Comparison with Exact WKB and NS Periods

Page 6 ends with a numerical pseudoenergy and an evidence record. Neither is yet a WKB or Nekrasov–Shatashvili period. The same phrase—quantum period—is often attached to four different objects: a formal WKB cycle integral, its directional Borel sum, the logarithm of a TBA $Y$-function, and a derivative of an NS free energy. They can describe the same analytic coordinate in important examples, but only after their operator, cycle, normalization, flat coordinate, and chamber data have been matched. A numerical agreement without that passport is not evidence for a new correspondence.

This page turns the bridges proved earlier into a comparison protocol. The TBA–WKB lane is available, for example, in the minimal polynomial chambers of Page 4. The WKB–NS lane is available for specified quantum Seiberg–Witten curves such as the calibrated pure-$SU(2)$ modified-Mathieu operator of Chapter 10. These two facts do not imply that every TBA has an NS interpretation, or even that the two named lanes close on the same model. Page 8 will apply the protocol to a controlled example.

Status ledger

Claim	Status and limitation
Formal WKB period	Defined coefficientwise after the Riccati, cycle, and regularization passports are fixed
Directional or lateral WKB period	Analytic object only under the stated Borel-summability and continuation hypotheses
TBA extraction of a period	Conditional on a proved identification of the TBA $Y$-function with the same summed Voros symbol
Polynomial minimal-chamber TBA–WKB map	Established on Page 4 after cycle reversal, parity-dependent rays, bank translation, and uniqueness hypotheses
WKB–NS period map	Model-dependent; Chapter 10 calibrates a pure-$SU(2)$ operator formally and records the extra analytic qualifications
Direct TBA–NS equality	Not a primitive theorem; it exists only if both bridges close with one common passport
Large-rapidity coefficient extraction	Asymptotic consequence of the period map; finite-window fitting remains numerical evidence
Additive period comparison	Branch-sensitive because logarithms are defined modulo $2\pi\ii\hbar$
Voros-symbol comparison	Branch-invariant but loses the additive lattice integer
Period agreement implies spectral agreement	False without a common boundary determinant or quantization condition

Quantum-period labels hide six typed outputs

Type-check the objects before comparing their values.

Construction	Natural output	Mathematical type
TBA or NLIE	$\varepsilon_a(\theta)$ or $Y_a=\ee^{-\varepsilon_a}$	Dimensionless logarithmic or multiplicative coordinate
Formal WKB	$\Pi_\gamma^{\mathrm{form}}(\hbar)$	Formal additive action
Exact WKB	$\mathcal S_{\vartheta\pm}\Pi_\gamma$	Sectorial additive action
Voros	$\Pi_\gamma/\hbar$ and $\mathcal V_\gamma$	Dimensionless exponent and multiplicative coordinate
NS geometry	$a$ and $a_D^{\mathrm{NS}}$	Polarized flat coordinates in a declared scheme
Spectral problem	$D(E)$ or $\mathcal Q(E)$	Boundary-dependent analytic function

For a scalar quantum curve in the book’s normal form, let

$$\Omega(\hbar) = P_{\mathrm{ev}}(q;u,\hbar)\,\dd q$$

be the regularized half branch-difference WKB form, with $P_{\mathrm{ev}}=(P_+-P_-)/2$. An oriented closed cycle $\gamma$ carries the formal additive period

$$\Pi_\gamma^{\mathrm{form}}(u,\hbar) = \oint_\gamma\Omega(\hbar),$$

its directional or lateral sum

$$\Pi_{\gamma,\vartheta\pm}^{\mathrm{WKB}} = \mathcal S_{\vartheta\pm} \Pi_\gamma^{\mathrm{form}},$$

and the multiplicative Voros coordinate

$$\mathcal V_{\gamma,\vartheta\pm} = \exp\!\left( \frac{\Pi_{\gamma,\vartheta\pm}^{\mathrm{WKB}}}{\hbar} \right).$$

The additive period retains an integer logarithm lift; the Voros coordinate does not. Reversing the cycle inverts the latter and negates the former.

This is a closed-cycle Voros coordinate. An open-path Voros symbol instead contains a regularized integral from one local normalization sector to another and encodes connection data. It cannot be replaced by $\exp(\Pi_\gamma/\hbar)$ for some closed $\gamma$ merely because both objects undergo Stokes jumps.

A TBA solution instead returns pseudoenergies and $Y$-functions,

$$Y_a(\theta) = \ee^{-\varepsilon_a(\theta)}.$$

This expression is not yet a WKB period. It becomes one only when a model-specific theorem or derivation identifies $Y_a$ with the same Voros coordinate on a declared ray.

The NS construction supplies another kind of output. In the rank-one normalization of Chapter 10,

$$a_D^{\mathrm{NS}}(a,\hbar) = -\frac{1}{2\pi\ii} \frac{\partial \mathscr F_{\mathrm{NS}}^{\mathrm{loc}} }{\partial a}.$$

It is a derivative in a flat quantum coordinate and a chosen local scheme. It is not the value of the free energy, not a pseudoenergy, and not automatically a spectral condition.

Three equations called TBA are not one equation

The ODE/IM spectral-determinant TBA, the exact-WKB/GMN period equation, and the saddle or limit-shape equations that can arise in the NS limit have different unknowns, kernels, asymptotics, and observables. A shared thermodynamic vocabulary does not identify them. This page uses a TBA–period map only where Pages 3–4 established the corresponding RH or Voros coordinate.

One passport must survive both bridges

Before comparing any pair, freeze the following record.

Entry	Question that must have one answer
Operator	Which polarization, ordering, subprincipal term, gauge, and wavefunction bundle define the quantum curve?
Deformation	How are the ODE parameter, TBA rapidity, WKB $\hbar$, and gauge $\epsilon_1$ related, including phase?
Modulus	Is the operator energy $E_{\mathrm{op}}$ equal to $u$, a rescaling of it, or shifted by $O(\hbar^2)$ contacts?
Period form	Is the integrand the half branch difference, and how are poles, endpoints, and amplitude winding regularized?
Cycle	What is the oriented homology class, its intersection pairing, and its Gauss–Manin continuation path?
Node–charge map	Which TBA node represents that oriented cycle, and has a decaying charge reversal been made?
Normalization	Where do $2\pi$, $\ii$, and powers of $\hbar$ enter the additive and multiplicative coordinates?
Analytic lane	Which Borel direction, lateral side, logarithm lift, contour bank, and Stokes chamber are used?
Flat coordinate	Is the comparison made after solving the quantum mirror map $a=a^{\mathrm{WKB}}(u,\hbar)$?
Gauge scheme	Which perturbative subtraction, mass convention, Abelian factor, and quadratic counterterm define $\mathscr F_{\mathrm{NS}}^{\mathrm{loc}}$?
Observable	Are the objects additive periods, exponentials, connection data, or roots of a complete determinant?

The node label $a$, the cycle label $A$, and the gauge coordinate $a$ are unrelated symbols until this table supplies their maps. Dimensional analysis and the classical limit should be checked before any high-precision calculation.

The comparison has two conditional edges. TBA data reach exact WKB only through the node–cycle, ray, branch, and uniqueness passport; exact WKB reaches NS data only through the quantum-curve, mirror-map, and scheme passport. There is no universal primitive TBA–NS edge, and even a closed period comparison does not cross the spectral firewall by itself.

A TBA period is a lifted logarithm of Y

Assume the Page 4 gates have identified one TBA coordinate with a directional Voros symbol for a decaying charge $\delta_a$:

$$Y_a(\theta) = \mathcal V_{\delta_a} \bigl(\hbar_a(\theta)\bigr).$$

Taking the transported logarithm gives the additive extraction formula

$$\Pi_{\delta_a}^{\mathrm{TBA}} \bigl(\hbar_a(\theta)\bigr) = -\hbar_a(\theta)\varepsilon_a(\theta) + 2\pi\ii\hbar_a(\theta)n_a,$$

where $n_a\in\mathbb Z$ is constant until a declared logarithm event is crossed. This formula is exact within that identified analytic problem; it is not a definition that can be imposed on an unrelated TBA.

In the minimal polynomial calibration, parity also chooses the ray:

$$\hbar_{2j}(\theta) = \ee^{-\theta}, \qquad \hbar_{2j-1}(\theta) = \ii\ee^{-\theta},$$

with the odd nodes evaluated on the declared lateral banks. Comparing all nodes at the same real positive $\hbar$ would silently rotate half of the Borel directions.

The exact source dictionary is stronger than a large-$\theta$ fit. In the Ito–Mariño–Shu convention, before the Page 4 cycle and bank translations, it reads

$$\begin{aligned} -\ii\varepsilon_{2j}(\theta) &= \frac{1}{\hbar} \mathcal S\Pi_{\gamma_{2j}}^{\mathrm{IMS}}(\hbar), \\ -\ii\varepsilon_{2j-1} \!\left(\theta+\frac{\pi\ii}{2}\pm\ii0\right) &= \frac{1}{\hbar} \mathcal S_\pm \Pi_{\gamma_{2j-1}}^{\mathrm{IMS}}(\hbar), \qquad \ee^\theta=\hbar^{-1}. \end{aligned}$$

Their cycles obey

$$m_{2j-1} = \Pi_{\gamma_{2j-1}}^{(0),\mathrm{IMS}}>0, \qquad m_{2j} = \ii\Pi_{\gamma_{2j}}^{(0),\mathrm{IMS}}>0.$$

Page 4 translates this package by $\Pi^{\mathrm{book}}=\ii\Pi^{\mathrm{IMS}}$, $\delta_a=-\gamma_a$, and the IMS-to-book lateral involution. Only after all three operations does it become the compact formula $Y_a=\mathcal V_{\delta_a}^{\mathrm{book}}$ used here. The exact shifted formula reconstructs the sectorial function; the moments below recover only its formal coefficients.

For an even node on the unshifted positive ray, suppose the oriented period has the formal expansion

$$\Pi_{\delta_a}^{\mathrm{form}}(\hbar) = -m_a + \sum_{k\geq1}p_{a,k}\hbar^{2k}.$$

On an unshifted positive ray with $n_a=0$, the large-rapidity expansion is

$$\varepsilon_a(\theta) = m_a\ee^\theta - \sum_{k\geq1} p_{a,k}\ee^{-(2k-1)\theta}.$$

Thus the drive checks the oriented classical action, while the decaying tail contains the formal quantum corrections. For example,

$$p_{a,1} = -\lim_{\theta\to+\infty} \ee^\theta \left[ \varepsilon_a(\theta)-m_a\ee^\theta \right].$$

At finite cutoff, fit several predicted powers and repeat the fit while varying the window, grid, precision, and subtraction. Odd nodes must be fit in the complex variable $\hbar_a$, not by copying the even-node signs.

Weighted moments avoid subtracting the drive

For the minimal $A_r$ equation, set

$$F_a(\theta) = L_{a-1}(\theta)+L_{a+1}(\theta), \qquad L_0=L_{r+1}=0.$$

Assume the declared minimal chamber has no crossed kernel pole, the fixed logarithm branch has no divisor on the contour, and $F_a$ is bounded at $-\infty$ and sufficiently fast-decaying at $+\infty$. Through the desired order, require finite controlled exponential moments of $|F_a|$. Splitting the integral into a region well below $\theta$ and the superexponentially small driven right tail then justifies termwise use of the large-$\theta$ kernel expansion, and gives

$$\varepsilon_a(\theta) \sim m_a\ee^\theta + \sum_{n\geq1} m_a^{(n)}\ee^{(1-2n)\theta},$$

where

$$m_a^{(n)} = \frac{(-1)^n}{\pi} \int_{\mathbb R} \ee^{(2n-1)\theta'} F_a(\theta')\,\dd\theta'.$$

This follows from expanding $1/(2\pi\cosh(\theta-\theta'))$ in odd decaying exponentials and controlling the complementary integration tail. It is usually more stable than fitting the tiny pointwise difference $\varepsilon_a-m_a\ee^\theta$.

Write $t=\ee^{-\theta}$. For an even node, $\hbar_a=t$ and the decaying-charge period is

$$\Pi_{\delta_a}(\hbar_a) \sim -m_a - \sum_{n\geq1} m_a^{(n)}\hbar_a^{2n}.$$

For an odd node, $\hbar_a=\ii t$ and the transported lateral period is

$$\Pi_{\delta_a,\pm}(\hbar_a) \sim -\ii m_a - \ii \sum_{n\geq1} (-1)^n m_a^{(n)}\hbar_a^{2n}.$$

These phase formulas use the Page 4 decaying-charge convention. Returning to its original IMS cycles reverses the charge and changes the printed period convention as a package. For complex solutions, require uniform absolute moment bounds on the chosen banks. The two laterals share the same formal coefficients, but finite analytic values use the declared bank; Page 4’s convention involution sends book $\pm$ to IMS $\mp$. Their average is not taken unless a median prescription is separately declared.

If $E_{L_b}(\theta)$ bounds the nonlinear-log error, then the propagated moment error satisfies

$$E_{m_a^{(n)}} \leq \frac{1}{\pi} \int_{\mathbb R} \ee^{(2n-1)\theta} \left[ E_{L_{a-1}}(\theta)+E_{L_{a+1}}(\theta) \right]\dd\theta +E_{\mathrm{tail}}^{(n)} +E_{\mathrm{quad}}^{(n)} +E_{\mathrm{arith}}^{(n)}.$$

The last two terms bound the weighted quadrature and finite-precision accumulation. They become increasingly important because the exponential weight amplifies the right side of the numerical window as $n$ grows.

Folded quartic equations and general RH kernels require their own kernel expansion; this moment formula is not universal.

Finite-parameter comparisons need two residuals

Let $\Pi^{\mathrm{WKB}}$ be obtained by Borel–Padé or another justified directional summation on the same cycle and ray. Once the logarithm lift is frozen, an additive discrepancy is

$$\Delta_{\mathrm{add}} = \Pi^{\mathrm{TBA}}-\Pi^{\mathrm{WKB}}.$$

The branch-invariant multiplicative discrepancy is

$$\Delta_{\mathrm{mult}} = \frac{ \mathcal V^{\mathrm{TBA}} }{ \mathcal V^{\mathrm{WKB}} } -1.$$

A small $\Delta_{\mathrm{mult}}$ with a large additive mismatch usually signals a missing $2\pi\ii\hbar n$, not a failure of the analytic coordinate. Conversely, additive agreement on one branch tests more information than exponentiated agreement.

Page 6’s forward bound propagates directly. If $\hbar$ and $n_a$ are fixed and the pseudoenergy error is at most $E_\varepsilon$, then

$$E_{\Pi}^{\mathrm{TBA}} \leq |\hbar|E_\varepsilon.$$

For an odd-node lateral value, add the independently bounded analytic reconstruction and bank-displacement errors. A wrong node–cycle map, phase, or chamber fails the categorical passport; it is not an additional small $E_{\mathrm{map}}$.

If the independent WKB sum has error $E_{\Pi}^{\mathrm{WKB}}$, a claimed disagreement requires

$$|\Delta_{\mathrm{add}}| > E_{\Pi}^{\mathrm{TBA}} + E_{\Pi}^{\mathrm{WKB}}.$$

Below that combined floor, the calculation is inconclusive. The same principle applies to the multiplicative residual after propagating the exponential condition number.

Borel–Padé must sum the same typed object

Generate the WKB coefficients independently of the TBA moments. For

$$\Pi_\gamma^{\mathrm{form}} = \Pi_{\gamma,0} + \sum_{n\geq1} \Pi_{\gamma,2n}\hbar^{2n},$$

define the quantum part of its formal Voros exponent by

$$\widehat V_\gamma^{\mathrm q} = \frac{ \Pi_\gamma^{\mathrm{form}}-\Pi_{\gamma,0} }{\hbar}.$$

The additive period tail and $\widehat V_\gamma^{\mathrm q}$ use different shifted Borel transforms:

$$\begin{aligned} \mathcal B(\Pi_\gamma-\Pi_{\gamma,0}) &= \sum_{n\geq1} \Pi_{\gamma,2n} \frac{\xi^{2n-1}}{(2n-1)!}, \\ \mathcal B\widehat V_\gamma^{\mathrm q} &= \sum_{n\geq1} \Pi_{\gamma,2n} \frac{\xi^{2n-2}}{(2n-2)!}. \end{aligned}$$

Choose the transform for the object actually compared, build several near-diagonal Padé approximants at guard precision, and integrate on the same ray and lateral side encoded by the TBA bank. Vary coefficient order, Padé shape, ray offset, and Laplace quadrature. Equality of several formal coefficients remains weaker than equality of sectorial functions: terms $C\ee^{-A/\hbar}$ can share the entire power-series asymptotic expansion while changing the analytic answer.

Ito–Mariño–Shu print an equivalent unshifted convention for the full even series,

$$\begin{aligned} \widehat\Pi_\gamma(\xi) &= \sum_{n\geq0} \frac{\Pi_\gamma^{(n)}}{(2n)!}\xi^{2n}, \\ \mathcal S_\vartheta\Pi_\gamma &= \frac{1}{\hbar} \int_0^{\ee^{\ii\vartheta}\infty} \ee^{-\xi/\hbar}\widehat\Pi_\gamma(\xi)\,\dd\xi, \\ |\arg\hbar-\vartheta| &< \frac{\pi}{2}. \end{aligned}$$

The factorial shift is tied to whether the classical term and the outside $1/\hbar$ are included. Mixing the coefficients from one convention with the Laplace integral of the other changes the answer.

The NS comparison is made at fixed quantum A-period

Choose a transported symplectic basis $(A,B)$ with $A\circ B=+1$ and define

$$\begin{aligned} a^{\mathrm{WKB}}(u,\hbar) &= \frac{1}{2\pi\ii} \oint_A\Omega(\hbar), \\ a_D^{\mathrm{WKB}}(u,\hbar) &= \frac{1}{2\pi\ii} \oint_B\Omega(\hbar). \end{aligned}$$

Where $\partial_u a^{\mathrm{WKB}}\neq0$, solve the local quantum mirror map

$$a = a^{\mathrm{WKB}}(u,\hbar) \quad\Longleftrightarrow\quad u=u(a,\hbar).$$

Only then is the normalization-complete candidate comparison

$$a_D^{\mathrm{WKB}} \bigl(u(a,\hbar),\hbar\bigr) \mathrel{\widehat=} a_D^{\mathrm{NS}}(a,\hbar).$$

The symbol $\widehat=$ means conditional equality in the recorded model-specific passport. Comparing the two sides at the same printed classical $u$ generally compares different coordinates. Near a zero of $\partial_u a^{\mathrm{WKB}}$, inversion is ill-conditioned and a new flat-coordinate patch or symplectic frame is required.

Formal matching compares coefficients in $\hbar$ and, when appropriate, the gauge coupling. Analytic matching additionally needs compatible resummation, logarithm, and continuation data. A convergent instanton series in one gauge chamber does not by itself prescribe a WKB lateral sum across a Stokes wall.

Established examples occupy different edges

The book has not established one universal three-way example.

Example	Established comparison	What remains outside the claim
Polynomial Schrödinger equation with simple ordered turning points in its minimal chamber	TBA/Wronskian coordinate ↔ resummed WKB period after the complete Page 4 passport	Universal ODE/IM–GMN equivalence, NS dictionary, and spectrum
Pure quartic in the frozen symmetry-reduced chamber	Folded equation-level TBA ↔ exact-WKB reduction in the stated chart	Global complex uniqueness and an NS dictionary
Pure-$SU(2)$ modified Mathieu	Formal WKB ↔ NS periods through the orders checked in Chapter 10; published chamber-sensitive analytic evidence	A TBA arrow or chamber-independent analytic equality
Homogeneous radial ODE/IM problem	Normalized ODE spectral determinant ↔ BLZ $Q$-eigenvalue for the declared module state	Equality with a WKB or NS period
Arbitrary TBA and arbitrary NS theory	No map	Every comparison edge

Keep the rotation phase $q_{\mathrm{rot}}$ of an ODE/IM functional relation separate from the instanton fugacity $\mathfrak q_{\mathrm{inst}}=\Lambda^4$. Their common letter in parts of the literature is not a dictionary.

What the modified-Mathieu evidence establishes

In the Grassi–Gu–Mariño source convention, $E_{\mathrm{src}}=2u_{\mathrm{src}}$, $\Pi_A^{\mathrm{src}}=2\pi a_{\mathrm{src}}$, and $\Pi_B^{\mathrm{src}}=2\ii a_{D,\mathrm{src}}$. Chapter 10 gives the nontrivial translation to this book’s energy, phase, cycles, and period normalizations. On the positive source ray in the weak chamber $|u_{\mathrm{src}}|>1$ for $\Lambda=1$—equivalently $|u|>2\Lambda^2$ in the book’s variables—the Borel-summed A-period agrees with the instanton-resummed gauge A-period. The full B-period is not Borel summable there. Grassi–Gu–Mariño numerically verify that, after subtracting the non-Borel-summable Bernoulli tail associated with the $\gamma(a,\hbar)$ contribution, the Borel sum of the reduced B-period agrees with the corresponding reduced gauge period. They separately observe that the full exact B-period agrees with the average lateral sum for $E_{\mathrm{src}}>2$. In a strong chamber, the gauge A-period is a nonlinear combination of lateral WKB sums rather than one directional sum. Thus “NS periods equal exact-WKB periods” is too strong even for this benchmark.

The homogeneous anharmonic-oscillator lane needs a separate historical qualification. Dorey–Tateo define a zeta-regularized spectral determinant, whose zeros are spectral levels after a sign change, and derive an exact quartic determinant functional relation. Their 1999 extension to the massless $A_{2M-1}$ system for general $x^{2M}$ is presented as a conjectural identification supported by numerical checks. Neither output is a WKB period before a further model-specific dictionary is supplied.

Scheme and frame changes are visible, not errors

A finite local counterterm

$$\mathscr F_{\mathrm{NS}}^{\mathrm{loc}} \longmapsto \mathscr F_{\mathrm{NS}}^{\mathrm{loc}} +c_2a^2+c_1a+c_0$$

shifts the dual coordinate by

$$a_D^{\mathrm{NS}} \longmapsto a_D^{\mathrm{NS}} - \frac{2c_2a+c_1}{2\pi\ii}.$$

The constant $c_0$ disappears, but the affine shift does not. It must be calibrated from the classical dual asymptotic, perturbative logarithm, or a declared frame convention rather than fitted away after seeing the answer.

Likewise, for a rank-one symplectic change

$$\begin{pmatrix} \widetilde a_D\\ \widetilde a \end{pmatrix} = M \begin{pmatrix} a_D\\ a \end{pmatrix}, \qquad M\in SL(2,\mathbb Z),$$

the WKB and NS period vectors must both transform by the same $M$. Agreement of one component before the frame change is not supposed to survive if only one lane is transformed.

For the shear $B\mapsto B+kA$,

$$a_D\longmapsto a_D+ka.$$

The NS definition produces the same shift from the coordinated counterterm $\Delta\mathscr F_{\mathrm{NS}}=-\pi\ii k a^2$. Frame and scheme changes therefore move together.

Lateral jumps are part of the comparison

Across a simple type-I WKB wall of active cycle $\delta$, the book’s fixed-lattice convention gives

$$\mathcal V_{\gamma,+} = \mathcal V_{\gamma,-} \left( 1+\mathcal V_{\delta,-} \right)^{-\delta\mathbin{\cdot}\gamma}.$$

After a logarithm lift is chosen, the additive discontinuity is

$$\Pi_{\gamma,+}-\Pi_{\gamma,-} = -\hbar (\delta\mathbin{\cdot}\gamma) \Log\!\left( 1+\mathcal V_{\delta,-} \right) +2\pi\ii\hbar k.$$

An identified TBA or RH construction must reproduce the same jump with the same intersection sign, lower lateral input, and transported lattice. On the NS lane, a local formal derivative does not automatically contain this analytic completion. One must specify the resummation or nonperturbative completion being compared. Averaging two laterals is an additional median prescription, not a neutral default.

This formula is not a blanket theorem for every Stokes graph. The regular-saddle result of Iwaki–Nakanishi assumes, among other conditions, a compact base, simple turning points, poles of order at least two with controlled lower potential terms, one regular saddle trajectory, and saddle-free reductions on its two sides; the Borel-sum statement is made for sufficiently large positive exact-WKB parameter after rotating to their phase convention, equivalently large $|\hbar|^{-1}$ in the compatible decay sector. Their degenerate-saddle formula contains $1-\mathcal V_\delta$ instead, and its closed-cycle symbols do not jump. The graph type must therefore be classified before copying a DDP factor.

A period triangle closes only after both edges close

There are two legitimate workflows.

1. TBA versus WKB. Solve the Page 6 equation, reconstruct the declared $\Pi^{\mathrm{TBA}}$, and compare it with a separately Borel-summed WKB period on the same cycle and ray.
2. WKB versus NS. Construct the quantum mirror map, calibrate the gauge scheme, and compare the WKB dual period with the NS derivative at fixed $a$.

A TBA–NS number is meaningful only when both workflows refer to the same operator and the same analytic coordinate. If the first uses a quartic oscillator while the second uses the pure-$SU(2)$ modified-Mathieu curve, their separate successes do not form a triangle.

Even a closed period triangle stops at a spectral firewall. A spectrum requires a boundary subspace, a determinant or complete exact quantization condition, and a common energy map. Neither $\mathcal V_\gamma$ nor $\partial_a\mathscr F_{\mathrm{NS}}$ equals a quantization condition merely because it appears inside one in a named problem.

A comparison manifest makes disagreements diagnosable

For every reported value, store

$$\mathcal C = (u,a,\hbar,\vartheta,\pm, \gamma,n, \text{frame}, \text{scheme}, \text{chamber}, \text{precision}).$$

Then report the following checks separately.

Symptom	First diagnostic
Wrong classical limit	Cycle orientation, one-form factor, or energy rescaling
Correct leading action, wrong $\hbar^2$ term	Ordering, subprincipal term, amplitude winding, or quantum modulus shift
Additive mismatch by $2\pi\ii\hbar n$	Logarithm lift or continuation history
Exponentials agree but dual periods drift affinely	NS counterterm or symplectic frame
Even nodes agree, odd nodes do not	Missing $\pi/2$ ray rotation or wrong lateral bank
Agreement at fixed $u$ fails after instanton corrections	Quantum mirror map omitted or inverted on the wrong branch
Jump has the correct factor but inverse power	Cycle intersection or upper/lower convention reversed
Difference shrinks with the nonlinear tolerance and then stalls	Tail, quadrature, WKB summation, or NS truncation floor
Periods agree but levels do not	Boundary determinant, domain, or energy/accessory map differs

Refine TBA, WKB, and NS computations independently. Shared coefficients or code paths are useful for speed but do not constitute an independent check.

Common pitfalls

Declaring a TBA pseudoenergy to be a period by notation. The map $\Pi=-\hbar\varepsilon$ follows from an identified Voros coordinate and a transported logarithm. Without that bridge it is only a relabeling.

Comparing the NS derivative at fixed classical modulus. The NS dual coordinate is a function of the flat quantum A-period. Invert the mirror map before evaluating the WKB B-period.

Using exponentials to hide an unexplained additive mismatch. A Voros ratio correctly removes a known logarithm lattice, but the integer must still be recorded if additive periods are claimed.

Averaging laterals without naming the prescription. Upper, lower, and median sums are different analytic objects on a singular direction.

Turning period agreement into a spectrum. A boundary determinant and energy dictionary remain independent data.

Exercises

1. Lift a TBA logarithm. Suppose an established dictionary gives $Y(\theta)=\mathcal V_\delta(\hbar)$ with $Y=\ee^{-\varepsilon}$ and $\mathcal V_\delta=\exp(\Pi_\delta/\hbar)$. Recover the most general additive period and explain when its integer can change.

Solution

Taking logarithms on transported sheets gives

$$\frac{\Pi_\delta}{\hbar} = -\varepsilon+2\pi\ii n, \qquad n\in\mathbb Z,$$

and hence

$$\Pi_\delta = -\hbar\varepsilon+2\pi\ii\hbar n.$$

The integer is fixed under continuation that avoids a zero or pole of the coordinate and avoids a logarithm cut. It changes only at a declared branch event; choosing a fresh principal logarithm at each parameter value does not define a valid continuation.

2. Read a WKB coefficient from rapidity. Let

$$\Pi(\hbar) = -m+p_1\hbar^2+p_2\hbar^4+O(\hbar^6), \qquad \hbar=\ee^{-\theta},$$

and take the zero logarithm lift. Expand $\varepsilon(\theta)$ through $O(\ee^{-3\theta})$, give a limit for $p_2$ after $p_1$ is known, and express the general even-node $p_n$ as a weighted minimal-chain moment.

Solution

Since $\Pi=-\hbar\varepsilon$,

$$\varepsilon(\theta) = m\ee^\theta -p_1\ee^{-\theta} -p_2\ee^{-3\theta} +O(\ee^{-5\theta}).$$

After subtracting the drive and the first correction,

$$p_2 = -\lim_{\theta\to+\infty} \ee^{3\theta} \left[ \varepsilon-m\ee^\theta+p_1\ee^{-\theta} \right].$$

Numerically, one replaces the limit by a multi-window fit and includes the Page 6 tail and discretization errors.

For the minimal-chain kernel, $p_n=-m^{(n)}$, so

$$p_n = \frac{(-1)^{n+1}}{\pi} \int_{\mathbb R} \ee^{(2n-1)\theta} \left(L_{a-1}+L_{a+1}\right)\dd\theta.$$

This identity requires the fixed chamber and logarithm branch, the weighted integrability stated in the text, and control of the driven right tail when the kernel series is integrated term by term.

3. Separate an additive branch error from a functional error. Two computations give

$$\Pi_1-\Pi_2 = 2\pi\ii\hbar n+\delta.$$

Compute their Voros-coordinate ratio and state what it tests.

Solution

Exponentiation removes the integer lattice:

$$\frac{\exp(\Pi_1/\hbar)}{\exp(\Pi_2/\hbar)} = \ee^{\delta/\hbar}.$$

Thus the ratio tests the analytic coordinate independently of $n$. If it equals one within errors, the additive mismatch may be only a logarithm lift. The test does not determine which integer is correct.

4. Show why the mirror map cannot be skipped. Suppose

$$a^{\mathrm{WKB}}(u,\hbar) = a_0(u)+\hbar^2a_1(u)+O(\hbar^4)$$

with $a_0'(u)\neq0$. Find $u(a,\hbar)$ through order $\hbar^2$.

Solution

Write

$$u(a,\hbar) = u_0(a)+\hbar^2u_1(a)+O(\hbar^4), \qquad a_0(u_0)=a.$$

Expanding the defining equation gives

$$a_0'(u_0)u_1+a_1(u_0)=0,$$

so

$$u_1(a) = -\frac{a_1(u_0(a))}{a_0'(u_0(a))}.$$

Substituting $u_0$ alone into the B-period misses the induced order- $\hbar^2$ term $u_1\partial_u a_{D,0}$. That is why fixed-$u$ and fixed-$a$ comparisons differ.

5. Track an NS counterterm. Let

$$a_D^{\mathrm{NS}} = -\frac{1}{2\pi\ii}\partial_a\mathscr F_{\mathrm{NS}}^{\mathrm{loc}}.$$

Find the change caused by $\Delta\mathscr F=c_2a^2+c_1a+c_0$. Which part is invisible?

Solution

Differentiation gives

$$\Delta a_D^{\mathrm{NS}} = -\frac{2c_2a+c_1}{2\pi\ii}.$$

The constant $c_0$ is invisible to the period derivative. The quadratic and linear terms produce a genuine affine frame shift and must be matched, not discarded as numerical error.

6. Transport both period vectors. Let

$$M = \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix}.$$

Apply it to $(a_D,a)^{\mathsf T}$ and explain how a WKB–NS comparison changes.

Solution

The new frame is

$$\widetilde a_D=-a, \qquad \widetilde a=a_D.$$

Both the WKB and NS vectors must transform by $M$. If only the WKB cycles are relabeled, comparing $\widetilde a_D^{\mathrm{WKB}}$ with the old $a_D^{\mathrm{NS}}$ mixes different components and should fail.

7. Take the logarithm of a DDP jump. Starting from

$$\mathcal V_{\gamma,+} = \mathcal V_{\gamma,-} (1+\mathcal V_{\delta,-})^{-\delta\mathbin{\cdot}\gamma},$$

derive the additive discontinuity and identify its ambiguity.

Solution

On a transported logarithm branch,

$$\frac{\Pi_{\gamma,+}-\Pi_{\gamma,-}}{\hbar} = -(\delta\mathbin{\cdot}\gamma) \Log(1+\mathcal V_{\delta,-}) +2\pi\ii k.$$

Multiplying by $\hbar$ gives the formula in the text. The integer $k$ records the logarithm lift; the lower lateral value and intersection sign are fixed by the book’s DDP convention.

8. Audit a proposed three-way match. A calculation compares a quartic oscillator TBA pseudoenergy with a pure-$SU(2)$ NS derivative. It observes ten matching digits after independently rescaling each result and claims a universal TBA–NS identity. Give four decisive objections.

Solution

First, the two computations have not been shown to quantize the same operator or classical curve. Second, an arbitrary fitted rescaling does not fix the period-form and $2\pi\ii$ normalization passport. Third, no node–cycle map or TBA–Voros identification has been established. Fourth, the NS derivative has not been evaluated through a common quantum mirror map and scheme. One must also match $\hbar$ phases, cycles, chambers, lateral sides, and numerical error budgets. Decimal agreement after free rescaling cannot replace these gates.

References

• K. Ito, M. Mariño, and H. Shu, “TBA Equations and Resurgent Quantum Mechanics”, Journal of High Energy Physics 2019 (2019), 228, especially Eqs. (3.3)–(3.14) and (3.30)–(3.43). Derives the minimal-chamber $A_r$ TBA from both resummed periods and Wronskian $Y$-functions, with the parity-dependent Borel directions used in the extraction above.
• P. Dorey and R. Tateo, “Anharmonic Oscillators, the Thermodynamic Bethe Ansatz, and Nonlinear Integral Equations”, Journal of Physics A 32 (1999), L419–L425. Derives the exact quartic determinant relation and presents the broader homogeneous ODE/IM–TBA identification with its original conjectural and numerical status.
• K. Iwaki and T. Nakanishi, “Exact WKB Analysis and Cluster Algebras”, Journal of Physics A 47 (2014), 474009. Separates cycle and path Voros symbols and proves the stated simple-wall transformation under its saddle and summability hypotheses.
• N. A. Nekrasov and S. L. Shatashvili, “Quantization of Integrable Systems and Four Dimensional Gauge Theories”, in XVIth International Congress on Mathematical Physics (2010), 265–289. Defines the NS twisted superpotential and its Bethe/gauge interpretation while retaining model and boundary-condition data.
• A. Mironov and A. Morozov, “Nekrasov Functions and Exact Bohr–Sommerfeld Integrals”, Journal of High Energy Physics 2010 (2010), 040. Compares pure- $SU(2)$ WKB periods and the deformed prepotential coefficientwise; it does not establish a universal Borel-resummed theorem.
• K. Ito, S. Kanno, and T. Okubo, “Quantum Periods and Prepotential in N=2 SU(2) SQCD”, Journal of High Energy Physics 2017 (2017), 065. Computes WKB differential operators and quantum prepotentials for specified matter theories, supporting model-specific formal matching.
• A. Grassi, J. Gu, and M. Mariño, “Non-Perturbative Approaches to the Quantum Seiberg–Witten Curve”, Journal of High Energy Physics 2020 (2020), 106. Compares gauge-resummed and Borel-resummed modified-Mathieu periods and displays the strong-chamber lateral relations that prevent a chamber-free equality claim.
• G. Başar and G. V. Dunne, “Resurgence and the Nekrasov–Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lamé Systems”, Journal of High Energy Physics 2015 (2015), 160. Develops the all-orders WKB/Seiberg–Witten relation and shows why exponentially small band and gap widths lie beyond bare all-orders period data.
• A.-K. Kashani-Poor and J. Troost, “Pure N=2 Super Yang–Mills and Exact WKB”, Journal of High Energy Physics 2015 (2015), 160. Derives quantum periods and shows explicitly that exact Floquet monodromy also needs Stokes continuation matrices.
• D. Gaiotto, G. W. Moore, and A. Neitzke, “Wall-Crossing, Hitchin Systems, and the WKB Approximation”, Advances in Mathematics 234 (2013), 239–403. Constructs the WKB/Darboux-coordinate Riemann–Hilbert framework in which chamber data, active charges, and wall-crossing are part of the analytic object.