Stokes diagrams of symmetric irregular classes

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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=∞ in the Riemann sphere, and the solid line encodes the growth/decay of exp(± 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 (it is furthest from the dashed line). 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)=π/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=∞ 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 ∂ denote the circle of real directions at ∞ and let I=⟨x^3/2⟩→∂ denote the degree two covering map given by the germs at ∞ of the functions ± 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 ∈I lies over some direction 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 𝕊⊂∂ 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∈∂) has a well-defined dominance ordering. All of this is intrinsic, and the Stokes diagram is a (non-canonical) projection of the Stokes circle I to the plane near ∞, indicating these dominance orderings, and the directions along which they change.

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 ℐ, the exponential local system, equipped with a covering map ℐ→∂. Each component of ℐ is a circle of the form ⟨q⟩ for some expression

q = a₁x^k₁+ ⋯ +a_mx^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⟩→∂, 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).

Thus in Stokes' Airy example the slope is 3/2, the ramification is 2 and the irregularity is 3.
An irregular class Θ is a finite set of such circles equipped with an integer multiplicity. Thus Θ can be written

Θ= n₁⟨q₁⟩ + ⋯ + n_r⟨q_r⟩

The rank of the irregular class Θ is the integer

Rank(Θ)= n₁Ram(q₁) + ⋯ + n_rRam(q_r)

and the classical theory of linear differential equations implies that any order n algebraic linear differential equation (or algebraic connection on a rank n algebraic vector bundle on the affine line) has a rank n irregular class (at infinity) canonically associated to it, and any irregular class arises in this way. One can even define the exponential local system in a coordinate independent way and thus transfer this discussion to any point of any Riemann surface (see [BY2015] Rmk 3), leading to the notion of wild Riemann surface (i.e. a Riemann surface equipped with some marked points and a rank n irregular class at each marked point, for some n).

The support of an irregular class is the finite subcover

I=⋃⟨q_i⟩ ⊂ℐ

given by the union of the component circles present in Θ. Thus I →∂ is a finite cover of the circle of directions ∂.

Clicking on the arrows, the program above will draw Stokes diagrams for a simple class of irregular classes I(a:b) generalising Stokes' Airy example I(3:2). These are the symmetric irregular classes, which really means they are the pull-back of a circle of the form ⟨z^1/b⟩ for some integer b, under some cyclic covering z=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→∂ (computing the finite set of Stokes directions and the total order on the fibres of I between the Stokes directions).

We won't review the Stokes phenomenon here (see [B2021] and references therein) but it is worth pointing out that there are two different types of discontinuity that occur in this setting: the Stokes filtrations may jump along the Stokes directions, whereas the Stokes gradings may jump along the singular directions (sometimes called the anti-Stokes directions): for the Airy example they are the directions to the points of maximal decay, i.e those closest to the origin in the Stokes diagram (arg(x)=0,±2π/3). The singular directions are where the discontinuities of the constants occur in Stokes' 1857 paper.

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 𝓜_B(a:b) denotes the dimension of the symplectic leaves of the Poisson wild character variety 𝓜_B(a:b), determined by the (rank b) wild Riemann surface (ℙ¹,∞,I(a:b)), taking all the circles in I(a:b) to have multiplicity one. These leaves have codimension k-1 where k=GCD(a,b), obtained by fixing the formal monodromy (around each of the k circles in I(a:b), given that one 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