### Line Style

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Static Circle:Mobile Circle:

Tracing Stick:

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(click "Draw"! further explanations are at the bottom)

3:2

I(3:2)=⟨ x

Please enable JavaScript.

Mobile Circle:

Tracing Stick:

Leaf dim $\MB($3:2$) =\,\, $0

Inner radius: 60 Speed: 2

This is for illustration only and comes with no guarantees concerning accuracy etc.

Please email me any suggestions for improvements though
(boalch at imj-prg.fr).

**Background:**

Click on "draw", and the program will draw the Stokes diagram of the
exponential factor $\< x^{3/2} >$.

Here $x$ is a coordinate on the complex plane and this means
we consider the growth/decay of the

two branches
of the function $\exp(x^{3/2})$ as $x$ tends to infinity along any ray.
This picture appears in

Stokes' 1857 paper [S1857], and was reproduced on the title page of [BY2015].

Thus the dashed line is a (small!) circle around the point
$x=\infty$ in the Riemann sphere, and the solid line encodes the growth/decay of
$\exp(\pm x^{3/2})$. For example along the positive real axis, there are two real branches and the right-most curve indicates the branch
$\exp(+x^{3/2})$ has maximal growth there (one of the directions along which the solid curve is furthest from the dashed line is along the real axis).
The other branch $\exp(-x^{3/2})$ lies inside the dashed line, and has maximal decay along the positive real axis.
In contrast along the ray $\arg(x)=\pi/3$ the dominance of the two branches
changes; this is an *"oscillating"* or *"Stokes"* direction (in the terminology of [W1976]).
Away from the Stokes directions the two branches have a well-defined (dominance) order.

The Stokes diagram arose in Stokes' study of the linear
differential equation $y''=xy$ for the Airy functions, since
formal solutions to this equation at $x=\infty$ involve the
exponential functions $\exp((2/3)x^{3/2})$,
and the Stokes diagram of
these functions looks the same
(we can ignore the constant $2/3$ here).

Of course the Stokes diagram is not intrinsically defined,
but it is representing something that can be defined intrinsically.
Let $\d$ denote the circle of real directions at $\infty$
and let $\I=\< x^{3/2} > \to \d$
denote the degree two covering map given by the germs at
$\infty$
of the functions $\pm x^{3/2}$ along various directions.
Thus $\I=\< x^{3/2}>$ denotes a *circle*
(basically the germ of the Riemann surface of these functions).
A point $p\in \I$ lies over some direction
$d\in \d$, and $p$ "is"
a choice of one of the two branches of the function
$x^{3/2}$
(on a germ of an open sector spanning the direction $d$).
The Stokes directions $\IS\subset \d$
are well-defined
and for any direction $d$ that is not a Stokes direction
the set $\I_d$
(the two points in the fibre of $\I$ over $d\in \d$)
has a well-defined dominance ordering, given by the dominance ordering of the two functions $\exp(\pm x^{3/2})$.
All of this is intrinsic, and the Stokes diagram is a
(non-canonical) projection of the Stokes
circle $\I$ to the plane near $\infty$,
indicating these dominance orderings,
and the directions along which the dominance changes.

This intrinsic formalism works in general:
for example
one can list all the possible exponential factors that occur at infinity
for any algebraic linear differential equation on the complex plane.
They make up a huge collection of circles $\cI$, the
*exponential local system*,
equipped with a covering map $\cI\to \d$.
Each component of $\cI$ is a circle of the form
$\< q >$ for some expression
$$ q = a_1 x^{k_1}+\cdots + a_m x^{k_m}$$
for rational numbers $k_i>0$ and complex numbers $a_i$.
Any such Stokes circle $\< q >$
has three numbers attached to it:

- $\slope(q)$ is the largest $k_i$ present in $q$,
- Ramification $\Ram(q)$ is the degree of the covering map $\< q >\to \d$, i.e. the lowest common multiple of the denominators of all the $k_i$ present in $q$.
- Irregularity of $q$ is the nonnegative integer $\Irr(q)=\slope(q)\Ram(q)$.

An

The

Clicking on the arrows, the program above will draw Stokes diagrams for a simple class of irregular classes $\I(a\col b)$ generalising Stokes' Airy example $\I(3 \col 2)$. These are the symmetric irregular classes, which really means they are the pull-back of a circle of the form $\< w^{1/b} >$ for some integer $b$, under some cyclic covering $w=x^a$. In these cases it is thus easy to visualise the changes of dominance orderings. Note that in general, when the component circles have many different slopes it is not so easy to see the dominance orderings via such diagrams, and one should work directly with the cover $\I\to \d$ (computing the finite set of Stokes directions and the total order on the fibres of $\I$ between the Stokes directions). Another way to visualise the changing dominances is via the

We won't review the Stokes phenomenon here (see [B2021] and references therein) but it is worth pointing out that there are

The paper [B2021] gives more details of this formalism and references for a lot more of the background, how to use this set-up to define the Stokes data intrinsically (either as Stokes filtrations, Stokes gradings or wild monodromy/Stokes automorphisms) and for the modern applications of these ideas in $2d$ gauge theory, such as the construction of the symplectic/hyperkahler moduli spaces that occur when one considers connections and Higgs fields with irregular singularities on Riemann surfaces, starting with [B2001], [BB2004].

Leaf dim $\MB(a\col b)$ denotes the complex dimension of the symplectic leaves of the Poisson wild character variety $\MB(a\col b)$, determined by the (rank $b$) wild Riemann surface: $$\bSi \ = \ \left(\IP^1, \infty, \I(a\col b)\right)$$ with just one singularity with irregular class $\I(a\col b)$ on the Riemann sphere, taking all the Stokes circles in $\I(a\col b)$ to have multiplicity one. These leaves have codimension $k-1$ where $k=\text{GCD}(a,b)$, obtained by fixing the formal monodromy (around each of the $k$ circles in $\I(a\col b)$, given that one parameter is already fixed by the monodromy relation). If $k=b$ these are isomorphic to wild character varieties appearing in the approach of Birkhoff [Bi1913], and were shown to be Poisson/symplectic in [B2001], and in general they are (very) special cases of the algebraic construction in [BY2015].

[BB2004] O. Biquard and P. Boalch, Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004) no. 1, 179–204

[Bi1913] G.D. Birkhoff, The generalized Riemann problem for linear differential equations and allied problems for linear difference and q-difference equations, Proc. Amer. Acad. Arts and Sci., 49, 1913 531-205.

[B2001] P. Boalch, Symplectic Manifolds and Isomonodromic Deformations, Adv. in Math. 163 (2001) 137–205

[BY2015] P. Boalch and D. Yamakawa, Twisted wild character varieties arXiv/1512.08091

[B2021] P. Boalch, Topology of the Stokes phenomenon, Proc. Symp. Pure Math. 103 (2021) 55-100 arXiv:1903.12612

[S1857] G.G. Stokes, On the discontinuity of arbitrary constants which appear in divergent developments, Cam. Phil. Trans., X.1, 1857, 105-128

[W1976] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Wiley Interscience 1976

(version α4, 12/10/2022. Previous version: version α3)