields in H. So write this way, couple to the annual section with some coupling constant and couple it to the, there is a potential, trace mu squared, where mu, okay, so the vector fields which generate the action of H. So since they preserve B, means that the linearity of B is 0 and the actions come in, when this, when the contraction of vector field with B is actually exact, it's d of some function. So this is the moment map function. So we'll put it here. And the covariant derivative of X is simply derivative plus the gauge field. So A is the H gauge field. And so that's the way to describe roughly the sigma model on a quotient with the target space, the quotient X over H. But it's not exactly the quotient because we, unless we force, well, we either, there are two interesting limits when E goes to 0, E goes to infinity, unfortunately E is the massive, it's the dimensional full parameter. So having, well, anyway, so when E goes to infinity, sorry, the Bogomolny trick which works again in this case and you end up now with equations which are, which say that first of all their map is covariantly holomorphic. And then there's something like stability condition that the curvature of the connection A plus E squared times mu times the volume form on sigma is equal to 0. So you, so when you can must rewrite the section, rearrange the section to look, to make it look like a sum of the squares, of normed squares of these two equations. And when E goes to infinity, this equation tells you that mu should be 0 almost everywhere. So it means that, and so, so this equations will actually define for you a map from sigma to the simplex equation of x. But it's not, unfortunately, it's not always the case, so you cannot all, you cannot achieve this map to pass through the locus. So sometimes, so there are some points, there are solutions where, where the image does not pass through the locus mu equals to 0. And so then, so these are some kind of degenerate instant solutions of record instantons. And so that's the important difference between the instantons-engaged linear sigma model, well, could be linear if x was vector space. So this is versus nonlinear. So the modulate spaces of solutions of the, of, so pseudocolomorphic holomorphic maps or pseudocolomorphic maps into this space is not the same as the modular space of solutions of these equations. But the, I mean, the difference between them is kind of local. So some, in some calculations, you can actually model this difference by certain change of couplings by some kind of contact term, the redefinition. And that's known as a mirror map in, in the context of a models in two dimension. So how this h acts on mu inverse 0. Is it G at the quotient? Or is it by isotope? What do you mean? h preserves mu, mu equals to 0? Well, how mu is it? It's a Hamiltonian action. You're assuming that. From this equation, it follows that the locus mu equals to 0 is h invariant. So you can, so you can quotient by h. Mu transforms in joint representation, in the co-joint representation of h in general. All right, so I think that's enough for today. We'll continue tomorrow.