Resurgence through Path Integrals
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Kontsevich, Maxim
I will review the approach to the resurgence phenomenon via integration overrapid decay cycles (Lefschetz thimbles) in path integrals. Examples include WKB asymptotics, heat kernels, WZW models and Chern-Simons theory.
 Okay. Thank you very much. Yeah. So I will try to present some little bit, some perspective on resurgence through algebraic geometry. So I recall that if you have series sum over n, which is divergent, but coefficients grow like n factorial, then we make Laplace transform, take sum over n factorial z to the power n, z to goes to zero, and it will be germ of analytic function near zero, and the property of resurgence has endless analytic continuation. In the talk of Jean-Nacal, we heard some word otarchy and the isographic form. And now the main guess. This thing means that you have polarized exponential hodge structure, let's play a second, of infinite rank. Yeah. So kind of like infinite dimensional break geometry, and it looks that the words have this meanings. So what is, and this exponential hodge structure of infinite rank, it's something if you make Laplace transform, we'll get a variation of hodge structure on a fine line, maybe with infinitely many points. So it's a usual variation of hodge structures, of polarized pure hodge structures. On c minus, in general, infinite set, maybe everywhere dense, some countable set. And to get representation of fundamental group of this two, some J-Linfinity of integers, which will be integrality of this desolation coefficients roughly. Yeah. So what is this exponential hodge structure of infinite rank? It's a finite rank, because for infinite rank, we don't have yet rigorous theory. Sir? This is pure hodge structure. Pure, yeah. No, why did you do that? Mixed maybe, yeah, maybe. Yeah, right, yeah, but yeah. Yeah, let me recall you what are usual hodge structures. Let's start with pure hodge structure. Use pure hodge structure. It's a vector space over c. This hodge filtration, sounds like an N of VH, and lattice, so you multiply not compatible with this filtration. And you have a property such that if you consider real in evolution, so it lattice gives you real structure on H. And now consider F bar and NF will be opposite filtrations up to shift. So you get decomposition of H into sum of HPQ. That's a P plus Q is equal to N. And yeah, that's a usual hodge structure, which appears in the commod of, let's say, complex projective variety. And then there's something called polarization, which come from the ample bundle. You get non-degenerate baleenar form on integral lattice, which is symmetric or skew symmetric, depending on parity of weight. And such that it induce baleenar forms on, it induce pseudo-hermitian form on HPQ for HP and Q. Namely, you consider the pairing of alpha and alpha bar for alpha belonging to HPQ. And this has signature minus 1 to power p. So it will be either positive or negative, depending on the things. And if you take sum of this appropriate sum, you get hermitian form. So you get Hilbert space structure. So sum of p minus 1p of this form on HPQ will be positive definite to get Hilbert space structure. Okay, that's a usual hodge structure, which appears on commod of projective varieties. And exponential hodge structures, it's some notion which we introduced about 10 years ago with Kacarkov and Pantiev, studying Mir Simitiv of Landau-Gisburg models. It's related to exponential integrals. This is a rate-to-easual integrals, and the definition of exponential hodge structures is the following. We get a holomorphic vector bundle. On C, this coordinate, we denote it U. It's the same as Planck constant or small parameter zeta in this one, or zeta in this situation. So we get this holomorphic bundle, so we get just a family of spaces depending on U. And then on U-plane, we have a special point, U equal to zero, and we get a connection on this bundle. And the holomorphic connection has second order pole on bundle with second order pole at U equal to zero. And in particular, you have certain monodromes, and we assume that flat sections of the bundle long-race and flat sections on rays grow like exponent minus some constant times U. In principle, they can have more different singularities. Then, on each array, we get filtration by order of growth. So this is the DeLine-Montgarnes filtration. And what is assumed that we have certain lattice. So the data is given, extra data is given to have a lattice, HUZ, sitting in HU, for U non-equal to zero, which is coverently constant with respect to filtration, with that connection. And all terms of filtration are also integer subspaces. I define it over PoZ. So this will be analog of HOD structure. Actually, the usual HOD structure is particular case when this is not the second order pole, and the monodromes plus minus one. So the analog of HOD structure and polarization is given by the following data. You get for U non-equal to zero, you get non-degenerate pairing. It's opposite side source filtration, which is coverently constant, and also compatible with integer lattice on HUZ. And then there is a way to formulate sign of polarization, which eventually gives you some Hilbert space structure on HU. And also from this synchrony, it can deduce certain filtrations on this HU without Z, labeled not only by integers, but maybe some fractional numbers. So that's its abstract story in where it appears. Appears in exponential intervals. So your H is finite dimensional. Yeah, here it will be finite dimensional, and what I want to say is that at least some past intervals one can see is infinite dimensional examples of this. Appears in exponential intervals, namely imagine the following situation. Suppose X is a final algebraic variety. One can think about just coordinate space, and suppose we have some polynomial, some function. And I'll just give you the simplest possible example. Assume that this polynomial has isolated points, isolated critical points. And there's some condition. I forgot what is the name. It says that something that this polynomial has no critical points at infinity. Same, yeah. Suppose you have this polynomial. What will be this H of U? We can consider space of algebraic conforms on X, and mod out by the quotient space by the image d minus U d plus multiplication by ds, applied to n minus one forms. It will be finite dimensional space, and the dimension of this H U will be number of critical points. So we get a vector bundle. Now I want to say what is the lattice for unit not equal to zero. Notice it will be commod of the following set. We consider X and consider pre-image of the domain when real part divide one over U s. This also function. So we take pre-image of the domain when real part of very big and take commod of the centrifugal fusions. So the dual space, or dual lattice, can be sorted certain cycles over which real part of s divided by U goes to plus infinity. And now what we can do, if you have a pairing between H of U and such cycles, we can integrate over such cycle of the function to integrate s minus U, and times some algebraic conform. We represent a class in this commod. So the centrifugal will be convergent, and this centrifugal gives you a family of lattices, it gives you a certain form of Fortis connection, and has a second order pole. But the main story is that this guy has definition when U equal to zero as well. And that's like analog of Hodge filtration. What is this pairing? I want to describe the pairing on kind of, the pairing is also completely natural in this situation. I will describe pairing on level of homology, and I'm going to go to dual, get pairing on level of homology. Namely, let's assume the target of U is generic. Then we get the basis of HUZ dual, consisting of called left-shift symbols. Namely, what you do? You have critical values of our polynomial. Now we draw a straight path along which this s of U minus some constants will be real and goes to plus infinity. We get the straight path, and over the straight path in ambient manifold we draw a family of, nearby we have kind of vanishing cycles, some small sphere, s minus one dimensional sphere. And then one continuously extends this family of s minus one dimensional sphere, we get a copy of Rn embedded in our space. And the bending depends on this critical point. So there are critical x alpha will be critical points. And then you get s of x alpha. And for each critical point you get the left-shift symbols, you get the convergent integral. So you get this convergent integral, and then we change the direction, argument of your U goes to minus U, you just get left-shift dual cycle going to the opposite direction. It acclaims and it's easy to see that it's kind of currently constant and you get duality between, it's just one kind of pairing between homogenous and fiber U and fiber minus U. So that's kind of scientific way to formulate this integral of left-shift symbols. And here's the whole story one can reformulate in kind of more elementary ways, namely for each alpha and in generic argument of U. So we can write what is the integral over left-shift symbol. It will be integral, we choose again some expression. We integrate over this left-shift symbol starting in point x alpha. We integrate some n-form which you write something like polynomial function of x multiplied by some reference or form which you write dnx, yeah. And f will be some polynomial function. So we get this integral to calculate and then when U goes to zero along the ray, we have this symmetric expansion to be exponent of my critical value. S of x alpha divided by U. Then we multiply by square root of 2 pi h bar to the dimension over 2, 2 pi U, sorry, dimension over 2. Then we divide by determinant of second derivative point x alpha for quadratic approximation theory. And then start with series starting from f of U x alpha plus you get certain series in form, power, series and U. It will be value of this function at this point and then we get extra correction. And this series has factorial growth. Maybe this series I denote by, it's a series which has factorial growth but it's actually equal to certain number which I kind of denote by square root of determinant, yeah. And this I denote by something like g alpha of U. Now it's actual function which has a symmetric expansion. So we remove all irrelevant terms. So, oops, I'm sorry, I'm sorry, I'm sorry, I'm sorry. We get just a bunch of functions g alpha of U, alpha of N, for one to number of critical points, which have defined outside of some stock's race, when the direction, race. And which have, but this function have the same asymptotic expansion which doesn't depend on direction. Formal expansion is the same. Stock's, stock's, formal expansion doesn't depend on the sector. So we get just a bunch of formal power series which you can calculate on computer using only formal expansion at this point. But you get actual functions as well. Okay, you get actual function, yeah. And this function satisfies kind of jump property, jump, if you go through some ray where argument ray, when it says it's argument of U, is equal to argument of difference between two critical values, yeah. And also, it's a self-prope. One minus self-prope, say. When you get this array, then what we see, it's kind of standard picture. What happens if you, on the one side of the ray, when you, on one side of the ray, and then we get rotation, we get different homologic classes. And you see that one homologic class doesn't change, so one function stays analytic. G alpha 2 is analytic. So jump, from the ray, g alpha 2 is 0. And jump of g alpha 1 is a certain multiple, integer multiple of some integer, and alpha 1, alpha 2, multiplied by g alpha 1, alpha 2. But because of this factor, we multiply by exponentially small term. Take exponent s of minus s of x alpha 2 minus s of s alpha 1 divided by h bar, which is very, very small. So it has no exponentially small. So it has trivial expansion. Sorry? Ah, yeah, sorry, yeah, because I kind of, it's notation I have in mind, this h bar and u, this was the same. Maximum, what is the upper vector? Delta, it's jump. It kind of jump of, I have function defined on two sides on a ray. It can see the difference. Yeah, so it's equivalent, yeah, so it's kind of in completely elementary mathematical terms. All this hot structure, besides this positivity process of polarizations, it's very elementary object. The polarization itself, it's kind of tricky stuff. It's called TT star equations and we'll not talk about this thing at all. Okay, yeah, so we got this completely clear picture in finite dimensions. Now go to infinite dimensions. Yeah, suppose I have certain remaining manifold, maybe two points of this remaining manifold. And I assume that manifold can be complicitized to some complex manifold. Maybe an algebraic manifold and metric will be algebraic tensor on this manifold. Now, so get this big complex manifold. Now, and m is, okay, as example, you can try to see like m is a sphere or some, some sequence could write explicitly formulas. Sphere or ellipsoid or hyperbolic space, maybe flat space even, yeah. Taurus are something, it's a very nice concrete formula. Now this, then have infinite dimensional complex manifold. X, which will be the space of path connecting this to points m0 and 1. Maps phi from 0, 1 to mc, m4 0 is m0, m4 1 is m1. It's infinite dimensional complex manifold, it's, it contains kind of half infinite dimensional things past sitting in real path, in real locals. And this is the same boundary condition. And now the Feynman integral, which you can see that the action functional will be one half of integral from 0 to 1 d phi, that's dT squared dT. Yeah, so you get this functional integral and now when we integrate over Xr, expand of S of phi divided, let's say by u, but now maybe I denote it by Planck constant. And defies, what expression means in path integral. So it's analog of this situation. We'll find a dimensional exponential integral. And all this, so it should have some kind of critical points, former power series expansion, and then actual functions and all this geometry. Let's me explain you kind of basic trick here. So we integrate how to calculate this integral. When h bar is equal to 1, so I have just, so you get this usual linear measure, it should be brown and motion on your manifold and you start from point m0 and point m1 and time 1. And this should be like, it should be equal by value of heat kernel, exponent minus one half of Laplacian on X real and you use physics notations. So consider heat kernel for time one between points zero and one m0, m1 and what to do for general h bar. The idea is it's very simple. So we see you multiply the thing divided by one of h bar. Let's keep on half. So you get this, this thing, and you see that it's in time, this thing is homogeneous of degree minus one. So one says that it's put it time bar is equal to ht. So tau bar will belongs to interval zero h bar, we will identify 10 points and then it will be equal to just equal df over dt bar. So you see that this integral is equal to the heat kernel for small time. To h bar, yeah. So you get this nice formula. So it will be heat kernel for small time. So the prediction is the following. If consider heat kernel for small time and divide by this leading term, you get some asymptotic expansion, then it should be resurgence series. This should be resurgence series and now, yeah, so it has some exponential small term. So it should divide by exponent minus, I think one over h bar times some cyclic square of lengths distance between those points. It should be leading growth of the portion. So it should be kind of like series in h bar and this should be resurgence. So this is actually completely open question for general like sinfinti remaining main fault we really don't know. Resurgence and I think it's true only for some special like algebraic varieties so things close to them. So let me show this example which I found extremely striking is this is example when consider this manifold will be hyperbolic plane. Yeah, that's maybe the simplest example which could be because in this case you have just only one geodesic into point and for some classical formula for the heat kernel. One can express that this m0 exponent minus Laplacian and you multiply by h bar is equal to the following thing. So it's written constant. It will be small s not capital S but it's related to the critical values of my function. So the L is distance between my two points. No negative real number. So the sink is given by certain integral and what here goes on. You can see the critical values of C on the space of compressive height pass. And consider what are critical values. The critical values are the following. You get certain points on a parabola, integer points in the parabola, and zero points on parabola. And you can see you get a bunch of lefty symbols. And this is my integral of the lefty symbols. And in fact if you look on this integral you will get certain finite dimensional. And actually you say that you identify up to the simple factors in the front. You identify the infinite dimension integral with finite dimension integral but over remaining surface which is infinite genus remaining surface. Remified this countable set of critical values. You can see this function of small s, this square root of hypervolic cosine minus hypervolic cosine. It's something which is too valued. On a sphere we have closed the jadez. Yes, yes. And that is the sphere. On a sphere if you have two points you have not only one jadez but you go around, around, around. And to length you add this length of the periodic jadez which is purely imaginary. You get some kind of length plus all these points have the following meaning. You consider this length plus 2 pi i n where n is an integer. One is real and the other one is just adding the energy in this sphere. But of imaginary radius i. So the critical value should be... Yes, all these points is a set of all these points which I consider. So you see you get kind of these integer points on a parabola. Moreover, this example I think is already very interesting because from the answer you can see, you have a prediction how many... You have prediction how this function jumps when you rotate zeta. These are the integer numbers n alpha 1, alpha 2 for each two critical points. And it's a very simple number to calculate for this interval. Plus minus 1. And the claim that should be coincided with something very interesting. Because if you look what is alpha 1, alpha 2 in general, there is some kind of number of certain gradient lines from one critical point to another. And gradient lines will be gradient lines in the space of paths. So it means that you should solve certain paths in the space of paths. It's a map from surface. And it means that the gradient lines will be a number of certain surfaces satisfying certain kind of Kashi-Rieman equation in a complexifying space. So it's kind of a very non-trivial prediction that from this answer we can see that everything fits together. We get kind of convergent answer only if you know this number of paths from power series. And this will be given by this picture. Now, so that's the main example, non-trivial example, which I have to tell you. About this heat kernel on hyperbolic space. One can ask similar stuff. What in small modification I can study the case of sphere, but for example, in this case it will be case of ellipsoid. It's non-equal axis in R3. It's ellipsoid. Actually, of any dimension. Why ellipsoids are nice? For ellipsoids, this geodesic flow is integral. So one can write lengths of geodesics to some set of functions, but also eigenvalues of Laplacian also kind of known. It's because it belongs to family of commuting operators. For example, one can do kind of really extreme case when ellipsoid became a flat, it will became like double of disk. Imagine disk glued to itself along the boundary. So geodesics will be this polygons labeled by pair of integers. So it will be lengths of geodesics. And calculate like close things. It will be Q times 2 cosine 2 pi P of Q something like this. And eigenvalues of Laplacian will be because decomposed by 0 of some basic functions. So in this case, there will be certain great identities, certain infinite sum, personality combination, along these zeroes of basic functions. It will be interesting to calculate all data explicitly. In this case. So here what was the trick? This action for this free particle was homogeneous in time, degree minus 1. Because of df over dt squared times dt to the homogeneous of degree minus 1. And then we can make this calculation. One can do make kind of similar exercise. It's related to very old work by Andrea Warrows. So consider a curator. In other example, consider such instead of Laplacian, consider secretary plus potential. And the whole story is the main thing is should be also homogeneous function. Like one can generate many variables. I just thought this one variable. So you have this homogeneous function. And when we calculate trace of exponent minus lambda h bar, how to calculate it? Again, by Feynman-Catz formula, it's will be integral of the space of pass. Is periodic boundary condition to R1, this coordinate x. I have a separator and I multiply by exponent minus action. You denote by s lambda of phi, d phi. S lambda of phi is just integral 0 to 1 and half dt squared. dT. It's a discrete spectrum. It's a discrete spectrum because the spectrum is discrete because its potential goes to plus infinity. It has a val. It's not the integral from 0 to lambda in s lambda. Sorry, you're right. The lambda. And now I try to rewrite the integral from 0 to 1. I want to rescale time and rescale everything. Time and phi. To make it integral from 0 to 1. Namely, I take now, tau lambda is lambda 1 minus t. And then when rate phi lambda is equal to. If you make it rescaling, then s lambda will be integral of 0 to 1. I'm going to go to the variables. I multiply by lambda to certain power, which is... You get just a homogeneity game. Nothing tricky here. And then the conclusion will be the following. Then if you take consider trace function, whatever zeta goes to, trace exponent minus zeta h to power n over n plus 1 should admit an endless analytic continuation. That's what's under the word predicted. But it's not on the level of pass integral. It's this simple homogeneity game with pass integrals. So there are two basic examples when it's very, very concrete. So it was a case of free particle and also this homogenous potential. There is still another example. Suppose we get... use the same name, but now... One can write another pass integral very badly defined. Suppose it has a complex manifold, like a tangent model, or more generally, holomorphic simpleactic manifold, which contains two complex Lagrangian sub-manifolds in 0 and 1. And what I want to write, I want to integrate... The space xc will be no real space at all here. So my infinite dimensional complex manifold will be space of pass. So that f of 0 belongs to L0, f of 1 belongs to L1, and it's my two tangent bundle. So in the action I choose... I have the infinite dimensional space. The action function, which I consider here, is defined only up to a constant and only locally. So I have to write what is the differential of these things. It will be a homomorphic one form, which will be integral of two forms of my pass. When I have one parameter of family of pass, I get surface, and to integrate it, I get... we'll define it. So roughly, in mechanics, you write something like this things plus boundary terms. It's called first-order formalism. And plus boundary terms. Yeah, in principle, one can include this random work. I think certain terms, like you add certain functions, it depends on three parameters. So it will be a function on cotangent bundle m times some time variable to c, and it's function h. You can add this such term, and if the thing is quadratic in p, this will be the equivalent to pass integral one, can integrate over p variable, and you get exactly random work. But now I ignore these things completely. So it seems it's completely geometric. I have this integral of one form. So the question, what is it, and what we'll have here? Yeah, in general, it's pretty unclear. How to interpret it? How to interpret it, what is the integral, yeah, it's... And what is the property of what we should expect from this integral? But at least it looks that for certain kind of class of example, this could be the following. L0, we'll get some L0, city can cotangent bundle, here draw fibers of my cotangent bundle, and it will be like this, this is L0, and L1 will be cotangent far by its certain point. Then we get finitely many intersection points, and kind of the guess is if L0 kind of corresponds like spectral curve, again, it's long story how to identify parameters to algebraic bundle with a connection. Then this asymptotic series, which you should get here, should be like WKB series for formal solutions of the models. You write L0, it's graph of differential, graph of D, maybe called F0, some multivalent function, and then it will be... you write a solution, you remove this main term to get a asymptotic series in each bar. So that's something which with Yan we can... part of general project we have with Yan on wall crossing, and what is going on, here's some kind of really interesting effect here, right to the following story. On the space of paths, we don't have really one valid function, we get functions defined up to constant, we have closed one form, and that means the following, we can... in the whole integral, we can go to universal cover, and then we go down, we can twist this arbitrary rank one local system on space of paths. So we get certain torus, and C star two kind of... by which you can twist your integral. Yeah, for example, if the substation can be in finite dimension of the story, you can twist your... a real connection by rank one local system, consider again commode of pair and so on, so you get things depending on a point on a torus. And what happens in infinite dimension? There are certain directions of Planck constant, for which you change parameterization of space, you kind of apply nonlinear automorphism of the torus. So you get things which are kind of very close to kind of a Calvaronian picture, you have some nonlinear changes along the rays. It's kind of new infinite dimensional effect. There are kind of new walls, new rays. So we're going to... you can apply analytic defomorphism of torus, some domain of torus depending on small parameter. So you get more involved picture with some kind of nonlinear changes as well. Before we had just vector bundle with basis and transformational grace, but now you have vector bundle on torus and parameterization... vector bundle of rings, which is sort of... Yes, yes. But eventually it should give again the same resurgent properties of the whole story. So I didn't had a time... That's essentially all one dimensional examples. And in principle, what was completely untouched, one can try to make, I guess, to imitate this, at least the question for two dimensional series, like the Minovitan or three dimensional with Sharon Simons. So again one get actual values of integrals from physics and then the same picture predicts your resurgence structure. Thank you. A question, Lesha, so you consider this free particle on hyperbolic space, you can think of it on circles, here or... Any varieties here? So for the general case, the general group, there is a formula called deus thermo-techmann, which essentially we write this as a sum over geodesics on this space, but as far as I remember... No, no, this is certain... No, there was a sudden sub-deus thermo-techmann for any remaining manifold. If you consider trace of exponent... If you consider some trace of exponent minus E t, that's a square root of Laplacian, it has singularities on a major axis equal to lengths of geodesics, yeah. So, for example, if I just found the particle on a circle, again I can write a sum over geodesics, there are no fluctuations around them. Yes, yes, you got the exact formula, yeah. It's exact formula. You got C to function, yeah, I think. So there is no asymptotic expansion. Yes, it's trivial, in this case it's trivial, yeah. And in these general cases where this deus thermo-techmann applies, it's also the same structure, right? No, but the conjecture is for more general remaining manifold with algebraic metric, then we get very complicated expansions, if it's not... if it's not homogeneous, yeah. Can you describe the whole structure of infinite rank in terms of this function functional, related to homology? Yeah, no, there's kind of... one can write this convulsion of perivot, what is it? But you see this number of... this gradient lines integer numbers in alpha 1 or 2, gives you the example of the day, kind of step-by-step construction of this infinite... should be space generated by all geodesics, yeah. I think it should be pure in this case, yeah. More questions? Yeah, I would like to... I didn't understand everything obviously, but how far are we, or you, from effective computation on the finite path integral for say a non-homogeneous problem? Yeah, for general potential... I think it's... for general potential, yeah. In principle, it looks that if it's... if it's not homogeneous, this gives you kind of few steps procedure, which is completely mechanical, one can calculate critical points in complex domain, then one should count how many gradient lines, it's a number of solutions of certain... like, pseudo-holographic disks somewhere, or some integer numbers which one can calculate, and then you get form power series which you do at each point, and then after that it should produce the actual numbers. Can you replace it by so many numbers? Yeah, because it should... yeah, the story is the following. This integers and alpha... and those jump formals for J, which depends only on critical values and integer numbers. So it gives you a way to glue certain holomorphic vector bundle, sterilize the infinity structure at zero, and then there's form power series after you solve the problem. It will be convergence series, yeah. Yeah, yes, yes, yes, it's completely general procedure. From this perspective it will be... It's effective, yeah, it will get to actual number, yeah. It will get it, convergent. Thank you. Is it clear that if you write for... if you write the formal WKB solution, such as in the classical... or is the same as you get a finite integral, a diagonal exception. So is it clear or... No, no, no, no, no, it's not clear. Okay, comment on your question. So from the final integral point of view, that's very difficult, but if you do Bender-Wohl type analysis, you can actually computerize... Yeah, you can... yeah, because you substitute the things to your expansion and you get to... It's pretty computational things here. It's a computer quality of... I think even the determinational part is not completely clear. Yeah. Yeah, but there are many things here. For example, here, the dimension, yeah, it will zeta function regularization then, sir. This will be zeta prime of zero, yeah, yeah, so it's... In field theory, all terms make sense here. No question. Thank you.
 Okay. Thank you very much.  Yeah. So I will try to present some little bit,  some perspective on resurgence through algebraic geometry.  So I recall that if you have series sum over n,  which is divergent,  but coefficients grow like n factorial,  then we make Laplace transform,  take sum over n factorial z to the power n,  z to goes to zero,  and it will be germ of analytic function near zero,  and the property of resurgence has endless analytic continuation.  In the talk of Jean-Nacal,  we heard some word otarchy and the isographic form.  And now the main guess.  This thing means that you have polarized exponential hodge structure,  let's play a second, of infinite rank.  Yeah. So kind of like infinite dimensional break geometry,  and it looks that the words have this meanings.  So what is, and this exponential hodge structure of infinite rank,  it's something if you make Laplace transform,  we'll get a variation of hodge structure on a fine line,  maybe with infinitely many points.  So it's a usual variation of hodge structures,  of polarized pure hodge structures.  On c minus, in general, infinite set,  maybe everywhere dense, some countable set.  And to get representation of fundamental group of this two,  some J-Linfinity of integers,  which will be integrality of this desolation coefficients roughly.  Yeah. So what is this exponential hodge structure of infinite rank?  It's a finite rank,  because for infinite rank, we don't have yet rigorous theory.  Sir?  This is pure hodge structure.  Pure, yeah.  No, why did you do that?  Mixed maybe, yeah, maybe.  Yeah, right, yeah, but yeah.  Yeah, let me recall you what are usual hodge structures.  Let's start with pure hodge structure.  Use pure hodge structure.  It's a vector space over c.  This hodge filtration,  sounds like an N of VH,  and lattice,  so you multiply not compatible with this filtration.  And you have a property such that if you consider real in evolution,  so it lattice gives you real structure on H.  And now consider F bar and NF will be opposite filtrations up to shift.  So you get decomposition of H into sum of HPQ.  That's a P plus Q is equal to N.  And yeah, that's a usual hodge structure,  which appears in the commod of, let's say, complex projective variety.  And then there's something called polarization,  which come from the ample bundle.  You get non-degenerate baleenar form on integral lattice,  which is symmetric or skew symmetric, depending on parity of weight.  And such that it induce baleenar forms on,  it induce pseudo-hermitian form on HPQ for HP and Q.  Namely, you consider the pairing of alpha and alpha bar for alpha belonging to HPQ.  And this has signature minus 1 to power p.  So it will be either positive or negative, depending on the things.  And if you take sum of this appropriate sum, you get hermitian form.  So you get Hilbert space structure.  So sum of p minus 1p of this form on HPQ will be positive definite to get Hilbert space structure.  Okay, that's a usual hodge structure, which appears on commod of projective varieties.  And exponential hodge structures, it's some notion which we introduced about 10 years ago  with Kacarkov and Pantiev, studying Mir Simitiv of Landau-Gisburg models.  It's related to exponential integrals.  This is a rate-to-easual integrals, and the definition of exponential hodge structures is the following.  We get a holomorphic vector bundle.  On C, this coordinate, we denote it U.  It's the same as Planck constant or small parameter zeta in this one, or zeta in this situation.  So we get this holomorphic bundle, so we get just a family of spaces depending on U.  And then on U-plane, we have a special point, U equal to zero, and we get a connection on this bundle.  And the holomorphic connection has second order pole on bundle with second order pole at U equal to zero.  And in particular, you have certain monodromes, and we assume that flat sections of the bundle long-race  and flat sections on rays grow like exponent minus some constant times U.  In principle, they can have more different singularities.  Then, on each array, we get filtration by order of growth.  So this is the DeLine-Montgarnes filtration.  And what is assumed that we have certain lattice.  So the data is given, extra data is given to have a lattice, HUZ, sitting in HU, for U non-equal to zero,  which is coverently constant with respect to filtration, with that connection.  And all terms of filtration are also integer subspaces.  I define it over PoZ.  So this will be analog of HOD structure.  Actually, the usual HOD structure is particular case when this is not the second order pole,  and the monodromes plus minus one.  So the analog of HOD structure and polarization is given by the following data.  You get for U non-equal to zero, you get non-degenerate pairing.  It's opposite side source filtration, which is coverently constant, and also compatible with integer lattice on HUZ.  And then there is a way to formulate sign of polarization, which eventually gives you some Hilbert space structure on HU.  And also from this synchrony, it can deduce certain filtrations on this HU without Z, labeled not only by integers, but maybe some fractional numbers.  So that's its abstract story in where it appears.  Appears in exponential intervals.  So your H is finite dimensional.  Yeah, here it will be finite dimensional, and what I want to say is that at least some past intervals one can see is infinite dimensional examples of this.  Appears in exponential intervals, namely imagine the following situation.  Suppose X is a final algebraic variety.  One can think about just coordinate space, and suppose we have some polynomial, some function.  And I'll just give you the simplest possible example.  Assume that this polynomial has isolated points, isolated critical points.  And there's some condition. I forgot what is the name. It says that something that this polynomial has no critical points at infinity.  Same, yeah.  Suppose you have this polynomial. What will be this H of U?  We can consider space of algebraic conforms on X, and mod out by the quotient space by the image d minus U d plus multiplication by ds, applied to n minus one forms.  It will be finite dimensional space, and the dimension of this H U will be number of critical points.  So we get a vector bundle.  Now I want to say what is the lattice for unit not equal to zero.  Notice it will be commod of the following set. We consider X and consider pre-image of the domain when real part divide one over U s.  This also function.  So we take pre-image of the domain when real part of very big and take commod of the centrifugal fusions.  So the dual space, or dual lattice, can be sorted certain cycles over which real part of s divided by U goes to plus infinity.  And now what we can do, if you have a pairing between H of U and such cycles, we can integrate over such cycle of the function to integrate s minus U, and times some algebraic conform.  We represent a class in this commod.  So the centrifugal will be convergent, and this centrifugal gives you a family of lattices, it gives you a certain form of Fortis connection, and has a second order pole.  But the main story is that this guy has definition when U equal to zero as well.  And that's like analog of Hodge filtration.  What is this pairing? I want to describe the pairing on kind of, the pairing is also completely natural in this situation.  I will describe pairing on level of homology, and I'm going to go to dual, get pairing on level of homology.  Namely, let's assume the target of U is generic.  Then we get the basis of HUZ dual, consisting of called left-shift symbols.  Namely, what you do? You have critical values of our polynomial.  Now we draw a straight path along which this s of U minus some constants will be real and goes to plus infinity.  We get the straight path, and over the straight path in ambient manifold we draw a family of, nearby we have kind of vanishing cycles, some small sphere, s minus one dimensional sphere.  And then one continuously extends this family of s minus one dimensional sphere, we get a copy of Rn embedded in our space.  And the bending depends on this critical point.  So there are critical x alpha will be critical points.  And then you get s of x alpha.  And for each critical point you get the left-shift symbols, you get the convergent integral.  So you get this convergent integral, and then we change the direction, argument of your U goes to minus U, you just get left-shift dual cycle going to the opposite direction.  It acclaims and it's easy to see that it's kind of currently constant and you get duality between, it's just one kind of pairing between homogenous and fiber U and fiber minus U.  So that's kind of scientific way to formulate this integral of left-shift symbols.  And here's the whole story one can reformulate in kind of more elementary ways, namely for each alpha and in generic argument of U.  So we can write what is the integral over left-shift symbol.  It will be integral, we choose again some expression.  We integrate over this left-shift symbol starting in point x alpha.  We integrate some n-form which you write something like polynomial function of x multiplied by some reference or form which you write dnx, yeah.  And f will be some polynomial function.  So we get this integral to calculate and then when U goes to zero along the ray, we have this symmetric expansion to be exponent of my critical value.  S of x alpha divided by U.  Then we multiply by square root of 2 pi h bar to the dimension over 2, 2 pi U, sorry, dimension over 2.  Then we divide by determinant of second derivative point x alpha for quadratic approximation theory.  And then start with series starting from f of U x alpha plus you get certain series in form, power, series and U.  It will be value of this function at this point and then we get extra correction.  And this series has factorial growth.  Maybe this series I denote by, it's a series which has factorial growth but it's actually equal to certain number which I kind of denote by square root of determinant, yeah.  And this I denote by something like g alpha of U. Now it's actual function which has a symmetric expansion.  So we remove all irrelevant terms.  So, oops, I'm sorry, I'm sorry, I'm sorry, I'm sorry.  We get just a bunch of functions g alpha of U, alpha of N, for one to number of critical points,  which have defined outside of some stock's race, when the direction, race.  And which have, but this function have the same asymptotic expansion which doesn't depend on direction.  Formal expansion is the same.  Stock's, stock's, formal expansion doesn't depend on the sector.  So we get just a bunch of formal power series which you can calculate on computer using only formal expansion at this point.  But you get actual functions as well.  Okay, you get actual function, yeah.  And this function satisfies kind of jump property, jump, if you go through some ray where argument ray, when it says it's argument of U,  is equal to argument of difference between two critical values, yeah.  And also, it's a self-prope. One minus self-prope, say.  When you get this array, then what we see, it's kind of standard picture.  What happens if you, on the one side of the ray, when you, on one side of the ray, and then we get rotation, we get  different homologic classes. And you see that one homologic class doesn't change, so one function stays analytic.  G alpha 2 is analytic. So jump, from the ray, g alpha 2 is 0.  And jump of g alpha 1 is a certain multiple, integer multiple of some integer, and alpha 1, alpha 2, multiplied by g alpha 1, alpha 2.  But because of this factor, we multiply by exponentially small term.  Take exponent s of minus s of x alpha 2 minus s of s alpha 1 divided by h bar, which is very, very small.  So it has no exponentially small. So it has trivial expansion.  Sorry?  Ah, yeah, sorry, yeah, because I kind of, it's notation I have in mind, this h bar and u, this was the same.  Maximum, what is the upper vector? Delta, it's jump. It kind of jump of, I have function defined on two sides on a ray.  It can see the difference.  Yeah, so it's equivalent, yeah, so it's kind of in completely elementary mathematical terms.  All this hot structure, besides this positivity process of polarizations, it's very elementary object.  The polarization itself, it's kind of tricky stuff. It's called TT star equations and we'll not talk about this thing at all.  Okay, yeah, so we got this completely clear picture in finite dimensions.  Now go to infinite dimensions.  Yeah, suppose I have certain remaining manifold, maybe two points of this remaining manifold.  And I assume that manifold can be complicitized to some complex manifold.  Maybe an algebraic manifold and metric will be algebraic tensor on this manifold.  Now, so get this big complex manifold.  Now, and m is, okay, as example, you can try to see like m is a sphere or some, some sequence could write explicitly formulas.  Sphere or ellipsoid or hyperbolic space, maybe flat space even, yeah.  Taurus are something, it's a very nice concrete formula.  Now this, then have infinite dimensional complex manifold.  X, which will be the space of path connecting this to points m0 and 1.  Maps phi from 0, 1 to mc, m4 0 is m0, m4 1 is m1.  It's infinite dimensional complex manifold, it's, it contains kind of half infinite dimensional things past sitting in real path, in real locals.  And this is the same boundary condition.  And now the Feynman integral, which you can see that the action functional will be one half of integral from 0 to 1 d phi, that's dT squared dT.  Yeah, so you get this functional integral and now when we integrate over Xr, expand of S of phi divided, let's say by u, but now maybe I denote it by Planck constant.  And defies, what expression means in path integral.  So it's analog of this situation.  We'll find a dimensional exponential integral.  And all this, so it should have some kind of critical points, former power series expansion, and then actual functions and all this geometry.  Let's me explain you kind of basic trick here.  So we integrate how to calculate this integral.  When h bar is equal to 1, so I have just, so you get this usual linear measure, it should be brown and motion on your manifold and you start from point m0 and point m1 and time 1.  And this should be like, it should be equal by value of heat kernel, exponent minus one half of Laplacian on X real and you use physics notations.  So consider heat kernel for time one between points zero and one m0, m1 and what to do for general h bar.  The idea is it's very simple.  So we see you multiply the thing divided by one of h bar.  Let's keep on half. So you get this, this thing, and you see that it's in time, this thing is homogeneous of degree minus one.  So one says that it's put it time bar is equal to ht.  So tau bar will belongs to interval zero h bar, we will identify 10 points and then it will be equal to just equal df over dt bar.  So you see that this integral is equal to the heat kernel for small time.  To h bar, yeah.  So you get this nice formula.  So it will be heat kernel for small time.  So the prediction is the following.  If consider heat kernel for small time and divide by this leading term, you get some asymptotic expansion, then it should be resurgence series.  This should be resurgence series and now, yeah, so it has some exponential small term.  So it should divide by exponent minus, I think one over h bar times some cyclic square of lengths distance between those points.  It should be leading growth of the portion.  So it should be kind of like series in h bar and this should be resurgence.  So this is actually completely open question for general like sinfinti remaining main fault we really don't know.  Resurgence and I think it's true only for some special like algebraic varieties so things close to them.  So let me show this example which I found extremely striking is this is example when consider this manifold will be hyperbolic plane.  Yeah, that's maybe the simplest example which could be because in this case you have just only one geodesic into point and for some classical formula for the heat kernel.  One can express that this m0 exponent minus Laplacian and you multiply by h bar is equal to the following thing.  So it's written constant.  It will be small s not capital S but it's related to the critical values of my function.  So the L is distance between my two points.  No negative real number.  So the sink is given by certain integral and what here goes on.  You can see the critical values of C on the space of compressive height pass.  And consider what are critical values. The critical values are the following. You get certain points on a parabola, integer points in the parabola, and zero points on parabola.  And you can see you get a bunch of lefty symbols.  And this is my integral of the lefty symbols.  And in fact if you look on this integral you will get certain finite dimensional.  And actually you say that you identify up to the simple factors in the front. You identify the infinite dimension integral with finite dimension integral but over remaining surface which is infinite genus remaining surface.  Remified this countable set of critical values.  You can see this function of small s, this square root of hypervolic cosine minus hypervolic cosine. It's something which is too valued.  On a sphere we have closed the jadez.  Yes, yes. And that is the sphere. On a sphere if you have two points you have not only one jadez but you go around, around, around.  And to length you add this length of the periodic jadez which is purely imaginary. You get some kind of length plus all these points have the following meaning.  You consider this length plus 2 pi i n where n is an integer.  One is real and the other one is just adding the energy in this sphere.  But of imaginary radius i.  So the critical value should be...  Yes, all these points is a set of all these points which I consider.  So you see you get kind of these integer points on a parabola.  Moreover, this example I think is already very interesting because from the answer you can see, you have a prediction how many...  You have prediction how this function jumps when you rotate zeta.  These are the integer numbers n alpha 1, alpha 2 for each two critical points.  And it's a very simple number to calculate for this interval.  Plus minus 1.  And the claim that should be coincided with something very interesting.  Because if you look what is alpha 1, alpha 2 in general, there is some kind of number of certain gradient lines from one critical point to another.  And gradient lines will be gradient lines in the space of paths.  So it means that you should solve certain paths in the space of paths.  It's a map from surface.  And it means that the gradient lines will be a number of certain surfaces satisfying certain kind of Kashi-Rieman equation in a complexifying space.  So it's kind of a very non-trivial prediction that from this answer we can see that everything fits together.  We get kind of convergent answer only if you know this number of paths from power series.  And this will be given by this picture.  Now, so that's the main example, non-trivial example, which I have to tell you.  About this heat kernel on hyperbolic space.  One can ask similar stuff.  What in small modification I can study the case of sphere, but for example, in this case it will be case of ellipsoid.  It's non-equal axis in R3.  It's ellipsoid.  Actually, of any dimension.  Why ellipsoids are nice?  For ellipsoids, this geodesic flow is integral.  So one can write lengths of geodesics to some set of functions, but also eigenvalues of Laplacian also kind of known.  It's because it belongs to family of commuting operators.  For example, one can do kind of really extreme case when ellipsoid became a flat, it will became like double of disk.  Imagine disk glued to itself along the boundary.  So geodesics will be this polygons labeled by pair of integers.  So it will be lengths of geodesics.  And calculate like close things.  It will be Q times 2 cosine 2 pi P of Q something like this.  And eigenvalues of Laplacian will be because decomposed by 0 of some basic functions.  So in this case, there will be certain great identities, certain infinite sum, personality combination, along these zeroes of basic functions.  It will be interesting to calculate all data explicitly.  In this case.  So here what was the trick?  This action for this free particle was homogeneous in time, degree minus 1.  Because of df over dt squared times dt to the homogeneous of degree minus 1.  And then we can make this calculation.  One can do make kind of similar exercise.  It's related to very old work by Andrea Warrows.  So consider a curator.  In other example, consider such instead of Laplacian, consider secretary plus potential.  And the whole story is the main thing is should be also homogeneous function.  Like one can generate many variables.  I just thought this one variable.  So you have this homogeneous function.  And when we calculate trace of exponent minus lambda h bar, how to calculate it?  Again, by Feynman-Catz formula, it's will be integral of the space of pass.  Is periodic boundary condition to R1, this coordinate x.  I have a separator and I multiply by exponent minus action.  You denote by s lambda of phi, d phi.  S lambda of phi is just integral 0 to 1 and half dt squared.  dT.  It's a discrete spectrum.  It's a discrete spectrum because the spectrum is discrete because its potential goes to plus infinity.  It has a val.  It's not the integral from 0 to lambda in s lambda.  Sorry, you're right.  The lambda.  And now I try to rewrite the integral from 0 to 1.  I want to rescale time and rescale everything.  Time and phi.  To make it integral from 0 to 1.  Namely, I take now, tau lambda is lambda 1 minus t.  And then when rate phi lambda is equal to.  If you make it rescaling, then s lambda will be integral of 0 to 1.  I'm going to go to the variables.  I multiply by lambda to certain power, which is...  You get just a homogeneity game.  Nothing tricky here.  And then the conclusion will be the following.  Then if you take consider trace function, whatever zeta goes to,  trace exponent minus zeta h to power n over n plus 1 should admit an endless analytic continuation.  That's what's under the word predicted.  But it's not on the level of pass integral.  It's this simple homogeneity game with pass integrals.  So there are two basic examples when it's very, very concrete.  So it was a case of free particle and also this homogenous potential.  There is still another example.  Suppose we get... use the same name, but now...  One can write another pass integral very badly defined.  Suppose it has a complex manifold, like a tangent model,  or more generally, holomorphic simpleactic manifold,  which contains two complex Lagrangian sub-manifolds in 0 and 1.  And what I want to write, I want to integrate...  The space xc will be no real space at all here.  So my infinite dimensional complex manifold will be space of pass.  So that f of 0 belongs to L0, f of 1 belongs to L1,  and it's my two tangent bundle.  So in the action I choose...  I have the infinite dimensional space.  The action function, which I consider here,  is defined only up to a constant and only locally.  So I have to write what is the differential of these things.  It will be a homomorphic one form,  which will be integral of two forms of my pass.  When I have one parameter of family of pass, I get surface,  and to integrate it, I get... we'll define it.  So roughly, in mechanics, you write something like this things plus boundary terms.  It's called first-order formalism.  And plus boundary terms.  Yeah, in principle, one can include this random work.  I think certain terms, like you add certain functions,  it depends on three parameters.  So it will be a function on cotangent bundle m times some time variable to c,  and it's function h.  You can add this such term, and if the thing is quadratic in p,  this will be the equivalent to pass integral one,  can integrate over p variable, and you get exactly random work.  But now I ignore these things completely.  So it seems it's completely geometric.  I have this integral of one form.  So the question, what is it, and what we'll have here?  Yeah, in general, it's pretty unclear.  How to interpret it?  How to interpret it, what is the integral, yeah, it's...  And what is the property of what we should expect from this integral?  But at least it looks that for certain kind of class of example,  this could be the following.  L0, we'll get some L0, city can cotangent bundle,  here draw fibers of my cotangent bundle,  and it will be like this, this is L0,  and L1 will be cotangent far by its certain point.  Then we get finitely many intersection points,  and kind of the guess is if L0 kind of corresponds like spectral curve,  again, it's long story how to identify parameters to algebraic bundle with a connection.  Then this asymptotic series, which you should get here,  should be like WKB series for formal solutions of the models.  You write L0, it's graph of differential,  graph of D, maybe called F0, some multivalent function,  and then it will be...  you write a solution,  you remove this main term to get a asymptotic series in each bar.  So that's something which with Yan we can...  part of general project we have with Yan on wall crossing,  and what is going on, here's some kind of really interesting effect here,  right to the following story.  On the space of paths, we don't have really one valid function,  we get functions defined up to constant, we have closed one form,  and that means the following, we can...  in the whole integral, we can go to universal cover,  and then we go down, we can twist this arbitrary rank one local system on space of paths.  So we get certain torus,  and C star two kind of...  by which you can twist your integral.  Yeah, for example, if the substation can be in finite dimension of the story,  you can twist your... a real connection by rank one local system,  consider again commode of pair and so on,  so you get things depending on a point on a torus.  And what happens in infinite dimension?  There are certain directions of Planck constant,  for which you change parameterization of space,  you kind of apply nonlinear automorphism of the torus.  So you get things which are kind of very close to kind of a Calvaronian picture,  you have some nonlinear changes along the rays.  It's kind of new infinite dimensional effect.  There are kind of new walls, new rays.  So we're going to...  you can apply analytic defomorphism of torus,  some domain of torus depending on small parameter.  So you get more involved picture with some kind of nonlinear changes as well.  Before we had just vector bundle with basis and transformational grace,  but now you have vector bundle on torus and parameterization...  vector bundle of rings, which is sort of...  Yes, yes.  But eventually it should give again the same resurgent properties  of the whole story.  So I didn't had a time...  That's essentially all one dimensional examples.  And in principle, what was completely untouched,  one can try to make, I guess,  to imitate this, at least the question for two dimensional series,  like the Minovitan or three dimensional with Sharon Simons.  So again one get actual values of integrals from physics  and then the same picture predicts your resurgence structure.  Thank you.  A question, Lesha,  so you consider this free particle on hyperbolic space,  you can think of it on circles, here or...  Any varieties here?  So for the general case, the general group,  there is a formula called deus thermo-techmann,  which essentially we write this as a sum over geodesics on this space,  but as far as I remember...  No, no, this is certain...  No, there was a sudden sub-deus thermo-techmann for any remaining manifold.  If you consider trace of exponent...  If you consider some trace of exponent minus E t,  that's a square root of Laplacian,  it has singularities on a major axis equal to lengths of geodesics, yeah.  So, for example, if I just found the particle on a circle,  again I can write a sum over geodesics,  there are no fluctuations around them.  Yes, yes, you got the exact formula, yeah.  It's exact formula.  You got C to function, yeah, I think.  So there is no asymptotic expansion.  Yes, it's trivial, in this case it's trivial, yeah.  And in these general cases where this deus thermo-techmann applies,  it's also the same structure, right?  No, but the conjecture is for more general remaining manifold with algebraic metric,  then we get very complicated expansions,  if it's not... if it's not homogeneous, yeah.  Can you describe the whole structure of infinite rank  in terms of this function functional, related to homology?  Yeah, no, there's kind of...  one can write this convulsion of perivot, what is it?  But you see this number of...  this gradient lines integer numbers in alpha 1 or 2,  gives you the example of the day, kind of step-by-step construction of this infinite...  should be space generated by all geodesics, yeah.  I think it should be pure in this case, yeah.  More questions?  Yeah, I would like to... I didn't understand everything obviously,  but how far are we, or you, from effective computation on the finite path integral  for say a non-homogeneous problem?  Yeah, for general potential... I think it's... for general potential, yeah.  In principle, it looks that if it's... if it's not homogeneous,  this gives you kind of few steps procedure,  which is completely mechanical, one can calculate critical points in complex domain,  then one should count how many gradient lines,  it's a number of solutions of certain... like,  pseudo-holographic disks somewhere,  or some integer numbers which one can calculate,  and then you get form power series which you do at each point,  and then after that it should produce the actual numbers.  Can you replace it by so many numbers?  Yeah, because it should... yeah, the story is the following.  This integers and alpha... and those jump formals for J,  which depends only on critical values and integer numbers.  So it gives you a way to glue certain holomorphic vector bundle,  sterilize the infinity structure at zero,  and then there's form power series after you solve the problem.  It will be convergence series, yeah.  Yeah, yes, yes, yes, it's completely general procedure.  From this perspective it will be...  It's effective, yeah, it will get to actual number, yeah.  It will get it, convergent.  Thank you.  Is it clear that if you write for... if you write the formal WKB solution,  such as in the classical... or is the same as you get a finite integral,  a diagonal exception.  So is it clear or...  No, no, no, no, no, it's not clear.  Okay, comment on your question.  So from the final integral point of view, that's very difficult,  but if you do Bender-Wohl type analysis, you can actually computerize...  Yeah, you can... yeah, because you substitute the things to your expansion  and you get to...  It's pretty computational things here.  It's a computer quality of...  I think even the determinational part is not completely clear.  Yeah.  Yeah, but there are many things here.  For example, here, the dimension, yeah,  it will zeta function regularization then, sir.  This will be zeta prime of zero, yeah, yeah, so it's...  In field theory, all terms make sense here.  No question.  Thank you.
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