The Heavy–Light Classical Limit

The classical limit used for the CFT–ODE dictionary is a coordinated double scaling, not the instruction “make the central charge large.” The four background primaries and the selected intermediate channel become heavy, with weights of order $b^{-2}$, while the degenerate insertion $V_{-b/2}(z)$ stays light, with weight of order one. The heavy part exponentiates into a classical four-point block; the light part survives as a probe wavefunction whose two degenerate fusion branches span a two-dimensional local solution space.

That factorization supplies the missing step in the five-insertion BPZ equation. The modulus derivative is not discarded. Acting on the heavy exponential, it becomes multiplication by a $z$-independent accessory residue. Only its action on the finite probe factor is subleading.

The limit used on this page

Datum	Scaling as $b\to0$	Limiting role
$c_{\mathrm{Vir}}$	$6b^{-2}+O(1)$	Classical Virasoro regime
Four background weights	$\Delta_i^{\mathrm{CFT}}=\delta_i b^{-2}+O(1)$	Double-pole coefficients of the oper
Selected internal weight	$\Delta_{0t}^{\mathrm{CFT}}=\delta_{0t}b^{-2}+O(1)$	Selects a classical-block branch
$V_{-b/2}(z)$	$\Delta_{\mathrm d}=O(1)$	Light degenerate probe
Cross-ratio and probe point	$t,z=O(1)$	Independent variables before the limit

This is not the analytic $c_{\mathrm{Vir}}=1$ regime, and it is not the two-heavy/two-light holographic vacuum-block limit. Here all four background weights and the exchanged weight are heavy; the single level-two degenerate insertion is light.

Central charge and weights must scale together

For the simplest real reading, take $b\to0^+$. For complex $b$, fix a sector and an asymptotic branch; then $c_{\mathrm{Vir}}$ diverges in magnitude rather than approaching $+\infty$ in an ordered sense.

With

$$Q_{\mathrm L} = b+b^{-1}, \qquad c_{\mathrm{Vir}} = 1+6Q_{\mathrm L}^2,$$

one has the exact expansion

$$c_{\mathrm{Vir}} = \frac{6}{b^2} +13 +6b^2.$$

Thus $b\to0$ implies $c_{\mathrm{Vir}}\to\infty$, but that alone does not specify a classical block. The external and internal weights must scale so that their ratios to $c_{\mathrm{Vir}}$ remain finite.

For every heavy label

$$j\in \left\{ 0,t,1,\infty,0t \right\},$$

use the centered momentum from the book conventions and choose

$$a_{\mathrm L,j} = \frac{\theta_j}{2b}.$$

Since $\Delta^{\mathrm{CFT}}(\alpha) =Q_{\mathrm L}^2/4-a_{\mathrm L}^2$,

$$\begin{aligned} \Delta_j^{\mathrm{CFT}} &= \frac{1-\theta_j^2}{4b^2} +\frac12 +\frac{b^2}{4} \\ &= \frac{\delta_j}{b^2} +O(1), \qquad \delta_j := \frac{1-\theta_j^2}{4}. \end{aligned}$$

More generally, $2b\,a_{\mathrm L,j}=\theta_j+O(b^2)$ gives the same leading result. In the uncentered momentum, write $\eta_j:=\lim_{b\to0}b\alpha_j$. Then

$$\eta_j = \frac{1+\theta_j}{2}, \qquad \delta_j = \eta_j(1-\eta_j).$$

Alternatively, the exact finite-$b$ lift $\alpha_j=\eta_j/b$ gives

$$\Delta_j^{\mathrm{CFT}} = \frac{\eta_j(1-\eta_j)}{b^2} +\eta_j.$$

Holding $\eta_j$ exactly fixed and holding $\theta_j$ exactly fixed are two different finite-$b$ lifts with the same leading $\delta_j$. The leading oper is insensitive to that distinction; its first correction need not be. The centered lift is especially useful here because it exposes the ODE exponent difference immediately.

The chosen degenerate field really is light

A convenient fixed-weight light representative has uncentered momentum

$$\alpha_\ell = b\sigma_\ell,$$

or by its reflection $Q_{\mathrm L}-\alpha_\ell$. Its weight is

$$\Delta_\ell^{\mathrm{CFT}} = \sigma_\ell +b^2\sigma_\ell(1-\sigma_\ell) .$$

By contrast, holding a generic uncentered momentum $\alpha_\ell=O(1)$ gives $\Delta_\ell^{\mathrm{CFT}}=O(b^{-1})$. That is small relative to $c_{\mathrm{Vir}}=O(b^{-2})$, but it is not the fixed-weight light scaling used here.

For the book’s $(2,1)$ presentation,

$$V_{2,1}(z) := V_{-b/2}(z),$$

and

$$\Delta_{\mathrm d} = \Delta^{\mathrm{CFT}} \left( -\frac b2 \right) = -\frac12 -\frac{3b^2}{4}.$$

Its weight remains finite while the background weights grow as $b^{-2}$. Its null relation,

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

is precisely the presentation whose BPZ equation retains a finite second derivative after multiplication by $b^2$.

The dual degenerate field behaves differently in this same limit:

$$\Delta^{\mathrm{CFT}} \left( -\frac{1}{2b} \right) = -\frac12 -\frac{3}{4b^2}.$$

It is heavy, and its null relation contains $b^{-2}L_{-2}$. It becomes the light presentation only after the dual exchange $b\leftrightarrow b^{-1}$; the Kac label alone does not license swapping the two fields.

Fusion powers become oper exponents

The finite-$b$ fusion power adjacent to a heavy insertion is

$$\rho_{i,\epsilon_i} = \frac{bQ_{\mathrm L}}2 + \epsilon_i b\,a_{\mathrm L,i}, \qquad \epsilon_i=\pm1.$$

Under the centered heavy scaling,

$$\rho_{i,\epsilon_i} = \frac{ 1+b^2+\epsilon_i\theta_i }{2} \longrightarrow \frac{1+\epsilon_i\theta_i}{2}.$$

This agrees with the indicial equation of a normal-form oper having double-pole coefficient $\delta_i$:

$$r(r-1)+\delta_i=0, \qquad r_\pm=\frac{1\pm\theta_i}{2}.$$

The CFT fusion branches therefore become the two local Frobenius powers of the limiting ODE. This is an independent check of both the sign in $\delta_i=(1-\theta_i^2)/4$ and the branch order fixed on the degenerate-fusion page.

A classical block is a logarithmic limit

Fix the $0t$ channel, the internal heavy weight, the branch of $\Log t$, and the normalization of the four-point chiral block $\mathcal V_{0t}(t;b)$. Its classical block is defined by

$$f_{0t}(t) := \lim_{b\to0} b^2 \Log \mathcal V_{0t}(t;b),$$

when this limit exists on the chosen branch. At minimum, the definition implies

$$\Log\mathcal V_{0t}(t;b) = \frac{f_{0t}(t)}{b^2} +o(b^{-2}).$$

On a regular generic semiclassical branch one often has the stronger expansion

$$\Log\mathcal V_{0t} = \frac{f_{0t}}{b^2} +g_{0t} +O(b^2).$$

This sign convention is a definition. Sources that write $\exp[-c_{\mathrm{Vir}}S/6]$ use the opposite leading convention, $f_{0t}=-S$, because $c_{\mathrm{Vir}}/6\sim b^{-2}$. Exponentiation of a generic four-heavy block is established independently of the mixed degenerate probe. The five-point factorization used below is a selected-branch semiclassical statement, explicitly realized in the references, rather than a theorem of global uniformity over the entire $t$-plane.

Recall the exact normalization

$$\mathcal V_{0t}(t;b) = t^{ \Delta_{0t}^{\mathrm{CFT}} - \Delta_0^{\mathrm{CFT}} - \Delta_t^{\mathrm{CFT}} } \widehat{\mathcal V}_{0t}(t;b), \qquad \widehat{\mathcal V}_{0t}(0;b)=1.$$

Define the normalized classical block by

$$\widehat f_{0t}(t) := \lim_{b\to0} b^2\Log \widehat{\mathcal V}_{0t}(t;b).$$

On the declared branch of $\Log t$,

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

Consequently,

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

This elementary shift is why an accessory formula is meaningless until the OPE prefactor has been stated.

The first sewing coefficient checks the scaling

The level-one coefficient derived on the Virasoro-block page gives

$$\widehat f_{0t}(t) = \kappa_1t +O(t^2),$$

where

$$\kappa_1 = \frac{ \left( \delta_{0t}+\delta_t-\delta_0 \right) \left( \delta_{0t}+\delta_1-\delta_\infty \right) }{ 2\delta_{0t} }.$$

Indeed, the quantum level-one coefficient grows as $b^{-2}\kappa_1$. At higher levels the claim is much stronger: powers more singular than $b^{-2}$ must cancel in the logarithm of the block. Semiclassical exponentiation is this nontrivial reorganization, not a termwise replacement of every weight by $\delta/b^2$.

The degenerate insertion factors off the heavy background

Choose one local degenerate branch and compatible chiral-vertex normalizations for the five-point block $\mathscr B_{5,\epsilon}(z,t;b)$ and the background four-point block. The heavy–light statement is that their ratio has a finite leading limit,

$$\psi_\epsilon(z;t) := \lim_{b\to0} \frac{ \mathscr B_{5,\epsilon}(z,t;b) }{ \mathcal V_{0t}(t;b) }.$$

Write the exact ratio as $\mathscr B_{5,\epsilon} =\mathcal V_{0t}\Psi_{b,\epsilon}$. The required statement is $\Psi_{b,\epsilon}\to\psi_\epsilon$. When a regular even-power expansion exists, this refines to $\Psi_{b,\epsilon} =\psi_\epsilon+b^2\psi_{\epsilon,1}+\cdots$. The heavy exponential remains entirely inside $\mathcal V_{0t}$, including any finite $z$-independent correction $\ee^{g_{0t}}$.

On compact probe domains away from the punctures, assume that $\Psi_{b,\epsilon}\to\psi_\epsilon$ locally uniformly in $z$. Holomorphy then gives convergence of its first two $z$-derivatives. In the modulus direction, the reduction below requires

$$\begin{aligned} b^2\partial_t \Log\mathcal V_{0t} &\longrightarrow \partial_t f_{0t}, \\ \Psi_{b,\epsilon} &\longrightarrow \psi_\epsilon, \\ b^2\partial_t\Psi_{b,\epsilon} &\longrightarrow 0. \end{aligned}$$

Pointwise convergence of $b^2\Log\mathcal V_{0t}$ alone does not imply the first derivative limit.

A vanishing momentum shift can leave a finite factor

In a branch in which the degenerate probe fuses with the insertion at $z=0$,

$$a_{\mathrm L,0}^{[\epsilon]} = a_{\mathrm L,0} -\epsilon\frac b2.$$

For the scaled exponent difference this is

$$\theta_0^{[\epsilon]} = 2b\,a_{\mathrm L,0}^{[\epsilon]} = \theta_0-\epsilon b^2.$$

Although the shift vanishes as $b\to0$, it occurs inside an exponential of order $b^{-2}$:

$$\begin{aligned} f_{0t} \left( \theta_0-\epsilon b^2 \right) &= f_{0t}(\theta_0) -\epsilon b^2 \partial_{\theta_0}f_{0t} +O(b^4), \\ \exp \left[ \frac{ f_{0t}(\theta_0-\epsilon b^2) -f_{0t}(\theta_0) }{b^2} \right] &\longrightarrow \exp \left( -\epsilon \partial_{\theta_0}f_{0t} \right). \end{aligned}$$

This is one contribution to the normalization of $\psi_\epsilon$. Local-coordinate powers and chiral-vertex conventions supply others. The leading oper needs only $\partial_t f_{0t}$, but connection coefficients can remember these finite momentum derivatives.

Power counting turns the BPZ PDE into an oper

Write the exact finite-$b$ equation derived on the BPZ-equations page as

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

where

$$p(z) = \frac1z+\frac{1}{z-1}, \qquad K(z,t) = \frac{ t(t-1) }{ z(z-1)(z-t) },$$

and

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

Multiply the equation by $b^2$ and insert the derivative-compatible factorization $\mathscr B_{5,\epsilon} =\mathcal V_{0t}\Psi_{b,\epsilon}$. After division by $\mathcal V_{0t}$, the exact rescaled equation is

$$\begin{aligned} 0={}& \partial_z^2\Psi_{b,\epsilon} -b^2p\,\partial_z\Psi_{b,\epsilon} +b^2U_b\Psi_{b,\epsilon} \\ &+ K \left( b^2\partial_t\Log\mathcal V_{0t} \right) \Psi_{b,\epsilon} +b^2K\,\partial_t\Psi_{b,\epsilon}. \end{aligned}$$

Every contribution now has an unambiguous order:

Contribution after multiplication by $b^2$	Leading action on the factorized block	Order
$\partial_z^2$	$\partial_z^2\psi_\epsilon$	$1$
$b^2U_b$	Heavy weights give the classical pole potential	$1$
$b^2K\partial_t$ on $\mathcal V_{0t}$	$K(\partial_t f_{0t})\psi_\epsilon$	$1$
$b^2K\partial_t$ on $\Psi_{b,\epsilon}$	$b^2K\partial_t\Psi_{b,\epsilon}\to0$	$o(1)$; $O(b^2)$ under the regular expansion
$-b^2p\partial_z$	$-b^2p\partial_z\psi_\epsilon$	$b^2$
The light weight inside $b^2U_b$	$b^2\Delta_{\mathrm d}$	$b^2$

Defining the finite-$b$ ratio $\Psi_{b,\epsilon}:=\mathscr B_{5,\epsilon}/\mathcal V_{0t}$ separates the heavy block from the probe before the limit is taken. Here $U_0(z,t):=\lim_{b\to0}b^2U_b(z,t)$ is the classical pole potential; only the three solid lanes assemble the leading oper.

The modulus derivative survives at leading order through $\partial_t f_{0t}$. What disappears is only its action on the finite probe factor. The leading equation is

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

with

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

where, for the unhatted block convention on this page,

$$c_t^{\mathrm{op}} = \partial_t f_{0t}.$$

For the hatted block,

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

The next page will compare this fixed-coordinate oper residue with other classical-block and standard-Heun accessory conventions. No such crosswalk should be inferred from the symbol “accessory” alone.

What the limiting statement does and does not guarantee

The derivation is local in parameter space. A safe version fixes:

• a channel, internal weight, branch of $\Log t$, and branch of the classical block;
• $t$ away from $0$, $1$, and $\infty$;
• $z$ in compact subsets away from $0$, $t$, $1$, and $\infty$;
• a domain avoiding zeros of the chosen block when its logarithm is used;
• generic weights away from singular Kac or resonant limits;
• a derivative-compatible asymptotic expansion in one sector of the $b$-plane.

Near a collision divisor, a $b$-dependent boundary-layer scaling can change the power counting. Analytic continuation in $t$ can also exchange classical saddles or branches. Such saddle walls are not automatically the Stokes curves of an irregular ODE.

The internal weight of a fixed chiral block remains prescribed; no saddle over that weight is being taken. An internal-momentum saddle belongs instead to a full correlator after its structure constants, antiholomorphic pairing, and integration contour have been supplied.

For each fixed admissible $t$, $\psi_\epsilon(z;t)$ solves an ODE in $z$ and may still depend parametrically on $t$. Its modulus-derivative contribution is subleading; under the regular even-power expansion above, it enters two orders later. At every nonzero $b$, the original five-point equation remains a PDE.

Common pitfalls

Taking only $c_{\mathrm{Vir}}\to\infty$. A classical block requires the external and selected internal weights to scale with $c_{\mathrm{Vir}}$. Keeping them fixed defines a different large-central-charge regime.

Calling both degenerate presentations light. Under $b\to0$, $V_{-b/2}$ is light and $V_{-1/(2b)}$ is heavy. They are exchanged only with the simultaneous duality $b\leftrightarrow b^{-1}$.

Deleting the modulus derivative. The term $b^2K\partial_t$ acts at order one on the heavy exponential. Only its action on the probe factor is suppressed.

Dropping the OPE prefactor silently. The hatted and unhatted classical blocks differ by an explicit logarithm, so their derivatives differ by a simple pole.

Ignoring a vanishing shift inside the exponent. A momentum shift of order $b^2$ changes $f/b^2$ by order one. Such terms affect normalized probe solutions and later connection coefficients.

Treating a local asymptotic as a global identity. The classical block has channel and analytic branches, and a fixed saddle need not dominate after arbitrary continuation in $t$.

Exercises

1. Separate the light and dual-degenerate weights

Expand $c_{\mathrm{Vir}}$, $\Delta(-b/2)$, and $\Delta(-1/(2b))$ as $b\to0$. Which degenerate field is light?

Solution

Direct substitution gives

$$\begin{aligned} c_{\mathrm{Vir}} &= 6b^{-2}+13+6b^2, \\ \Delta(-b/2) &= -\frac12-\frac{3b^2}{4}, \\ \Delta(-1/(2b)) &= -\frac12-\frac{3}{4b^2}. \end{aligned}$$

Thus $V_{-b/2}$ has finite weight and is the light probe in this limit. The dual field has a weight of the same order as the heavy background.

2. Recover the oper coefficient and exponents

Let $a_{\mathrm L}=\theta/(2b)$. Derive $\delta$ and the two limiting fusion powers.

Solution

The heavy weight is

$$\Delta^{\mathrm{CFT}} = \frac{Q_{\mathrm L}^2}{4} -\frac{\theta^2}{4b^2} = \frac{1-\theta^2}{4b^2} +O(1),$$

so

$$\delta=\frac{1-\theta^2}{4}.$$

The two fusion powers are

$$\rho_\epsilon = \frac{1+b^2+\epsilon\theta}{2} \longrightarrow \frac{1+\epsilon\theta}{2}.$$

They solve $r(r-1)+\delta=0$, as required by the normal-form oper.

3. Take the classical limit of the level-one coefficient

Use the exact level-one sewing coefficient to recover $\kappa_1$ in the small-$t$ expansion of $\widehat f_{0t}$.

Solution

Substitute $\Delta_j^{\mathrm{CFT}}=\delta_j/b^2+O(1)$ into

$$\mathcal V_1 = \frac{ \left( \Delta_{0t}+\Delta_t-\Delta_0 \right) \left( \Delta_{0t}+\Delta_1-\Delta_\infty \right) }{ 2\Delta_{0t} }.$$

The result is

$$\mathcal V_1 = \frac{\kappa_1}{b^2} +O(1),$$

with

$$\kappa_1 = \frac{ \left( \delta_{0t}+\delta_t-\delta_0 \right) \left( \delta_{0t}+\delta_1-\delta_\infty \right) }{ 2\delta_{0t} }.$$

Since $\widehat{\mathcal V}_{0t} =\exp(\widehat f_{0t}/b^2+O(1))$, its coefficient of $t$ gives $\widehat f_{0t}(t)=\kappa_1t+O(t^2)$.

4. Audit the OPE-prefactor shift

Starting from $\mathcal V_{0t} =t^{\Delta_{0t}-\Delta_0-\Delta_t}\widehat{\mathcal V}_{0t}$, derive the relation between the two classical accessory derivatives.

Solution

Take a logarithm, multiply by $b^2$, and pass to the heavy limit:

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

Therefore

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

The difference is fixed and explicit, but only after the branch of $\Log t$ and the block normalization have been declared.

5. Keep the correct part of the modulus derivative

Under the regular ansatz $\mathscr B_5=\ee^{f/b^2}(\psi+b^2\psi_1+\cdots)$, compute $b^2\partial_t\mathscr B_5$ through relative order $b^2$.

Solution

Differentiation gives

$$\begin{aligned} b^2\partial_t\mathscr B_5 = \ee^{f/b^2} \Bigl[ (\partial_t f)\psi \\ \qquad +b^2 \left( \partial_t\psi +(\partial_t f)\psi_1 \right) +O(b^4) \Bigr]. \end{aligned}$$

The first term survives and generates the accessory contribution. The derivative $\partial_t\psi$ is subleading; setting the whole modulus derivative to zero would remove the leading term as well.

6. Extract a finite factor from a vanishing fusion shift

Show that $\theta_0\mapsto\theta_0-\epsilon b^2$ produces a finite exponential factor in a classical block.

Solution

Taylor expansion gives

$$f(\theta_0-\epsilon b^2) = f(\theta_0) -\epsilon b^2\partial_{\theta_0}f +O(b^4).$$

Dividing the corresponding block by the unshifted block yields

$$\exp \left[ \frac{ f(\theta_0-\epsilon b^2)-f(\theta_0) }{b^2} \right] \longrightarrow \ee^{-\epsilon\partial_{\theta_0}f}.$$

The shift is invisible in the leading heavy weight but remains in the finite normalization of the probe solution.

7. Identify where the reduction is nonuniform

Give three situations in which the fixed-$z$, fixed-$t$ power counting does not by itself justify the limiting oper.

Solution

Examples include:

1. $z$ approaches $0$, $t$, $1$, or $\infty$ on a $b$-dependent scale, so derivative and pole terms can rebalance.
2. $t$ approaches a degeneration point $0$, $1$, or $\infty$, requiring a separate collision or matched limit.
3. Analytic continuation crosses a classical saddle or branch wall, so the selected exponent $f_{0t}$ no longer describes the same asymptotic contribution.

Resonant or Kac-degenerate parameter limits also require coordinated basis limits. None of these effects changes the exact finite-$b$ PDE; they restrict where the displayed asymptotic reduction is uniform.

8. Show why pointwise exponentiation is insufficient

Consider a family defined locally by

$$\Log B_b(t) = \frac{f(t)}{b^2} + \sin \left( \frac{t}{b^2} \right).$$

Show that $b^2\Log B_b\to f$ pointwise while the scaled logarithmic derivative need not converge.

Solution

Multiplication by $b^2$ gives

$$b^2\Log B_b(t) = f(t) + b^2 \sin \left( \frac{t}{b^2} \right) \longrightarrow f(t).$$

However,

$$b^2\partial_t\Log B_b(t) = f'(t) + \cos \left( \frac{t}{b^2} \right),$$

and the oscillating term has no limit for generic fixed $t$. The PDE-to-oper reduction therefore needs convergence of the scaled logarithmic derivative, or sufficient locally uniform derivative bounds, not only pointwise convergence of the logarithm.

References

• 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. Level-two null-vector decoupling and BPZ equations.
• M. Beşken, S. Datta, and P. Kraus, “Semi-Classical Virasoro Blocks: Proof of Exponentiation”, Journal of High Energy Physics 2020 (2020), 109. Direct exponentiation of Virasoro blocks with heavy external and exchanged weights.
• A. Litvinov, S. Lukyanov, N. Nekrasov, and A. Zamolodchikov, “Classical Conformal Blocks and Painlevé VI”, Journal of High Energy Physics 2014 (2014), 144. Degenerate-probe factorization, accessory derivatives, and the classical monodromy problem. Its Kac labels are interchanged relative to the convention in this book; the light field is identified by its null relation and weight.
• D. Harlow, J. Maltz, and E. Witten, “Analytic Continuation of Liouville Theory”, Journal of High Energy Physics 2011 (2011), 071. Heavy and fixed-weight light momentum scalings, complex saddles, and continuation-sector caveats.
• M. Piątek and A. R. Pietrykowski, “Solving Heun’s Equation Using Conformal Blocks”, Nuclear Physics B 938 (2019), 543–570. Heavy–light factorization of five-point degenerate blocks and the Heun reduction.
• J. Teschner, “Classical Conformal Blocks and Isomonodromic Deformations”, 2017. Classical asymptotics, branches, and the Garnier/isomonodromic interpretation.
• G. Bonelli, C. Iossa, D. Panea Lichtig, and A. Tanzini, “Irregular Liouville Correlators and Connection Formulae for Heun Functions”, Communications in Mathematical Physics 397 (2023), 635–727. Normalization-aware semiclassical blocks, BPZ-to-Heun reduction, and finite momentum-derivative factors.