Regular Punctures: HeunG, Liouville Blocks, and SU(2) with Four Flavors

The four-punctured Riemann sphere is the smallest example in which all three sides of the ODE/CFT/gauge dictionary are nontrivial. In the declared AGT/Nekrasov sewing scheme, its one cross-ratio is the ultraviolet coupling of four-dimensional $SU(2)$ theory with four flavors. A nondegenerate Virasoro four-point block in a chosen sewing channel matches the normalized instanton series. After an additional degenerate insertion and a controlled heavy or NS limit, the same background supports a four-singularity Heun oper.

Those statements involve different functions and different independent variables. The bulk block depends on the modulus; the degenerate block depends on the modulus and a probe position; HeunG is one normalized local germ in the probe coordinate. This page fixes one complete mass scheme and carries the distinction all the way to the standard Heun accessory parameter.

Collision-free notation

Symbol	Meaning on this page	Symbol deliberately avoided
$t=q_{\mathrm{UV}}$	Four-puncture cross-ratio, sewing variable, and instanton fugacity in the declared scheme	The standard-Heun singularity parameter $a$
$z$	Degenerate-probe and eventual ODE coordinate	A gauge coupling
$a_{\mathrm C}$	Dimensionful Coulomb modulus	The Heun singularity position
$\mathfrak q_{\mathrm H}$	Accessory parameter in the house standard Heun equation	The gauge fugacity $q_{\mathrm{UV}}$
$\alpha_{0t}$	Internal Liouville momentum in the $0t$ channel	A standard-Heun exponent parameter

Thus the house Heun singularity parameter is $a_{\mathrm H}=t$, whereas the gauge-theory Coulomb modulus is $a_{\mathrm C}$. They are not two names for one datum.

Four regular punctures give one modulus and three cusps

Fix the ordered background punctures at

$$(z_0,z_t,z_1,z_\infty) = (0,t,1,\infty), \qquad t\in\mathbb C\setminus\{0,1\}.$$

Throughout this chart,

$$t = q_{\mathrm{UV}} = \exp(2\pi\ii\tau_{\mathrm{UV}})$$

is the original AGT/Nekrasov instanton coordinate near the $t=0$ cusp. A finite renormalization can define another ultraviolet coordinate; $t$ is also not the infrared nome or effective coupling.

A circle separating $\{0,t\}$ from $\{1,\infty\}$ selects the $0t$ sewing channel. Near $t=0$ the circle becomes a long thin tube. The same tube carries an intermediate Virasoro module and the weakly coupled $SU(2)$ gauge node:

$$\begin{gathered} \text{tube label} \quad\longleftrightarrow\quad \alpha_{0t} \quad\longleftrightarrow\quad a_{\mathrm C}, \\ \text{tube modulus} \quad\longleftrightarrow\quad t \quad\longleftrightarrow\quad q_{\mathrm{UV}}. \end{gathered}$$

The other two collision points of the cross-ratio are equally physical. They describe different weakly coupled frames:

Cusp	Pair separated by the long tube	Local weak-coupling coordinate
$t\to0$	$\{0,t\}\mid\{1,\infty\}$	$t$
$t\to1$	$\{t,1\}\mid\{\infty,0\}$	$1-t$
$t\to\infty$	$\{0,1\}\mid\{t,\infty\}$	$t^{-1}$

These are the three pants decompositions behind the familiar triality of the $N_f=4$ theory. A change of cusp changes the sewing channel and the weakly coupled Lagrangian frame. It does not merely replace one small number by another while leaving all labels fixed.

The regular four-puncture prototype. The background modulus $t$ is the ultraviolet gauge fugacity, the tube label is the Coulomb or internal-momentum datum, and $z$ appears only after the extra degenerate probe is inserted. The bulk four-point block is therefore not a HeunG wavefunction. The lower arrow is the proposed defect extension followed by a conditional heavy or NS reduction.

Status ledger

Statement	Status
$C_{0,4}$ in the $0t$ frame describes $SU(2)$ with $N_f=4$ near $t=0$	Class-$S$ construction
The Heisenberg-stripped $U(2)$ instanton series equals the unit-leading Virasoro block	Regular AGT identity: originally conjectured and checked as a formal series; supported by later algebraic constructions
A generic nondegenerate four-point block is a HeunG wavefunction	False: it has no independent coordinate $z$
A five-point block with $V_{-b/2}(z)$ satisfies a BPZ equation	Exact CFT statement in a compatible fusion channel
That degenerate insertion is represented by a gauge surface defect	Proposed, evidence-supported extension of AGT
The finite-$b$ five-point equation is an ordinary Heun ODE	False: a modulus derivative survives
Its heavy or NS shadow is a four-pole Heun oper	Conditional on derivative-compatible factorization, branches, and normalization
The oper solution is literally HeunG	Only after a scalar gauge and a unit-leading local branch have been fixed

The bulk equality uses four nondegenerate primaries

Let

$$\widehat{\mathcal V}_{0t} \left( t; \alpha_\infty,\alpha_1,\alpha_t,\alpha_0; \alpha_{0t} \right) = 1+\sum_{k\geq1}\mathcal V_k t^k$$

be the unit-leading Virasoro block with external fields at $(\infty,1,t,0)$ and internal momentum $\alpha_{0t}$. All four external modules are generic here. There is no null vector and no probe coordinate.

On the gauge side, use the localization representative with Coulomb eigenvalues $(a_{\mathrm C},-a_{\mathrm C})$, two antifundamental masses $\mu_1,\mu_2$, and two fundamental masses $\mu_3,\mu_4$. Write $\boldsymbol a_{\mathrm C}:=(a_{\mathrm C},-a_{\mathrm C})$; then

$$\begin{aligned} Z_{\mathrm{inst}}^{U(2),N_f=4} ={}& \sum_{\boldsymbol Y} t^{|\boldsymbol Y|} z_{\mathrm{vec}} \left( \boldsymbol a_{\mathrm C},\boldsymbol Y \right) \\ &\times \prod_{r=1}^{2} z_{\mathrm{antifund}} \left( \boldsymbol a_{\mathrm C},\boldsymbol Y;\mu_r \right) \prod_{r=3}^{4} z_{\mathrm{fund}} \left( \boldsymbol a_{\mathrm C},\boldsymbol Y;\mu_r \right). \end{aligned}$$

The split into two antifundamentals and two fundamentals is a convenient $U(2)$ fixed-point bookkeeping representative. The $SU(2)$ doublet is pseudoreal; changing the printed orientation is allowed only together with the corresponding equivariant mass reflection and Heisenberg factor.

The detailed box factors were defined in Chapter 10. What matters here is the precise regular AGT identity

$$\begin{aligned} Z_{\mathrm{inst}}^{U(2),N_f=4} \left( a_{\mathrm C},\boldsymbol\mu;t \right) ={}& (1-t)^{\kappa_{\mathcal H}} \\ &\times \widehat{\mathcal V}_{0t} \left( t; \boldsymbol\alpha_{\mathrm{ext}}; \alpha_{0t} \right), \end{aligned}$$

after the mass map below is imposed. The factor

$$Z_{\mathcal H}(t) = (1-t)^{\kappa_{\mathcal H}}$$

is the Heisenberg or conventionally named $U(1)$ block. It is Coulomb-independent but not modulus-independent. The AGT-normalized traceless representative in this convention is the quotient

$$Z_{\mathrm{inst}}^{SU(2),\mathrm{AGT}} := \frac{ Z_{\mathrm{inst}}^{U(2),N_f=4} }{ (1-t)^{\kappa_{\mathcal H}} } = \widehat{\mathcal V}_{0t}.$$

This notation records a normalization prescription, not the instruction “set the Abelian Coulomb coordinate to zero.” The original AGT paper presented the equality as a conjecture and checked its instanton expansion; later Virasoro–Heisenberg bases explain the factorized Nekrasov matrix elements algebraically. The displayed identity is used here as the standard generic formal-series correspondence in its declared scheme.

The unhatted Virasoro block additionally contains

$$t^{ \Delta_{0t}-\Delta_0-\Delta_t }.$$

That power matches a classical gauge factor times an external sewing normalization, not the unit-leading instanton sum.

The first coefficient is an immediate audit

Write

$$\Delta(\alpha) = \alpha(Q_{\mathrm L}-\alpha), \qquad \Delta:=\Delta(\alpha_{0t}).$$

For a generic internal Verma module with $\Delta\neq0$,

$$\mathcal V_1 = \frac{ (\Delta+\Delta_t-\Delta_0) (\Delta+\Delta_1-\Delta_\infty) }{ 2\Delta }.$$

Expanding the Heisenberg factor gives the concrete $N_f=4$ prediction

$$Z_1^{U(2),N_f=4} = \mathcal V_1-\kappa_{\mathcal H}.$$

The two one-box fixed points also compute the left-hand side directly. In the printed matter convention, write $\epsilon_\Sigma=\epsilon_1+\epsilon_2$ and define

$$\begin{aligned} P_+ &:= \prod_{r=1}^{2}(a_{\mathrm C}+\mu_r) \prod_{r=3}^{4} (a_{\mathrm C}-\mu_r+\epsilon_\Sigma), \\ P_- &:= \prod_{r=1}^{2}(-a_{\mathrm C}+\mu_r) \prod_{r=3}^{4} (-a_{\mathrm C}-\mu_r+\epsilon_\Sigma). \end{aligned}$$

Then

$$\begin{aligned} Z_1^{U(2),N_f=4} = -\frac{1}{\epsilon_1\epsilon_2} \left[ \frac{P_+} {2a_{\mathrm C}(2a_{\mathrm C}+\epsilon_\Sigma)} + \frac{P_-} {2a_{\mathrm C}(2a_{\mathrm C}-\epsilon_\Sigma)} \right]. \end{aligned}$$

The Weyl reflection $a_{\mathrm C}\mapsto-a_{\mathrm C}$ exchanges the two colored-box contributions, and every summand is dimensionless. These are useful checks before any conformal-block substitution.

After substituting the mass passport below, this rational identity checks the external-label order, the equivariant shifts, the Coulomb normalization, and the Heisenberg exponent at once. Agreement of pole locations without agreement of this numerator is not a complete check.

One four-flavor mass scheme makes every shift visible

Retain the book convention

$$\begin{gathered} \varepsilon_\star^2 = \epsilon_1\epsilon_2, \qquad b^2 = \frac{\epsilon_2}{\epsilon_1}, \\ \epsilon_\Sigma := \epsilon_1+\epsilon_2 = \varepsilon_\star Q_{\mathrm L}. \end{gathered}$$

Some original AGT formulas denote this sum by $\epsilon_+$. This book reserves $\epsilon_\Sigma$ because other sources use $\epsilon_+$ for half the sum.

The original AGT four-flavor formula sets the common scale to one and uses the reciprocal $b$ convention. Restoring dimensions and translating to this book exchanges $b$ with $b^{-1}$ but leaves $Q_{\mathrm L}$ unchanged.

Take $\mu_{1,2}$ to be antifundamental Nekrasov masses and $\mu_{3,4}$ to be fundamental Nekrasov masses in that representative. Define four dimensionful centered puncture masses by

$$\begin{aligned} p_\infty &= \frac{\mu_1-\mu_2}{2}, & p_1 &= \frac{ \mu_1+\mu_2-\epsilon_\Sigma }{2}, \\ p_t &= \frac{ \mu_3+\mu_4-\epsilon_\Sigma }{2}, & p_0 &= \frac{\mu_3-\mu_4}{2}. \end{aligned}$$

The external and internal Liouville momenta are then

$$\begin{aligned} \alpha_f &= \frac{Q_{\mathrm L}}2 +\frac{p_f}{\varepsilon_\star}, && f\in\{\infty,1,t,0\}, \\ \alpha_{0t} &= \frac{Q_{\mathrm L}}2 +\frac{a_{\mathrm C}}{\varepsilon_\star}. \end{aligned}$$

The inverse mass map is equally important:

$$\begin{aligned} \mu_1 &= p_1+p_\infty+\frac{\epsilon_\Sigma}{2}, & \mu_2 &= p_1-p_\infty+\frac{\epsilon_\Sigma}{2}, \\ \mu_3 &= p_t+p_0+\frac{\epsilon_\Sigma}{2}, & \mu_4 &= p_t-p_0+\frac{\epsilon_\Sigma}{2}. \end{aligned}$$

Thus the four external momenta are not the four printed $\mu_i$ one by one. Two use differences; two use shifted sums. The shift $\epsilon_\Sigma/2$ is exactly the equivariant centering that disappears if one copies only the classical mass relation.

For comparison with the source’s dimensionless notation, put $\nu_i=\mu_i/\varepsilon_\star$ and define

$$\begin{aligned} s_{\mathrm L} &= \frac{\nu_1+\nu_2}{2}, & d_{\mathrm L} &= \frac{\nu_1-\nu_2}{2}, \\ s_{\mathrm R} &= \frac{\nu_3+\nu_4}{2}, & d_{\mathrm R} &= \frac{\nu_3-\nu_4}{2}. \end{aligned}$$

The source map becomes

$$\begin{array}{c|cccc|c} \text{position} & \infty & 1 & t & 0 & \text{internal} \\ \hline \text{Liouville momentum} & \dfrac{Q_{\mathrm L}}2+d_{\mathrm L} & s_{\mathrm L} & s_{\mathrm R} & \dfrac{Q_{\mathrm L}}2+d_{\mathrm R} & \dfrac{Q_{\mathrm L}}2+ \dfrac{a_{\mathrm C}}{\varepsilon_\star} \end{array}$$

and fixes the Heisenberg exponent to

$$\kappa_{\mathcal H} = 2s_{\mathrm L} \left( Q_{\mathrm L}-s_{\mathrm R} \right) = 2\alpha_1 \left( Q_{\mathrm L}-\alpha_t \right).$$

The word mass has three levels

The $\mu_i$ are masses printed by one $U(2)$ localization convention. The $p_f$ are centered Cartan data attached to punctures. The $\alpha_f$ are dimensionless Liouville momenta. Replacing a fundamental by an antifundamental uses $\mu\mapsto\epsilon_\Sigma-\mu$ in this convention and changes the printed map. A different representative can be equally correct, but it must carry its transformed Heisenberg factor.

Two elementary reflection checks are already visible. Exchanging $\mu_1\leftrightarrow\mu_2$ sends $p_\infty\mapsto-p_\infty$ and therefore $\alpha_\infty\mapsto Q_{\mathrm L}-\alpha_\infty$ while leaving $p_1$ fixed. Similarly, $\mu_3\leftrightarrow\mu_4$ realizes Liouville reflection at $0$. The Coulomb Weyl reflection $a_{\mathrm C}\mapsto-a_{\mathrm C}$ sends $\alpha_{0t}\mapsto Q_{\mathrm L}-\alpha_{0t}$.

The fifth insertion creates the Heun coordinate

The bulk relation above has only the modulus $t$. To obtain a second-order equation in a new coordinate, insert $V_{-b/2}(z)$ and choose an adjacent channel satisfying the degenerate fusion rule. The chiral object becomes

$$\mathscr B_5(z,t) = \left\langle V_\infty(\infty) V_1(1) V_t(t) V_{-b/2}(z) V_0(0) \right\rangle_{\mathrm{ch}}.$$

If $\alpha_{\mathrm L}$ and $\alpha_{\mathrm R}$ label the channels immediately across the probe, compatibility requires

$$\alpha_{\mathrm R}-\alpha_{\mathrm L} = \pm\frac b2$$

up to Liouville reflection. Merely assigning the degenerate numerical weight to a generic five-point sewing graph does not impose null-state decoupling.

After the global Ward identities are used, the exact equation can be written

$$\left[ b^{-2}\partial_z^2 -A_{\mathrm{BPZ}}(z)\partial_z +K(z,t)\partial_t +U_b(z,t) \right] \mathscr B_5(z,t) = 0,$$

where

$$A_{\mathrm{BPZ}}(z) = \frac1z+\frac1{z-1}, \qquad K(z,t) = \frac{ t(t-1) }{ z(z-1)(z-t) },$$

and

$$\begin{aligned} U_b(z,t) ={}& \frac{\Delta_0}{z^2} + \frac{\Delta_t}{(z-t)^2} + \frac{\Delta_1}{(z-1)^2} \\ &+ \frac{ \Delta_\infty-\Delta_0-\Delta_t-\Delta_1-\Delta_{\mathrm d} }{ z(z-1) }. \end{aligned}$$

Here

$$\Delta_{\mathrm d} = -\frac12-\frac{3b^2}{4}.$$

The modulus derivative is nonzero. Even after a numerical value of $t$ has been chosen, $\partial_t\mathscr B_5$ is not set to zero. At finite $b$ this is a two-variable BPZ PDE, often called a nonstationary Heun equation, rather than the ordinary HeunG equation.

Under the proposed defect extension of AGT, the new object is represented by a surface-defect partition function:

$$Z_{\mathrm{defect}}(z,t) \quad\longleftrightarrow\quad \mathscr B_5(z,t).$$

This proposed gauge identification must be kept distinct from the exact CFT null equation.

Translate the degenerate label

AGGTV writes $\epsilon_1=b_{\mathrm{src}}\hbar$ and $\epsilon_2=\hbar/b_{\mathrm{src}}$. Hence $b_{\mathrm{src}}=b^{-1}$ in the convention of this book, and its $\Phi_{2,1}=V_{-b_{\mathrm{src}}/2}$ is our dual field $V_{-1/(2b)}$, not our light $V_{-b/2}$. The light presentation used above corresponds to exchanging the two Omega planes, or equivalently to the dual Kac label in that source. The central charge is unchanged, but the null relation and the heavy–light role must be translated.

The NS shadow is a four-pole oper

Take the book’s NS direction

$$\epsilon_2\to0 \quad\text{at fixed}\quad \epsilon_1=\hbar, \qquad b\to0, \qquad \varepsilon_\star=b\hbar.$$

Use the centered-mass NS path: hold the dimensionful puncture masses $p_f$, the Coulomb modulus $a_{\mathrm C}$, and $\hbar$ fixed. The equivariant labels $\mu_i$ in the inverse passport then acquire the corresponding $O(\epsilon_2)$ centering shift as the limit is taken. This path gives the oriented exponent differences

$$\theta_f := 2b \left( \alpha_f-\frac{Q_{\mathrm L}}2 \right) = \frac{2p_f}{\hbar}, \qquad \theta_{0t} := \frac{2a_{\mathrm C}}{\hbar}.$$

This is not only a dimensional mnemonic. At finite $b$ one has the exact identity

$$b^2\Delta_f = \frac{ (1+b^2)^2-\theta_f^2 }{4},$$

and therefore

$$b^2\Delta_f \longrightarrow \frac{1-\theta_f^2}{4} \qquad (b\to0).$$

Define

$$\delta_j = \frac{1-\theta_j^2}{4}, \qquad j\in\{0,t,1,\infty,0t\}.$$

The four $\delta_f$ become the local double-pole coefficients of the oper. The internal quantity $\delta_{0t}$ has a different job: it selects the monodromy or classical-block branch and thereby helps determine the accessory residue.

Let

$$\mathcal V_{0t}(t;b) = t^{\Delta_{0t}-\Delta_0-\Delta_t} \widehat{\mathcal V}_{0t}(t;b)$$

be the full background block in the same channel. After fixing a fusion-compatible local normalization, write the five-point block as

$$\mathscr B_5(z,t;b) = \mathcal V_{0t}(t;b)\,\Psi_b(z,t).$$

On a fixed logarithm branch, assume the derivative-compatible limits

$$\begin{aligned} b^2\Log\mathcal V_{0t} &\longrightarrow f_{0t}(t), & b^2\partial_t\Log\mathcal V_{0t} &\longrightarrow \partial_t f_{0t}, \\ \Psi_b(z,t) &\longrightarrow \psi(z;t), & b^2\partial_t\Psi_b(z,t) &\longrightarrow 0. \end{aligned}$$

Equivalently, the five-point block has the leading factorization

$$\mathscr B_5(z,t;b) \sim \exp \left[ \frac{f_{0t}(t)}{b^2} \right] \left[ \psi(z;t)+o(1) \right].$$

The derivative limits are genuine hypotheses: pointwise convergence of $b^2\Log\mathcal V_{0t}$ alone does not justify differentiating its limit. This is one reason the oper statement is conditional rather than an exact finite-Omega identity.

Then the leading equation is

$$\left[ \partial_z^2 +T_{\mathrm{op}}(z;t) \right] \psi(z;t) = 0,$$

with

$$\begin{aligned} T_{\mathrm{op}}(z;t) ={}& \frac{\delta_0}{z^2} + \frac{\delta_t}{(z-t)^2} + \frac{\delta_1}{(z-1)^2} \\ &+ \frac{\Lambda}{z(z-1)} + \frac{ t(t-1)c_t^{\mathrm{op}} }{ z(z-1)(z-t) }, \end{aligned}$$

where

$$\Lambda = \delta_\infty-\delta_0-\delta_t-\delta_1, \qquad c_t^{\mathrm{op}} = \partial_t f_{0t}.$$

The last equality uses the full, unhatted block. If

$$f_{0t}(t) = \left( \delta_{0t}-\delta_0-\delta_t \right)\Log t +\widehat f_{0t}(t),$$

then

$$c_t^{\mathrm{op}} = \frac{ \delta_{0t}-\delta_0-\delta_t }{t} +\partial_t\widehat f_{0t}.$$

The Heisenberg factor survives the NS limit

The classical/NS conversion uses

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

Define both NS quantities on the centered-mass path used above:

$$\left. \mathcal W_{\mathrm{NS}}^X \right|_{\boldsymbol p} := \lim_{\substack{ \epsilon_2\to0\\ p_f,a_{\mathrm C},\hbar\ \mathrm{fixed} }} \epsilon_2\Log Z_X, \qquad X\in\{U(2)\text{ inst},\mathcal H\}.$$

The hatted bulk equality then implies

$$\widehat f_{0t} = \frac1\hbar \left( \left.\mathcal W_{\mathrm{NS}}^{U(2),\mathrm{inst}}\right|_{\boldsymbol p} - \left.\mathcal W_{\mathrm{NS}}^{\mathcal H}\right|_{\boldsymbol p} \right),$$

including the same logarithm branch on both sides. Chapter 10 declares its raw localization limit at fixed printed $\mu_i$ instead. Translating a result from that path requires recentering the mass passport before the coupling derivative is compared; the resulting finite or contact terms belong to the normalization ledger. Derivatives from the two paths must not be mixed silently.

The survival can be seen in closed form. The mass passport gives

$$\kappa_{\mathcal H} = \frac{ 2(p_1+\epsilon_\Sigma/2) (\epsilon_\Sigma/2-p_t) }{ \epsilon_1\epsilon_2 },$$

so on the centered path

$$\left. \mathcal W_{\mathrm{NS}}^{\mathcal H} \right|_{\boldsymbol p} = \frac{2}{\hbar} \left( p_1+\frac\hbar2 \right) \left( \frac\hbar2-p_t \right) \Log(1-t).$$

Since $\kappa_{\mathcal H}$ itself grows on the heavy branch, the Heisenberg contribution need not vanish after multiplication by $\epsilon_2$. Therefore an accessory extracted from the raw $U(2)$ series must subtract the Heisenberg derivative and restore the chosen OPE power before it can equal $\partial_t f_{0t}$.

This is the regular four-flavor instance of the Matone/accessory warning: a coupling derivative becomes a printed ODE coefficient only after its contact, Abelian, and scalar-gauge terms have been fixed.

HeunG requires one final scalar gauge

The oper is already a general four-singularity equation, but HeunG refers to the exponent-zero solution of the house first-derivative form. Define oriented standard-Heun parameters by

$$\begin{aligned} \gamma_{\mathrm H} &= 1-\theta_0, & \delta_{\mathrm H} &= 1-\theta_1, & \epsilon_{\mathrm H} &= 1-\theta_t, \\ \alpha_{\mathrm H} &= \frac{ 2-\theta_0-\theta_1-\theta_t+\theta_\infty }{2}, & \beta_{\mathrm H} &= \frac{ 2-\theta_0-\theta_1-\theta_t-\theta_\infty }{2}. \end{aligned}$$

They obey

$$\alpha_{\mathrm H} +\beta_{\mathrm H} +1 = \gamma_{\mathrm H} +\delta_{\mathrm H} +\epsilon_{\mathrm H},$$

and

$$\alpha_{\mathrm H}-\beta_{\mathrm H} = \theta_\infty.$$

The compact normal-form accessory is

$$\mathcal K_{\mathrm H} = t\Lambda -t(t-1)c_t^{\mathrm{op}}.$$

The standard Heun accessory is the affine coordinate

$$\mathfrak q_{\mathrm H} = \mathcal K_{\mathrm H} + \frac{\gamma_{\mathrm H}}2 \left( t\delta_{\mathrm H} +\epsilon_{\mathrm H} \right).$$

Consequently

$$\begin{aligned} \mathfrak q_{\mathrm H} ={}& t\Lambda -t(t-1)c_t^{\mathrm{op}} \\ &+ \frac{\gamma_{\mathrm H}}2 \left( t\delta_{\mathrm H} +\epsilon_{\mathrm H} \right). \end{aligned}$$

This is the promised separation of roles:

$$\text{distinct roles:} \qquad t, \quad a_{\mathrm C}, \quad c_t^{\mathrm{op}}, \quad \mathfrak q_{\mathrm H}.$$

They are respectively a position modulus, a Coulomb coordinate, a normal-form residue, and a standard-form accessory parameter.

Now set

$$\begin{aligned} y(z) ={}& z^{-\gamma_{\mathrm H}/2} (1-z)^{-\delta_{\mathrm H}/2} \\ &\times \left( 1-\frac zt \right)^{-\epsilon_{\mathrm H}/2} \psi(z). \end{aligned}$$

Then $y$ satisfies

$$\begin{aligned} y'' &+ \left( \frac{\gamma_{\mathrm H}}z + \frac{\delta_{\mathrm H}}{z-1} + \frac{\epsilon_{\mathrm H}}{z-t} \right)y' \\ &+ \frac{ \alpha_{\mathrm H}\beta_{\mathrm H}z -\mathfrak q_{\mathrm H} }{ z(z-1)(z-t) }y = 0. \end{aligned}$$

Choose the oper branch $\psi\sim z^{(1-\theta_0)/2}$, the displayed branches of the scalar gauge, and the multiplicative constant for which $y(0)=1$. Provided

$$\gamma_{\mathrm H} \notin \mathbb Z_{\leq0},$$

the resulting local germ is

$$y(z) = \operatorname{HeunG} \left( t,\mathfrak q_{\mathrm H}, \alpha_{\mathrm H},\beta_{\mathrm H}, \gamma_{\mathrm H},\delta_{\mathrm H}; z \right).$$

The unit-leading normalization can be checked before any global continuation:

$$y(z) = 1+ \frac{\mathfrak q_{\mathrm H}} {t\gamma_{\mathrm H}}z +O(z^2).$$

Positive-integer $\gamma_{\mathrm H}$ is resonant but does not obstruct this exponent-zero normalized germ; it is generally the second local solution that develops a logarithm. When $\gamma_{\mathrm H}\in\mathbb Z_{\leq0}$, the unit-leading recurrence can instead be obstructed or nonunique and needs a compatibility condition or a parameter limit. HeunG is not a name for the full solution space or for the ungauged defect wavefunction.

The regular dictionary is reversible only with its passport

The central entries can now be read in either direction.

ODE or CFT datum	Gauge-theory datum	Convention and excluded locus	Source or status
Cross-ratio $t$ in the $0t$ channel	$q_{\mathrm{UV}}=\exp(2\pi\ii\tau_{\mathrm{UV}})$	Declared local coordinate near $t=0$; not the infrared nome or effective coupling	Gaiotto §2; AGT App. B.1, Eq. (B.1), and App. B.3
Internal $\alpha_{0t}=Q_{\mathrm L}/2+a_{\mathrm C}/\varepsilon_\star$	Coulomb modulus $a_{\mathrm C}$	Weyl/reflection sign fixed; generic away from Kac poles	AGT §3.2
External $\alpha_f=Q_{\mathrm L}/2+p_f/\varepsilon_\star$	Shifted sums and differences of $\mu_1,\ldots,\mu_4$	Two antifundamentals plus two fundamentals in the printed representative	AGT Eqs. (3.7)–(3.13), Apps. B.1–B.2
$\widehat{\mathcal V}_{0t}$	$Z_{\mathrm{inst}}^{U(2)}/Z_{\mathcal H}$	Generic formal series; OPE power excluded	AGT §3.2; AFLT Proposition 2.1
Five-point block $\mathscr B_5(z,t)$	Proposed defect function $Z_{\mathrm{defect}}(z,t)$	Compatible fusion channel; not a universal defect theorem	AGGTV §1.2 and §§2.1–2.2
$c_t^{\mathrm{op}}=\partial_t f_{0t}$	Normalized NS coupling derivative	Heavy/NS factorization, log branch, Heisenberg and OPE factors fixed	Piątek–Pietrykowski
HeunG local germ	Gauge-fixed local defect branch	$\gamma_{\mathrm H}\notin\mathbb Z_{\leq0}$ for the direct normalized germ	DLMF §§31.2(i)–(ii), 31.3(i)

Given the right column and the declared mass convention, the table constructs a CFT block and then an ODE probe. Given a Heun oper, the reverse direction first extracts oriented exponent differences, a modulus, and an accessory. It does not by itself choose the fundamental versus antifundamental representative, a Coulomb sign, a defect realization, or a spectral boundary condition.

S-duality changes the chart and flavor basis

The four-flavor theory has an enhanced $SO(8)$ flavor symmetry, while the chosen pants decomposition displays only an $SU(2)^4$ subgroup. The three pairings of the four punctures correspond to three weakly coupled frames and to the vector, spinor, and conjugate spinor presentations related by triality.

For the CFT block, changing the pairing is a fusion transformation. For the gauge theory, it is an electric–magnetic duality together with a reorganization of the flavor masses. For the Heun equation, it is a Möbius transformation of the singularities followed by an exponent gauge and an affine transformation of the accessory parameter.

Therefore neither

$$t\mapsto1-t$$

nor

$$t\mapsto t^{-1}$$

may be performed while keeping $(p_f,a_{\mathrm C},\mathfrak q_{\mathrm H})$ numerically fixed. The transformed channel has a different internal coordinate, a permuted or triality-rotated mass basis, and a transformed local solution normalization. What is invariant is the underlying theory or global continuation problem after the complete dictionary has been transported.

Interpretation and limitations

This regular prototype explains why AGT is powerful without making it a universal solution formula. The bulk equality identifies a chiral Virasoro block with the protected instanton series in the modulus and gives a highly effective expansion for the classical block. A full Liouville four-point correlator would additionally require structure constants, holomorphic–antiholomorphic pairing, and an internal-momentum integral. After the heavy or NS reduction, the degenerate extension supplies a two-component local system in the probe coordinate; at finite $b$ it remains the two-variable BPZ PDE above. Its limiting accessory is fixed by a logarithmic derivative of the background block, so instanton coefficients can determine a Heun equation order by order in the weak-coupling coordinate.

The construction is local in parameter space. It assumes a chosen sewing disk, logarithm branch, reflection representatives, and generic nonresonant data. Analytic continuation can cross block singularities or semiclassical saddle walls. The limiting Heun equation still does not choose a self-adjoint domain, a scattering contour, a quasinormal condition, or any other spectral realization.

Finally, $t\to0$ at fixed masses is the nodal weak-coupling boundary of the regular four-puncture theory; by itself it does not create an irregular puncture. A confluent Heun equation requires a coordinated collision or flavor-decoupling limit in which masses or momenta, the coupling, the accessory, and the wavefunction normalization are rescaled. The next page derives that irregular limit in the same matter and Heisenberg normalization.

Common pitfalls

Using one symbol for three jobs. The gauge fugacity $t$, Coulomb modulus $a_{\mathrm C}$, and Heun accessory $\mathfrak q_{\mathrm H}$ occupy distinct roles and are not interchangeable coordinates. Within a matched AGT family the accessory is determined from the modulus, masses, Coulomb or branch data, and the classical-block derivative. A formula that calls several of these objects $q$ or $a$ must be renamed before it is compared.

Calling the bulk block HeunG. The nondegenerate four-point block depends on $t$ and matches the bulk instanton series. The Heun coordinate $z$ enters only with the additional degenerate or defect observable.

Dropping the equivariant mass shift. In the displayed convention, $p_1$ and $p_t$ contain $-\epsilon_\Sigma/2$. Omitting it corrupts the external weights even when the classical mass relation looks plausible.

Discarding the Abelian factor before differentiating. The Heisenberg factor is independent of $a_{\mathrm C}$ but depends on $t$ and heavy masses. It changes the NS coupling derivative and hence the accessory relation.

Freezing the modulus inside the finite-$b$ PDE. Choosing a value of $t$ does not make $\partial_t\mathscr B_5$ vanish. Only the heavy or NS factorization converts its leading action into multiplication by an accessory coefficient.

Treating S-duality as a bare cross-ratio substitution. A new cusp comes with a new channel, Coulomb coordinate, flavor basis, and Heun normalization. Transport the whole passport.

Exercises

1. Invert the mass passport

Starting from the four $p_f$ in terms of $\mu_i$, recover all four $\mu_i$ and identify where the equivariant shift enters.

Solution

The difference and shifted-sum equations give

$$\begin{aligned} \mu_1 &= p_1+p_\infty+\frac{\epsilon_\Sigma}{2}, & \mu_2 &= p_1-p_\infty+\frac{\epsilon_\Sigma}{2}, \\ \mu_3 &= p_t+p_0+\frac{\epsilon_\Sigma}{2}, & \mu_4 &= p_t-p_0+\frac{\epsilon_\Sigma}{2}. \end{aligned}$$

The shift occurs in the average of each pair. It is absent from the differences $p_\infty$ and $p_0$, so checking only those two punctures cannot detect a missing equivariant centering.

2. Match Weyl and puncture reflections

Show what $\mu_1\leftrightarrow\mu_2$ and $a_{\mathrm C}\mapsto-a_{\mathrm C}$ do to the corresponding Liouville momenta.

Solution

The mass exchange fixes $p_1$ and sends $p_\infty\mapsto-p_\infty$. Hence

$$\alpha_\infty = \frac{Q_{\mathrm L}}2 +\frac{p_\infty}{\varepsilon_\star} \longmapsto Q_{\mathrm L}-\alpha_\infty.$$

Likewise,

$$\alpha_{0t} = \frac{Q_{\mathrm L}}2 +\frac{a_{\mathrm C}}{\varepsilon_\star} \longmapsto Q_{\mathrm L}-\alpha_{0t}$$

under the Coulomb Weyl reflection. Both transformations preserve the corresponding conformal weights.

3. Recover the one-instanton shift

First check that the displayed one-box expression is dimensionless and Weyl invariant. Then use $Z_{\mathrm{inst}}^{U(2)}=(1-t)^{\kappa_{\mathcal H}} (1+\mathcal V_1t+O(t^2))$ to derive the coefficient of $t$. Finally, substitute the mass passport and verify the direct localization result against $\mathcal V_1-\kappa_{\mathcal H}$.

Solution

Each $P_\pm$ has mass dimension four, while $\epsilon_1\epsilon_2a_{\mathrm C}(2a_{\mathrm C}\pm\epsilon_\Sigma)$ also has dimension four. Under $a_{\mathrm C}\mapsto-a_{\mathrm C}$, the $P_+$ and $P_-$ summands exchange, including their denominators, so their sum is Weyl invariant.

Next,

$$(1-t)^{\kappa_{\mathcal H}} = 1-\kappa_{\mathcal H}t+O(t^2),$$

multiplication gives

$$Z_{\mathrm{inst}}^{U(2)} = 1+ \left( \mathcal V_1-\kappa_{\mathcal H} \right)t +O(t^2).$$

Thus $Z_1^{U(2)}=\mathcal V_1-\kappa_{\mathcal H}$. Dividing by the Heisenberg factor restores the unit-leading Virasoro coefficient.

For the full audit, use

$$\Delta_j = \frac{Q_{\mathrm L}^2}{4} - \frac{p_j^2}{\varepsilon_\star^2}, \qquad \Delta = \frac{Q_{\mathrm L}^2}{4} - \frac{a_{\mathrm C}^2}{\varepsilon_\star^2},$$

together with $Q_{\mathrm L}=\epsilon_\Sigma/\varepsilon_\star$. After the four inverse mass formulas are inserted, both $\mathcal V_1-\kappa_{\mathcal H}$ and the two-box sum reduce to the same rational function with possible poles only at $a_{\mathrm C}=0$ and $2a_{\mathrm C}=\pm\epsilon_\Sigma$. Bringing the difference over that common denominator gives an identically zero numerator. This checks the mass ordering and the equivariant centering, not only the pole set.

4. Extract the surviving modulus derivative

Why can the nondegenerate four-point block not solve the Heun equation in $z$? What is the minimum CFT replacement? For that replacement, use $\mathscr B_5=\mathcal V_{0t}\Psi_b$ to identify which part of $b^2K(z,t)\partial_t\mathscr B_5$ survives the heavy limit.

Solution

The bulk block has the single variable $t$, the modulus of $C_{0,4}$. A Heun wavefunction requires an independent coordinate $z$ moving among the four fixed singularities. Inserting $V_{-b/2}(z)$ in a channel whose adjacent labels differ by $\pm b/2$, up to reflection, produces a five-point block $\mathscr B_5(z,t)$ with both variables and a level-two null equation. At finite $b$ that equation is still a PDE.

For its modulus derivative,

$$\partial_t\mathscr B_5 = \mathcal V_{0t} \left[ (\partial_t\Log\mathcal V_{0t})\Psi_b +\partial_t\Psi_b \right].$$

After the common heavy exponential is divided out, the declared limits give

$$b^2K\, \frac{\partial_t\mathscr B_5}{\mathcal V_{0t}} \longrightarrow K(z,t)(\partial_t f_{0t})\psi.$$

The derivative of the finite probe ratio is suppressed, while the derivative of the heavy background becomes the accessory term.

5. Read ODE exponents from masses

Use $\varepsilon_\star=b\hbar$ and $\alpha_f=Q_{\mathrm L}/2+p_f/\varepsilon_\star$ to show that the oper exponent difference at puncture $f$ is $2p_f/\hbar$.

Solution

The centered heavy momentum gives

$$2b \left( \alpha_f-\frac{Q_{\mathrm L}}2 \right) = 2b\frac{p_f}{\varepsilon_\star} = \frac{2p_f}{\hbar}.$$

Therefore $\theta_f=2p_f/\hbar$ and the two local normal-form powers are $(1\pm\theta_f)/2$. No factor of $\varepsilon_\star$ remains in the NS oper.

6. Check the Heun Fuchs relation

Verify that the five oriented Heun exponent parameters defined above satisfy the Fuchs relation.

Solution

The finite-point sum is

$$\gamma_{\mathrm H} +\delta_{\mathrm H} +\epsilon_{\mathrm H} = 3-\theta_0-\theta_1-\theta_t.$$

Meanwhile,

$$\alpha_{\mathrm H} +\beta_{\mathrm H} = 2-\theta_0-\theta_1-\theta_t.$$

Adding one to the latter gives the former. The difference $\alpha_{\mathrm H}-\beta_{\mathrm H}=\theta_\infty$ fixes the oriented exponent difference at infinity.

7. Convert the oper residue to the Heun accessory

Starting from

$$c_t^{\mathrm{op}} = \frac{ t\Lambda-\mathcal K_{\mathrm H} }{ t(t-1) },$$

and

$$\mathcal K_{\mathrm H} = \mathfrak q_{\mathrm H} - \frac{\gamma_{\mathrm H}}2 \left( t\delta_{\mathrm H}+\epsilon_{\mathrm H} \right),$$

solve for $\mathfrak q_{\mathrm H}$. Then insert $y=1+c_1z+O(z^2)$ into the standard Heun equation and recover $c_1$.

Solution

The first equation gives

$$\mathcal K_{\mathrm H} = t\Lambda-t(t-1)c_t^{\mathrm{op}}.$$

Substitution into the second yields

$$\mathfrak q_{\mathrm H} = t\Lambda -t(t-1)c_t^{\mathrm{op}} + \frac{\gamma_{\mathrm H}}2 \left( t\delta_{\mathrm H} +\epsilon_{\mathrm H} \right).$$

The shift is affine and depends on the local exponent gauge. Thus the standard accessory is not the moving-pole residue.

For the local series, the terms of order $z^0$ after multiplication by $z(z-1)(z-t)$ give

$$t\gamma_{\mathrm H}c_1 - \mathfrak q_{\mathrm H} = 0.$$

Hence

$$c_1 = \frac{\mathfrak q_{\mathrm H}} {t\gamma_{\mathrm H}},$$

which is the unit-leading HeunG checksum.

8. Move to another weak-coupling cusp

Set $w=1-z$ and $t'=1-t$. Relabel the four puncture masses so that the transformed points again appear as $(0,t',1,\infty)$, and compute the four new printed masses from the same passport. How does the moving-pole oper residue transform? Explain what is still not fixed by this relabeling.

Solution

The map sends the old points $(1,t,0,\infty)$ to $(0,t',1,\infty)$, so

$$p'_0=p_1, \qquad p'_{t'}=p_t, \qquad p'_1=p_0, \qquad p'_\infty=p_\infty.$$

Applying the same two-antifundamental/two-fundamental passport gives

$$\begin{aligned} \mu'_1 &=p_0+p_\infty+\frac{\epsilon_\Sigma}{2}, & \mu'_2 &=p_0-p_\infty+\frac{\epsilon_\Sigma}{2}, \\ \mu'_3 &=p_t+p_1+\frac{\epsilon_\Sigma}{2}, & \mu'_4 &=p_t-p_1+\frac{\epsilon_\Sigma}{2}. \end{aligned}$$

Because $dz/dw=-1$ and the Möbius Schwarzian vanishes, the simple-pole residue at the moving point changes orientation:

$$c_{t'}^{\prime\,\mathrm{op}} = -c_t^{\mathrm{op}}.$$

The new $0t'$ tube represents the old $t1$ pairing. Its internal momentum or electric Coulomb coordinate $a'_{\mathrm C}$ must be obtained by fusion or S-duality; it is not fixed by permuting the external masses. The scalar-gauge branches and the standard accessory $\mathfrak q'_{\mathrm H}$ must then be recomputed in the transformed chart. Thus replacing $t$ by $1-t$ alone is incomplete.

References

• D. Gaiotto, “N=2 Dualities”, Journal of High Energy Physics 08 (2012) 034. Section 2, especially equations (2.7)–(2.12) and Figure 3, identifies the four-punctured sphere, its pants decompositions, the $N_f=4$ theory, and the three weakly coupled duality frames.
• L. F. Alday, D. Gaiotto, and Y. Tachikawa, “Liouville Correlation Functions from Four-dimensional Gauge Theories”, Letters in Mathematical Physics 91 (2010), 167–197. Section 2 explains the four-puncture flavor triality. Section 3.2, especially equations (3.5)–(3.13), fixes the fundamental/antifundamental mass basis, external and internal momenta, and the four-point Heisenberg factor. Appendix B.1 fixes the ultraviolet instanton coordinate and matter orientation; Appendix B.2 explains the equivariant mass-zero shift; Appendix B.3 distinguishes the ultraviolet coordinate from the infrared nome.
• 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”, Journal of High Energy Physics 01 (2010) 113. Section 2 proposes the relation between a degenerate insertion and a surface defect. Section 1.2 labels this a conjecture or working hypothesis. Section 2.1, equation (2.6), records the source’s reciprocal-$b$ convention, while Section 2.2 records the necessary fusion restriction.
• V. A. Alba, V. A. Fateev, A. V. Litvinov, and G. M. Tarnopolsky, “On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture”, Letters in Mathematical Physics 98 (2011), 33–64. Proposition 2.1 constructs the generic Virasoro–Heisenberg basis whose factorized matrix elements reproduce the Nekrasov weights.
• A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, “Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory”, Nuclear Physics B 241 (1984), 333–380. The level-two null-state relation gives the exact differential equation for the degenerate probe.
• M. Piątek and A. R. Pietrykowski, “Solving Heun’s Equation Using Conformal Blocks”, Nuclear Physics B 938 (2019), 543–570. The paper analyzes heavy–light factorization of five-point degenerate blocks and the resulting Heun normal form and local solutions.
• NIST Digital Library of Mathematical Functions, Chapter 31, “Heun Functions”. Sections 31.2(i)–(ii) and 31.3(i), especially equations (31.2.1)–(31.2.4) and (31.3.1)–(31.3.4), define the standard equation, normal form, exponent ledger, and normalized local germ used in the final crosswalk. DLMF’s $W''=UW$ convention corresponds to $T_{\mathrm{op}}=-U$ here; the site calls DLMF’s normalized $H\ell$ germ HeunG.