Just-in-Time Mathematical Toolkit

This toolkit develops only the topology and spectral theory used repeatedly in the book. Its purpose is operational: after reading it, a loop should mean a monodromy matrix, a lifted contour should mean a period or Voros symbol, and a boundary condition should mean part of an operator domain. The later chapters return to each construction where more precision is needed.

Fix a connected Riemann surface $\Sigma$, a finite singular set $D$, and the punctured surface

$$X=\Sigma\setminus D.$$

The same space supports two related descriptions of a linear ODE:

• topology records how solutions continue along paths in $X$;
• analysis records which continued solutions satisfy the boundary or growth conditions of a spectral problem.

Keeping these roles separate prevents two common category errors: treating monodromy as a basis-independent matrix, and treating every zero of a connection coefficient as an eigenvalue of a self-adjoint operator.

Paths, loops, and the fundamental group

A path from $a$ to $b$ is a continuous map $\gamma:[0,1]\to X$ with $\gamma(0)=a$ and $\gamma(1)=b$. Two such paths are equivalent when one can be continuously deformed into the other while both endpoints remain fixed. Analytic continuation on $X$ depends on this fixed-endpoint homotopy class, not on the particular parametrization.

Choose a base point $z_*\in X$. Homotopy classes of loops beginning and ending at $z_*$ form the fundamental group $\pi_1(X,z_*)$. The identity is the constant loop, and the inverse loop traverses the same curve backwards. We use the path-product convention fixed on the notation page: $\gamma_1\gamma_2$ traverses $\gamma_2$ first and then $\gamma_1$.

For the sphere punctured at $n$ points $a_1,\ldots,a_n$, one may choose positively oriented based loops with a single relation

$$\gamma_1\gamma_2\cdots\gamma_n=1.$$

The ordering is part of the choice of cuts and generators. For $X=\mathbb C\setminus\{0,1\}$, the loops about $0$ and $1$ freely generate $\pi_1(X,z_*)$; the loop about infinity is then fixed by the sphere relation.

Why the base point matters

A chosen path between two base points gives an isomorphism between their fundamental groups by attaching that path and its inverse to each loop. Changing the connecting path changes this isomorphism by an inner automorphism. This is the topological origin of simultaneous conjugation of monodromy matrices.

Paths between several endpoints form the fundamental groupoid rather than a single group. This is the natural language for connection problems: monodromy uses loops at one base point, whereas a connection matrix compares frames normalized in two different regions.

Local systems and monodromy

A rank-$r$ complex local system $\mathcal L$ on $X$ is, informally, a collection of $r$-dimensional solution spaces that can be canonically identified along sufficiently short paths. A nonsingular linear system

$$\frac{\dd Y}{\dd z}=A(z)Y$$

defines such a local system: its local horizontal sections are its solutions. Continuation around a based loop gives a representation

$$\rho:\pi_1(X,z_*)\longrightarrow GL(r,\mathbb C), \qquad \rho(\gamma)=M_\gamma.$$

With the book’s right-action convention, $\Phi^\gamma=\Phi M_\gamma$, the chosen path product gives

$$M_{\gamma_1\gamma_2}=M_{\gamma_1}M_{\gamma_2}.$$

A fundamental matrix $\Phi(z_*)$ is a framing of the fiber at the base point. Replacing it by $\Phi(z_*)G$ changes every monodromy matrix by

$$M_\gamma\longmapsto G^{-1}M_\gamma G.$$

Consequently, an unframed local system determines a representation only up to simultaneous conjugation. Traces, characteristic polynomials, and conjugacy classes are invariant; individual matrix entries are not.

At an irregular singularity, the local system on the punctured neighborhood remembers ordinary monodromy but not the full asymptotic classification. Formal exponential factors, formal monodromy, sectorial normalizations, and Stokes matrices are additional data developed in Chapters 1 and 2.

If $\Phi_\alpha$ and $\Phi_\beta$ are normalized frames in two regions, then

$$\Phi_\beta=\Phi_\alpha C_{\alpha\beta}.$$

The matrix $C_{\alpha\beta}$ belongs to groupoid data: it depends on the continuation path and both endpoint frames. Monodromy data and connection matrices are related, but neither is a basis-free synonym for the other.

A three-puncture check

For a Fuchsian equation on $\mathbb P^1\setminus\{0,1,\infty\}$, choose generators satisfying

$$\gamma_0\gamma_1\gamma_\infty=1.$$

Then

$$M_0M_1M_\infty=I.$$

This identity checks matrix order, but it does not determine the connection matrix between a basis at $0$ and a basis at $1$. That comparison also needs the local normalizations and a continuation path.

Riemann surfaces and multivalued functions

A Riemann surface is a one-complex-dimensional manifold with holomorphic coordinate changes. It turns a multivalued expression on the $z$-plane into a single-valued object on a suitable covering surface.

For example, $y^2=z$ defines a two-sheeted cover of the punctured plane. Continuing once around $z=0$ exchanges the sheets; continuing twice returns to the original value. A branch cut is a planar bookkeeping device that selects one sheet over a cut domain. It is not an intrinsic boundary of the Riemann surface.

In WKB analysis, let

$$\varphi=Q(z)(\dd z)^2$$

be the quadratic differential in a chosen normal-form coordinate. The coordinate-invariant spectral cover over $U=\Sigma\setminus\operatorname{Poles}(\varphi)$ is the subspace of $T^*U$ defined by

$$\widehat\Sigma: \qquad \lambda^2=\varphi.$$

Writing $\lambda=y\,\dd z$ gives the familiar local equation

$$y^2=Q(z),$$

One then normalizes this cover and, when useful, compactifies it over the poles. A zero or pole of $\varphi$ of odd order is a branch point of the normalized double cover. Even-order zeros and poles are not branch points. Zeros of any positive order remain turning points or higher WKB critical points, while an even-order pole generally lifts to poles of the WKB one-form $\lambda$.

If a degree-two cover $\pi:\widehat\Sigma\to\Sigma$ has $r$ simple branch points, the Riemann–Hurwitz formula gives

$$2g(\widehat\Sigma)-2 =2\bigl(2g(\Sigma)-2\bigr)+r.$$

For a cover of the sphere, this reduces to $g(\widehat\Sigma)=(r-2)/2$. Thus a generic quartic curve

$$y^2=\prod_{j=1}^{4}(z-a_j)$$

has genus one, while $y^2=z^2-1$ has genus zero after compactification.

Compact curve versus punctured curve

Compactification supplies useful homology cycles, but WKB differentials may have poles at the added points. Periods and contour deformations must therefore remember the punctures, residues, and any regularization.

Absolute and relative cycles

Let $S$ be the page-specific set of lifted poles and other points removed from the normalized cover, and set

$$\widehat X=\widehat\Sigma\setminus S.$$

An oriented closed curve on $\widehat X$ determines a class in the absolute homology group

$$H_1(\widehat X,\mathbb Z).$$

Two closed curves represent the same class when their difference is the boundary of an oriented two-chain. If a meromorphic one-form $\lambda$ is holomorphic throughout the swept region, homologous cycles have equal periods:

$$\oint_{\gamma_1}\lambda =\oint_{\gamma_2}\lambda.$$

If the deformation crosses a pole, the periods differ by the appropriate $2\pi\ii$ times residue. Homology on the compact surface alone does not erase this analytic information.

Open paths whose endpoints lie in a marked set $P$ define relative classes

$$H_1(\widehat X,P;\mathbb Z).$$

The boundary map records their endpoints:

$$\partial: H_1(\widehat X,P;\mathbb Z) \longrightarrow H_0(P;\mathbb Z).$$

For connected $\widehat X$, the relevant part of the long exact sequence is

$$\begin{aligned} H_1(\widehat X;\mathbb Z) &\longrightarrow H_1(\widehat X,P;\mathbb Z)\\ &\xrightarrow{\ \partial\ }H_0(P;\mathbb Z) \longrightarrow H_0(\widehat X;\mathbb Z). \end{aligned}$$

Thus a relative path has a signed endpoint sum of zero. Closed cycles lie in the kernel of $\partial$ and come from absolute homology, up to the preceding map. In exact WKB, a closed cycle typically supports a quantum period, while a path between turning points, poles, or asymptotic ends is relative and may require endpoint subtraction.

For a path-connected space,

$$H_1(\widehat X;\mathbb Z) \cong \pi_1(\widehat X,z_*)^{\mathrm{ab}}.$$

Homology therefore remembers winding and intersection data but forgets the order of noncommuting loops. It cannot replace the full fundamental group in a general monodromy representation.

There is no universal deleted set for every WKB problem. One common exact-WKB choice uses absolute cycles in $H_1(\widehat\Sigma\setminus\widehat P;\mathbb Z)$ and paths in a group of the form $H_1(\widehat\Sigma\setminus\widehat P_0,\widehat P_\infty;\mathbb Z)$, where the lifted turning points, simple poles, higher poles, and asymptotic ends are assigned roles dictated by the differential. Singular endpoints may instead require a real blow-up or a regularized path space. Each later page declares the group it actually uses.

A contour drawn only in the $z$-plane is incomplete. One must also state:

1. its lift to the chosen sheet;
2. how it crosses the branch cuts;
3. its orientation;
4. whether its endpoints or punctures are fixed;
5. which poles lie inside a deformation.

For a two-sheeted cover with involution $\iota:y\mapsto-y$, the WKB differential changes sign:

$$\iota^*(y\,\dd z)=-y\,\dd z.$$

The same projected loop on opposite sheets therefore has the opposite classical action.

Intersection pairings

Two transverse oriented cycles on an oriented surface have an algebraic intersection number. At each crossing, assign $+1$ when the ordered tangent vectors of the first and second cycles agree with the surface orientation, and $-1$ otherwise; then add the local signs. The result is bilinear, homology-invariant, and antisymmetric:

$$\langle\gamma_1,\gamma_2\rangle =-\langle\gamma_2,\gamma_1\rangle.$$

For a compact surface of genus $g$, choose a symplectic basis $A_1,\ldots,A_g,B_1,\ldots,B_g$ with

$$\begin{aligned} \langle A_i,A_j\rangle&=0, & \langle B_i,B_j\rangle&=0,\\ \langle A_i,B_j\rangle&=\delta_{ij}, & \langle B_i,A_j\rangle&=-\delta_{ij}. \end{aligned}$$

Periods in a different symplectic basis are related by an integral symplectic transformation. A formula written in terms of “the $A$ period” therefore has no invariant meaning until the cycle basis and orientation are declared.

Intersection numbers govern later discontinuity and wall-crossing formulae. Those chapters state the precise sign convention again, because reversing either a cycle or the order in the pairing reverses the exponent or jump.

There is also an absolute–relative pairing

$$H_1(\widehat X;\mathbb Z) \times H_1(\widehat X,P;\mathbb Z) \longrightarrow\mathbb Z,$$

defined by perturbing representatives so their interior crossings are transverse and away from $P$. It is the pairing most directly used when a closed WKB cycle acts on an open Voros path. On a punctured surface the absolute pairing can be degenerate—for example, small loops around punctures may lie in its radical—so a “symplectic basis” is asserted only for the nondegenerate quotient or for compact absolute homology.

The minimum spectral dictionary

A differential expression becomes a spectral problem only after its function space and domain have been specified. For a real Sturm–Liouville expression, write

$$\tau y =-\frac1{w(x)} \left[\bigl(p(x)y'(x)\bigr)'-q(x)y(x)\right], \qquad p>0,\quad w>0.$$

It acts naturally in $L^2((a,b),w(x)\,\dd x)$, but the expression $\tau$ alone is not yet an operator. An operator $L$ also needs a domain $\mathcal D(L)$ encoding regularity and boundary conditions.

For sufficiently regular $f$ and $g$ and interior points $a<c<d<b$, the Lagrange identity is

$$\begin{aligned} &\int_c^d \left[ (\tau f)\,\overline g -f\,\overline{\tau g} \right]w\,\dd x\\ &\qquad =\left[ p\left( f\,\overline{g'} -f'\,\overline g \right) \right]_c^d. \end{aligned}$$

Self-adjoint boundary conditions make the boundary form on the right vanish in the appropriate endpoint limits for every pair in the domain and make the domain equal to that of the adjoint. At a singular endpoint, the individual limits need not exist until the maximal domain and boundary form have been defined. Merely having real coefficients is not enough.

At a singular endpoint, the limit-point/limit-circle alternative decides whether one must impose a separated boundary condition. The endpoint is limit-circle when every solution of $\tau y=\lambda y$ for one—and hence every—nonreal $\lambda$ is locally square-integrable there; otherwise it is limit-point.

• in the limit-point case, no boundary condition is imposed there for a separated self-adjoint realization;
• in the limit-circle case, one boundary condition is required there.

The precise test depends on the differential expression and measure. “Choose the regular solution” is not a substitute when both local solutions lie in the Hilbert space or when a physical boundary condition is not self-adjoint. More general self-adjoint domains can couple the two endpoints.

Spectrum and resolvent

For a closed densely defined operator $L$, the resolvent set consists of $\lambda$ for which

$$(L-\lambda)^{-1}$$

exists as a bounded operator on the whole Hilbert space. Its complement is the spectrum. An eigenvalue is a spectral point with a nonzero solution of

$$(L-\lambda)y=0, \qquad y\in\mathcal D(L).$$

A self-adjoint operator has real spectrum, but it need not have only eigenvalues. A purely discrete spectrum follows under additional compactness conditions, such as compact resolvent. For nonselfadjoint operators, the spectrum may be complex, eigenvectors need not be orthogonal or complete, and geometric and algebraic multiplicities can differ.

An analytic operator pencil $L(\lambda)$ is more general than $L-\lambda$. Its spectral points are values at which $L(\lambda)$ fails to be invertible. Before applying the analytic Fredholm theorem, one needs a holomorphic Fredholm family on a fixed domain, normally of index zero, that is invertible for at least one parameter value.

Boundary Wronskians and resonances

Suppose $y_L(z,\lambda)$ and $y_R(z,\lambda)$ are solutions selected by the left and right boundary conditions, analytically normalized in $\lambda$. At a fixed ordinary match point $z_m$, their boundary function is

$$D(\lambda) =\Wr[y_L,y_R]\big|_{z=z_m}.$$

Abel’s identity determines how this Wronskian varies with $z$; in normal form it is independent of $z$. In Sturm–Liouville form the naturally constant quantity is $p(z)\Wr[y_L,y_R]$. Suppose each boundary condition selects a nonzero one-dimensional solution subspace. Then

$$D(\lambda)=0$$

means that the two one-dimensional solution spaces coincide, so one solution satisfies both boundary conditions.

Rescaling the boundary solutions by nonvanishing analytic factors gives

$$y_L\mapsto a(\lambda)y_L, \qquad y_R\mapsto b(\lambda)y_R, \qquad D\mapsto a(\lambda)b(\lambda)D.$$

The zero set is unchanged, but the function and its derivatives are not. This is why a Jost or Evans function is usually defined only up to a nonvanishing analytic factor unless a normalization is fixed.

For scattering resonances and black-hole quasinormal modes, the selected solutions are typically outgoing, ingoing, or decaying in different asymptotic regions. The resulting problem is generally nonselfadjoint. Resonances are often defined as poles of a meromorphically continued resolvent or scattering matrix; a Wronskian zero represents the same resonance only after that equivalence and the continuation sheet have been established.

Three distinct claims

A formal ODE solution satisfying two asymptotic descriptions, a zero of a boundary Wronskian, and a pole of a continued resolvent are not automatically the same statement. Later applications spell out the hypotheses linking them.

A translation table for later chapters

Later object	Toolkit interpretation	Extra data still required
Monodromy matrix	Parallel transport around a based loop	A framing and loop convention
Connection matrix	Transport between two normalized frames	Endpoint bases and continuation path
Spectral curve	Branched cover carrying the globally defined one-form $\lambda$	Normalization, compactification, and punctures
Closed classical or quantum period	Integral along an absolute cycle	Sheet, orientation, differential, and regularization
Voros path or cycle symbol	Exponential of a regularized WKB integral along a relative path or absolute cycle	Endpoint treatment, subtraction, and lateral sum
Boundary Wronskian or Jost–Evans function	Analytic Wronskian of selected solutions	Boundary normalizations and operator interpretation
Eigenvalue	Spectral point with a vector in the operator domain	Hilbert space, closed operator, and boundary conditions
Resonance or QNM	Pole or equivalent boundary zero on a continued sheet	Continuation, radiation conditions, and multiplicity convention

Common pitfalls

Confusing a loop with its drawing. Two curves that look similar in a chosen cut plane can lift to different paths or sheets. Record the based homotopy class and the lift, not only the sketch.

Calling a monodromy matrix invariant. A matrix requires a framed solution basis. The unframed data are a conjugacy class of representations, together with any additional local conjugacy constraints.

Integrating on the base plane. The square root in a WKB differential is single-valued on the spectral cover. A projected contour without sheet and cut-crossing data does not determine the sign of its period.

Replacing monodromy by homology. First homology is the abelianization of the fundamental group. It is ideal for additive period data but discards the ordering information carried by noncommuting monodromy matrices.

Equating an ODE with an operator. The differential expression does not specify a Hilbert space, domain, or boundary conditions. Spectral language is used only after those choices are stated.

Exercises

1. Monodromy at three punctures. On $\mathbb P^1\setminus\{0,1,\infty\}$, suppose the chosen generators satisfy $\gamma_0\gamma_1\gamma_\infty=1$. Express $M_\infty$ in terms of $M_0$ and $M_1$, and show that a change of framing preserves the relation.

Solution

The representation property gives

$$M_0M_1M_\infty=I, \qquad M_\infty=(M_0M_1)^{-1}.$$

Under $\Phi\mapsto\Phi G$, every matrix changes to $\widetilde M_j=G^{-1}M_jG$. Hence

$$\widetilde M_0\widetilde M_1\widetilde M_\infty =G^{-1}(M_0M_1M_\infty)G =I.$$

2. Genus of a polynomial spectral curve. Let $Q$ be a polynomial of degree $d\geq1$ with simple zeros. Compactify and normalize $y^2=Q(z)$. Show that

$$g=\left\lfloor\frac{d-1}{2}\right\rfloor.$$

Solution

Every finite zero is a simple branch point. Infinity is a branch point exactly when $d$ is odd, so the number of branch points is

$$r= \begin{cases} d, & d\ \text{even},\\ d+1, & d\ \text{odd}. \end{cases}$$

Riemann–Hurwitz for a double cover of the sphere gives $g=(r-2)/2$, which is $\lfloor(d-1)/2\rfloor$ in both cases.

3. Absolute or relative? Compactify $y^2=z^2-1$, and then remove the two points $\infty_+$ and $\infty_-$ above infinity. Classify a lifted path joining the two branch points and the closed lift obtained by going along the cut on one sheet and returning on the other. Explain what changes when the orientation is reversed.

Solution

The path joining the branch points has nonzero boundary and is a relative class with endpoints in $P=\{-1,1\}$. Traversing from one branch point to the other on one sheet and returning on the other produces a closed absolute cycle. It separates $\infty_+$ from $\infty_-$ and generates $H_1(\widehat\Sigma\setminus\{\infty_+,\infty_-\};\mathbb Z)\cong\mathbb Z$. On the compact genus-zero cover it becomes null homologous. Reversing either path negates its class in the relevant absolute or relative group and therefore negates the integral of any fixed one-form along it.

4. Boundary-function normalization. If $D(\lambda)=\Wr[y_L,y_R]$, show that rescaling $y_L$ and $y_R$ by nonvanishing analytic functions preserves the zeros of $D$. Why can such a rescaling still matter in a residue or norming-constant formula?

Solution

Bilinearity gives

$$\widetilde D(\lambda) =a(\lambda)b(\lambda)D(\lambda).$$

Because $a$ and $b$ never vanish, $D$ and $\widetilde D$ have the same zeros with the same multiplicities. Their derivatives at a zero $\lambda_n$ satisfy

$$\widetilde D'(\lambda_n) =a(\lambda_n)b(\lambda_n)D'(\lambda_n),$$

so any residue or norming formula involving $D'(\lambda_n)$ must use the same normalization as its numerator.

5. One expression, two operators. On $[0,\pi]$, compare $L=-\dd^2/\dd x^2$ with Dirichlet conditions $y(0)=y(\pi)=0$ and with Neumann conditions $y'(0)=y'(\pi)=0$. Derive boundary functions and their spectra.

Solution

Write $k=\sqrt{\lambda}$. For the Dirichlet problem, the solution normalized by $y(0)=0$ and $y'(0)=1$ is

$$y_D(x,\lambda)=\frac{\sin(kx)}{k},$$

with its removable value at $k=0$. A boundary function is

$$D_D(\lambda) =\frac{\sin(\pi\sqrt\lambda)}{\sqrt\lambda}.$$

Its zeros are $\lambda=n^2$ for $n=1,2,\ldots$. For the Neumann problem, use $y_N(x,\lambda)=\cos(kx)$ and impose the right derivative condition:

$$D_N(\lambda) =y_N'(\pi,\lambda) =-\sqrt\lambda\sin(\pi\sqrt\lambda).$$

Its zeros are $\lambda=n^2$ for $n=0,1,2,\ldots$. The differential expression is the same, but the domains—and therefore the operators and spectra—are different.

References

• A. Hatcher, Algebraic Topology, Chapters 1 and 2, for fundamental groups, covering spaces, and homology.
• O. Forster, Lectures on Riemann Surfaces, for covering surfaces, analytic continuation, and meromorphic differentials.
• K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida, From Gauss to Painlevé, for local systems, monodromy representations, and Fuchsian equations.
• K. Iwaki and T. Nakanishi, “Exact WKB analysis and cluster algebras”, for the absolute and relative homology groups, intersection pairing, and paths used by Voros symbols.
• K. Iwaki, Les Houches Lectures on Exact WKB Analysis and Painlevé Equations, for the spectral cover, critical-point deletions, WKB paths, and cycles.
• G. Teschl, Mathematical Methods in Quantum Mechanics, for self-adjointness, resolvents, spectral types, and one-dimensional Schrödinger operators.
• E. B. Davies, Linear Operators and Their Spectra, for closed operators, nonselfadjoint spectra, and analytic spectral questions.
• M. Zworski, “Mathematical study of scattering resonances”, for meromorphic continuation and the resonance-pole definition.