Okay, thank you very much. Yeah, so I will try to present some little bit, some kind of 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 you make Laplace transform,

take sum over a n over n factorial z to the power n when z2 goes to zero, and it will be germ of analytic function near zero, and the property of resurgence

says that it has endless analytic continuation. And in the talk of Jean Nicole, we heard something called autarky and isographic form. And now the main guess. This thing means that

you have a polarized exponential Hodge structure of infinite rank. Yeah, so kind of like infinite

dimensional algebraic geometry, and it looks that the words have these 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. It will be 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 to some g infinity over integers, which will be integrality of this tessellations coefficients roughly. Yeah, so what is this exponential Hodge structure of infinite rank? It's of, let's say, a finite rank,

because for infinite rank we don't have yet rigorous theory.

Let me recall to you what are usual Hodge structures. Let's start with pure Hodge structure. It's a vector space over C, this Hodge filtration

and lattice, essentially not compatible with filtration, and you have a property

such that if you consider real involution, so lattice gives you real structure on H, and now consider f bar and f will be opposite filtrations, up to shift, so you get

decomposition of H into sum of Hpq, let's say p plus q is equal to n, and yeah, that's a usual hot structure, which appears on cohomology of, let's say, complex projective variety, and then there's something called polarization, which comes from

ample bundle, you get non-degenerate bilinear form on integer lattice,

which is symmetric or skew-symmetric, depending on parity of weight, and such that it induce bilinear forms on, it induce a Hermitian, pseudo-Hermitian form Hpq, for each being q,

namely you consider the pairing of alpha and alpha bar, for alpha belonging to Hpq,

and this has signature minus one to power p, so it will be either positive or negative, depending on the things, and if you take sum of its appropriate sign, you get Hermitian form, so you get Hilbert space structure, so on p of this form on Hpq,

it will be positive definite, you get Hilbert space structure. Okay, that's a usual hot structure, which appears on cohomology of projective varieties, and exponential hot structures, it's some notion, which we introduced about 10 years ago with Katsarkov and Panthiev,

studying mere symmetry of Landau-Gisbert models, it's related to exponential integrals, this I wrote to usual integrals since the definition of exponential hot structures

is the following. You get a holomorphic vector bundle on c, this coordinate, we denote it u, but it's the same as Planck constant or small parameter,

zeta in this one or z in this situation, so we get this holomorphic bundle, so we get this family of spaces depending on u, and then on u-plane we have special point u equal to zero,

and we get a connection on this bundle, holomorphic connection, it has second order pole, on bundle, the second order pole at u equal zero, and in particular you will have certain

monodromy, and we assume that flat sections of the bundle on rays grow like exponent minus

some constant times u. In principle they can have more different singularities, it will be simplest exponential type singularities.

Then what we get on each ray, we get filtration by order of growth, we get this Deling-McGrange filtration, and what is assumed that we have certain

lattice, so the data is given, the extra data is given, you have a lattice h u z sitting in h u for u not equal to zero, which is covariantly constant with respect to filtration,

with respect to connection, and all terms of filtration are also integer subspaces,

I defined over z. So this will be analog of hot structure, actually the usual hot structure, it's a particular case when this is not a second order pole, but first order pole in Rome is plus minus one. So this analog of hot structure and polarization

is given by the following data, you get for u not equal to zero, you get a non-degenerate

pairing at opposite sides of filtration, which is covariantly constant and also compatible with

integer lattice on h u z, is integer valued, and then there is a way to formulate

property of, like sign of polarization, which eventually gives you some Hilbert space structure on h u. And also from this synchron it can deduce certain filtrations on this h u

without z, labeled not only by integers but maybe some fractional numbers. So that's its abstract story and 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 fast intervals, at least some fast intervals, one can see as infinite dimensional examples of this appears in exponential integrals, namely imagine that x, the following situation,

suppose x is a fine algebraic variety, one can think about just coordinate space, and suppose we have some polynomial, some function, 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's polynomial has no critical points at infinity.

Same, yeah. Suppose we have this polynomial, what will be this h of u?

We consider space of algebraic n-forms on x, and mod out by, let's say the quotient space by the image d minus u d plus multiplication by d s, applied to n minus 1 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 u not equal to zero, the lattice, it will be a cohomology of the following, so to consider x, and consider pre-image of the

domain when real part, we divide 1 over u s, as a function, and we take pre-image of the domain when real part is very big, and take cohomology with integer coefficients.

So the dual space, or dual lattice, can be sorted in certain cycles, 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, namely we can integrate over such cycle of the following, integrate s minus u,

and times some algebraic n-form, representing class in this cohomology.

So this integral will be convergent, and this integral gives you a family of lattices, it's covariantly constant, it gives a certain formula for this connection, it 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 code filtration. What is this pairing? I want to describe the pairing on kind of, the pairing is also completely natural in this situation,

I'll describe pairing on level of cohomology, and we're going to go to dual, get pairing on level of cohomology. Namely, let's assume that argument of u is generic,

then we get the basis of h u z dual, consisting of codes called Lefshitz timbels. Namely, what you do, you have critical values of our polynomial,

now we draw a straight path along which this s over u minus some constants will be real and goes to plus infinity, so we get the straight path, and over the straight path, in ambient

manifold we draw a family of, nearby we have a vanishing cycle, some small sphere, s minus one-dimensional sphere, and we get a family of, 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 meaning depends on these critical points, so the critical f x alpha will be critical points, and then you get s of x alpha,

and for each critical point you get this Lefshitz symbol, it gets converted to integral, so you get this converted to integral, and then if we change direction, argument of your u goes to minus u, you just get Lefshitz dual cycle going to this opposite direction,

I claim it is, and it's easy to see that it's kind of currently constant and you get duality between, it's just Poincare pairing between a module in fiber u and fiber minus u. Okay, yeah, so that's kind of scientific way to formulate this integral over Lefshitz 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 Lefshitz symbol,

it will be integral, we choose again some expression, we integrate over this Lefshitz symbol starting at point x alpha,

we integrate some n form which you write, something like polynomial function of x multiplied by some reference form which you write dn x here, 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 asymptotic expansion to the 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,

or 2 pi u, sorry, to dimension over 2, then we divide by determinant of the second derivative at 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 in u, it will be value of this function at this point, and then we get extra correction, and this series has factorial growth, and 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?

You're right, and this I denote by something like g alpha of u, now it's actual function which has asymptotic expansion, I remove all irrelevant terms,

so, you get just a bunch of functions, g alpha of u,

alpha of alpha 1 to number of critical points, which have defined outside of some stocks race, direction race,

and which have, but these functions have the same asymptotic expansion, which doesn't depend on direction, formal expansion is the same, stocks, stocks, formal expansion, doesn't depend on the sector,

so you get just a bunch of formal power series which you can calculate on computer using formal expansion at this point, but you get actual functions as well.

Yeah, you get actual function, yeah, and this function satisfies kind of jump property, jump, if you go through some ray, we are argument ray,

then says that argument of u is equal to argument of difference between two critical values, yeah? When you get this ray, then what we see, it's kind of standard picture,

what happens if you, on the one side of the ray, and then we get a rotation, we get slightly different homology classes,

and you see that one homology 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, it has no exponentially small, so has trivial Taylor expansion,

sorry? Oh, yeah, sorry, yeah, because I kind of, it's notation I have in mind this h bar, and use this for the same, eventually. Delta, it's jump, a kind of jump of, I have function defined on two sides,

so on the ray, you can see the difference, I think, yeah. Yeah, so get, 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 will 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 a certain remaining manifold, maybe two points of this remaining manifold, and my,

and I assume that manifold is, can be complexified to some complex manifold, maybe algebraic manifold, and metric will be algebraic tensor on this manifold.

I also get this big complex manifold, and m is, okay, as example, you can try to see it like m is a sphere,

or something you can write explicitly formulas, sphere or ellipsoid or hyperbolic space,

maybe flat space even, yeah, torus, or something, it's a very nice concrete formula. Now this, then we have infinite dimensional complex manifold, X, which will be the different space of path connecting this to points m0 and 1,

maps 5 from 0, 1 to mc, f of 0 is m0, f of 1 is m1. It's infinite dimensional complex manifold,

it's, it contains kind of half infinite dimensional, six paths sitting in real path, in real locus, which is the same boundary condition.

And now the Feynman integral, which we consider, the action functional will be one half of integral from 0 to 1, d phi, that's over dt squared, dt.

Yeah, so you get this functional integral, and now when we integrate over X, r, X under s of phi divided, let's say by u, but now maybe I denote it by Planck constant,

and it defies what the expression means in path integral. So it's analog of this situation, of finite dimensional exponential integral.

And all this, so it should get some kind of critical points,

form of power series expansion, and then actual functions and all this geometry. Let 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 will be brownian motion

on your manifold, and you start from point m0 and point m1 in time 1. And this should be like, should be equal by value of hit kernel exponent minus

maybe one half of Laplacian on X real, and you use physics notations. So consider hit kernel for time 1 between points 0 and 1, m0 and m1. And what to do for general h-bar? The idea is, it's very simple.

So we see, you multiply this thing, divide by 1 over h-bar. Let's keep one half. Now so we get this, this thing, and you see that it's in time, this thing is homogeneous of degree minus 1.

So we say that it's, put it, time-bar is equal to ht, so tau-bar belongs to interval 0 h-bar.

We identify 10 points, and then it will be equal to just equal df over dt-bar. Yeah, so you see that this integral is equal to exponent,

to the hit kernel for small time, to h-bar, yeah, okay.

Yeah, so you get this nice formula, so it will be hit kernel but for small time. Yeah, so the prediction is the following. If you consider hit kernel for small time and divide by this leading term, you get some asymptotic expansion, then it should be resurgent series.

This should be resurgent series, and now,

yeah, so it should have some exponential small term, so it should divide by exponent minus I think 1 over h-bar times something like square of length distance between those points,

which will be leading growth of explosion, should be kind of like series in

h-bar, and this should be resurgent. Yeah, that's actually a completely open question. For general, like C-infinities remain in main fault, we really don't know

whether this guy is a resurgence, and I think it's true only for some special like algebraic varieties, so things close to them. So let's make this example, which I found extremely striking.

This is example, then consider this main fault will be hyperbolic plane.

Yes, it's maybe the simplest example, which could be, because in this case you have just only one geodesic entering two points, and from some classical formula for the hit kernel, one can express

that this m0 x-point minus Laplacian of Laplacian one, and you multiply by h-bar, is equal to the following, I think, it's a certain constant, it's something,

I think, and then you get a certain integral, it will be small s, not capital S, but it's

related to the critical values of my function, I'll get this formula,

where l is distance between my two points in a negative real number. So this thing is given by

certain integral, and what here goes on? You can see the critical values of C on space of compressive height path.

Pass in a complexified hyperbolic plane. 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 z-points on parabolas, as you can see,

you get a bunch of left shift symbols, and this is my integral of the left shift symbols. And in fact, if you look on this integral, you'll get certain finite dimensional, actually,

you said that you identify up to the simple factors in the front, you identify your infinite dimension integral with finite dimension integral, over Riemannian surface, which is infinite genus Riemannian surface, ramified as this countable set of critical values. You can see this function of small s, this one of square root of hyperbolic cosine minus

hyperbolic cosine, it's something which is too valued. If it's not on a sphere, you have to call it a sphere. Yes, yes. Yeah, and the idea is that on a sphere, if you have two points, you have not only one geodesic, but we go around, around, around, so it means that two lengths, you add

this length of the periodic geodesic, which is purely purely imaginary, you get some kind of like 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 ones are just adding the point in this sphere. Yeah, but of imaginary, of radius i. Right, so the critical value should be... Yeah, 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. And 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.

Those are integer numbers n alpha 1, alpha 2 for each two critical points. And these are very simple numbers to calculate for this interval, kind of plus minus one. And the claim that should be coincide with something very interesting,

because if you look what is alpha 1, alpha 2 in general stations, 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 path. So it means that you should solve certain paths in the space of path.

It's a map from surface. And it means it's gradient lines, it will be number of certain surfaces satisfying certain kind of Cauchy-Riemann equation. In a complex defined space, yes. So it's kind of a very non-trivial prediction that from this answer, we can see that everything fits together and

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. So that's main example, non-trivial example, which I have to tell you.

It's about this heat kernel on hyperbolic space. One can ask similar stuff.

What in small modification, I can study cases here. But for example, more interesting case will be case of ellipsoid

with 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 C2 functions. But also, eigenvalues of Laplacians also kind of known 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 be kind of like double of disk. Imagine disk glued to itself along the boundary. Then geodesics will be this polygons labeled by pair of integers.

So it will be a length of geodesics on completely closed things. It will be whatever Q times 2 cosine 2 pi p of Q, something like this. And eigenvalues of Laplacian will be

because they're decomposed by zeros of some Bessel functions.

So in this case, there will be certain great identities, certain infinite sum, there's no decontination along these zeros of Bessel 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 one, because of dF over dt squared times dt. It will be homogeneous of degree minus one. And then it means that we can make this a calculation. One can make kind of similar exercise.

It's related to very old work by Andrea Voros. So consider a operator, consider in this example,

instead of Laplacian, consider secretivity plus potential. And the whole story is the main thing is that it should be also homogeneous function. Like, if one can generalize to many variables, I just start with one variable. So we have this homogeneous function.

And when we calculate trace of exponent minus lambda h-bar, how to calculate it? Again, by Feynman-Kutts formula, it will be integral of the space of path,

this periodic boundary condition, 2R1 with coordinate x, where I have a separator. And I multiply by exponent minus action,

we denote by S lambda of phi d phi. S lambda of phi is just integral 0 to 1, one half d phi over dt squared, phi to the power of n, times dt.

It's a discrete spectrum because the spectrum is discrete because its potential goes to plus infinity.

And now I try to rewrite as 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 I write phi lambda is equal to,

if you make it rescaling, then S lambda will be integral of 0 to 1. When I go to do the variables,

multiply by lambda to a certain power, which is, you get just a homogeneity game,

nothing tricky here. And then the conclusion will be the following. Then if you consider trace function, whatever zeta goes to, trace exponent minus zeta h to power n over n plus 1,

should admit endless analytic continuation.

Yeah, that's what Andrei Evorov predicted, but as far as it's set not on level of past interval. Yeah, so it's this simple homogeneity game with past integrals.

Yeah, so there are these two basic examples. It's very, very concrete.

So it was a case of free particle

and also this homogeneous potential. There is still another example. So suppose we get, you use the same name, but now one can write another past integral,

very badly defined.

Suppose I have a complex manifold, like a tangent bundle, or more generally holomorphic simply active manifold, which contains two complex Lagrangian sub-manifolds n0, l1.

And what I want to write, I want to integrate the space Xc, there will be no real space at all here,

it will be just not my infinite dimensional complex manifold, it will be space of paths, that f of 0 belongs to l0, f of 1 belongs to l1, and f of 2 is a tangent bundle,

and as the action I choose, the action function which I consider here is defined only up to a constant,

and only locally, so I just have to write what is the differential of these things, it will be holomorphic one form, which will be integral of two form of my paths.

When I have one parameter family of paths, I get surface and to integrate I get the defined thing. Yeah, so roughly think it's, like in mechanics you write something like

these things plus boundary terms. It's called first order formalism, plus boundary terms. Yeah, in principle one can include this

random walk, adding certain terms, like you add certain function, it depends on three parameters,

so it will be function on cotangent bundle m times some time variable to c, you can add this such term, and if the thing is quadratic in p, this is equivalent to pass integral, one can integrate over p variable, and we get exactly random walk.

But now I ignore these things completely, so it thinks 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 interpreted what is the integral, yeah, and what resurgence property of what we should expect from this integral,

but at least it looks at some, for certain kind of class of example, this could be the following. Suppose L0, we get some L0,

ceticon cotangent bundle, here draw fibers of my cotangent bundle, and it will be like 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 is asymptotic series,

which you should get here, you should be like WKB series for formal solutions of the models, which write like, you write L0, it's graph of differential,

graph of D, maybe called f0, some multivariate function, and then it will be, you write a solution, you remove this main term

to get asymptotic series in each bar. So that's something which with Yanvi kind of part of general product 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 path, we don't have really one valued function, we get functions defined up to constant, we have closed 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 path. So we get certain torus, and C star two, kind of first beta numbers of the space of path,

by which you can twist your integral. Yeah, for example, if the substation can be in finite dimensional story, you can see the, you can twist your

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 dimensional? There are certain direction of Planck constant,

for which you change parameterization space, you kind of apply non-linear automorphism of the torus. So you get things which kind of very close to kind of a Calibaronian picture, you have some non-linear changes along the rays.

It's kind of new infinite dimensional effect, there are kind of new walls, new rays, set it, we're going through this, you can apply diffeomorphism, analytic diffeomorphism of torus, of some domain of torus depending on small parameter.

Yeah, so we get a more involved picture with some kind of non-linear changes as well. Before we had just vector bundle this basis and transformational grace, but now you have vector bundle on torus and parameterization.

should be something like a bundle of rings, which is sort of Yes, yes. Yeah, but eventually it should give, again, the same sort of properties of the whole story. So I didn't have the time. Yeah, that's essentially all one-dimensional examples.

And in principle, and what is completely untouched, one can try to make, I guess, imitate this, at least, equation for two-dimensional theories, like there's the Mino-Witten, or three-dimensional with Shannon-Simons. Yeah, yeah, so again, one get actual values of integrals from physics,

and then the same picture predicts your resurgence structure. Thank you. I see, yeah.

It's a question, question mark. So you consider this free particle on a 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 Dewey's Theorem of Thecman,

which essentially rewrite this as a sum over geodesics on the spaces. But as far as I remember... No, no, no, this is certain... No, there was some way to determine the Hecun for any remaining manifold. If you consider a trace of exponent minus et squared of Laplacian,

it has singularities on imaginary axis equal to lengths of geodesics, yeah. So, for example, if I just constant part on a circle, again, I can write a sum over geodesics. Yes, yes.

There are no fluctuations around them. Yes, yes, you've got exact formula, yeah. It's exact formula. Yes, exactly. You've got C2 function, yes. Yeah, I see. Okay, so there is no asymptotic expansion. Yes, it's trivial, yes, in this case it's trivial, yeah. Okay, and in these general cases where this Dewey's Theorem of Thecman applies,

it's also same structure, right? No, but the conjecture is for more or less general remaining manifold with algebraic metric. Then you get very complicated expansions, if it's not homogeneous, yeah.

Can you describe which structure of infinite rank in terms of description functional, relative to homogeneous? Yeah, no, there is kind of abstract one can write, it's combinator of pair. What is it? But you see this number of this gradient lines, integer numbers in alpha 1 or alpha 2,

gives you at the end of the day, kind of step by step construction of this infinite. It should be space generated by all geodesics. I think it should be pure in this case, yeah. More questions? Yeah, I would like to understand everything obviously,

but how far are we or you from effective computation on the Feynman path integral for, say, a non-homogeneous problem? For general potential, I think it's for general potential, yeah.

In principle, it looks at, 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 solution of certain like pseudo holomorphic disks. So there are some integer numbers which one can calculate. And then you get form power series you should do at each point. And then after that, it should produce actual numbers. Can you replace it by solving the Feynman formula?

Yeah, yeah, because it should, yeah, the story is the following. This integers and alpha, and those jump formulas for j, which depends only on critical values and integer numbers. So it gives you a way to glue certain holomorphic vector bundle, with trivialized infinity structure at zero.

And then this form power series after you solve Feynman's problem, will be convergent series, yeah. Yeah, yes, yes, yes. So it's completely general procedure. It's from this perspective, it will be, it's effective. Yeah, it will get to actual number, yeah. It will get convergent.

Is it clear that if you write for, it's 400 polynom, if you write the formal WKB solution, such as in the classical, or is the same as you get Feynman integral, diagonal expansion? So is it clear? No, no, no, no, no, no, no, it's not clear.

Can I comment on your question? So from the Feynman integral point of view, that's very difficult. But if you do Van der Voole type analysis, you can actually computerize. Yeah, you can, yeah, because you substitute the things to your expansion, you get to...

It's purely computational. It's purely computational since, yeah, it's a computer, Even the determinantal part is not completely clear, though. Right. So it's part of infinite dimension itself. Yeah, no, but this also, no, there are many things here. For example, here, this dimension, yeah? It's with zeta function regularization, then, so. This will be zeta prime of zero, yeah, of Laplace, yeah.

So in field theory, all terms make sense, yeah.