Appendix A: Notation and Normalization Dictionary

The notation and status page fixes the house conventions used throughout this book. This appendix has a different job: it is a translation desk. Given a formula from another source, it identifies which changes are mere renamings, which invert or conjugate matrices, which rescale periods, and which alter the normalized analytic object.

The governing principle is simple:

Translate the operator and its normalized bases before translating the name of a special function, period, determinant, or correspondence.

The fastest reliable check is a round trip. Convert the source data into the book’s conventions, reconstruct one invariant—an exponent difference, Wronskian ratio, monodromy trace, first series coefficient, or period intersection—and then convert back.

The five-entry translation passport

Record the differential operator, the coordinate and scalar gauge, the ordered local or sectorial bases, the continuation paths or cycles, and the parameter tuple with its dimensions. A dictionary missing any one of these entries may preserve an equation class while changing the function being computed.

Four forms of the same local operator

Start from the scalar equation

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

With $Y=(y,y')^{\mathsf T}$, its companion system is

$$\frac{\dd Y}{\dd z} = \begin{pmatrix} 0&1\\ -q&-p \end{pmatrix} Y.$$

Removing the first derivative by

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

gives the local oper or normal form

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

The same equation in the book’s Schrödinger convention is

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

Many asymptotics sources instead write $\psi''=Q\psi$. Their coefficient is

$$Q=\frac{V-E}{\hbar^2}=-T.$$

This minus sign is not cosmetic: on a real classically allowed interval, $E>V$ gives $T>0$ and oscillatory solutions.

Source form	Book translation	Data that still require a check
$y''+py'+qy=0$	Primary scalar form	Coordinate, scalar normalization, and branches
$Y'=AY$	Specify whether solutions are columns; for the companion choice, $A=\bigl(\begin{smallmatrix}0&1\\-q&-p\end{smallmatrix}\bigr)$	Left gauge versus right basis action
$\psi''+T\psi=0$	Local oper/normal form	The half-density branch and global single-valuedness of the gauge
$\psi''=Q\psi$	$T=-Q$	Turning-point and Stokes-ray phase conventions
$[-\hbar^2\partial_z^2+V]\psi=E\psi$	$T=(E-V)/\hbar^2$	Ordering, domain, endpoint lines, and spectral sheet

A gauge transformation of a column system acts on the left:

$$\widetilde\Phi=G(z)\Phi, \qquad \widetilde A = GAG^{-1}+G'G^{-1}.$$

A constant change of the ordered solution basis acts on the right: $\widetilde\Phi=\Phi H$. Confusing these two actions is one of the most common sources of reversed connection matrices.

The normal-form gauge is local. If its exponential factor or half-density square root is multivalued, it preserves the local projective equation but does not automatically identify the global local system or its physically normalized endpoint solutions.

The scalar–system crosswalk and Liouville-to-oper derivation give the global gauge and half-density details suppressed by this desk card.

Wronskians and connection matrices

The book uses

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

and the connection direction

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

These two declarations determine most sign and inverse conversions:

A source declares	Translation into the book
$W_{\mathrm s}[f,g]=f'g-fg'$	$W_{\mathrm s}[f,g]=-\Wr[f,g]$
$\Phi_\alpha=\Phi_\beta\widehat C_{\beta\alpha}$	$\widehat C_{\beta\alpha}=C_{\alpha\beta}^{-1}$
$\widetilde\Phi_\nu=\Phi_\nu H_\nu$	$\widetilde C_{\alpha\beta}=H_\alpha^{-1}C_{\alpha\beta}H_\beta$
Row solutions with monodromy or connection acting on the left	Rebuild the matrix relation from the declared row basis; a blind transpose is not a general conversion rule
An unweighted “constant Wronskian” for $p\neq0$	The constant object is $\exp(\int p\,\dd z)\Wr[f,g]$ on the chosen branch

For an ordered basis $(f_1,f_2)$ and $g=c_1f_1+c_2f_2$, the book’s coefficient formulas are

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

They provide an immediate audit: reverse the source’s Wronskian sign or basis order and verify that the reconstructed $g$ is unchanged.

The full derivation, cocycle law, and resonant limiting prescription are on the connection-matrix and Wronskian page.

For spectral locations, two boundary functions may be equivalent on a chart $U$ through

$$\widetilde D=uD, \qquad u\in\mathcal O^\times(U).$$

This preserves zeros and their orders. It does not by itself preserve a reported residue: the numerator and endpoint normalization must transform with the same change of frame.

Monodromy, paths, and Stokes crossings

For a positively oriented loop, the book continues a column fundamental matrix on the right:

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

The path product $\gamma_1\gamma_2$ traverses $\gamma_2$ first and then $\gamma_1$, so

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

When a source traverses products from left to right, its written matrix order is reversed before comparison. The safe procedure is to follow one test column around two named loops rather than infer the order from notation alone.

Near a nonresonant regular singularity, scalar exponents $\rho_\pm$ give monodromy eigenvalues $\ee^{2\pi\ii\rho_\pm}$. The exponent difference $\theta=\rho_+-\rho_-$ fixes only their ratio. After choosing a determinant-one half-density or traceless-system lift, introduce the central sign $s_{\mathrm{lift}}\in\{+1,-1\}$ and write

$$\operatorname{spec}M = \left\{ s_{\mathrm{lift}}\ee^{\pi\ii\theta}, s_{\mathrm{lift}}\ee^{-\pi\ii\theta} \right\}, \qquad \operatorname{tr}M = 2s_{\mathrm{lift}}\cos(\pi\theta).$$

The natural scalar normal form has local powers $(1\pm\theta)/2$, hence $s_{\mathrm{lift}}=-1$. A residue-normalized traceless Fuchsian system with local eigenvalues $\pm\theta/2$ has $s_{\mathrm{lift}}=+1$. They represent the same projective class but not the same $SL(2)$ lift. Replacing $\theta$ by $-\theta$ exchanges the ordered eigenlines; replacing an exponent by an integer shift leaves its eigenvalue unchanged. At integral exponent difference, inspect the Frobenius obstruction before introducing a logarithmic basis.

Chapter 15 also uses a half-difference convention. Its translation is

$$\widehat\theta=\frac{\theta}{2}, \qquad \operatorname{tr}M = 2s_{\mathrm{lift}}\cos(\pi\theta) = 2s_{\mathrm{lift}}\cos(2\pi\widehat\theta),$$

with the same $s_{\mathrm{lift}}$ on both sides. The cosine equality does not choose that central sign, exponent lift, or ordered eigenline.

At an irregular singularity, the book writes a formal solution as

$$\widehat\Phi = \widehat H\,\ee^{Q_{\mathrm{irr}}}z^\Lambda$$

and crosses adjacent sectors by

$$\Phi_{k+1}=\Phi_kS_k.$$

Reversing the crossing gives $S_k^{-1}$. Reordering exponential factors conjugates by the permutation matrix and may exchange upper- and lower-triangular Stokes matrices. The labels “Stokes ray” and “anti-Stokes ray” are not translated by name: compare the source’s explicit phase condition on the exponential difference.

With $m$ sector crossings, $\Phi_{j+m}=\Phi_jM_{\mathrm f}$, and the preceding right-action convention, positive local monodromy is

$$M_{\mathrm{loc}} = M_{\mathrm f} S_{m-1}^{-1}\cdots S_0^{-1}.$$

Changing the crossing direction, sector numbering, or side of matrix action changes this ordered word; the factors must be rederived together.

Quantity	Convention-dependent presentation	More stable comparison
Local exponent	Sign, ordering, and integer lift	Eigenvalue ratio and the ordered local basis
Full versus half exponent difference	$\theta$ versus $\widehat\theta=\theta/2$	$2s_{\mathrm{lift}}\cos(\pi\theta)=2s_{\mathrm{lift}}\cos(2\pi\widehat\theta)$
Monodromy matrix	Base point, loop product, left/right action	Conjugacy class plus framing
Stokes multiplier	Sector order and exponential ordering	Oriented sectorial connection
Composite monodromy	Choice and order of puncture loops	Trace on a declared lift for a declared loop class
Logarithmic solution	Frobenius normalization	Nilpotent/Levelt data and the actual obstruction

For the detailed path algebra, see connection problems and monodromy. For irregular endpoints, use the wild-monodromy ledger, which keeps formal monodromy, sector order, and Stokes matrices separate.

Liouville, CFT, and Omega-background coordinates

The book’s Liouville conventions are

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

The centered momentum

$$a_{\mathrm L} = \alpha-\frac{Q_{\mathrm L}}2$$

gives

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

Thus a source using $\alpha=Q_{\mathrm L}/2+\ii P$ has $a_{\mathrm L}=\ii P$. Liouville reflection $\alpha\mapsto Q_{\mathrm L}-\alpha$ becomes $a_{\mathrm L}\mapsto-a_{\mathrm L}$; it does not by itself choose an ordered Frobenius exponent.

The ordered Omega-background convention is

$$b^2=\frac{\epsilon_2}{\epsilon_1}, \qquad \hbar=\epsilon_1, \qquad \epsilon_2\longrightarrow0.$$

The NS limit therefore has $b\to0$. A source using $b_{\mathrm s}^2=\epsilon_1/\epsilon_2$ is translated by $b_{\mathrm s}=b^{-1}$ after choosing the square-root branch. Its two degenerate Kac labels are correspondingly exchanged. The book’s second-order branch has momentum $-b/2$, fusion shifts $\pm b/2$, and null vector

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

For the finite-Omega AGT dictionary, choose the sign of $b$ and fix the compatible square root by

$$\begin{aligned} \varepsilon_\star &:= b\epsilon_1 = \frac{\epsilon_2}{b}, & \varepsilon_\star^2 &= \epsilon_1\epsilon_2,\\ \epsilon_\Sigma &= \epsilon_1+\epsilon_2 = \varepsilon_\star Q_{\mathrm L},\\ \alpha_f &= \frac{Q_{\mathrm L}}2 + \frac{p_f}{\varepsilon_\star}, & \alpha_{0t} &= \frac{Q_{\mathrm L}}2 + \frac{a_{\mathrm C}}{\varepsilon_\star}. \end{aligned}$$

Here $p_f$ is a centered dimensionful puncture mass and $a_{\mathrm C}$ is the dimensionful Coulomb modulus in the $0t$ channel. The two degenerate probes read two different exponent coordinates:

$$\theta_f^{[1]} = \frac{2p_f}{\epsilon_1}, \qquad \theta_f^{[2]} = \frac{2p_f}{\epsilon_2}.$$

These relations are exact only after the square-root, reflection representative, flavor basis, and channel are fixed. The mass–momentum crosswalk develops their inverse maps and resonant limits.

Source notation	Book notation or required declaration
$\alpha=Q_{\mathrm L}/2+\ii P$	$a_{\mathrm L}=\ii P$
$b_{\mathrm s}^2=\epsilon_1/\epsilon_2$	$b_{\mathrm s}=b^{-1}$ and exchange the two degenerate branches
A dimensionful Coulomb modulus $a$	Divide by the declared Omega-background scale and include the page-specific sign or imaginary unit before comparing with $a_{\mathrm L}$
$Z_{\mathrm{Nek}}$	The book means $Z_{\mathrm{cl}}Z_{\mathrm{1-loop}}Z_{\mathrm{inst}}$; many sources use the same name for $Z_{\mathrm{inst}}$ alone
$\mathcal W$	The book’s local NS object is $\mathcal W_{\mathrm{NS}}^{\mathrm{loc}}=\lim_{\epsilon_2\to0}\epsilon_2\log Z_{\mathrm{Nek}}$ after declared subtractions; check retained perturbative and $U(1)$ factors
A “conformal block”	Declare chiral block versus full correlator, normalization of three-point factors, channel, and analytic continuation

Thus setting $\epsilon_2=0$ inside a finite-Omega function is not the NS operation. One first forms the logarithm in a declared chamber and extracts its connected simple-pole coefficient.

The word “block” also labels three different regimes. Their exact monodromy-weight crosswalk is:

$$\begin{array}{c|c|c} \text{regime} & \text{momentum lift} & \text{weight} \\ \hline c=1 & a_{\mathrm L}=\ii\theta/2 & \Delta^{(1)}=\theta^2/4 \\ b\to0 & b\,a_{\mathrm L}\to\theta/2 & \delta=(1-\theta^2)/4 \end{array}$$

In the analytic $c=1$ chart,

$$Q_{\mathrm L}=0, \qquad b=\pm\ii, \qquad \vartheta=\frac{\theta}{2}, \qquad \rho=\frac{\sigma}{2}$$

for the common source variables $\vartheta$ and $\rho$. In the classical chart, one chosen branch has

$$\eta := \lim_{b\to0}b\alpha = \frac{1+\theta}{2}, \qquad \delta = \eta(1-\eta).$$

In particular, $b=1$ gives $c=25$, not $c=1$.

Block object	Meaning in this book
Analytic $c=1$ block	Ingredient in a Fourier expansion of an isomonodromic tau function; not a real-$b$ unitary Liouville correlator
Classical block $f$	A declared large-central-charge limit such as $f=\lim_{b\to0}b^2\log\mathcal F$ with heavy weights scaled simultaneously
Degenerate BPZ block	A chiral block with an extra degenerate coordinate; its fusion branch and normalization produce an ODE solution only after the BPZ gauge is fixed

The analytic $c=1$ and classical-block constructions solve different connection problems. Neither label can be inferred from the word “Liouville” alone. The two-regime comparison keeps their tau-function and accessory outputs separate.

The book deliberately adds subscripts to recurrent symbol collisions:

Symbol	Its mathematical role
$Q(z)$	Coefficient in the source form $\psi''=Q(z)\psi$
$Q_{\mathrm L}$	Liouville background charge
$Q(E)$	Baxter function or spectral determinant in an ODE/IM model
$q_{\mathrm H}$	Heun accessory parameter in the numerator of the ODE
$\mathfrak q$ or $q_{\mathrm{UV}}$	Exponentiated gauge coupling or sewing coordinate
$q_{\mathrm{IR}}$	A period-defined effective coupling
$a_{\mathrm H}$	Position of the fourth Heun singularity
$a_{\mathrm C}$	Gauge-theory Coulomb modulus
$a_{\mathrm L}$	Centered Liouville momentum
$\delta_i$	Classical conformal weight $(1-\theta_i^2)/4$
$\delta$	General- or confluent-Heun exponent parameter
$\theta_i$	Oriented local exponent difference
$\theta_{\mathrm B}$	Borel–Laplace summation direction

An equality such as $q_{\mathrm{UV}}=t$ is a chart choice in a declared AGT dictionary, not a universal identification. The AGT passport explains the required finite-Omega, defect, mass, and normalization entries.

Four accessory-like quantities also occupy different rungs:

$$\begin{aligned} u_{\mathrm M} &:= -\mathfrak q\, \partial_{\mathfrak q} \mathscr F_{\mathrm{NS}}, & c_t^{\mathrm{op}} &:= \operatorname*{res}_{z=t} T_{\mathrm{op}}(z),\\ H &:= t(t-1)c_t^{\mathrm{op}}, & q_{\mathrm H} &:= \text{standard-Heun accessory}. \end{aligned}$$

The first is a Matone coupling derivative, the second a moving-pole oper residue, and the third a compact four-puncture coefficient. The final $q_{\mathrm H}$ is obtained only after the coordinate and scalar-gauge conversion, generally by an affine map. The Matone–accessory ledger states which equalities are definitions, defect consequences, or model-dependent calibrations.

WKB, Seiberg–Witten, and monodromy cycles

All period comparisons begin by naming the surface on which the cycle lives. For the WKB spectral cover

$$\Sigma:\quad y_{\mathrm{cl}}^2=V(z)-E,$$

the book chooses an oriented symplectic pair with

$$\langle A,B\rangle=+1.$$

Reversing a cycle reverses its period. If a source declares $\langle B,A\rangle=+1$, one convenient translation is $A_{\mathrm{book}}=A_{\mathrm s}$ and $B_{\mathrm{book}}=-B_{\mathrm s}$.

The book’s closed WKB period is

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

In sources using $\psi=\exp(\int S\,\dd z)$,

$$S_{\mathrm{odd}} = \frac{P_{\mathrm{even}}}{\hbar}, \qquad \Pi_\gamma = \hbar\oint_\gamma S_{\mathrm{odd}}\,\dd z.$$

On the quantum Seiberg–Witten pages the canonical coordinate is $Q$, and the same normalization becomes the compact ladder

$$\begin{aligned} P_{\mathrm{ev}} &= P_{\mathrm{even}} = \frac{P_+-P_-}{2}, & \Pi_\gamma &= \oint_\gamma P_{\mathrm{ev}}\,\dd Q,\\ a_\gamma^{\mathrm{WKB}} &= \frac{\Pi_\gamma}{2\pi\ii}, & V_\gamma &= \frac{\Pi_\gamma}{\hbar}, \qquad \mathcal V_\gamma=\ee^{V_\gamma}. \end{aligned}$$

Here $P_\pm$ are the two Riccati branches. The displayed identities define formal closed-cycle objects after any required pole regularization. A sectorial analytic Voros symbol additionally needs a Borel direction, lateral side, and Stokes chamber.

In the book’s rank-one Seiberg–Witten frame,

$$\Pi_{\mathrm{SW}} = \begin{pmatrix} a_D\\ a \end{pmatrix}, \qquad a=\oint_A\lambda_{\mathrm{SW}}, \qquad a_D=\oint_B\lambda_{\mathrm{SW}},$$

with $A\circ B=+1$. The conditional quantum dictionary uses

$$\mathscr F_{\mathrm{NS}} = \hbar\mathcal W_{\mathrm{NS}}, \qquad a_D^{\mathrm{NS}} = -\frac{1}{2\pi\ii} \partial_a\mathscr F_{\mathrm{NS}}.$$

These formulas assume the local subtraction scheme and electric frame of the source page. They do not assert that every WKB period equals an NS derivative. That comparison additionally fixes the operator, curve differential, quantum mirror map, cycle, regulator, and analytic chamber. On Chapter 10’s real-cosh mechanical continuation, the Planck constants are related by

$$\hbar=\epsilon_1=-\ii\hbar_{\mathrm m},$$

not by $\epsilon_1=\hbar_{\mathrm m}$ on the real mechanical slice. The conditional WKB–SW dictionary derives every rung, while the monodromy–SW–WKB cycle crosswalk tracks the associated lattices and factors of two.

Object	Lives on	Normalization questions
Puncture or composite-monodromy loop	Base punctured curve	Base point, loop product, $SL(2)$ lift
WKB cycle $\gamma$	Double cover $\Sigma$	Sheet, orientation, pole subtraction, $\arg\hbar$
SW cycle $A$ or $B$	SW curve	Differential, electric/magnetic polarization, and factors such as $2\pi\ii$
Open connection path	Relative homology group	Endpoint tangents, regularization, and lateral side
TBA charge/cycle	Declared charge lattice	Intersection pairing, chamber, and kernel convention

A monodromy loop, an SW cycle, and a WKB cycle may correspond after a specific lift and degeneration, but their common letter does not establish that map. Compare intersection forms and differentials before comparing numbers.

Heun reference cards

The book’s general Heun equation is

$$\begin{aligned} y'' &+ \left( \frac{\gamma}{z} + \frac{\delta}{z-1} + \frac{\epsilon}{z-a} \right)y'\\ &+ \frac{\alpha\beta z-q} {z(z-1)(z-a)} y =0, \end{aligned}$$

with

$$\epsilon = \alpha+\beta+1-\gamma-\delta.$$

Its exponent-zero germ at $z=0$ is

$$H\!\ell(a,q;\alpha,\beta,\gamma,\delta;z) = 1+\frac{q}{a\gamma}z+O(z^2).$$

DLMF, Wolfram, and Maple use this ordered six-parameter tuple directly for their generic local general-Heun germ. The identity is local and assumes the unit-normalized recurrence is unobstructed; continued principal values still depend on compatible cuts and paths.

The DLMF and Wolfram confluent-Heun equation used here is

$$\begin{aligned} y'' &+ \left( \frac{\gamma}{z} + \frac{\delta}{z-1} + \epsilon \right)y'\\ &+ \frac{\alpha z-q}{z(z-1)} y =0, \end{aligned}$$

with normalized germ

$$C_0(z)=1-\frac q\gamma z+O(z^2).$$

Maple instead calls

$$\operatorname{HeunC} \left( \alpha_{\mathrm M}, \beta_{\mathrm M}, \gamma_{\mathrm M}, \delta_{\mathrm M}, \eta_{\mathrm M}, z \right).$$

The forward map to the preceding DLMF tuple is

$$\begin{aligned} \epsilon&=\alpha_{\mathrm M},& \gamma&=\beta_{\mathrm M}+1,\\ \delta&=\gamma_{\mathrm M}+1,& q&=\frac12\left( \alpha_{\mathrm M}-\beta_{\mathrm M}-\gamma_{\mathrm M} \right.\\ &&&\qquad\left. +\alpha_{\mathrm M}\beta_{\mathrm M} -\beta_{\mathrm M}\gamma_{\mathrm M} \right)-\eta_{\mathrm M},\\ \alpha &= \alpha_{\mathrm M} \left( 1+\frac{\beta_{\mathrm M}+\gamma_{\mathrm M}}2 \right) +\delta_{\mathrm M}.&& \end{aligned}$$

The inverse map is

$$\begin{aligned} \alpha_{\mathrm M}&=\epsilon,& \beta_{\mathrm M}&=\gamma-1,\\ \gamma_{\mathrm M}&=\delta-1,& \delta_{\mathrm M} &= \alpha-\frac{\epsilon(\gamma+\delta)}2,\\ \eta_{\mathrm M} &= \frac12+\frac{\gamma(\epsilon-\delta)}2-q.&& \end{aligned}$$

The first derivative crosses with the parameters. Directly in Maple’s tuple,

$$\begin{aligned} \operatorname{HeunC}'(0) &= -\frac q\gamma\\ &= \frac{ (1+\gamma_{\mathrm M}-\alpha_{\mathrm M})\beta_{\mathrm M} +\gamma_{\mathrm M}-\alpha_{\mathrm M}+2\eta_{\mathrm M} }{ 2(\beta_{\mathrm M}+1) }. \end{aligned}$$

The first expression is the DLMF-tuple slope and the second is the native Maple slope. Their equality, together with $\gamma=\beta_{\mathrm M}+1$, is a faster audit than comparing distant principal values. The displayed local normalization excludes the resonant denominator $\beta_{\mathrm M}=-1$.

Family	Reference tuple and book object	Compact software crosswalk
General Heun	$(a,q,\alpha,\beta,\gamma,\delta)$; $H\!\ell$ at $0$	DLMF, Wolfram, and Maple are direct on a generic local chart
Confluent Heun	$(q,\alpha,\gamma,\delta,\epsilon)$; $C_0$ at $0$	DLMF/Wolfram direct; Maple uses the map above
Doubly confluent Heun	$(q,\alpha,\gamma,\delta)$; ordinary-point or sectorial bases	Wolfram rescales and moves the base point; Maple also changes coordinate and gauge
Biconfluent Heun	$(q,\alpha,\gamma,\delta)$; $B_0$ at $0$	Wolfram direct specialization; Maple uses a square-root rescaling
Triconfluent Heun	$(q,\alpha,\gamma)$; $(T_0,T_1)$ at $0$	Wolfram direct specialization; Maple uses an affine and cubic-root normalization

The exact DCHE, BHE, and THE transformations, including their base-point and derivative transport, are tabulated on the Heun parameter-crosswalk page. Confluence changes local categories and sectorial normalizations; a family name alone never identifies a named function.

The canonical-basis atlas specifies which unit-leading, logarithmic, ordinary-point, or sectorial object each book symbol denotes.

Minimal translation certificate

A research-grade translation should make the following record reversible:

1. Write both differential operators, not only their family names.
2. Give the coordinate map $z=\phi(w)$ and scalar gauge $y=\chi(w)u$.
3. State the nonzero multiplier relating the two operator expressions.
4. Map every regular singular point, irregular direction, base point, and excluded divisor.
5. Transport unit-leading, logarithmic, or sectorial basis normalizations.
6. Record Wronskian sign, connection direction, continuation action, loop product, and cycle orientation.
7. Give both ordered parameter tuples, dimensions, inverse map, and branch choices.
8. Check one local coefficient and one global invariant in each direction.

For a claimed physical correspondence, add the operator domain, boundary flags, polarization, and observable normalization. An ODE crosswalk can be exact while the spectral or response-function identification remains conditional.

Common pitfalls

Equating normal form with global gauge equivalence. The first-derivative removal may be multivalued and changes the natural endpoint normalization. Carry its branch and leading factor into the connection problem.

Transposing a left-action formula mechanically. Row versus column solutions, reversed basis order, and reversed paths lead to different matrix operations. Reconstruct one continued basis vector to determine the correct inverse or transpose.

Equating dimensionless and dimensionful momenta. Liouville momentum, Coulomb modulus, and WKB action can differ by $\sqrt{\epsilon_1\epsilon_2}$, $\hbar$, signs, imaginary units, and additive mass shifts. Dimensional analysis is the first dictionary check.

Trusting a software function name at an exceptional parameter. A local unit-normalized recurrence may fail at resonant parameters, and a confluent family may require a sectorial rather than ordinary-point normalization. Translate the ODE and initial or asymptotic data, then test a first coefficient.

Exercises

1. Convert three simultaneous convention changes

A source writes $\psi''=Q\psi$ with $Q=x^2-E$, defines $W_{\mathrm s}[f,g]=f'g-fg'$, and uses $\Phi_0=\Phi_1\widehat C_{10}$. Translate all three declarations into the book’s conventions for $\hbar=1$.

Solution

The Schrödinger potential is $V=x^2$ and

$$T=E-x^2=-Q.$$

The Wronskian changes sign:

$$\Wr[f,g]=-W_{\mathrm s}[f,g].$$

Since the book writes $\Phi_1=\Phi_0C_{01}$, the source matrix is its inverse:

$$C_{01}=\widehat C_{10}^{-1}.$$

These conversions preserve the solution space. Endpoint normalizations and the spectral domain still have to be compared.

2. Exchange the Omega planes

A source uses $b_{\mathrm s}^2=\epsilon_1/\epsilon_2$ and calls the field of momentum $-1/(2b_{\mathrm s})$ the $(1,2)$ degenerate field. Translate its parameter, Kac label, and NS limit into the book’s conventions.

Solution

With the same ordered Omega pair,

$$b_{\mathrm s}=b^{-1}.$$

Therefore

$$-\frac1{2b_{\mathrm s}} = -\frac b2.$$

This is the branch called $(2,1)$ in the book. The source limit $b_{\mathrm s}\to\infty$ becomes the book’s $b\to0$, equivalent to $\epsilon_2\to0$ at fixed $\epsilon_1=\hbar$. The null-vector equation and fusion shifts, rather than the Kac label alone, certify the conversion.

3. Test the Maple confluent-Heun map

Convert the Maple tuple

$$(\alpha_{\mathrm M},\beta_{\mathrm M},\gamma_{\mathrm M}, \delta_{\mathrm M},\eta_{\mathrm M}) =(2,1,3,5,7)$$

to the DLMF tuple $(q,\alpha,\gamma,\delta,\epsilon)$ and compute the first derivative of the normalized germ.

Solution

The map gives

$$\epsilon=2, \qquad \gamma=2, \qquad \delta=4,$$

and

$$q=-\frac{17}{2}, \qquad \alpha=11.$$

Hence

$$C_0'(0) = -\frac q\gamma = \frac{17}{4}.$$

The inverse formulas return $\delta_{\mathrm M}=5$ and $\eta_{\mathrm M}=7$, completing the round trip.

4. Reverse a symplectic convention

A source declares $\langle B_{\mathrm s},A_{\mathrm s}\rangle=+1$. Keep its $A$-cycle as the book’s $A$-cycle and translate the $B$-cycle so that $\langle A,B\rangle=+1$. What happens to the dual period?

Solution

Antisymmetry gives $\langle A_{\mathrm s},B_{\mathrm s}\rangle=-1$. Choose

$$A=A_{\mathrm s}, \qquad B=-B_{\mathrm s}.$$

Then $\langle A,B\rangle=+1$, while every $B$-period changes sign:

$$\oint_B\lambda = -\oint_{B_{\mathrm s}}\lambda.$$

If a quantization condition also changes the orientation of an open path or the sign of the differential, those transformations must be applied separately.

5. Round-trip a half exponent and a Stokes crossing

A source writes a local half exponent difference $\widehat\theta=3/10$, uses the traceless-system lift $s_{\mathrm{lift}}=+1$, and reports

$$\operatorname{tr}M = 2\cos(2\pi\widehat\theta).$$

It also relates two adjacent sectorial bases by $\Phi_k=\Phi_{k+1}\widehat S$. Translate both declarations into the book’s conventions, then convert back.

Solution

The book’s full exponent difference is

$$\theta=2\widehat\theta=\frac35,$$

so on the same declared traceless-system lift

$$\operatorname{tr}M = 2\cos(\pi\theta) = 2\cos\left(\frac{3\pi}{5}\right).$$

In the natural scalar-oper lift, $s_{\mathrm{lift}}=-1$ instead, so the trace of the corresponding central lift is

$$-2\cos\left(\frac{3\pi}{5}\right).$$

The book records a forward sector crossing as $\Phi_{k+1}=\Phi_kS_k$. Inverting the source relation gives

$$S_k=\widehat S^{-1}.$$

The reverse translation restores $s_{\mathrm{lift}}=+1$, divides $\theta$ by two, and inverts $S_k$, returning $\widehat\theta=3/10$ and $\widehat S$. The round trip assumes the same loop, $SL(2)$ lift, ordered exponential factors, and oriented sector crossing; none of those data follows from the trace alone.

References

• NIST DLMF, §1.13 Differential Equations, §2.7 Differential Equations, §31.2 Heun’s Equation, and §31.12 Confluent Forms, for Liouville reduction, Abel’s identity, Frobenius theory, and the Heun reference forms.
• K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida, From Gauss to Painlevé, for monodromy, character varieties, and isomonodromic conventions.
• M. Jimbo, T. Miwa, and K. Ueno, “Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coefficients. I. General Theory and Tau-Function”, Physica D 2 (1981), 306–352, for formal, Stokes, and tau-function data.
• A. Ronveaux, ed., Heun’s Differential Equations, Oxford University Press (1995), for the classical Heun-family parameterizations.
• Maplesoft, “HeunG—The Heun General Function” and “HeunC—The Heun Confluent Function”, and Wolfram Research, “HeunG—General Heun Function” and “HeunC—Confluent Heun Function”, for the software tuples and origin normalizations translated above.
• A. Beilinson and V. Drinfeld, Opers, for the geometric oper convention underlying the scalar normal form.
• S. Ribault, Conformal Field Theory on the Plane, for Liouville momenta, reflection, and degenerate-field conventions.
• O. Gamayun, N. Iorgov, and O. Lisovyy, “Conformal Field Theory of Painlevé VI”, JHEP 10 (2012) 038, for the analytic $c=1$ conformal-block expansion of the Painlevé VI tau function.
• L. F. Alday, D. Gaiotto, and Y. Tachikawa, “Liouville Correlation Functions from Four-Dimensional Gauge Theories”, and L. F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa, and H. Verlinde, “Loop and Surface Operators in N=2 Gauge Theory and Liouville Modular Geometry”, for the finite-Omega AGT and degenerate-defect dictionaries.
• N. Seiberg and E. Witten, “Electric–Magnetic Duality, Monopole Condensation, and Confinement in N=2 Supersymmetric Yang–Mills Theory”, for the original rank-one curve, differential, and electric–magnetic period framework.
• N. Nekrasov and S. Shatashvili, “Quantization of Integrable Systems and Four Dimensional Gauge Theories”, for the NS limit, quantum periods, and twisted-superpotential conventions.
• L. Hollands and O. Kidwai, “Higher Length–Twist Coordinates, Generalized Heun’s Opers, and Twisted Superpotentials”, for convention-complete comparisons between oper coordinates, twisted superpotentials, and exact WKB periods.
• K. Iwaki and T. Nakanishi, “Exact WKB Analysis and Cluster Algebras”, for Voros symbols, lateral summation, and Stokes automorphisms.
• D. Gaiotto, “Opers and TBA”, for the relation among opers, WKB charge lattices, and TBA coordinates in a fixed convention.