Notation and Status Conventions

A formula for global ODE data is inseparable from its conventions. This page fixes the default choices used throughout the book. A later page may override one of them for a named source or model, but the override is stated in a normalization ledger before the formula is used.

Equation before function name

Names such as “HeunC,” “spectral determinant,” or “quantum period” do not determine a mathematical object by themselves. Always record the equation, normalization, branches or cycles, parameter map, and status of the claimed identity.

Scalar equations and fundamental matrices

The default scalar equation is

$$y''(z)+p(z)y'(z)+q(z)y(z)=0,$$

with $p$ and $q$ meromorphic on a specified Riemann surface or domain. Primes mean differentiation with respect to the displayed independent variable.

For a first-order system we use column vectors and write

$$\frac{\dd Y}{\dd z}=A(z)Y, \qquad \frac{\dd\Phi}{\dd z}=A(z)\Phi.$$

The columns of the fundamental matrix $\Phi$ form an ordered solution basis. Two transformations play different roles:

• A gauge transformation acts on the left, $\widetilde\Phi=G(z)\Phi$, and gives $$\widetilde A =GAG^{-1}+G'G^{-1}.$$
• A constant change of solution basis acts on the right, $\widetilde\Phi=\Phi H$ with $H\in GL(2,\mathbb C)$, and leaves $A$ unchanged.

The book discusses traceless $\mathfrak{sl}_2$ connections locally. Three global questions are kept distinct:

• A single-valued gauge of a given $GL(2)$ flat bundle to trace zero requires a horizontal trivialization of its determinant local system. In the present trivial-bundle setting, this is equivalent to $\det\rho=1$.
• An $SL(2)$ lift of the projectivized local system is a different problem. Starting from a $GL(2)$ lift $\rho$, one seeks a rank-one local system $\chi$ with $\chi^2=(\det\rho)^{-1}$ and twists $\rho$ by $\chi$. This is not a gauge transformation of the original global system.
• A theta characteristic $K^{1/2}$ enters when a projective connection is represented globally as a scalar half-density operator. It is a square root of the canonical bundle, not of the determinant trivialization above.

Wronskians and connection matrices

The Wronskian convention is

$$\Wr[f,g]=f g'-f'g.$$

Thus

$$\det \begin{pmatrix} f & g\\ f' & g' \end{pmatrix} =\Wr[f,g], \qquad \Wr[f,g]'=-p\,\Wr[f,g].$$

For two ordered fundamental matrices, the connection direction is always

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

If $\widetilde\Phi_\alpha=\Phi_\alpha H_\alpha$ and $\widetilde\Phi_\beta=\Phi_\beta H_\beta$, then

$$\widetilde C_{\alpha\beta} =H_\alpha^{-1}C_{\alpha\beta}H_\beta.$$

For $g=c_1f_1+c_2f_2$, the compatible Wronskian ratios are

$$c_1=\frac{\Wr[g,f_2]}{\Wr[f_1,f_2]}, \qquad c_2=\frac{\Wr[f_1,g]}{\Wr[f_1,f_2]}.$$

These signs are used in every later boundary and Green-function formula.

Normal form and spectral sign

Locally, away from branch obstructions, set

$$y(z)= \exp\left(-\frac12\int^z p(\zeta)\,\dd\zeta\right)\psi(z).$$

The scalar equation becomes

$$\psi''(z)+T(z)\psi(z)=0, \qquad T=q-\frac12p'-\frac14p^2.$$

This is the book’s normal form or local oper form. The exponential gauge factor can be multivalued, so a local calculation does not by itself prove a global equivalence of local systems.

The principal spectral convention is

$$\left[-\hbar^2\frac{\dd^2}{\dd z^2}+V(z)\right]\psi(z) =E\psi(z), \qquad y_{\mathrm{cl}}^2=V(z)-E.$$

The same equation appears in three common notations:

Form	Coefficient dictionary
$[-\hbar^2\partial_z^2+V]\psi=E\psi$	Primary spectral convention
$\psi''=Q\psi$	$Q=(V-E)/\hbar^2$
$\psi''+T\psi=0$	$T=(E-V)/\hbar^2=-Q$

This sign check is useful on the real axis: where $E>V$, one has $T>0$ and the leading solutions are oscillatory.

Under a coordinate change $z=z(w)$, normal form is preserved by the half-density transformation

$$\widetilde\psi(w) =\left(\frac{\dd z}{\dd w}\right)^{-1/2}\psi(z(w)),$$

after choosing a local branch of the square root. The coefficient transforms as a projective connection:

$$\begin{aligned} \widetilde T(w) &=\left(\frac{\dd z}{\dd w}\right)^2 T(z(w))\\ &\quad+\frac12\{z,w\}. \end{aligned}$$

Here

$$\{z,w\} =\frac{z'''}{z'}-\frac32\left(\frac{z''}{z'}\right)^2$$

is the Schwarzian derivative. The Schwarzian term is precisely why $T(z)(\dd z)^2$ does not transform as an ordinary quadratic differential.

Local exponents and monodromy

Near a regular singular point $z_0$, write $x=z-z_0$ and

$$p(z)=\frac{p_{-1}}{x}+O(1), \qquad q(z)=\frac{q_{-2}}{x^2}+O(x^{-1}).$$

The local exponents $\rho_\pm$ are the roots of

$$\rho(\rho-1)+p_{-1}\rho+q_{-2}=0.$$

Their difference is

$$\theta=\rho_+-\rho_-,$$

defined only up to sign unless an ordering is specified. For a positively oriented loop and a nonresonant Frobenius basis, the scalar monodromy eigenvalues are $\ee^{2\pi\ii\rho_\pm}$. The exponents themselves, their difference, and the monodromy eigenvalues are related but are not identical data.

In local traceless normalization we often write

$$M_i\sim \begin{pmatrix} \ee^{\pi\ii\theta_i} & 0\\ 0 & \ee^{-\pi\ii\theta_i} \end{pmatrix}.$$

This displays a chosen determinant-one lift of the scalar projective monodromy. The exponent difference fixes only the eigenvalue ratio:

$$\operatorname{spec}(M_i) =\left\{ \sigma_i\ee^{\pi\ii\theta_i}, \sigma_i\ee^{-\pi\ii\theta_i} \right\}, \qquad \sigma_i\in\{+1,-1\}.$$

The two lifts differ by $-I$. For example, the normal-form exponents $(1\pm\theta_i)/2$ naturally give the choice $\sigma_i=-1$.

The sign here is the ambiguity of one local conjugacy class. In a global lift, signs assigned to different generators are constrained by the fundamental-group relations and together form a $\{\pm1\}$-valued character.

At resonance, $\theta_i\in\mathbb Z$, logarithmic solutions may occur and the monodromy need not be diagonalizable. An integer exponent difference alone does not prove that a logarithm is present.

Analytic continuation acts on the right:

$$\Phi^\gamma=\Phi M_\gamma.$$

Loops are based at a declared point and positive orientation is counterclockwise. We use the composition convention in which $\gamma_1\gamma_2$ traverses $\gamma_2$ first and then $\gamma_1$; with this choice,

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

Whenever a source uses the opposite path convention, its matrix products are reversed before comparison.

Branches and continuation paths

Unless a page states otherwise,

$$-\pi<\operatorname{Arg}z\leq\pi, \qquad \operatorname{Log}z =\log|z|+\ii\operatorname{Arg}z.$$

Consequently $z^\lambda=\exp(\lambda\operatorname{Log}z)$ has its principal cut on the negative real axis. This is only a default for elementary expressions. A connection problem supplies its own cuts, base point, and continuation path, and those problem-specific choices take precedence.

A path is specified up to homotopy in the punctured domain. Saying only “continue from $0$ to $1$” is insufficient when the path can pass a singularity or cut on either side.

Formal solutions and Stokes data

At an irregular singular point, a formal fundamental solution is written schematically as

$$\widehat\Phi(z) =\widehat H(z)\,\ee^{Q(z)}z^\Lambda,$$

where $\widehat H$ is generally a divergent formal series, $Q$ contains the exponential factors, and $z^\Lambda=\ee^{\Lambda\operatorname{Log}z}$ fixes the formal monodromy branch.

Canonical analytic solutions are attached to sectors and asymptotic normalizations. Sectors are indexed counterclockwise in the asymptotic coordinate declared on the page. At infinity, a page distinguishes an order in the $z$-plane from an order in the local coordinate $\xi=1/z$. Across a common boundary, the default right-action convention is

$$\Phi_{k+1}=\Phi_k S_k,$$

where $S_k$ is the Stokes matrix for that oriented crossing. Reversing the crossing replaces $S_k$ by $S_k^{-1}$. Reordering the exponential factors can exchange upper- and lower-triangular Stokes matrices.

The phrases “Stokes ray” and “anti-Stokes ray” are not used without an explicit phase condition because the terminology is reversed in parts of the literature. Exact-WKB pages state the relevant condition on $\int\sqrt{Q(z)}\,\dd z$ directly.

WKB series, cycles, and lateral sums

For

$$\left[-\hbar^2\partial_z^2+V(z)\right]\psi=E\psi,$$

use

$$\psi(z)= \exp\left(\frac1\hbar\int^z P(\zeta,\hbar)\,\dd\zeta\right).$$

The Riccati equation is

$$P^2+\hbar P'=V-E,$$

with leading branches

$$P^{(\pm)}_0=\pm y_{\mathrm{cl}}, \qquad y_{\mathrm{cl}}^2=V-E.$$

Let $P^{(+)}$ and $P^{(-)}$ denote the two formal Riccati branches:

$$P^{(\pm)2} +\hbar {P^{(\pm)}}' =V-E.$$

When $V-E$ is independent of $\hbar$, they obey

$$P^{(-)}(z,\hbar) =-P^{(+)}(z,-\hbar).$$

The even part is

$$P_{\mathrm{even}} =\frac{P^{(+)}-P^{(-)}}{2}.$$

It is not generally a solution of the nonlinear Riccati equation by itself. Closed quantum periods use this even part unless a page explicitly chooses another convention:

$$\Pi_\gamma(\hbar) =\oint_\gamma P_{\mathrm{even}}(z,\hbar)\,\dd z.$$

In the common exact-WKB notation $\psi=\exp(\int S\,\dd z)$, its odd part is $S_{\mathrm{odd}}=P_{\mathrm{even}}/\hbar$.

Cycles live on the spectral cover, not merely in the base $z$-plane. Their orientation is drawn in each example. For a symplectic pair we choose $\langle A,B\rangle=+1$; reversing a cycle reverses its period. Open paths are relative cycles and may require endpoint subtraction or pole regularization.

For a formal series with a singular Borel direction $\theta$, lateral sums are denoted

$$\mathcal S_{\theta+}, \qquad \mathcal S_{\theta-},$$

where $+$ approaches the ray from argument $\theta+0$ and $-$ from $\theta-0$. The discontinuity convention is

$$\operatorname{Disc}_\theta =\mathcal S_{\theta+}-\mathcal S_{\theta-}.$$

Every exact-WKB formula also states $\arg\hbar$, the summation direction, Stokes chamber, turning-point and pole assumptions, and the normalization of open and closed cycles.

CFT and Omega-background parameters

Liouville parameters use

$$Q_{\mathrm L}=b+b^{-1}, \qquad c=1+6Q_{\mathrm L}^2, \qquad \Delta(\alpha)=\alpha(Q_{\mathrm L}-\alpha).$$

The dimensionless centered Liouville momentum $a_{\mathrm L}=\alpha-Q_{\mathrm L}/2$ gives

$$\Delta =\frac{Q_{\mathrm L}^2}{4}-a_{\mathrm L}^2.$$

Its relation to a dimensionful gauge-theory Coulomb modulus requires the page-specific $\sqrt{\epsilon_1\epsilon_2}$ rescaling, together with any sign or imaginary-unit convention.

For the Omega background, the book fixes

$$b^2=\frac{\epsilon_2}{\epsilon_1}, \qquad \hbar=\epsilon_1.$$

Thus the Nekrasov–Shatashvili limit $\epsilon_2\to0$ at fixed $\epsilon_1$ corresponds to $b\to0$. Sources using $b^2=\epsilon_1/\epsilon_2$ are translated by $b\leftrightarrow b^{-1}$, together with the corresponding exchange of the two degenerate Kac labels.

The second-order BPZ branch uses the degenerate primary conventionally called $(2,1)$, with uncentered momentum $-b/2$ and fusion shifts by $\pm b/2$. Because some sources call the dual field $(2,1)$, the null-vector equation and fusion shifts—not the label alone—determine the map.

In the convention used here, the null-vector relation is

$$\left(L_{-1}^2+b^2L_{-2}\right)V_{-b/2}=0.$$

Keep the following objects separate:

$$Z_{\mathrm{Nek}}(\epsilon_1,\epsilon_2), \qquad \mathcal W_{\mathrm{NS}} =\lim_{\epsilon_2\to0} \epsilon_2\log Z_{\mathrm{Nek}}(\epsilon_1,\epsilon_2).$$

An all-orders NS expression is not automatically a nonperturbatively complete spectrum. Operator ordering, mass shifts, polarization, quantum mirror map, cycle basis, and boundary conditions remain model-dependent inputs.

Determinant terminology

The word “determinant” is qualified throughout:

Name	Construction
Boundary Wronskian, Jost function, or Evans function	Wronskian of solutions satisfying two boundary normalizations
Fredholm determinant	$\det(I+K)$ for trace-class $K$; another Schatten class requires a named regularized determinant such as $\det_p$
Zeta-regularized determinant	Defined when the spectral zeta function, with a spectral cut when needed, continues regularly to the origin
Canonical-product spectral determinant	Entire function reconstructed from zeros with stated genus and normalization
Isomonodromic Fredholm determinant	Operator determinant representing an isomonodromic tau function

These objects can coincide up to explicit factors in special problems, but the coincidence is a theorem to prove, not a naming convention.

Status and cross-check labels

Advanced statements use the following fixed meanings:

Label	Meaning
Theorem	Proved under the hypotheses stated with the result
Proposition or derivation	Established directly from earlier results in the text
Formal identity	Equality in a ring of formal series; no convergence claim
Conditional exact statement	Exact after the listed summability, topology, analyticity, and boundary assumptions
Conjectural correspondence	Supported but not proved in the stated generality
Numerical observation	Reproducible evidence with precision, choices, and error checks recorded

When feasible, a substantive formula receives two checks chosen from:

1. an exactly solvable or hypergeometric limit;
2. a Wronskian or determinant identity;
3. direct numerical integration;
4. a recurrence or continued fraction;
5. an independent period calculation;
6. spectral data;
7. confluence, symmetry, or coordinate covariance.

Global KaTeX macros

The site defines only four global convenience macros:

Source	Rendered meaning
\dd	differential $\dd z$
\ee	exponential base $\ee$
\ii	imaginary unit $\ii$
\Wr	Wronskian operator $\Wr$

Pages do not define local macros. This keeps copied formulas readable and prevents the same command from changing meaning between chapters.

Exercises

1. Basis covariance. Starting from $\Phi_\beta=\Phi_\alpha C_{\alpha\beta}$ and $\widetilde\Phi_\nu=\Phi_\nu H_\nu$, derive the transformation law for the connection matrix.

Solution

Substitute $\Phi_\nu=\widetilde\Phi_\nu H_\nu^{-1}$:

$$\widetilde\Phi_\beta H_\beta^{-1} =\widetilde\Phi_\alpha H_\alpha^{-1} C_{\alpha\beta}.$$

Multiplication on the right by $H_\beta$ gives

$$\widetilde\Phi_\beta =\widetilde\Phi_\alpha \left(H_\alpha^{-1}C_{\alpha\beta}H_\beta\right),$$

so $\widetilde C_{\alpha\beta} =H_\alpha^{-1}C_{\alpha\beta}H_\beta$.

2. Spectral translation. Translate $-\hbar^2\psi''+V\psi=E\psi$ into both $\psi''=Q\psi$ and $\psi''+T\psi=0$. Check the sign in a classically allowed real interval.

Solution

Rearranging gives

$$\psi''=\frac{V-E}{\hbar^2}\psi,$$

so $Q=(V-E)/\hbar^2$ and $T=-Q=(E-V)/\hbar^2$. If $E>V$ and $\hbar$ is real, then $T>0$ and $\psi''+T\psi=0$ has oscillatory leading behavior, as expected.

3. Exponents versus monodromy. Explain why replacing a local exponent $\rho$ by $\rho+n$, $n\in\mathbb Z$, leaves the monodromy eigenvalue unchanged, and why this does not settle the resonant logarithm question.

Solution

For a positive loop,

$$\ee^{2\pi\ii(\rho+n)} =\ee^{2\pi\ii\rho}\ee^{2\pi\ii n} =\ee^{2\pi\ii\rho}.$$

Monodromy eigenvalues therefore remember exponents only modulo integers. When two exponents differ by an integer, the Frobenius recurrence may or may not force a logarithmic second solution. That information is contained in the local equation and the Jordan form of monodromy, not in the eigenvalues alone.

References

• NIST DLMF, Differential equations, for Fuchs–Frobenius theory, irregular singularities, and classical WKB conventions.
• K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida, From Gauss to Painlevé, for monodromy and isomonodromic conventions.
• K. Iwaki, Les Houches Lectures on Exact WKB Analysis and Painlevé Equations, for current exact-WKB notation and hypotheses.
• S. Ribault, Conformal Field Theory on the Plane, for Liouville and degenerate-field conventions.
• N. Nekrasov and S. Shatashvili, “Quantization of integrable systems and four dimensional gauge theories”, for the NS limit and twisted superpotential.