Deligne’s regular solution in dimension 1

In this post, I want to recall the elements of the regular Riemann-Hilbert Correspondence (but only in dimension 1, on our small disk around the origin). We’ll talk more about the category of Meromorphic Connections on X with singularities at 0, and how they’re just a different way of phrasing linear ODEs whose solutions have singularities at 0. From there, we examine the simplest class of solutions, those with regular singularities. These are, in general, multi-valued functions of z that satisfy an analytic condition called moderate growth. From there, we can state Deligne’s solution to the regular R-H correspondence, and start to understand more about the failure in the irregular setting. We end with some emergent phenomena that occur only for irregular singularities, which add to the many difficulties in proving the irregular R-H correspondence. Lots of things from this post are from chapter 5 of “D-modules, perverse sheaves, and representation theory“.

Meromorphic Connections = Fancy ODE’s with singularities

Let X be a small open ball around the origin in \mathbb{C}, \mathscr{O} := \mathscr{O}_{X,0}, and \mathscr{O}(*0) the field of fractions of \mathscr{O}, representing holomorphic functions with possible poles at 0. If z is a local coordinate on X, then \mathscr{O} \cong \mathbb{C}\{z\}, and \mathscr{O}(*0) \cong \mathbb{C}\{z\}[z^{-1}]. Then, recall that a meromorphic connection consists of the data of a free \mathscr{O}(*0)-module M and a \mathbb{C}-linear map \nabla : M \to M satisfying the Leibniz rule \nabla(fm) = \frac{df}{dz}u + f\nabla(u) for elements f \in \mathscr{O}(*0), u \in M. A morphism of meromorphic connections is just a map \phi : (M,\nabla) \to (N,\nabla) that is \mathscr{O}(*0)-linear from M to N and commutes with the action of \nabla. This is actually a bit of a simplification of connections–normally we’d specify \nabla as a map

M \xrightarrow{\nabla} M \otimes_\mathscr{O} \Omega_{X/\mathbb{C}}^1

Contracting this “universal” \nabla with elements of \mathscr{D}_X produces the original definition I gave.

These objects then form an Abelian category (since we’re really just restricting to a thick subcategory of Holonomic D-modules). Again, the point of all this is just to formalize both our fast-and-loose simplifications from last time, and to phrase the problem functorially. This category of meromorphic connections is just a simplified way of talking about (complex) linear ODE’s with (possible) singularities at 0. Now, what does a general meromorphic connection look like?

On X with our local coordinate z, choosing a system of generators of M over \mathscr{O}_X(*0), say e_1(z),\cdots ,e_n(z) (these always exist globally on X because X is contractible). Then, looking at the action of \nabla on this basis gives us a connection matrix with values in \mathscr{O}(*0)

\nabla e_j =- \sum_{i=1}^n a_{ij}(z) e_i,

the negative sign is just a convention, to make the next expression nicer. Thus, if u(z) = \sum_i u_i(z)e_i(z) is a general element of M, then the above expression and the Leibniz rule gives us

\nabla (\sum_{i=1}^n u_i e_i) = \sum_{i=1}^n \left (\frac{du_i}{dz}-\sum_{j=1}^n a_{ij}u_j \right )e_i

Hence, the collection of flat sections of \nabla (those \vec{u} = \sum_i u_ie_i that satisfy \nabla u = 0) correspond to solutions of the system of linear ordinary differential equations

\frac{d \vec{u}}{dz} = A(z)\vec{u}

Likewise, any such differential equation gives rise to a connection–let \widetilde M be the space of solutions to the above differential equation. Then, this equation is a rule for how the symbol \frac{d}{dz} acts on the elements of \widetilde{M}, taken as the definition of \nabla.

The simplest case that arises from having meromorphic coefficients is when the connection matrix A(z) has at worst poles of degree one, i.e., if A(z) = \frac{A}{z} for some constant matrix A \in M_n(\mathbb{C}) (to compensate for maybe choosing a bad generating set for M over \mathcal{O}(*0), we only care if A(z) is “gauge equivalent” to a matrix of the form \frac{A}{z}). In the rank one case, this is a differential equation of the form

\frac{du}{dz} = \frac{\alpha}{z}u(z)

for some \alpha \in \mathbb{C}, and has fundamental solution u(z) = z^\alpha. What sort of function is this? When \alpha = 0,1,2,3,\cdots, z^\alpha is a globally defined holomorphic function (it’s just a monomial!), and when \alpha = -1,-2,-3,\cdots, z^\alpha is a globally defined meromorphic function on X that is holomorphic on X^*=X-\{0\}. When \alpha \notin \mathbb{Z}, we have to use the definition z^\alpha := \exp(\alpha \log(z)), where \log(z) is only a well-defined function on the complement of a choice of branch cut. Moreover, while this is a perfectly valid solution on any open simply connected subset of X^*, as we travel around the origin and analytically continue z^\alpha, the value of this function jumps due to monodromy. Precisely, if we let z \mapsto e^{2\pi \theta i}z as 0 \leq \theta \leq 1 varies, we find

z^\alpha \mapsto e^{2\pi i \alpha}z^\alpha := \exp\left \{\alpha(\log |z| +i(\arg(z)+2\pi)\right \}.

By cutting up the domain of log, we can make it single-valued

This is an example of a multi-valued function on X, and doesn’t lie in \mathcal{O} or \mathcal{O}(*0), and naturally form a ring (after fixing a universal cover of X^*). We don’t need this entire ring quite yet (it contains other weird things like functions with essential singularities and Whitney functions), but we’ll come back to it.

Regular Singularities and Moderate Growth

Meromorphic connections which have connection matrices gauge equivalent to one of the form \frac{A}{z} (we can even fudge a bit to allow A(z) \in M_n(\mathcal{O})) are said to have a regular singularity at 0. This is a bad choice of terminology, since sometimes “regular” means “non-singular” in algebraic geometry, so perhaps “tame” would’ve been a better name since they are classified by the rank of the connection and the monodromy matrix. These are the mildest sorts of singularities that appear in the theory of meromorphic connections, and are the the subject of the classical Riemann-Hilbert correspondence (in D-module language) of Kashiwara and Mebkhout. The solutions to these sorts of differential equation are very similar to our example of z^\alpha given above; if M has rank n, we can find by the Frobenius method n linearly independent solutions of the form

z^\alpha_i \phi_i(z)

where \alpha_i \in \mathbb{C} and \phi_i(z) is a holomorphic function on X with f(0) \neq 0 (the expression is slightly more complicated if an exponent \alpha_i is repeated, or if some pair \alpha_i and \alpha_j differ by an integer). These functions may be only well-defined single-valued functions on certain arc neighborhoods of 0, i.e., open sets of the form

S_\epsilon(a,b) := \{z \in X^*  |  0<|z|<\epsilon, a < \arg(z) < b\}

for some \epsilon > 0 and 0 \leq a < b \leq 2\pi.

arc neighborhoods on which z^{\frac{1}{3}} is single-valued

Solutions to ODE’s with regular singularities on X, even multi-valued ones, always behave “like” meromorphic functions in an appropriate arc neighborhood of any particular angle \theta \in S^1. When we say “behaves like a meromorphic function”, we mean that as we approach 0 from inside that arc neighborhood, the norm of the solution u(z) grows only polynomially in |z|^{-1}, as if it were a Laurent series with only finitely many terms of negative degree.

More precisely, solutions u(z) to ODEs with regular singularities are said to have moderate growth at \theta \in S^1, in the sense that there are 0< \epsilon,\delta \ll 1, constant C_\theta > 0 and exponent N_\theta \in \mathbb{N} for which

|u(z)| \leq C_\theta|z|^{-N_\theta} (1)

on the arc neighborhood S_\epsilon(\theta-\delta,\theta+\delta). As \theta travels around S^1, the values of the constants C_\theta and N_\theta may change. We just say u(z) has moderate growth at 0 if it has moderate growth at every \theta \in S^1.

For single-valued functions on X, moderate growth at 0 is equivalent to being meromorphic along 0.

This analytic characterization is actually a necessary and sufficient condition; a meromorphic connection (M,\nabla) has a regular singularity at 0 if and only if all of its flat sections have moderate growth at 0. Algebraically, this can be characterized by Fuch’s Criterion, but we will not focus on this perspective.

Aside from this interesting growth condition, or the simplicity of their solutions, why do we care about regular singularities? Let Conn^{reg}(X;0) be the category of Meromorphic Connections with regular singularities at 0, and Conn(X^*) the category of flat connections on X^*.

Theorem (Deligne): The restriction functor M \mapsto M_{|_{X^*}} induces an equivalence of categories

Conn^{reg}(X;0) \xrightarrow{\thicksim} Conn(X^*)

To show that we can always extend a given (N,\nabla) \in Conn(X^*) of rank n to an element of Conn^{reg}(X;0), we note that (N,\nabla) is uniquely determined by a monodromy representation \rho : \pi_1(X^*) \to Gl_{n}(\mathbb{C}) defined by the local system \ker \nabla. If we let \gamma correspond to the element 1 in the identification \pi_1(X^*) \cong \mathbb{Z}, then \rho is determined by the matrix C = \rho(\gamma) . We can then always find a matrix \Gamma \in M_{n}(\mathbb{C}) such that \exp(2 \pi i \Gamma)= C, which we then use to define a connection matrix on \widehat{N} := \mathscr{O}(*0)^{n} via

\nabla e_q = -\sum_{1 \leq p \leq n} \frac{\Gamma_{pq}}{z} \otimes e_p

where \{e_1,\cdots,e_n\} is the standard basis for \widehat{N}. The resulting meromorphic connection (\widehat{N},\nabla) clearly has a regular singularity at 0 and restricts to N on X^*.

Essentially, the singularities of connections with regular singularities are so mild that they are completely determined by their monodromy around the singular point. Recalling the previous post, we obtain the following:

Corollary (Deligne): There is an equivalence of categories

Conn^{reg}(X;0) \xrightarrow{\thicksim} Loc(X^*)

(M,\nabla) \mapsto (M_{|_{X^*}},\nabla_{|_{X^*}}) \mapsto \ker \nabla_{|_{X^*}}

Where Loc(X^*) is the category of finite-rank \mathbb{C}-local systems on X^*.

Irregular Singularities and Stokes Phenomena

We say a meromorphic connection (M,\nabla) has an irregular singularity at 0 if it doesn’t have a regular singularity at 0; hence, flat sections uof \nabla do not have moderate growth at 0. What does this mean? First off, the absence of regularity doesn’t necessarily mean u doesn’t satisfy (1) everywhere, just that there are some \theta \in S^1 which have no arc neighborhood on which u(z) has polynomial growth in |z|^{-1}.

For example, take (\mathscr{O}(*0),d-\frac{1}{z^2}), whose associated linear ODE has fundamental solution u(z)=e^{\frac{1}{z}}. Then, u(z) has moderate growth at every \frac{\pi}{2} < \theta < \frac{3\pi}{2}. Why? On this region, \textnormal{Re} \frac{1}{z} < 0, and so |u(z)| = e^{\textnormal{Re} \frac{1}{z}} decays to 0 exponentially as z \to 0. u(z) does not have moderate growth on (3\pi/2,\pi/2) (\mod 2\pi). When \textnormal{Re} \frac{1}{z} > 0, say \theta = 0 for simplicity, then

e^{1/z} = \sum_{n =0}^\infty \frac{1}{z^n n!} > z^{-N}

for every exponent N > 0.

Aside from no longer having moderate growth, one of the other reasons Deligne’s theorem no longer holds in the irregular setting is that we can’t uniquely specify a general meromorphic connection on X by its restriction to X^*. We actually saw this in the last post–the Meromorphic Connections (\mathscr{O}(*0),d-df) are all non-isomorphic (for f \in \mathscr{O}(*0)/\mathscr{O}), but their local systems of flat sections are all isomorphic to the constant local system \mathbb{C}_{X^*}. In essence, there are too many local systems with the same monodromy, if we have no restrictions on the singularities.

So, why do we care about these ODEs if they’re so difficult to deal with? Many interesting special functions arising in engineering and physics arise as solutions to ODEs with irregular singularities. Take, for example, the Airy equation

\frac{d^2 u}{dz^2} = zu

which has an irregular singularity at z= \infty, and its fundamental solutions Ai(z) and Bi(z) are approximated (in a rigorous sense, using asymptotic analysis) by linear combinations of the functions u_\pm(z) = z^{-1/4}e^{\pm \frac{2}{3}z^{3/2}} for large values of z. Now, here is the interesting thing: a solution Ai(z) of the Airy equation is an entire function of z, since the coefficient function (which is just z) is entire, but the functions u_\pm are multi-valued functions. Hence, as we let z \mapsto z e^{2\pi i}, around z=0, the “true solution” Ai(z) will return to its original value, but u_+ and u_- will not. Hence, Ai(z) and A(z e^{2\pi I}) cannot be represented by the same linear combination of u_+ and u_-.

This is known as the Stokes Phenomenon, where solutions to linear ODE’s can undergo drastic qualitative changes as \theta \in S^1 passes through certain angles. Specifically, these qualitative changes happen to the asymptotic behavior of the solutions in the different regions bounded by these special angles (sometimes called Stokes angles, or Stokes lines (or anti-Stokes lines, if you’re a physicist)). This is a phenomenon unique to the world of irregular singularities. We go through all of this extra work because sometimes it can be exceedingly difficult to express exact solutions in terms of elementary functions (or even special functions!), and the approximations in terms of exponential factors can often tell us much more about the “physical” properties the true solutions that are difficult to see from their formulas. We recommend Meyer’s “A Simple Explanation of the Stokes Phenomenon” for this perspective.

In the next post, I’ll talk some more about the formalism of approximation and asymptotic analysis, and how it naturally leads to a way of characterizing solutions of with irregular singularities via the Stokes filtration.