MSc Physics
Track: Theoretical Physics
Master Thesis
Getting a grip on the Stokes
Phenomenon
Stokes Constants and their relations
by
Bruno Eijsvoogel
6048307
May 2017 60 EC 01/02/2016 - 01/02/2017Supervisor
Second Examiner
dr. M.L. Vonk
dr. D.M. Hofman
Abstract
The asymptotic behaviour of a function as it approaches a limit point can depend on the direction it is taken in. The presence of a discontinuity in this dependence is called the Stokes phenomenon and its precise form can be characterized by a set of constants known as Stokes constants.
This master’s thesis describes the calculation of Stokes constants for the Painlev´e I equation in a pedagogical manner. Starting from a treatment of some summation techniques for asymptotic series, we introduce the necessary concepts within resurgence theory in order to calculate the Stokes constants for Painlev´e I. Previously uncalculated S1(3) turns out to equal −4
3Se
(4)
1 . This result supports the conjecture n eS (n)
1 = −(n − 1)S (n−1)
1 .
The origins of this and other conjectured relations between Stokes constants remains unknown. Finding more relations could lead to a better understanding of the Stokes phenomenon.
Finally we briefly consider resurgence in the context of topological string theory, to motivate Stokes constant calculation using the Holomorphic Anomaly Equations.
Contents
1 Introduction 3
1.1 Structure of the Thesis . . . 3
1.2 Research Aims . . . 3
1.3 Physical Context . . . 4
2 Series Expansions 4 2.1 Power Series . . . 5
2.1.1 Asymptotic Expansions . . . 5
2.1.2 Divergence and non-perturbative contributions . . . 7
2.2 Transseries . . . 8
2.3 Expanding about ∞ . . . 8
3 Stokes Phenomenon & Stokes Constants 9 3.1 Borel Summation Method . . . 10
3.2 Singularities in the Borel Plane . . . 11
3.3 Alien Calculus . . . 13
3.4 Pointed Alien Derivative . . . 15
3.5 The Bridge Equation . . . 16
4 Painlev´e I 20 4.1 Asymptotic Expansion Solution . . . 20
4.2 Transseries Solution . . . 21
4.3 Large Order Calculations . . . 23
4.3.1 Stokes constant calculation strategy . . . 28
5 Topological Strings 29 5.1 Complex Structure . . . 29
5.1.1 Linear Complex Structure . . . 29
5.1.2 Complex Structure on the Torus . . . 31
5.1.3 Cycles and Periods of the Torus . . . 33
5.2 Calabi-Yau manifolds . . . 35
5.2.1 Cycles and Periods on the Calabi-Yau . . . 35
5.3 The Holomorphic Anomaly Equations . . . 36
6 Summary and Outlook 37 7 Acknowledgements 38 Appendices 39 A Approximation Methods 39 A.1 Borel-Pad´e Summation . . . 39
A.2 Richardson Transform . . . 41
B More Large Order relations 43 B.1 (3|1) Large Order . . . 43
B.2 (1|3) Large Order . . . 52
1
Introduction
1.1
Structure of the Thesis
The Stokes phenomenon can be described as the sudden change in asymptotic behaviour of functions in different regions in the complex plane. This thesis has firstly the aim to give a pedagogical treatment of the Stokes phe-nomenon. A treatment that is both intuitive but also presents some detailed calculations. We will mainly look at the Stokes phenomenon in the context of Borel summation in which the Stokes phenomenon has a fairly clear explicit formulation in simple examples. We will be tackling the two-parameter transseries solution of the Painlev´e I equation which exihibits the Stokes phenomenon in a way that is still unclear.
The second objective is to present this difficulty and the research that was done in the name of this master project, aimed at further understanding the Stokes phenomenon in this context.
Since the majority of the thesis will be this first pedagogical review, the present introduction will be a sketch of the second part: the research question. This way we can keep in mind what the review is all for.
The research done in this thesis can be viewed as a small addition to the body of work which has [11] as its foun-dation. The work done is however more closely related to the progress made in [1]. We have tested the methods used in [1] pertaining to sections regarding Painlev´e I and used them to make further calculations.
Initially the goal of this project included conducting calculations of Stokes constants in the context of the Holomor-phic Anomaly Equations. Time proved to be too short, but the literature review persists near the end of the thesis in section 5.
1.2
Research Aims
We will be looking at the Stokes phenomenon in the context of the so-called transseries solution of the Painlev´e I equation. They will be introduced in sections 2.2 and 4 respectively. Such a solution is divergent, and to make sense of it we need to perform a Borel summation, see section 3.1. To do this we will need to evaluate an integral through the so-called Borel-plane. The point is that there are singularities in the Borel-plane resulting in more than one way to do this integral. This ambiguity is just the Stokes phenomenon in a different guise.
In the simplest cases (one-parameter transseries) we can explicitly describe how the solution changes if we integrate around the other side of a singularity. But for an even slightly more complicated case (two-parameter transseries) the change is not clear at all.
What we do know is that these changes are encoded by the so-called Stokes constants. Let’s sketch what this looks like. Denote a one-parameter transseries as Φ(z, σ) where z is the expansion parameter and σ is the integration constant that parametrizes the family of solutions and denote the Borel summation operator along a path θ as Sθ. In case we have a singularity on the positive real axis and θ+ and θ− are paths passing above and below the
singularity respectively, the two different Borel summations are related by
Sθ+[Φ](z, σ) = Sθ−[Φ](z, σ + S1) (1.1)
where S1 is a Stokes constant. For a two-parameter transseries Φ(z, σ1, σ2) so far we only know that it should look
like
Sθ+[Φ](z, σ1, σ2) = Sθ−[Φ] (z, ˆσ1(σ1, σ2), ˆσ2(σ1, σ2)) , (1.2)
where the ˆσiare power series in σ1 and σ2 and whose coefficients contain the Stokes constants.
A list of these constants (Si, eSi)1have been calculated before in [1], and in this thesis we calculate one more. These
Stokes constants have various relations amongst themselves, some of which can be derived relatively straightfor-wardly. The origins of some other relations however are not yet understood. The upshot is that these non-trivial relations should lead us to understanding the form of these functions ˆσi and through this provide a better
under-standing of the Stokes phenomenon.
In this light it should be clear that it would have been exceedingly interesting to produce a similar analysis in another context than that of Painlev´e I. This was the original plan for this project. Namely to see if similar non-trivial relations existed between the Stokes constants of for example the Holomorphic Anomaly Equations. For the
1The fact that we are dealing with a two-parameter transseries gives us two classes of Stokes constants that we distinguish with the
Figure 1: The two integration contours around a singularity corresponding to different Borel summations.
research done in this area see [8] [9] [10].
Within the context of Painlev´e I however, we did work out some long calculations and produced a new Stokes constant S1(3). The result supported a conjectured relation between Stokes constants:
n eS1(n)= −(n − 1)S1(n−1). (1.3)
1.3
Physical Context
Throughout most of this thesis the objectives might often seem very far from any physical application. The Stokes phenomenon is a mathematical phenomenon but it occurs in approximations to solutions of many physical prob-lems. Airy’s equation that describes the diffraction of light see [3] has an asymptotic series solution that exihibts the Stokes phenomenon, as does a geometric model that describes crystal growth [14] and the first of the Painlev´e transcendents, the last of which will be our main focus in section 4. Painlev´e I shows up among other things in 2D quantum gravity [16] while Painlev´e II can be used to mode the electric field inside a semiconductor [15].
Aside from physical problem that are directly effected by the Stokes phenomenon, analyzing this problem is phys-ically interesting because of its link to resurgence theory: a mathematical framework that offers a bridge between perturbative and non-perturbative information.
We will however not allude to any of these physical examples. This thesis will be about mathematical methods. That this is related to physical problems is only in the back of our mind.
2
Series Expansions
In this chapter we will start by discussing power series as representations of functions. More specifically, the method of assuming solutions to differential equations are of the form of a power series.
We will encounter power series that diverge and this will lead us to a more general assumption for solutions to differential equations called a transseries. After that we will specifically look at solutions of this form that are resurgent.
Unfortunately a full introduction to the intricacies of resurgence goes beyond the scope of this thesis. So instead of starting out with proper definitions, we will introduce the concepts we will need as we go along by their most important properties and refer to more advanced texts for interested readers.
2.1
Power Series
Let’s take an ordinary differential equation (ODE) as an example to see how we can use power series to describe its solutions.
df
dx = f (x) (2.1)
This ODE is very simple and the solution is obvious: f (x) = Cex. But what if we hadn’t thought of this? We
could have made the assumption that the solution was of the form:
f (x) =X
n≥0
anxn. (2.2)
Inserting this form into the ODE gives us an expression for the thus far undetermined coefficients: X n≥0 nanxn−1= X n≥0 anxn X n≥1 nanxn−1= X n≥0 anxn X n≥0 (n + 1)an+1xn= X n≥0 anxn. (2.3)
Equating the equal powers of x we get
(n + 1)an+1= an for n ≥ 0, (2.4)
resulting in the following series
f (x) = a0 1 + x +x 2 2 + x3 6 + x4 24+ . . . . (2.5)
This gives us exactly the expected solution with a0= C. This is the power series method at its best and easiest.
The expansion describes the solution completely and the power series converges for all x both real and complex. This is due to an shrinking extremely fast. We will make the concept of fast growing and shrinking of coefficients
more precise in section 2.1.2.
2.1.1 Asymptotic Expansions
Now let’s look at an example that doesn’t go as smoothly.
(1 − x)df
dx = f (x) (2.6)
Following the same procedure of inserting a power series solution, we find the recursion for the coefficients
an+1= an. (2.7)
So our solution is the geometric series times an arbitrary factor: ˜
f (x) = a0
X
n≥0
xn. (2.8)
This sum converges to a function of closed form f (x) = a0
1−x, but only for values |x| < 1. The sum does fully
describe the solution within this region, but outside the sum diverges and we cannot assign values to it in the usual fashion. However the function f (x) is valid in the whole complex plane except at x = 1. It also remains a solution to our ODE for these values.
Expanding the domain of f (x) outside the unit disc is an example of analytic continuation.
to the singularity at the point x = 1.
So far we have considered series that, when they converge, are expressible in a closed form. This isn’t always necessarily the case. For a converging series without such an expression, one can calculate more and more coefficients in order to achieve better precision. Alternatively one can also take values of x closer and closer to 0 to make the approximation better. We obviously prefer to add terms, since this leaves us free to keep x where it is.
There are however expansions for which adding terms doesn’t improve the approximation because they don’t converge! But if for a fixed number of terms the approximation does improve as we take x closer to the origin it is called an asymptotic expansion.1 Putting this more formally:
f (x) and g(x) are asymptotic as x → x0 if
lim x→x0 f (x) g(x) = 1, (2.9) which we denote as f (x) ∼= g(x) (x → x0). (2.10)
An asymptotic expansion is then just a series that is asymptotic to the function under consideration with a specific limit. We will often omit this limit because it will be obvious from the context what it is (most often 0 or ∞). The direction in the complex plane from which we take this limit is of crucial importance! In fact for a given function f (x) one can generally find different asymptotic expansions when taking the limit to x0 from different
directions. This is called the Stokes phenomenon. We will come back to what this effect looks like in the first section of chapter 3.
For now let’s do a few numerical calculations with the geometric series to see in practice that divergent series can be asymptotic. Dropping a0 and starting at x = 12 our true value will be
f 1 2 = 1 1 −1 2 = 2, (2.11)
and truncating our expansion at n = 5 we approximate it by
˜ f5 1 2 = 5 X n=0 1 2 n = 1.96875 with an error 5= 0.03125. (2.12)
If we add another term (n = 6) the error will be smaller (6 = 0.015625), but we could also reduce the error by
taking a smaller x like
f 1 4 = 1 1 − 14 = 4 3, (2.13)
and truncating our expansion at n = 5 we approximate it by
˜ f5 1 4 = 5 X n=0 1 4 n = 1.3330078125 with an error 5= 0.000325521. (2.14)
But why should we consider this second method of minimizing the error? It alters which values we can look at of our original function, while adding extra terms doesn’t. The important part is that adding extra terms doesn’t make your approximation better when the expansion is divergent !
Divergent expansions however can still be asymptotic. Let’s take our truncated geometric series outside its radius of convergence
f (3) = 1 1 − 3 = −
1
2, (2.15)
and truncating our expansion at n = 5 we approximate it by
˜ f5(3) = 5 X n=0 3n= 364 with an error 5= 364.5. (2.16)
1This means that every convergent expansion is also asymptotic. But since this property isn’t very relevant in convergent cases one
Looks pretty bad, but it gets better when we go slightly closer to the origin
f (2) = 1
1 − 2 = −1, (2.17)
and truncating our expansion at n = 5 we approximate it by
˜ f5(2) = 5 X n=0 2n= 63 with an error 5= 64. (2.18)
At the risk of overemphasizing, we repeat that adding extra terms here is going to make our approximation worse. We have the number of terms fixed and bring x closer to 0. This example may seem a little silly since the values the truncated expansion gives seem completly unrelated to the true values.1 There are however examples of expansions
that for small enough x seem to converge, but after a number of terms diverge anyway. Truncating these expansions at the seemingly converging part (ie the term with the smallest absolute value.) gives good approximations and is called optimal truncation. In the next section we will look more closely at the role of the expansion coefficients, how their increase and decrease affects the behaviour of the truncated asymptotic expansions and in which cases the use of optimal truncation is a justified method. There are however more sophisticated ways to extract values from asymptotic expansions. One of them is called Borel summation which we will discuss in section 3.1.
2.1.2 Divergence and non-perturbative contributions
Let’s take a closer look at divergence. Whenever a series diverges we can try to put a bound on the growth the sequence of its coefficients.
In the last example take
˜ f (1) ∼= a0 ∞ X n=0 1 = a0+ a0+ a0+ . . . (2.19)
We can bound the sequence of coefficients by a large enough constant C
an< C ∀n ∈ N (2.20)
since the coefficients do not grow. In another case we have
˜ f (2x) ∼= ∞ X n=0 2nxn ⇒ an= 2n (2.21)
These coefficients do grow, so there is no constant large enough to bound all coefficients but we can bound the growth as
an< CA−n ∀n ∈ N. (2.22)
The minus sign in the exponent is just a convention because of the interpretation A will have later. We will be most interested in series whose coefficients are bounded by
an< CA−nn! (2.23)
These are called Gevrey-1 formal series. Formal because the series don’t necessarily equal a number in the usual sense right now for any x 6= 0. This is because we are allowing the coefficients grow so fast that the radius of convergence could be 0.
In [3] it is shown that in these cases the value at which you should optimally truncate is n = Nopt ≡
Ax
. This seems reasonable since the larger A, the slower the coefficients grow and the smaller x, the slower the terms grow as well, so the larger Nopt should be.
At the end of section 3 will show that the error in this procedure is proportional to a non-analytic function
error ∼ e−|A/x| (2.24)
This is very important because an expression like this exponential is not expressible as a power series around 0. If you try Taylor expand this expression about the origin each coefficient will turn out to be 0. It is a contribution lurking in the function that an asymptotic series is ”trying” to represent, but can’t. Because asymptotic series are ubiquitous in perturbation theory, we call this error a non-perturbative contribution.
This intuitively leads us to a more general form for a solution of an ODE than our original power series. We could add this non-perturbative contribution with a power series to get
∞ X n=0 anxn+ e−|A/x| ∞ X m=0 bmxm, (2.25)
in an attempt to better represent the solution. In the very simplest case we might assume that these coefficients bn
grow like the anso the that the total error our new expression would be proportional to e−2|A/x|. Now we could add
another series with this exponential as a factor in front and continue to do this. We are now building the simplest case of a transseries.
2.2
Transseries
Instead of assuming the solution to an ODE is a power series, we could use a more general expression called a transseries. Transseries are very general but we will start of with an extremely simple form just to get an idea.1
Φ(x, σ) ∼=X
n
σne−An/xxnβX
m
a(n)m xm (2.26) where σ is called the instanton counting parameter and should be thought of as the integration constant that parametrizes a family of solutions to an ODE. A and β are constants that depend on the ODE. The significance of these constants will be made more clear in section 4. It is important to note that the innermost sum is just a power series. The n = 0 term of the outer sum is really just a power series with nothing added. This sum is called the perturbative expansion or perturbative contribution. Each of the other sums is called the n-instanton contribution. This is also due to that instanton contributions to a quantum mechanical amplitude are of this form.
The reason for naming things after instantons is that again in the context of perturbation theory the exponentially suppressed contributions are attributed to instantons.
The exponential is the main ingredient here, since it is not expressible as a power series.
The importance of this object in the theory of resurgence is that a so-called resurgent transseries has the crucial but not defining property that its coefficients are asymptotically related for large values of m! Such a relation is called a large order relation and we will derive the simplest kind for the Painlev´e I equation in section 4 and two more complicated examples in appendix B.
This is a very interesting property since it implies we can calculate parts of the non-perturbative contributions from just the perturbative expansion. This is the most important property of resurgence and also the origin of its name. Contributions to one power series resurge in another.
2.3
Expanding about ∞
So far we have been looking at expansions about the origin. Taking powers of (x − a) with a = 0. We could have just as well taken a 6= 0. Perturbative expansions about 0 are studied when the true parameter value (eg coupling) will be small. If however our relevant parameter is actually very big we might want to expand about ∞. Of course we can’t plug in infinity for a, but we can take powers of 1
x instead of x see (2.27). That way if we take x → ∞ all
the terms of the expansion except the constant term vanish, just like setting x = 0 would in an expansion about 0.
∞
X
n=0
anx−n (2.27)
3
Stokes Phenomenon & Stokes Constants
Whenever we approximate a function we would like to be able to categorize the subsequent contributions to this approximation, most preferably from most important to continually less important.
For example take a power series in a small parameter x:
3x + 10x2+ x3+ . . . (3.1) or in a large parameter z:
1 + 5z−1+ 6z−2+ . . . . (3.2) We might call the first term the leading contribution and the second term the subleading contribution.
Or in the case of a transseries like in (2.26) we might, for small x, call the perturbative series leading and the other series proportional to a power of
e−A/x (3.3)
subleading. It does however matter how we take this limit of x to 0. Just looking at the exponentials below we already notice lim x↓0e −A/x= 0 lim x↑0e −A/x→ ∞ (3.4)
that we can only call the first exponential a subleading contribution. And if we approach the origin from the positive and negative imaginary axis we also get very different results. In fact those limits don’t exist since they oscillate at an ever increasing rate.
lim x↑0e −A/(ix) lim x↓0e −A/(ix) (3.5)
In the complex plane we have a whole 2π of different directions in which to approach 0.
If we have a large parameter z we look at e−Az which has different behaviour depending in which direction we go to infinity.
The formal way as presented in [4] to compare contributions in asymptotic analysis is as follows: f (x) is negligible compared to g(x) as x → x0 if lim x→x0 f (x) g(x) = 0, (3.6) and is denoted f (x) << g(x) (x → x0).
To make this slightly more tangible we briefly look at the asymptotic expansions for large values of the Airy function Ai(x). But for a more thorough treatment the author strongly recommends examples concerning this function in [3]. The function Ai(x) has a asymptotic expansion in 1x
Ai(x) ∼= e −2x3/2 3 x1/4√π 1 2− 5 96x3/2+ 385 9216x3− 85085 1327104x9/2 + . . . . (3.7)
We have here an asymptotic relation between a function and a series, but we haven’t been specific about the limit. Large z implies the limit should be x → ∞, but does the direction matter? In fact the expansion does agree fairly well both along the positive and negative real axis for an arbitrary truncation. However this is not what it means for to functions to be asymptotic. Taking the limits we have that
lim
x→+∞
f(N )(x)
while the limit
lim
x→−∞
f(N )(x)
Ai(x) (3.9)
does not exist, where we have denoted f(N )(x) as the truncated sum. The true Airy function oscillates towards −∞ while the asymptotic expansion does not. We can find another expansion that is asymptotic to Ai(x) at ∞ along the negative real axis.1
Ai(−x) ∼=√ 1 πx1/4 sin π 4 + 2x3/2 3 1 − 385 4608x3 + . . . + cos π 4 + 2x3/2 3 − 5 48x + 85085 663552x4 + . . . (3.10) where we
We then say that the Airy function has different asymptotic behaviour in different sectors of the complex plane. By which we mean that the limiting behaviour towards complex infinity of the function is captured by different power series for different directions. And the most striking part of it is that the change is discontinuous in that direction! In a more in depth analysis we can find specific angles at which the asymptotic behaviour suddenly changes. The rays at these angles are called Stokes line and anti Stokes lines. A Stokes line is a boundary between two directions where a a contribution that is negligible on one side that is no longer negligible on the other, while an anti-Stokes line is a direction where the asymptotic behaviour of the function is oscillatory.
3.1
Borel Summation Method
Whenever the coefficients of a power series an grow as n! the radius of convergence is 0. However there is a method
to still extract values from this series called Borel summation. The essence of this method is outlined by the following steps.
1. Transform the divergent sum to a different sum that has a finite radius of convergence. 2. Analytically continue the converged sum outside this radius.
3. Transform the analytic continuation back using the inverse transformation.
The transformation in step 1 is called the Borel Transform B. For a formal series about infinity without constant term ˜ φ(z) ∼= ∞ X n=0 anz−n−1 B[ ˜φ](s) ≡ ∞ X n=0 an n!s n (3.11)
A shorthand for B[ ˜φ] is ˆφ. Then the inverse transformation is the generalized Laplace transform Lθ, which we first only apply on a single term of the Borel transform series
Lθhan n!s ni(z) ∼ = Z eiθ∞ 0 an n!s ne−szds (3.12) =an n!z −n−1Z eiθ∞ 0 sne−sds (3.13) = anz−n−1, (3.14)
because the inverse transformation might not be well defined for the whole Borel transform. We have to analytically continue the Borel transform outside its radius of convergence in a certain direction in the s-plane, also referred to
1Notice that we don’t mean that positive and negative infinity are different limit points. In the complex plane there is one unique
as the Borel plane. We have a certain freedom to choose θ for our inverse transformation since we get the same value for the following integral
n! = Z eiθ∞
0
sne−sds for |θ| < π
2, (3.15)
but to use Lθ on the Borel transform we also need to avoid its singularities! A formal series eφ(z) is called
fine-summable in the direction θ if it satisfies the following requirements:
• Its Borel transform can be analytically continued to complex infinity in the direction θ • The series of the inverse transforms of the Borel transform terms
∞ X n=0 Lθhan n!s ni(z), (3.16) converges.
The final result is then valid for z | Re(zeiθ) > c , for some constant c. Note that this is a half plane centered
around the ray in the −θ direction. For a formal approach to the question when this is allowed, for which θ’s it is allowed and in which part of the z-plane the result holds see [18]. It is a mathematically rigorous treatment that is still very legible for a mathematically inclined physics graduate.
Now that everything is defined we can walk through the steps of the Borel summation method. 0. We first have a Gevrey-1 formal series ˜φ(z) ∼=P∞
n=0anz−n−1
1. We take the Borel Transform ˆφ(s) = B[ ˜φ](s) ∼=P∞
n=0 an
n!s n
So that we now have a finite radius of convergence.
2. We analytically continue ˆφ(s) outside its radius of convergence out to eiθ∞ along a path that avoids any
singularities, see figure 2. We discuss possible singularities in the next section. 3. We preform the Laplace transformation along this path.
The whole procedure can be contained inside the Borel summation operator Sθ. It takes an asymptotic series and
maps it to a function. This isn’t some sort of magic trick that randomly asigns values to asymptotic series. If for example the series under consideration was the formal solution to a differential equation, then the Borel summation will also be a solution.
We will sometimes use similar notation like SC to signify analytically continuing along a specified path C that
connects the origin to infinity. And in case there are singularities in the direction θ we have the so-called lateral Borel Summations Sθ+ and Sθ− to indicate a path in the θ direction that avoids them on the left or on the right
respectively, see figure 1 page 4.
We have so far excluded formal sums with a constant term. This causes some difficulty at the Borel transform step, but not in the entire procedure of Borel summation. So we can just extend the definition1.
SC[a0+ ˜φ] ≡ a0+ SC[ ˜φ] (3.17)
3.2
Singularities in the Borel Plane
If we now have a look at an asymptotic series ˜φ(x) of the Gevrey-1 kind that has coefficients that satisfy (2.23), we will now take its Borel transform.
B[ ˜φ] = ∞ X n=0 an n!s n with a n< CA−nn! (3.18)
1We can then also extend the definition of the Borel transform by B[a
Figure 2: We can analytically continue the Borel transform outside the radius of convergence as long as we aviod singularities.
The new coefficient an
n! grows less fast and gives us a radius of convergence of at least A. If the radius is not infinite
this must mean that we have at least one singularity in the Borel plane a distance A away and that it is the closest singularity to the origin.
If for example an = A−nn! ˜ φ(z) = ∞ X n=0 A−nn!z−n−1 (3.19) B[ ˜φ] = ∞ X n=0 s A n (3.20)
Our Borel transform is just the geometric series, which we saw above has a finite radius of convergence
B[ ˜φ](s) = 1
1 − As for |s| < |A|. (3.21) We can analytically continue B[ ˜φ](s) outside the unit disc. The function has a singularity at s = A though. For now let’s say A ∈ R>0. Then if we want to transform back by integrating over the positive real axis we will have to
choose a direction in which to avoid this point. If we take the difference of these lateral Borel summations we get a clockwise closed contour integral that encloses s = A.
S0+[ ˜φ] − S0−[ ˜φ] = I e−sz 1 −Asds = − I e−Asz s − 1A ds = 2πiAe −Az (3.22)
So Borel summation ambiguity tells us that the same asymptotic expansion can be related to different functions and the Stokes phenomenon tells us that the the same functions can be related to different asymptotic expansions. It is also notable that this ambiguity in Borel summations is of the same form of the minimum error we found hand-wavingly in (2.24). Another reminder that divergence of asymptotic series is directly linked to non-perturbative contributions.
see the Stokes phenomenon there. In section 4 we will consider the specific case Painlev´e I but we will encouter an obstacle. Not only is the Borel transform not expressible in closed form, the expansion coefficients aren’t even expressible in closed form. We only have a recursion relation. This is why we wanted the Stokes constants in the first place: To get a grip on the Stokes phenomenon.
The analysis we have described so far is considered to be classical asymptotics [3]. The theory of resurgence goes beyond this. To do the central calculation of this thesis in section 4.3 we need some of its tools.
The definition of a resurgent function is quite involved so we refer the interested readers to [2], [3], [1] for introduc-tions within a physical perspective and to [18] for a more mathematically formal introduction which is still readable for the mathematically inclined physicist. Jean ´Ecalle, the inventor of resurgence theory, is the primary source for the complete theory of resurgence, so the truly invested mathematician is referred to [13].
Instead this chapter will be devoted to displaying a couple of necessary tools for the calculations of section 4.3 and to give some indication of what they represent.
3.3
Alien Calculus
The objective of alien calculus is to quantify and compare singular behaviour of functions. There are a few different types of Alien Operators1but we will only be discussing the alien derivative ∆ω. The index ω refers to a point in
the Borel plane which will singular or else the operator gives zero. In [20] this operator is implicitly defined as
Sθ+= Sθ−exp X ω∈Γθ e−ωz∆ω ! (3.23)
where Γθis the set of all singularities on the ray in the θ direction.
Notice the left hand side is just like the calculation of the non-perturbative contribution in (3.22). Hence apparently the Stokes phenomenon can be quantified by these alien derivatives. So we would very much like to calculate them.
Remark
Before looking at an example let’s display two key objects in (3.23) that we will encounter later. First of all the outermost exponential Sθ= exp X ω∈Γθ e−ωz∆ω ! (3.24)
is called the Stokes Automorphism. As pointed out in [7], this can be intuitively compared to generating finite translations of a function from inifitesimal ones.
a d dxf (x) ∼ e −ωz∆ ωφ(z)˜ (3.25) f (x + a) = exp(a d dx)f (x) ∼ exp X ω e−ωz∆ω ! ˜ φ(z) (3.26)
So where the Stokes Automorphism encodes the change in asymptotic behaviour across a direction θ in the complex plane, the alien derivative is like an infinitesimal change in asymptotic behaviour due to a single singular point in that direction.
As such we will also be interested in the analogy to f (x) − f (x + a) = 1 − exp(adxd ) f (x): The Discontinuity
Discθ= 1 − Sθ (3.27)
The use of the discontinuity will be our means of introducing resurgence into the calculations in section 4. We will then need the bridge equation from section 3.5 to compute the alien derivatives.
Example
Let’s start out by looking at an easy example. Starting by immediately restricting ω to be a so-called simple singularity.
A function ˆφ(s) has a simple singularity at s = ω if ”close to ω” it ”looks like”1 a
2πi(s − ω) + ˆψ(s − ω)
log(s − ω)
2πi + ˆχ(s − ω), (3.28) for some functions ˆψ(s) and ˆχ(s) that are holomorphic near the origin and a constant a. In other words our singularities are only allowed to be poles and logarithmic singularities. No log2(s − ω) or crazier things. We will make our example even easier by requiring ω be the only singularity in the arg(ω) direction.
This last assumption makes the exponential on the r.h.s. of (3.23) much more manageable
exp e−ωz∆ω = 1 + ∞ X n=1 e−nωz n! (∆ω) n . (3.29)
Remember that ∆ω commutes with any function that doesn’t have a singularity at ω. So plugging this into (3.23)
we can now rewrite this as our familiar difference in lateral Borel summations.
Sθ+− Sθ− = Sθ− ∞ X n=1 e−nωz n! (∆ω) n (3.30)
Now lets take an asymptotic series ˜φ(z) whose Borel transform looks like (3.28) and let the l.h.s. of (3.30) work on it, Sθ+[ ˜φ](z) − Sθ−[ ˜φ](z) = I Cω ˆ φ(s)e−szds = I Cω a 2πi(s − ω) + ˆψ(s − ω) log(s − ω) 2πi + ˆχ(s − ω) e−szds = ae−ωz+ I Cω ˆ ψ(s − ω)log(s − ω) 2πi e −szds (3.31)
where Cω is the path that goes from 0 to infinity along θ+ and back along θ− so that we effectively have a closed
path around the singularity ω. To simplify the last integral it is useful to separate it into two integrals from 0 to eiθ∞. And while we’re at it we can change shift2 the integration variable s 7→ s + ω
= ae−ωz+ e−ωz Z ei(θ+)∞ 0 ˆ ψ(s)log(s) 2πi e −szds − e−ωzZ ei(θ−)∞ 0 ˆ ψ(s)log(s) 2πi e −szds (3.32)
As goes to 0 the only difference between these two integrals is their logarithms have are on different branches. Their difference being 2πi. So we can combine them as
ae−ωz+ e−ωz Z eiθ∞
0
ˆ
ψ(s)e−szds = ae−ωz+ e−ωzSθ[ ˜ψ](z) (3.33)
Now compare with r.h.s. of (3.29) which seems to imply that
Sθ[∆ωφ](z) = a + S˜ θ[ ˜ψ](z) (3.34)
1The mathematically precise way to formulate this is that we can find a small enough open neighbourhood around ω where ˆφ
coincides with the above expression (3.28).
2Note we don’t have to change the integration interval since these contours are deformable anyway as long as we don’t cross the only
and that all higher powers of ∆ω vanish. This last fact should come as no surprise since we have taken ˆψ to be
holomorphic at ω. If we now asymptotically expand the Borel summations again we have a more explicit expression
∆ωφ(z) = a + ˜˜ ψ(z) (3.35)
This example closely followes [20] where more examples are treated.
3.4
Pointed Alien Derivative
One very important property of the alien derivative is how it commutes with the normal z-derivative. We will derive its commutation relation for this specific example. It holds true in general and one can find a proper derivation in [18]. After that we will introduce the pointed alien derivative that does commute with the z-derivative.
We want to calculate the commutator
d dz, ∆ω = d dz∆ω− ∆ω d dz (3.36)
The first term is easy since we can just take the z-derivative of (3.35).
d dz∆ω
˜ φ =d ˜ψ
dz (3.37)
But for the second term we need show how Bhd ˜dzφirelates to Bh ˜φi. Let’s take ˜ φ(z) = ∞ X n=0 anz−n−1 with Bh ˜φ i = ∞ X n=0 an n!s n (3.38) d ˜φ dz = − ∞ X n=0 (n + 1)anz−n−2= − ∞ X n=1 nan−1z−n−1 (3.39) B " d ˜φ dz # = − ∞ X n=1 nan−1 n! s n = − ∞ X n=0 an n!s n+1= −s Bh ˜φi (3.40)
So taking the derivative in the z-plane corresponds to multiplication by −s in the s-plane. So since we are considering
d ˜φ
dz its Borel Transform near ω will behave as (3.28) times (−s)
− as
2π(s − ω)+ (−s ˆψ(s − ω))
log(s − ω)
2πi − s ˆχ(s − ω) (3.41) So we again operate with the l.h.s. of (3.30) to get
Sθ+ " d ˜φ dz # − Sθ− " d ˜φ dz # = I Cω − ase −sz 2πi(s − ω)ds + I Cω −s ˆψ(s − ω)e−szlog(s − ω) 2πi ds = −aωe−ωz+ e−ωz I Cω −(s + ω) ˆψ(s)e−szlog(s) 2πi ds = −aωe−ωz+ e−ωz Z eiθ∞ 0 −(s + ω) ˆψ(s)e−szds = −aωe−ωz− ωSθh ˜ψ i e−ωz+ e−ωz Z eiθ∞ 0 −s ˆψ(s)e−szds = −ωa + Sθh ˜ψ i e−ωz+ e−ωz Z eiθ∞ 0 B " d ˜ψ dz # e−szds ! = −ωSθ h a + ˜ψie−ωz+ Sθ " d ˜ψ dz # e−ωz = −ωSθ h ∆ωφ˜ i e−ωz+ Sθ d dz∆ω ˜ φ e−ωz (3.42)
Where in the last line we used (3.35) and (3.37) to rewrite everything in terms of alien derivatives. Now comparing to the r.h.s. of (3.30) we must have that all derivatives higher than first order must vanish. The only term that can contribute is Sθ " ∆ω d ˜φ dz # e−ωz. (3.43)
So now equating the above and dropping the Borel summation and exponential we finally get our commutator.
∆ω d ˜φ dz = −ω∆ω ˜ φ + d dz∆ω ˜ φ d dz, ∆ω ˜ φ = ω∆ωφ˜ (3.44)
With this commutation in mind we can define the pointed alien derivative
•
∆ω≡ e−ωz∆ω (3.45)
which exactly commutes with d dz since d dz, • ∆ω ˜ φ = −ω • ∆ωφ + e˜ −ωz d dz, ∆ω ˜ φ = 0 (3.46)
The reason this commuting operator is important is that we can use it to derive (a specific case of) an important equation: The bridge equation. This allows us to express alien calculus in ordinary calculus. We will take some time to show its derivation in the next subsection so we can obtain some simple expressions for Alien derivatives in the case of Painlev´e I in section 3.5.
3.5
The Bridge Equation
Now we bring the transseries back to the discussion. We will start with the one-parameter transseries in (2.26). But our main goal is the two-parameter transseries because the general solution to Painlev´e I will be of this form.
We assume the one-parameter transseries u(z, σ) is the solution to some (possibly non-linear and inhomogeneous) first order ODE1 in z, but that has coefficients that have vanishing∆•
l derivative.
F (u0(z, σ), u(z, σ), z) = 0 (3.47) If we now operate on this equation with the
•
∆l, thanks to its commutation with dzd we will get an linear,
homoge-neous, first order ODE in
•
∆lu(z, σ). The coefficients might depend on u(z, σ) and that’s fine. For example if we
have u(z, σ)3+du dz − cos(z) = 0 (3.48) then • ∆l u(z, σ)3+du dz − cos(z) = 0 3u(z, σ)2 • ∆lu(z, σ) + d dz • ∆ u(z, σ) = 0 (3.49)
where we have used that the alien derivative satisfies the Leibniz rule and that the cosine term vanishes because its Borel transform doesn’t have singularities for finite l. Now we’re not going to solve this equation for
•
∆lu(z, σ).
Instead we notice that this reasoning holds for any other derivatives as well, as long as it commutes with dzd. So if we do this with ∂σ∂ we would get exactly the same ODE for ∂σ∂u.
3u(z, σ)2 ∂u ∂σ + d dz ∂u ∂σ = 0 (3.50)
Since a linear homogeneous first order ODE has a one dimensional solution space, this must mean that they are proportional
•
∆lu(z, σ) = Sl(σ)
∂u
∂σ (3.51)
but of course this proportionality is only independent of z.
The transseries we have used thusfar is called one-parameter transseries because of the single parameter σ. It also only has one Instanton action A. We now introduce a specific kind of two-parameter transseries
u(z, σ1, σ2) = ∞ X n=0 ∞ X m=0 (σ1)n(σ2)me−(n−m)AzΦ(n|m)(z) (3.52)
There is a lot to unpack here. We discuss everything one at a time.
First of all and least surprising, we now have two parameters σ1 and σ2. Second our exponential looks different.
It is basically the multiplication of e−nAz, which we had in our old transseries, with a new emAz which is again a non-perturbative contribution with a new instanton action −A.
Remark
One may be concerned about this new instanton action having the opposite sign. The one-parameter transseries converged for z → ∞ along the positive real axis. Now we don’t have this anymore, but this is not a problem. This is a formal expansion. We are not interested in convergence right now and if we are, we will use methods like Borel summation to produce finite numbers. But for now we only interested in the expansion coefficients themselves. Then finally we have Φ(n|m)(z) which in some cases (eg n = m) is just an asymptotic series as in our old transseries.
1Strictly speaking this is a PDE since u also depends on σ, eventhough there are no derivatives with respect to σ. Making this
distinction is a bit of a nuisance because it implies that our equation is either ODE or PDE depending on our ansatz. An ODE if we solve for an asymptotic series and a PDE if we solve for a transseries. We will thus call it an ODE and even leave the derivatives with respect to z as non-partial derivatives.
For n 6= m it can contain powers of logarithms which we will discuss in the next section, but that we don’t have to worry about just yet. Just like we called the subsequent asymptotic series of the one-parameter transseries n-instanton series, we will call the Φ(n|m) the generalized (n|m)-instanton series.
For a two-parameter transseries satifying a second order ODE in z, it can be shown that its pointed derivative will satisfy linear, homogeneous, second order ODE. Again its ordinary derivatives with respect to the two parameters σ1,2will satisfy that same ODE. We will show this explicitly for Painlev´e I in section 4. Since we have three distinct
functions satisfying the same second order, linear ODE, we can therefore write the pointed alien derivative as a linear combination.1 • ∆lAu(z, σ1, σ2) = Sl(σ1, σ2) ∂ ∂σ1 u(z, σ1, σ2) + eSl(σ1, σ2) ∂ ∂σ2 u(z, σ1, σ2) (3.53)
This is then the bridge equation for the two-parameter transseries. We now define the degree of an expression as
deg(σn1σ2melAz) = n − m + l (3.54) so that deg (u(z, σ1, σ2)) = 0 deg • ∆lAu(z, σ1, σ2) = −l deg ∂ ∂σ1 u(z, σ1, σ2) = −1 deg ∂ ∂σ2 u(z, σ1, σ2) = +1 (3.55)
If we want the degree of both sides of (3.53) to match we need deg (Sl(σ1, σ2)) = 1 − l
degSel(σ1, σ2)
= −1 − l (3.56)
So if we assume that Sland eSl are expressible as power series
Sl(σ1, σ2) = ∞ X k1=0 ∞ X k2=0 S(k1,k2) l σ k1 1 σ k2 2 e Sl(σ1, σ2) = ∞ X k1=0 ∞ X k2=0 e S(k1,k2) l σ k1 1 σ k2 2 (3.57)
then with the requirement on their degree, they must be
Sl(σ1, σ2) = ∞ X k=max(0,−1+l) Sl(k−l+1,k)σk−l+11 σ2k e Sl(σ1, σ2) = ∞ X k=max(0,−l−1) e Sl(k,k+l+1)σk1σ2k+l+1 (3.58)
where the constants no longer depend on two indices so we can redefine Sl(k−l+1,k)≡ Sl(k−l+1) Se
(k,k+l+1) l ≡ eS
(k+l+1)
l (3.59)
1We use lA in the alien derivative this time because the singularities will be at multiples of A, but this doesn’t make our equation
less general. We also take l 6= 0 because that alien derivative should be 0 anyway. The Borel transform of u should have a finite radius of convergence (Gevrey-1 see 3.2).
to get what are called Stokes constants1.
Now we can start collecting the expressions that we will want to fill in the bridge equation (3.53).
• ∆lAu = X n,m σn1σm2 e−(n−m+l)Az∆lAΦ(n|m)(z) ∂u ∂σ1 =X n,m nσn−11 σ2me−(n−m)AzΦ(n|m)(z) ∂u ∂σ2 =X n,m mσn1σm−12 e−(n−m)AzΦ(n|m)(z) (3.60) Sl(σ1, σ2) ∂u ∂σ1 = X n,m,k nσ1n+k−lσ2m+ke−(n−m)AzSl(k−l+1)Φ(n|m)(z) (3.61) e Sl(σ1, σ2) ∂u ∂σ2 = X n,m,k mσ1n+kσ2m+k+le−(n−m)AzSe (k+l+1) l Φ(n|m)(z) (3.62)
So far our indices n and m run from 0 to ∞. But we want to compare the bottom two lines to the one of the alien derivative. So we will shift the indices of the former.
For (3.61): ˆ n = n + k − l ˆ m = m + k (3.63) For (3.62): ˆ n = n + k ˆ m = m + k + l (3.64) Making Sl(σ1, σ2) ∂u ∂σ1 = X ˆ n, ˆm,k (ˆn − k + l) σ1ˆnσ2mˆe−(ˆn− ˆm+l)AzSl(k−l+1)Φ(ˆn−k+l| ˆm−k)(z) = ∞ X ˆ n, ˆm=0 σ1nˆσm2ˆe−(ˆn− ˆm+l)Az min(ˆn+l, ˆm) X k=max(0,−1+l) (ˆn − k + l) Sl(k−l+1)Φ(ˆn−k+l| ˆm−k)(z) (3.65)
We have manipulated index values a bit. First off notice that since from the old indices satisfying n ≥ 0 and m ≥ 0 we can infer that k ≤ ˆn + l and k ≤ ˆm. This gives us the upper bound for k. Further our new indices ˆn and ˆm have starting values that can be greater than 0 depending on k and l, so this should mean we can’t pull those sums outside the k-sum. But we can set them to start at 0 anyway since the extra terms we get are the only ones with Φ(x|y) with x, y ≤ 0. We thus require those Φ to be 0.
Similarly for the other sum.
e Sl(σ1, σ2) ∂u ∂σ2 = X ˆ n, ˆm,k ( ˆm − k − l) σ1ˆnσ2mˆe−(ˆn− ˆm+l)AzSe (k+l+1) l Φ(ˆn−k| ˆm−k−l)(z) = ∞ X ˆ n, ˆm=0 σ1nˆσm2ˆe−(ˆn− ˆm+l)Az min(ˆn, ˆm−l) X k=max(0,−l−1) ( ˆm − k − l) eSl(k+l+1)Φ(ˆn−k| ˆm−k−l)(z) (3.66)
These have an identical form as the expression for the pointed alien derivative, so this finally gives us an easily calculable expression for alien derivatives of the generalized (n|m)-instanton series.
∆lAΦ(n|m)= min(n+l,m) X k=max(0,−1+l) (n − k + l) S(k−l+1)l Φ(n−k+l|m−k)(z) + min(n,m−l) X k=max(0,−l−1) (m − k − l) eS(k+l+1)l Φ(n−k|m−k−l)(z) (3.67)
4
Painlev´
e I
Now we move on to introduce the equation which will be the main focus of this thesis. Painlev´e I:
u(z)2−1 6
d2u
dz2 = z (4.1)
In this section we will briefly show how one goes about calculating the asymptotic series solution as well as the two-parameter transseries solution. The focus however will lie on the so-called large order relations. These are approximate expressions for the transseries coefficients that contain the Stokes constants.
The large order relations are of interest to us because they are our means to calculate Stokes constants.
4.1
Asymptotic Expansion Solution
This is an equation whose asymptotic series solution exhibits Gevrey-1 behaviour. We can calculate it with the following powerseries ansatz
u(0)(z) = zα1X
g≥0
u(0)g z−α2g with α
i∈ R, and α2> 0 (4.2)
where we have expanded about infinity instead of 0 this time. This is because in the context of 2D quantum gravity a small coupling corresponds to a large z [1].
First off we might want to determine α1. Let’s look at the forms u2 and u00 take
u0(0)= zα1X g≥0 (−α2g + α1) u(0)g z −α2g−1 u00(0)= zα1X g≥0 (−α2g + α1) (−α2g − 1 + α1) u(0)g z −α2g−2 u(0) 2 = z2α1X g≥0 c(0)g z−α2g (4.3) where c(0)g =Pgm=0u (0) mu (0) g−m.
First off notice that to equal the z on the r.h.s. of (4.1) we need the u2to provide it for us. Since u00only has more
negative powers of z1. Therefore we should set α
1 = 12 and require the succesive powers of z from u2 and u00 to
cancel.
If we want these contributions to cancel we need their powers in z to be comparable. This determines α2by equating
the powers in z
2α1− α2g = α1− α2g0− 2 (4.4)
1Note that we take α
where we need g = g0+ 1. This leads to α2= 52.
Now that our parameters are fixed we can start calculating the series coefficients. We can use our expressions from (4.3) to plug in (4.1) and compare equal powers. Firstly we will have
u(0)
2
= 1 (4.5)
which we normalize1 to u(0)
0 = 1. But subsequently the u00 term enters and the loose z on the r.h.s. leaves, giving
us c(0)g =1 6(−α2(g − 1) + α1) (−α2(g − 1) − 1 + α1) u (0) g−1 for g ≥ 1 (4.6) g X m=0 u(0)mu(0)g−m =1 6 −5 2(g − 1) + 1 2 −5 2(g − 1) − 1 + 1 2 u(0)g−1 (4.7) 2u(0)0 u(0)g + g−1 X m=1 u(0)mu(0)g−m =(5g − 6)(5g − 4) 24 u (0) g−1. (4.8)
Rewritting a few things and replacing u(0)0 = 1 we get our recursion u(0)g = 25(n − 1)2− 1 48 u (0) g−1− 1 2 g−1 X m=1 u(0)mu (0) g−m (4.9)
When we calculate these coefficients we see that they diverge factorially. Such that for large g
u(0)g ∼ CA−gg! (4.10)
In other words the expansion is asymptotic and of Gevrey-1 type. We know from section 2.1.2 that this constant A is related to the error of optimal truncation. We have approximated it by taking the ratio of coefficients
lim
g→∞(g + 1)
u(0)g
u(0)g+1
= A (4.11)
This convergence is relatively slow. A useful trick to speed up this procedure is the Richardson transform for which we have provided a short manual in appendix A.2.
We can find the exact value of A in a similar way to finding the parameters αiand β of the asymptotic series. By
looking at the transseries ansatz for Painlev´e I, one finds that we have
A = 8 √ 3 5 (4.12) see [1] [3].
4.2
Transseries Solution
The Painlev´e I equation also admits a one-parameter and two-parameter transseries solution. For the procedures needed to calculate series coefficients for these series we refer to [11] [1]. Here we just present and discuss the form of the two-parameter solution.
We first change the notation a bit according to the conventions of [1] x = z−5/4 ˆ u(x) = u(z)√ z z=x−4/5 , (4.13)
1We can make this choose it to be 1 instead of -1 because of the ambiguity in the sign of the leading√z term. It could have just as
and to keep the notation less messy, we won’t use the hat on the ˆu. So whenever we use the two-parameter transseries it will be the rescaled one.
Now our ansatz for the solution is
u(x, σ1, σ2) = ∞ X n=0 ∞ X m=0 (σ1)n(σ2)me−(n−m)A/xΦ(n|m)(x) (4.14) where Φ(n|m)(x) = knm X k=0 logk(x) 2k Φ [k] (n|m)(x) with Φ [k] (n|m)(x) = ∞ X g=0 u(n|m)[k] 2(g+β[k]nm) xg+βnm[k] (4.15)
There is quite a lot of notation to be explained here. Let’s start with Φ[k](n|m)(x). This is a power series in x with coefficients u. The coefficients obviously have labels n, m and k for the series they belong to and g as the basic summation index.1 β[k]
nm plays the same role as α1of determining the starting power in asymptotic series in (4.2).
We also shift the index by the amount βnm[k] so that index and the power of x are the same (besides the factor of 2).
The expression for βnm[k] is given below in (4.19).
Then Φ(n|m)(x) which is a sum of power series with powers2of 12log(x) that goes up until
knm≡ min(n, m) − mδnm. (4.16)
These log-terms are necessary due to a well known effect in power series solutions called Resonance. Below in (4.17) we will show a recursion for all the transseries coefficients. If you try to derive it for a transseries ansatz without log terms this leads to inconsistencies. In (4.17) for certain values of n and m the leading coefficient u(n|m)g will cancel
for all g. Sometimes this is fine and we can isolate u(n|m)g−2 , but this can also lead to equations like 0 = 0. If we had not taken the log terms this would have also led to equations of the form const= 0, which is why we need them. This is will not cause any other trouble though, because the log term coefficients can be easily calculated from the others see (4.20).
Now we can calculate as many coefficients as we want of each (n|m)-instanton series, by inserting our transseries ansatz back into the differential equation. This gives us an implicit recursion relation for the coefficients. This is a long and tedious calculation so we just give the result presented in [1].
δg0δ0nδ0mδ0k= g X ˆ g=0 n X ˆ n=0 m X ˆ m=0 k X ˆ k=0 u(ˆˆgn| ˆm)[ˆk]u(n−ˆg−ˆgn|m− ˆm)[k−ˆk] −25 96(n − m) 2A2u(n|m)[k] g + 25 96(m − n)(k + 1)Au (n|m)[k+1] g−2 +25 96(m − n)(g − 3)Au (n|m)[k] g−2 − 25 384(k + 2)(k + 1)u (n|m)[k+2] g−4 − 25 192(k + 1)(g − 4)u (n|m)[k+1] g−4 − 1 384(5g − 16)(5g − 24)u (n|m)[k] g−4 (4.17)
When we want to compute the coefficients for the (n|m)-instanton series for specific n and m, we need to rewrite this equation to get something of the form
u(n|m)[0]g = . . . (4.18) where the r.h.s. should only contain terms of lower generalized instanton coefficients.
The recursion is correct for any combination of four integers n, m, k, g. We just have to set the coefficients with negative g and k > knm to zero. When using this equation to calculate coefficients one will notice that for higher
n and m that first non-zero coefficient has a ever increasing index. This starting point is encoded by
βnm[k] = n + m 2 − knm+ k 2 I (4.19)
1The factor of 2 is again just a convention that comes from another choice of variables x = w2 that is also used in the literature. 2The power of 2 in x = w2 shows up here again. This time as a factor of 1
where the brackets [α]I stand for the integer part of α.
From (4.17) we can derive some important relations between coefficients such as
u(n|m)[k]2(g+β) = (−1)g+βnm[k]−(n+m)/2u(m|n)[k] 2(g+β) u(n|m)[k]2(g+β) = 1 k! 4(m − n) √ 3 u(n−k|m−k)[0]2(g+β) . (4.20)
The second equation holds similarly for the whole corresponding series
Φ[k](n|m)= 1 k! 4(m − n) √ 3 Φ[0](n−k|m−k), (4.21) since the factor in front does not depend on the summation index g.
4.3
Large Order Calculations
Now that we have the full two-parameter transseries solution at our disposal (or at least arbitrarily many coeffi-cients), we will derive a few relations between these coefficients for large values of g, called large order relations. Our tools will be alien calculus and the bridge equation that we discussed in section 3.5.
The important part is that these relations will include the Stokes constants we are after and are therfore a means to calculate them. In the present section we will show how this is done for u(0|0)[0], but u(3|1)[0] and u(3|3)[0] are
presented in appendix B. From these examples it should become clear how to produce similar calculations for any (n|m)-instanton coefficients. In Table 1 below we show the large order relations that we have calculated for this thesis project.
u(0|0)[0] u(2|0)[0] u(0|2)[0] u(1|1)[0] u(3|1)[0] u(3|3)[0] u(5|3)[0] Table 1: Calculated large order Relations
The initial trick that will pave our path back to resurgence theory is one that works for any complex function with a branch-cut along the θ direction, but that is analytic everwhere else. We rewrite the function with a slightly altered version of Cauchy’s Integral Theorem. See figure 3 for a graph of the integration contour.
2πif (z) = Z ei(θ+)∞ 0 dwf (w) w − z + Z θ−+2π θ+
dφ iReiφ f (Re
iφ) Reiφ− z + Z 0 ei(θ−)∞ dw f (w) w − z = Z ei(θ+)∞ 0 dwf (w) w − z − Z ei(θ−)∞ 0 dw f (w) w − z + Z θ−+2π θ+
dφ iReiφ f (Re
iφ) Reiφ− z R→∞ →0 = Z eiθ∞ 0 dwDiscθf (w) w − z (4.22)
In the last line we took the arc part of the contour integral to vanish for R → ∞. We can generalize this for a function with several branch-cuts such that
2πif (z) =X θ Z eiθ∞ 0 dwDiscθf (w) w − z (4.23)
The branch-cut in our case will be a Stokes line, the discontinuity is the one in (3.27) and the functions we will study are none other than the Φ(n|m)’s of the Painlev´e I two-parameter transseries solution. Fortunately we know
Figure 3: The blue line represents the integration contour in the w-plane with an aribitrary point z within the contour of integration. The solid black line is f ’s branch-cut.
Remark
We cannot use the expression for the Discontinuity in (3.27) as is. Its form depends on the expansion variable. We are now expanding in x about 0, but in section 3.3 we were effectively expanding in z = 1x about ∞1.
Let us briefly sketch some of the consequences of using different expansion variables. All we really need to look at is how the complex plane maps onto itself by these variable transformations.
z = 1
x (4.24)
If we look at this map in polar coordinates it should be clear that this inverts every point in the plane about the unit circle and then mirrors it in the real axis. The mirroring is the important part for us2, because it flips orientation.
Concretely, going counterclockwise in the z-plane coincides with going clockwise in the x-plane. So if we do go counterclockwise in the x-plane and want to cross the branch-cut, that is the same as crossing the z-plane branch-cut clockwise ie with an overall minus sign.
Sθ+− Sθ− (z-plane) 7→ Sθ−− Sθ+ (x-plane) (4.25)
Discθ (z-plane) 7→ −Discθ (x-plane)
Finally let’s also look at a different variable since it is also used sometimes used [1].
x = ω2 (4.26)
Here the new variable has the same orientation but a complete cycle around the origin of the ω-plane is equal to two such cycles in the x-plane. If we where to use the expression (3.27) only replacing z = ω12 we would get a result
with an incorrect sign and with an additional factor of 2, because we have crossed the branch-cut twice.
We are now ready to derive the large order relation of the perturbative coefficients u(0|0)[0] of the two-parameter transseries solution. Using (4.15) and (4.23)
Φ(0|0)(x) = 1 2πi Z +∞ 0 Disc0Φ(0|0)(w) w − x dw + 1 2πi Z −∞ 0 DiscπΦ(0|0)(w) w − x dw, (4.27)
1Not the same z as in (4.1).
we can asymptotically expand this as ∞ X g=0 u(0|0)[0]2g xg∼= 1 2πi Z +∞ 0 ∞ X g=0 xgw−g−1Disc0Φ(0|0)(w)dw + 1 2πi Z −∞ 0 ∞ X g=0 xgw−g−1DiscπΦ(0|0)(w)dw (4.28)
where we have simlutaneously expanded w−x1 = P∞
g=0x
gw−g−1 about x = 0 aswell. Now we can simply equate
equal powers of x to get an expression for the coefficient
u(0|0)[0]2g ∼= 1 2πi Z +∞ 0 w−g−1Disc0Φ(0|0)(w)dw + 1 2πi Z −∞ 0 w−g−1DiscπΦ(0|0)(w)dw (4.29)
Now we want to calculate the discontinuities using (3.27) but with the opposite sign as explained in the remark above. We will take the discontinuities up to first order in exponentials1 to keep things as simple as possible for
now and illustrate the main idea.
Disc0Φ(0|0)(x) = S (0) 1 e −A/xΦ (1|0)(x) + O[e−2A/x] DiscπΦ(0|0)(x) = eS (0)
−1eA/xΦ(0|1)(x) + O[e2A/x]
(4.30) with Φ(1|0)(x) = ∞ X h=0 u(1|0)[0]2h+1 xh+12 Φ (0|1)(x) = ∞ X h=0 (−1)hu(1|0)[0]2h+1 xh+12, (4.31)
where we have used (4.20) to switch the instanton indices
u(0|1)[0]2h+1 = (−1)hu(1|0)[0]2h+1 . (4.32) Plugging all of this back into (4.29) we get
u(0|0)[0]2g ∼=S (0) 1 2πi ∞ X h=0 u(1|0)[0]2h+1 Z +∞ 0 w−g+h−12e−A/wdw +Se (0) −1 2πi ∞ X h=0 (−1)hu(1|0)[0]2h+1 Z −∞ 0 w−g+h−12eA/wdw =S (0) 1 2πi ∞ X h=0 u(1|0)[0]2h+1 Γ(g − h − 1 2) Ag−h−1 2 − i(−1)gSe (0) −1 2πi ∞ X h=0 u(1|0)[0]2h+1 Γ(g − h − 1 2) Ag−h−12 =(S (0) 1 − i(−1)gSe (0) −1) 2πi ∞ X h=0 u(1|0)[0]2h+1 Γ(g − h − 1 2) Ag−h−1 2 (4.33)
Last thing to note about this is that from (4.17) we can infer that u(0|0)[0]2g must be 0 for odd g. This implies that we must have
e
S−1(0)= iS1(0). (4.34) For every Stokes constant we will calculate there will be another Stokes constant that is constrained by the others. The equation above is the simplest case but we will see this happen as well in appendix B. In fact all Stokes constants with negative lower index are constrained in the same way by using the symmetry of (4.20) on large order
Figure 4: A plot of the number of decimals that S1(3) for different g agrees with S1(3) for g = 72. The last few dots grow faster than linear because S1(3) for g close to 72, the approximation that S(3)1 for g = 72 is the true value does not hold anymore.
relations. We will elaborate on this topic at the end of the chapter.
We can now replace every g with 2g and get our final expression for the large order relation.
u(0|0)[0]4g ∼=2S (0) 1 2πi ∞ X h=0 u(1|0)[0]2h+1 Γ(2g − h − 1 2) A2g−h−1 2 (4.35)
We can calculate all the coefficients for arbitrarily high g and h using (4.17) and we can approximate the asymptotic series using the Borel-Pad´e procedure (see appendix A.1). So now we can calculate although it is the only Stokes constant for Painlev´e I that has actually been calculated exactly [21].
S1(0) = −i3
1/4
2√π (4.36)
So far this section was just to illustrate the large order calculation in its simplest form for Painlev´e I. In appendix B we extend this procedure to find further Stokes constants. In [1] this method was used to calculate a list of Stokes constants (not only for Painlev´e I), and for this project we have calculated an additional one, namely
S1(3)= −60.44560623823963943939349537039 . . . i (4.37) We have calculated this number to what seems to be an accuracy of 29 decimal places at (g = 72). The way we estimate this error is firstly by comparing much lower values of g and seeing that the number of decimal places of that agree with S1(3) for g = 72 rises linearly with g. Extrapolating this line gives us that estimate see figure 4. The linear behaviour only holds aslong as g is much lower than 72. That way the value at g = 72 is effectively the true value.
One of the tantalizing results of [1] was that many Stokes constants seemed to be related and a couple of conjectured relations were suggested such as
n eS(n)1 = −(n − 1)S1(n−1) (4.38) which the result of (4.37) affirms. In [1] the related Stokes constant
e
S1(4)= 45.334204678679729579545121527793918625213565893043296 . . . i, (4.39) was calculated to 108 decimals! Our calculation converges to the conjectured value
S1(3)→ −4 3Se
(4)
Therefore the relation is confirmed to 29 decimals for n = 4.
It is important to clarify there are relations between Stokes constants which immediately follow from the large order derivation, such as (4.34) that are entirely different from relations like (4.38). The conjectured relations are done only on the basis of the numerical results. Their origin is not explicitly understood, but it is exactly those non-trivial relations that we hope will lead to a better understanding of the Stokes phenomenon.
The form of the Stokes phenomenon we initially set out to understand looks like this
Sθ+[u](x, σ1, σ2) = Sθ−[u] (x, ˆσ1(σ1, σ2), ˆσ2(σ1, σ2)) , (4.41)
where u is the full two-parameter transseries solution to Painlev´e I. Recall the goal was to find hints on how to describe the ˆσi. This relation is non trivial for directions θ = 0, π. For simplicity let’s consider the θ = 0 case.
We can rewrite this relation in terms of alien calculus using (3.23). This then gives us
exp X
ω∈Γ0
e−ω/x∆ω
!
u(x, σ1, σ2) ∼= u (x, ˆσ1(σ1, σ2), ˆσ2(σ1, σ2)) (4.42)
which to first order in the outer exponential becomes
I + ∞ X l=1 e−lA/x∆lA ! u(x, σ1, σ2) ∼= u (x, ˆσ1(σ1, σ2), ˆσ2(σ1, σ2)) . (4.43)
Now we make the same approximation as before exactly because we made it before too. We only calculated first order Stokes constants so it only makes sense for now to calculate the Stokes phenomenon to first order in alien derivatives.
I + e−A/x∆A
u(x, σ1, σ2) ∼= u (x, ˆσ1(σ1, σ2), ˆσ2(σ1, σ2)) . (4.44)
We plug in the transseries for both u’s now X n,m=0 σn1σ2me−(n−m)A/xΦ(n|m)+ e−A/x∆AΦ(n|m) ∼= X n,m=0 ˆ σn1σˆm2 e−(n−m)A/xΦ(n|m) (4.45)
Now we use our alien derivative rules from (3.67) to get
∆AΦ(n|m)= min(n+1,m) X k=0 (n − k + 1) S1(k)Φ(n−k+1|m−k) + min(n,m−1) X k=0 (m − k − 1) eS1(k+2)Φ(n−k|m−k−1). (4.46)
We can now use the conjectured relation (4.38) that reads
e
S1(k+2)= −k + 1 k + 2S
(k+1)
1 (4.47)
in this case, to simplify this to
∆AΦ(n|m)= min(n+1,m) X k=0 n − k + 1 − k k + 1(m − k) S1(k)Φ(n−k+1|m−k), (4.48)
where we have also shifted the summation index of the second sum. This gives us X n,m=0 σ1nσ2me−(n−m)A/x Φ(n|m)+ e−A/x min(n+1,m) X k=0 f (n, m, k)S1(k)Φ(n−k+1|m−k) ∼= X n,m=0 ˆ σ1nσˆm2e−(n−m)A/xΦ(n|m), (4.49)
with f (n, m, k) ≡ n − k + 1 − k k + 1(m − k) . (4.50)
Equation (4.49) implicitly describes the behaviour of the ˆσi. We have it to first order in Stokes constants so far.
Using more eventual relations between Stokes constants would make it possible to encorporate more contributions to the l.h.s. of (4.49).
We conclude this section about the large order calculations with some practical notes concerning carrying out further Stokes constant calculations.
4.3.1 Stokes constant calculation strategy
Much of the very long calculations have been postponed to the appendices. These are surely insightful for anyone attempting to do similar calculations. But a sense of direction is also valuable. This is what the present section will be about.
This master’s project has mainly focussed on calculations of Stokes constants of the form S±1p and eS±1p . There is a recurring pattern in the specifics of the large order calculations that are necessary for these specific constants. Not every large order relation produces new Stokes constants.
In general the large order relation of u(n|n)[0] will enable one to calculate S(n)
1 , but will also give a constraint for
e
S−1(n) by observing that u(n|n)[0]2g+β = 0 for odd g, just like in (4.34).
Large order relations of u(n+2|n)[0] in combination with those of u(n|n+2)[0] will allow one to calculate eS1(n+2) and constrain S−1(n+2). The constraint is derived from comparing the large order relations of u(n+2|n)[0] and u(n|n+2)[0], by means of the switching the instanton indices as in (4.20)
u(n+2|n)[0]= (−1)n+[n/2]I+1u(n|n+2)[0]. (4.51)
The other constrained Stokes constants that we will come across during the appendices are
e S−1(1)= −iS(1)1 +√4π 3S (0) 1 e S−1(2)= i −√2π 3 2S1(1)+ eS1(2)− 2π2iS(0) 1 e S−1(3)= −iS(3)1 √2π 3 2S1(2)+ eS1(3)+2π 2i 3 3S(1)1 + 2 eS1(2)−16π 3 9√3S (0) 1 S−1(2)= −i eS(2)1 −√2π 3S (0) 1 S−1(3)= i eS1(3)+√2π 3S (1) 1 + 4π2i 3 S (0) 1 . (4.52)
The careful reader notices that there seem to be no constraints for S−1(n)for n = 0, 1. That is because we don’t have these at all. It is purely due to the labeling convention taken in (3.59). The same is true for the eS(n)1 with positive lower index. These also start at n = 2.
