Wall-crossing Formulas and Resurgence
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Transcript: English(auto-generated)
00:16
So, my talk is about the technique which we invented with Maxim for
00:22
the purposes of wall-crossing formulas, and today I will explain how it could be related to to the subject of resurgence. So, I start with motivations and it's wall-crossing formulas,
00:40
subject which was popular like ten years ago. So, let me start with an example. Consider the algebra of formal power series of two variables.
01:00
If you don't like rational numbers, you can replace them by complex numbers. It doesn't matter. And let's handle it with the Poisson bracket. So, then for a pair of non-negative integers a and b satisfying this condition,
01:29
let's define the automorphism of this algebra, b, which preserves the Poisson bracket,
01:48
which maps a pair x, y in the following expression.
02:05
All right, so then the claim, which can be checked directly,
02:26
is that this automorphism satisfies the following formula, which is essentially equivalent to the five-term identity for the logarithm function.
02:57
All right, so this is the simplest example of the wall-crossing formula.
03:03
And in the paper with Maxim in the archive, we proposed a big class of wall-crossing formulas,
03:21
for which this one is just the simplest possible example. Even here, we can introduce a number k in the Poisson bracket,
03:42
so this is case k equal one, and then define a collection of Poisson automorphisms of this algebra, and then do this bracket. The difference is that you put k here in the power,
04:01
and these automorphisms, Poisson automorphisms, they satisfy more complicated identities, wall-crossing formulas,
04:21
where in the left hand side you can still multiply two basic things, but the right hand side will be more complicated, sometimes it will be an infinite product of these automorphisms raised in some integer powers.
04:42
Here, product of these raised in integer powers. So, these wall-crossing formulas, they were invented for a purpose,
05:06
so they appear in mathematics in the theory of Donaldson-Thomas invariants, say, of three-dimensional Calabi-Yau categories, for example, Calabi-Yau three-folds,
05:26
and these integer powers, they are these DT invariants. In physics, the same numbers, and essentially for closely related reason, appeared in the four-dimensional n equal two gauge theories,
05:49
and basically this is a development of the last like 10, 15 years at most, but this simplest formula, five-term identity,
06:03
it appears long before this relatively recent development, it appeared, it can be found in the paper by Gillinger, Delaber, and Pham, 1993,
06:33
there was resurgence for the operator which appeared in the previous talk and in many other
06:46
talks, it's a Schrodinger operator with a small parameter, and V of X is a polynomial.
07:04
So, it seems to be a completely different topic at the first glance, so the question is, what is the underlying mathematical structure which naturally appears in both, in this wall-crossing story, and in the resurgence of the Schrodinger operator,
07:29
and so that's a structure which we proposed in that paper with Maxim, and that's one which I remind you as a motivation, so underlying structure,
07:50
the one which we called stability data on graded lie algebras.
08:06
So, by the way, it also underlies these wall crossings which appear in implicitly in Maxim's talk on exponential integrals, and I will comment later on that. So, what are the data? So, I'm a mathematician, so I would like to
08:23
give definitions and state results precisely. So, what are the data? So, three abelian group, so in physics it's called charge lattice,
08:50
gamma-graded Lie algebra, say over rationals, so any field of characteristic zero,
09:10
so in the condition gamma-graded means this, homomorphism of abelian groups,
09:36
so the notation is chosen in such a way that physicists translate immediately,
09:42
so this is the central charge, and collection of elements, graded component,
10:06
which basically in the sense these are Donaldson-Thomas invariants or BPS invariants up to something which some transformation which I explain later on.
10:25
So, these are the data, so four types of data, and there is just one axiom which we call support property, that's an axiom in this data.
10:48
So, basically this means that if we consider the support of this collection, set of all gammas for which A of gamma is non-zero, then this support is separated
11:11
from the kernel of the central charge. Yeah, better.
11:33
So, it can be formulated as an existence of certain quadratic form which is negative on the kernel and positive on the support.
11:43
So, this is an untrivial part which one should always check in concrete examples. So, this axiom, it allows to define certain, if you like, certain pro-neal potent group,
12:04
more precisely for every array on the plane, so it's a corollary of this support property, if you like. So, for every array with vertex in the origin, one can give a meaning
12:28
of the following expression. So, one can take exponent over all of the sum over all A of gamma with a condition that the central charge belongs to L,
12:49
and moreover from this property one can deduce that these gammas belong to a certain convex column C of L which belongs to gamma.
13:10
So, this can be thought as the race for which this A of L is non-trivial, can be treated as kind of analogs of, if you like, subgroups in some big, possibly infinite
13:26
dimensional group. In particular, if we take a bigger sector, but which has to be strict, which means less than 180 degrees,
13:44
it can contain many arrays, like countably many arrays for which A of L is non-trivial, and so we can define the corresponding group element which is a product, say, in the clockwise direction of these simplest inputs.
14:11
All right, so this is the axiomatic. On the set of this stability data,
14:22
there is a certain topology which basically means that if we have a continuous path in the space of central charges, which is entirely linear algebra data, then it can be lifted to a continuous path in the space of all stability data, in particular involving this
14:46
A of gamma, which in general they kind of behave in the locally constant way. So, in many examples, in practice, we have not just the lattice, this lattice gamma,
15:06
but we also have an integer-valued skew symmetric form, it can be symplectic or not.
15:22
And so, if this additional piece of data is given, then we can consider the algebra functions on the corresponding torus. This is a torus, so you can consider a torus of
15:46
characters, it's a matter of choice. Okay, let me replace, some of these are physicists. Okay, it's C*. So, this is a linear algebra, because this is...
16:08
Okay, I can take dual, so again, either of these, yeah, all right. So, anyway, so this is a torus, it's a Poisson torus, so therefore there is a bracket, and in fact,
16:39
if we treat it not as an algebra, but just as a Lie algebra, so it's a direct sum of
16:46
one-dimensional vector spaces, so graded components are one-dimensional, and so this, if you want to, on this, let's call it torus Lie algebra, if we want to handle it with a stability data,
17:10
so this a of gamma will be just numbers. So, then the...
17:24
integer numbers? No, in general, no. So, I put a C, I can, of course, return to gem, so at best I can kind of assume that these are rational, but I will tell you in a minute what one should do
17:45
in order to make them integers. All right, now, returning to this example of Schrodinger operator, so how to see this data? So, we have the spectral curve,
18:04
which is hyper elliptic in this case, so it projects 2 to 1 to the projective line, and there are ramification points, and there is also a singularity at infinity,
18:22
essential singularity, so let's take gamma to be the relative first homology, this is the set of all these ramification points and infinity, so there is a Poisson pairing
18:43
and inside sigma is this, s is the ramification point, and infinity is all. No, I think it should be just a homology, open curve will have a really Poisson pairing.
19:03
Yeah, it will have, so I pre... No, it's the first homology, you're right, you're right. Okay, but then I need to take the dual one, which is the first homology of the open curve,
19:24
and the central charge will be just a period map. Now, in this example, so this is gamma, there is a natural Poisson pairing,
19:43
and so then the analog of this, it's a map t of gamma, which maps
20:02
which maps with generator, inter... So this is an example of Poisson cluster transformation of Fock and Gensharov, and you can
20:22
put here instead of minus, you can put z2 value of casei code in general, I mean if you want to generate z2 value, one casei of sigma lambda. Now, returning to Stavros question, so
20:51
at best we can hope that A of gamma are integers, excuse me, are rationals, but if you have a skew symmetric form, then it turns out that the Möbius transform, inverse Möbius transform
21:06
of A of gamma tends to be integer. So if we write A of gamma in this way, then...
21:35
So these are BPS invariants in physics.
21:42
All right, so then the wall crossing formulas, which are basically just a direct corollary of the definition of the topology on the space of stability data, so it's a certain wall crossing formulas, these are identities between t of gamma raised in this integer powers.
22:19
So then in the example...
22:20
In example, in this five-term identity, the sum of gamma equal to one. Now the term wall crossing formula, just for those who have never seen it before, so the origin
22:41
is the following. If you consider the space of central charges, it's a vector space, and in the space of central charges there are real, could I mention one, walls where this one of gamma jumps. So one of gamma, in fact, it depends on the central charge,
23:11
and so if you have on one side of the wall, if you have
23:23
some collection of omega of gamma, so they are locally constant. If a set of walls is not dense subset, so when we cross the wall, we apply the wall crossing formula and can calculate these numbers on the other side of the wall if we
23:45
start it somewhere and move in the space of central charges. And that vanishes on the wall? No. Okay, thank you. Yeah, I should say what are these walls. So the walls,
24:05
generic walls, they are defined in the following way. You look in the lattice, you look for central charges which do the following. There is a lattice of rank two,
24:22
like two z independent vectors, which central charge is mapped to the line. So there are lattices which are collapsed, so then the central charges are proportional.
24:43
All right, now if we return to this example of the Schrodinger operator, which I use as a motivation for this wall crossing story.
25:05
So in that example, we have five term identity, omega of gamma equal to zero. So of course, like Schrodinger operator, it's the same as H connection of rank two, SL two connection. So what we have, say, SL two connection curve can be in fact any.
25:37
So then in the drum picture, we have
25:42
in the fine line in the space of connections and the monodromy data, Riemann-Hilbert correspondence, monodromy converts it into analytic path. Is AX monomorphic or any?
26:03
AFX can be monomorphic, it can have both. And if I use the notation for the generators of this Torus-Lee algebra, so when I have
26:23
analytic path, I'm interested only in the germ of this path as H goes to infinity. So, and in the terminology of 90s, so this gamma of H is called Varus symbol.
26:50
Wall crossing formulas actually, they ensure that you can glue global solutions from local WKB solutions. There is also
27:02
a Varus coefficient, well, and the theorem of a call that these Varus coefficients,
27:36
they are resurgent, and resurgent in H. And moreover,
27:52
a call introduced a notion which is called direction of singularities,
28:02
direction of singularities, which you can easily guess where it is. And so these directions or singularities are raised through, in my annotations, through V of gamma. Now, Varus coefficients, they are transformed slightly differently,
28:30
but it's a direct consequence. So I put here plus or minus, because I can, as I said,
28:42
I can introduce a cycle. So here, let me put omega of gamma, this is omega of gamma.
29:09
So and Varus symbols, they are transformed as cluster coordinates. And so then you can interpret, if you'd like, this observation that in the space of
29:30
framed local systems, which is a cluster variety, you have an analytic path near infinity.
29:42
And so then, in a sense, this appearance of H and all these resurgence, it's auxiliary to the fact that there are cluster coordinates in the space of framed local systems. Okay, now, this subject of 90s about Varus resurgence, it received some kind of boost
30:16
after this wall-crossing story. And there are papers devoted to the study of Varus resurgence
30:27
in more complicated geometries. So I can mention a paper by Iwaki and Nakanishi a few years ago,
30:42
everything is in archive. And even today, you see it's a non-stopping story. Even today, there is a paper by Hollands and Niedzke, exactly on the same topic, where they generalized the second-order Schrodinger operator story to higher order, including operators.
31:08
Okay, now, in this story, still omega of gamma, this BPS invariance, they are equal to 1, although the wall-crossing formulas can be more complicated, more than just five-term
31:28
identity. But you can, since, as you see, this is all about the geometry of the space of local systems, so you can ask other questions. In particular, this now well-known, famous
31:51
paper by Guyota, Moore and Niedzke, in fact, several papers, they wanted to construct hypercalor
32:06
metric on the space of local systems. So it's a different problem, but still very interesting in it. It involved the same geometry, it involved the geometry of the spectral curve
32:20
and triangulations with vertices and ramification points. So in that story, we see more complicated wall-crossing formulas with non-trivial
32:46
omega of gamma not necessarily equal to 1. In fact, the geometric meaning of this omega of gamma is a virtual number of geodesics on the spectral curve with a given homology class.
33:07
So it's another repeatable problem in geometry. Now, so in these examples, the Lie algebra is infinite dimensional.
33:24
It's a Lie algebra of functions on the torus, say on the Poisson torus, handled with a natural bracket. Now, in Maxim's talk, a different type of wall-crossing
33:44
formulas appeared. He didn't write them, but maybe he did, I don't remember. Oh no, he did. So he studied exponential integrals, say in finite dimension, and here some volume form.
34:16
So here the graded Lie algebra is just GLN, where N is...
34:26
So gamma here, if you remember, it's middle cohomology or middle homology of the space. So all this in some big space with respect to some boundary at infinity
34:46
with integer coefficients. So this is GLN and the grading is just by the root. So gamma belongs to the root like this. So the wall-crossing formulas here are more simple.
35:09
They are so-called Chukoteva wall-crossing formulas. So from physics perspective, this is a two-dimensional massive theory, supersymmetric, and this is four-dimensional.
35:25
But as I explain later, you can combine both things and still have a geometric meaning. Now, if you have a stability data, I removed, I erased the definition, but there is a central
35:43
charge and you can always introduce a small parameter by rescaling the central charge. So in particular, in this matrix integrals, excuse me, in this exponential integral story, you have a collection of critical values of the function s, and so then
36:12
this rescaled central charge is this one. So then the walls here, the walls which
36:23
are the central charge basically is given by the collection of these complex numbers, and so the wall is a collection of complex numbers where three of them belong to the same real line.
36:47
Yes, it's complex, yeah. And actually, yeah, so then you can just having a fixed stability data, fixed central charge, you can kind of rotate
37:03
this h, change the argument, and so you see some interesting events. So, okay. Is it possible to talk about the status? I mean, there's this sort of conjectural hyperkalometric, I mean, has that been proved now? No, no, it's not proved. So the thing is that
37:27
kind of they have some integral equation with generalized thermodynamical which depends on the additional parameter, additional positive parameter, some radius.
37:41
So have they proved that these functions do give the collection match for a twister space or not? No, no. So, you want me, I can give a talk on the work of Goyotomov and it will be a different talk, but roughly speaking, kind of, if you start with a
38:11
so you have a variation by a billion varieties, so you can try to construct, first you construct the naive metric, yeah, which is semi-flat. Then you add this
38:25
instanton corrections, which basically corresponds to, you consider, kind of, geometrically, they appear when you have pseudo-holomorphic disk with a boundary on some,
38:43
in this case, on the geodesic fully quadratic differential. So then you have some integral equation, you want to solve it, kind of, by iteration. So the question is whether the solution exists. So they proved that the solution exists as a series
39:01
or a small parameter, which is this r, which is, okay. So then, in a sense, it's a non-archimedean hyper-calor metric, yeah, but they didn't prove that it converges. Is that not the only possible, sort of, geometric interpretations of what these
39:21
plasmas actually are, though? I mean, is there a different geometric interpretation? No, no, no, I'm not sure it's a question about geometry. It's a question, you do iterations, so your integral equation, the answer is written as an infinite sum over some planar trees,
39:43
and so... The question is, is there a different geometric interpretation, apart from that, if we did want to understand geometrically what the functions actually are? No, I think it's just, it's just a technical problem of proving some estimates. And actually, it's a very good kind of a point,
40:06
because I am coming to these estimates which they cannot, I mean, which are not available in the case of a special class of stability data, or more generally, actually, later with Maxim,
40:25
we introduced the notion of low crossing structure, which serves also complex integrable systems, like Hitchens system, for which the lattice and the Lie algebra, they can depend
40:42
on some base, they form a local system. But anyway, so exponential stability data, and this is a class of stability data which, as we expect, is related to resurgence.
41:06
I will formulate a conjecture later. So then the setup, it's graded Lie algebra, and assume that it ends out with a norm,
41:29
so like every graded component carries a norm, Banach norm. Definition. So suppose that this norm
41:46
graded Lie algebra ends out with a stability data, in a sense which unfortunately I erased, but hopefully you remember. So it's central charge, a collection of elements, say, of gamma,
42:02
all this blah blah blah, satisfying the support property. So we say that stability data are of exponential type if there exists positive constants such that we have this
42:28
exponential estimate for BPS invariance, for Donaldson-Thomas invariance, depending on who you are, physicist or mathematician. All right, so this is the definition.
42:45
And so there are several geometrically defined stability data which we expect to be exponential. So if you like examples. So first, suppose x is a three-dimensional calabi-yau,
43:04
or complex numbers. So then the latest is third homology or co-homology, it doesn't matter if you assume that it's compact. Let's assume that. So the bilinear form is a Poincare pair,
43:34
and the central charge is the integral over holomorphic volume form,
43:43
check this on calabi-yau, or three-dimensional cycles. So this is one example. And omega of gammas are BT invariants. So second is complexified Chern-Simons,
44:17
it's kind of thinking in-dimensional type of example.
44:26
Example, what are gamma? No, no, gamma or omega of gamma, better. These are called BT invariants, like number of special Lagrangians
44:44
in the given class in three-dimensional, like virtual number of special Lagrangian sub-manifolds,
45:00
x with given class. I think it's even not known whether every class can be realized by special Lagrangian, but I mean we are all optimistic, so then we hope that this is an example not only every realized, but also there is an exponential bound on this number,
45:25
of omega. Because you can you can replace in that estimate a of gamma by omega gamma. So complexified Chern-Simons, but actually not probably not holomorphic Chern-Simons.
45:42
And the last example, complex integrable systems of hitching type. So for the discussion, how this omega of gamma appears naturally in the framework of complex integrable systems, I rather
46:09
give you a reference to our paper in archive. All right. Excuse me, so what is gamma in complexified Chern-Simons?
46:27
In complexified Chern-Simons, it is h3 of your three-dimensional real manifold. Well, it's z, or it can be zero, yeah, theoretically.
46:44
Yeah, but it's z, yeah, it's here. No, not for holomorphic. For holomorphic actually I don't know, because whether it's of exponential type, you have count the number of co-associative sub-manifolds, I have no idea
47:07
whether there are results. What is a of gamma? So given an integer gamma, what would be the or omega of gamma that you're counting? What will be the omega of gamma?
47:34
What is it? Is it number of harmonic bundles, harmonic local systems on the three dimensions? Some kind of like custom invariance, but for complex, you know.
47:48
For complexified, yeah. Yeah, but if you want to compute it, kind of, you can replace any complex representation by, I think you can, by UN. No, you cannot.
48:05
Yeah, yeah, yeah, you're right. Yeah, it's sort of a number in properly defined sense of representations in whatever gln C. Where gamma is n, gamma is z, so
48:33
gamma is z. Little gamma here is... Yeah, it's just integer, yeah. All right, so good, let me... Not only does positive integers, what about negative?
48:47
Sometimes there can be no negative integers, yeah. For example, if you consider representations of quiver, your lattice is like zn, dimension, whatever, dimensional lattice, but actually you
49:02
see only non-negative gammas, you see the colon. All right, now, how to approach resurgence? So the idea is to encode geometrically exponential stability data as construction of some complex
49:26
analytic manifold, which is in fact an analytification of some formal scheme, which exists always without any exponential bound. So let me kind of to be brief,
49:41
because I don't have too much time. So what's the idea? So suppose that you decompose a plane into the finite union of sectors, such that every sector intersects
50:06
only with the previous one and with the next one, no more. And then, if you have this support property, so we can construct collection of strict convex columns in this vector space.
50:40
I don't know, it can be rational. And if we cover the whole plane, so then we have a
50:50
column, what we call the view of column, this kind of picture. All right, so then the idea is actually it's very much similar to what appeared in
51:14
Maxim's talk on exponential integrals, when you started with some trivial bundle
51:30
over each sector Vi, and then you modified this trivial bundle with some wall crossing automorphism, so it becomes not necessarily non-trivial. And the main point was,
51:44
okay, so let me do this. So in this case, it will be a locally trivial bundle with the fiber, which is a torus. So let's consider first a small disk with
52:03
the coordinate h bar. And if we look for the intersection of these two things, we have from general wall crossing formalism, we have automorphism, which I denote by Gi. And so I have a collection automorphism,
52:31
Gi, from the completed algebra of functions on this torus. If I choose the basis,
52:47
maybe completed. Ah, okay, yeah, you're right. All right, so gamma is something which is given, which I choose, it's part of the data.
53:07
So my Lie algebra in this case is just the Lie algebra of, say algebraic vector fields on this torus, which is graded. So then the map, this infinite dimensional group,
53:28
it acts on the basis of monomials, and it acts in the following way.
53:54
And if we impose the, if we assume that the stability data is
54:02
exponential, and there is an exponential bound on the coefficients of this series, so they define analytic germ. And also, if you want, since I want to formulate the conjecture about the resurgence of some series, so I can also assume that in each cone, I have
54:29
an estimate that the real part of z of beta over h, when h is in the
54:41
sector and beta is in the cone, that the real part is positive. Okay, so then we can use this, if you'd like, that one parameter group to twist these automorphisms in the same way
55:08
as we went from various symbols to various coefficients. It's exactly the same procedure. So then we have maps for every sector, and so then we can use these automorphisms
55:45
to glue trivial bundles with the fiber key over each sector into something new. Now, because of the exponential behavior, exponential decay of these factors,
56:00
exactly the same as in Maxim's talk about exponential integrals, Taylor series do not depend on the sector, and so then we get a bundle for the punctured, say, disk, which can be continued to infinity and therefore analytically to the center.
56:25
And so then the conjecture is that this series, and it's a general conjecture, you can check it in the examples, this series is resurgent, and the same is true if you take the logarithm of this.
56:46
Okay, so I have assuming exponential bound.
57:00
Yeah, holomorphic section of the glue bundle, but this series, they are formal series, which are resurgent. Yes, that's what I mean. Yes, these cones are associated to the sector. It's not entirely anything, it's almost anything.
57:25
There are some compatibility conditions with the central charge. So the central charge is a map from the vector space, which contains these cones, to R2. So if you consider the dual one, you have an embedding of R2 to this, so it should be kind of a slice section which
57:47
contains, which sits inside, which kind of, I don't know, like imagine you have some pizza and you just push it in the center, so then the horizontal slice is the image of R2.
58:03
It's a technical condition which, as I said, I prefer not to be very technical. All right, so again, there are examples when
58:20
we expect, geometric examples when we expect that these conditions are satisfied, so we have resurgent series constructed just from the stability data. I didn't forget, I simply, I look, I look at the time and I see, yeah, yes, we, okay,
58:48
yeah, there are several, okay, so since there are just basically two minutes left, so many things are expected, that if we deform the central charges, exponential stability
59:00
property is stable. Moreover, there are some other very interesting properties which you can define just working with stability data on graded-ly algebras. For example, it was observed
59:20
both in mathematics and physics, like for example in the work of Cardova, that generating a series which appear in wall-crossing formula, they are algebraic, so they satisfy algebraic equations. So sometimes you can write down them, but we have a general proof of this algebraicity based on another property of this stability data, which
59:48
we call algebraic stability data. So the point is that we have this kind of basic block, which we call stability data.
01:00:00
and then we can impose additional properties. And from this kind of geometry, we can deduce analyticity, like in this case, or algebraicity, or maybe some deeper property
01:00:21
of the series which used to glue the corresponding global object. For example, if you want to combine it with this JLN wall crossing with this, which appear in these exponential integrals, instead of gluing
01:00:41
some complex analytic space using these automorphisms, where this is just an analytic path, similar to Varro's story. You build also a vector bundle over it. So then you have a mixture of two type of wall crossing formulas,
01:01:00
which are known in physics as 2D4D wall crossing formulas. It's a mixture of Czochrowie and what we've done. So anyway, my time is up.
01:01:21
Would it be possible to write similar wall crossing formulas for more complicated polynomial for some structures? You mean? I'm not saying it's not written by the algebra. For example, if instead of x, y, you write some polynomial, but write a quadratic polynomial. Homogeneous, but I don't know how to grade it.
01:01:43
You should be graded in very tendertral sections. It's not impossible, but I don't know how to do that. So this series is there that you say that this is the story, and I have two questions on it. First of all, is there any respect to which… Not this series, it's a subject, it's kind of misleading.
01:02:01
Oh, shit! Yeah, this… Yeah, you, it's, I mean, I was briefed because it was in complete analogy with the story about VAROS coefficients. Yeah, it's, you… In exponential terms, in one over x bar, and then z.
01:02:22
Ah, yeah, you can fix, you can put z equal to one, it doesn't matter. It's the same story with VAROS. When you write down WKB expansion in the parameter h, there is also this parameter on the curve. So you can fix it, and you still have research.
01:02:46
But are there Korma power series in h-bar? Yeah, that's a good question. They are expected to be in research? In h-bar? Yeah. Yes, yes, yes, yes. Yes, no, no, you glue these things on the bottom,
01:03:01
you can see the holomorphic section, it's an expand in h-bar, you get formal power series in h-bar. Okay. So here it's not an expansion in h-bar, but the point is that you glue… Okay, let me repeat again. So the logic is the same as with exponential integrals. You have a collection of sectors on the plane.
01:03:24
So in each of these you have some formal series, so you correct them. And then you glue some analytic vector bundle, but the expansion, this is h, this is h-plane, expansion at zero of the sections is the same,
01:03:45
it does not depend on the sector. So this expansion in h is a resurgent series. And how is this related to the standard grammar with engineering function, or Donaldson Thomas engineering function of Calabi-Yadov before?
01:04:01
If you have exponential estimate for the number of BPS states, Donaldson Thomas invariance, then you have resurgence, you can insert exactly in the way which I just described. It's quite general. I even didn't use a Poisson structure there.
01:04:21
For close Calabi-Yadov should have not exponential… No, no, no, for close… It's only for local Calabi-Yadov. Yeah, it's only for local. Like for example, I don't know, Konig bundle or spectral curve, something like that.
01:04:42
Let's thank Ian again.
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