Classical Conformal Blocks and Accessory Parameters from Isomonodromic Deformations

Classical conformal blocks naturally appear in the large central charge limit of 2D Virasoro conformal blocks. In the $AdS_{3}/CFT_{2}$ correspondence, they are related to classical bulk actions and are used to calculate entanglement entropy and geodesic lengths. In this work, we discuss the identification of classical conformal blocks and the Painlev\'e VI action showing how isomonodromic deformations naturally appear in this context. We recover the accessory parameter expansion of Heun's equation from the isomonodromic $\tau$-function. We also discuss how the $c = 1$ expansion of the $\tau$-function leads to a novel approach to calculate the 4-point classical conformal block.

Classical conformal blocks are defined by the conformal block exponentiation conjecture in the large central charge limit 1 [1,6,16] We denote the central charge by c and ∆ i = c 6 δ i , i = 0, x, 1, ∞, are the conformal dimensions of the chiral primary operators V ∆ i , with δ i being the classical dimensions. Π ∆ is the projection operator to the intermediate channel with weight ∆ = c 6 δ. In broad terms, we say that an operator O is light if its weight ∆ O c and it is called heavy if ∆ O ∼ c as c → ∞. The function f δ ({δ i }; x) is called the classical conformal block [2,3]. No closed CFT expression is known for this special function. It can be written as a series expansion in x using a direct CFT approach [18] or Zamolodchikov's recurrence formula [1,19]. The CFT approach quickly gets too cumbersome to find the explicit coefficients at higher orders in x. A formal resummation of the recurrence formula was presented in [20], but the full classical conformal block is still out of reach.
A more promising direction is to obtain classical conformal blocks via the AGT correspondence [21]. The Nekrasov partition function Z Nek encodes information about the moduli space of vacua and its non-perturbative corrections in supersymmetric gauge theories [22]. According to the AGT correspondence, for a certain class of N = 2 SUSY theories, Z Nek is identified with a 2D Liouville conformal block. One can show that the so-called Nekrasov-Shatashvili limit [23] is equivalent to the large central charge limit of Liouville theory. This fact was used in [24,25] to obtain an expression for the classical conformal block in terms of the N = 2 twisted superpotential, calculated at the saddle-point of the partition function. The saddle-point condition then has to be solved order-by-order in x. The twisted superpotential can be understood in terms of the symplectic geometry of the moduli space of SL(2, C) flat-connections and the Bethe/gauge correspondence [26], which gives extra hints on the deeper mathematical structure of classical conformal blocks. For a review on exact results in N = 2 field theories, see [27].
In a parallel development, Litvinov et al [3] discussed how the 4-point classical conformal block is related to the classical action of the Painlevé VI (PVI) equation [28]. In this approach, the derivative of the PVI action, evaluated on a PVI solution with certain boundary conditions, gives the accessory parameter of a Fuchsian differential equation with 4 regular singular points, also known as Heun's equation [29]. The Heun equation is obtained by the classical limit of the 5-point conformal block with a level-2 light degenerate insertion The level-2 null vector equation for ϕ L (z), also known as level-2 BPZ equation [18], reduces to the normal form of Heun's equation where the accessory parameter is given by On the other hand, the semiclassical limit of the 5-point conformal block with a level-2 heavy degenerate insertion obeys a BPZ equation equivalent to the Hamilton-Jacobi equation of the PVI action. This means that the BPZ equation implies that λ must be a solution of the PVI equation.
In order to recover the 4-point classical conformal block from the PVI action, the authors of [3] set λ = ∞ in (1.5) and obtain an integral formula for the classical block. Taking the derivative of this formula then leads to the accessory parameter of Heun's equation (1.4) in terms of the initial condition for λ, fixed by λ = ∞. With a clever usage of a double series expansion of the PVI solution [28], the authors of [3] managed to solve the condition λ = ∞ order by order in x and then substituted the result into the accessory parameter formula. However, their procedure relies on substituting the double series expansion into the PVI equation, which is a complicated second order non-linear differential equation, to obtain the doubles series expansion terms also order by order in x.
In this paper, we show how the isomonodromic τ -function [30][31][32], also known as PVI τ -function [33,34], can be used to find the accessory parameter expansion discussed in [3] in a more straightforward way. Both PVI solutions and the accessory parameter can be written in terms of the τ -function. Since the works of Sato, Jimbo and Miwa [35][36][37][38][39], it is known that the PVI τ -function is related to a c = 1 correlator of monodromy fields. In fact, the isomonodromic approach effectively solves the Riemann-Hilbert problem of SL(2, C) Fuchsian systems [30][31][32]. For a long time, only the asymptotics of the τ -function was known, limiting its scope of applications. Its full expansion was constructed only recently in [34] and proved in [40]. This expansion is given by a linear combination of c = 1 conformal blocks, which are written in closed form via AGT correspondence. There are two integration constants (σ, s) for the τ -function, labeling irreducible representations of the 4-point monodromy group, which we review in section 3.3. The relevant expansion in terms of c = 1 conformal blocks [34] is presented in 4.1. This τ -function expansion is what allows us to solve the initial condition for the isomonodromic flow and then find the accessory parameter expansion.
The way we solve the accessory parameter expansion clarifies two important things. First, the initial condition used by [3] can be uniquely defined in terms of isomonodromic deformations. Second, the recent result [34] on the c = 1 expansion of the PVI τ -function allow us to present the accessory parameter expansion in a more systematic way, providing a practical algorithm on how to fully solve this problem.
Let us now give the outline of this paper. In section 2, we review the relationship between the semiclassical limit of BPZ equations and Fuchsian equations. We introduce a slightly more general derivation than [3], using a 6-point conformal block with 2 degenerate insertions instead of a 5-point conformal block. This introduces the relationship between the classical conformal block (1.5) and the Painlevé VI action.
In Section 3, we review the standard setup of isomonodromic deformations, the τfunction definition and the associated ordinary differential equation (ODE) with one extra apparent singularity. We then review the connection between the semiclassical limit of CFT correlators and isomonodromic deformations [41,42]. We move on to discuss the relationship between Fuchsian equations and the monodromy group of the 4-punctured sphere, summarizing how the different parameters in this paper relate with each other. Finally, we show how the BPZ equations of section 2 can be encoded in a Fuchsian system when c = 1. The conclusion is that the monodromy data of a c = ∞ conformal block can be encoded in a c = 1 Fuchsian system. This makes the connection between classical conformal blocks, Painlevé VI and isomonodromic deformations explicit.
In section 4, we present our algorithm to calculate the accessory parameter (1.4) using the isomonodromic τ -function [33,34]. The algorithm consists of three steps, described in detail in this section. The crucial step is to solve a special initial condition to the isomonodromic flow. We can only solve this condition order-by-order in x, in a similar fashion to [3] or the AGT approach of [25]. This constrains the general moduli space of the monodromy group to a subspace with only one parameter, the composite monodromy σ. Our approach gives new analytic insights on classical conformal blocks compared to [3].
In Section 5, we show how the PVI action can be written only in terms of τ -functions. In principle, this gives a formula for the 4-point classical conformal block, up to the solution of the appropriate PVI initial condition. However, as we discuss in the linear dilaton case, the PVI τ -function simplifies in certain special cases [34,43]. We leave the study of the different special cases for future work.
The relationship between isomonodromic deformations and the CFT semiclassical limit has been previously explored in [41,42]. Although these papers recognize the importance of isomonodromic deformations, the relevance of the τ -function and its detailed implementation has only been discussed here. Therefore, our main contribution is to give a unified prescription on how to use the isomonodromic τ -function to obtain the accessory parameter of Heun's equation and the associated classical conformal block in 2D CFT.

Classical Conformal Blocks and Accessory Parameters
In this section, we review how the semiclassical limit of a special 6-point conformal block leads to a Fuchsian equation with 4 singular points and one apparent singularity. The method generalizes to arbitrary n-point conformal blocks with an appropriate number of extra degenerate insertions [41,42].
Our derivation relies only on the 2D conformal symmetry and the definition of conformal blocks. Although we use the standard Liouville notation for the conformal dimensions and central charge, we do not make any particular assumption about the spectrum. This comes a posteriori and it is the main point of the bootstrap program [2,6,7,[44][45][46][47]. For reviews on Liouville theory and CFT, we suggest [48,49].
A chiral primary operator V ∆(P ) has conformal dimension where P is the momentum of the operator and Q parametrizes the central charge as with b ∈ C. The spectrum is dual under b → 1/b, and we choose b → 0 to denote the semiclassical limit. Let us consider the chiral 6-point correlator with two degenerate insertions where ϕ ∆ L and ϕ ∆ H stands for light and heavy level-2 degenerate operators, respectively. ∆ = (∆ 1 , . . . , ∆ 4 ) stands for the conformal dimensions of generic heavy fields at positions z = (z 1 , · · · , z 4 ). The conformal dimensions can be written as C P represents the appropriate products of structure constants. With the ordering given in figure 1, the conformal block F P (∆ L , ∆ H , ∆|z, λ, z) is labeled by the intermediate momenta P = (P, P + is 1 2b , P + is 1 2b + is 2 b 2 ), s 1 , s 2 = ±1.
where we omitted the conformal dimensions in F P (z, λ, z) for convenience. We can simplify these equations using global conformal transformations and the Ward identity where T (w) is the 2D stress tensor, with z L = z and z H = λ. A straightforward consequence of the asymptotic behavior T (w) ∼ w −4 as w → ∞ is that [48]. Using (2.6) and (2.7) in (2.5a), we choose z 1 = 0, z 2 = t, z 3 = 1, z 4 = ∞ and relabeling the ∆'s accordingly, we get Let us now analyze the semiclassical limit. Assuming P k = iθ k /b and P = iσ/b as b → 0, we have Assuming heavy-light factorization and exponentiation 2 [2, 3, 16], we write the semiclassical limit of the 6-point conformal block when b → 0 as (2.11) We simplified the notation to P → (σ, ±) because the light field ϕ L (z) does not contribute to the intermediate momenta in the semiclassical limit. The function ψ(z, λ, t) encodes the fusion (monodromy) information of the light field, as we are going to see below. Fusing the light degenerate field with any of the other fields, we end up with the semiclassical limit of the 5-point block with all insertions being heavy Substituting (2.11) in (2.8) gives which is a Fuchsian equation with 5 singular points and accessory parameters (2.14) Following the same procedure for (2.5b), we get The semiclassical limit of this equation gives a constraint on the accessory parameters (2.14) This is exactly the condition for z = λ to be an apparent singularity of (2.13) [41,42,50]. This means that ψ(z, λ, t) has integer monodromy around z = λ but no logarithmic behavior. Thus z = λ is not a singular point of the solution. Moreover, (2.16) can be interpreted as a Hamilton-Jacobi equation for S ± σ (λ, t) The equation of motion for λ(t) obtained from this Hamiltonian is the PVI equation, therefore S ± σ (λ, t) is the PVI action. The heavy BPZ equation gives a saddle-point condition for S ± σ (λ, t) [3]. If we define the Hamiltonian system (λ(t), p(t)) evolving under the PVI Hamiltonian (2.18), it is possible to show that the monodromy data of the Fuchsian equation (2.13) does not change as we change t in the complex plane. Therefore, (2.13) is the isomonodromic deformation of a 4-point Fuchsian equation, a deformed Heun's equation, with z = λ being an apparent singularity and not contributing to the monodromy data [41,42,50]. This means that isomonodromic deformations naturally emerge in CFT. We will review the standard setup of isomonodromic deformations in the next section. We also discuss how isomonodromic deformations relate the monodromy group of the 4-punctured sphere and the moduli space of Fuchsian equations (also called opers in the literature [51]).

Isomonodromic Deformations and the Semiclassical Limit
In the previous section, we saw that the conformal block exponentiation (2.11) effectively transforms the light BPZ equation into a linear ODE for ψ(z, λ, t). The classical conformal block S ± σ (λ, t) turns out to be the PVI action. In this section, we review the isomonodromic setup from a Fuchsian system and how to obtain the semiclassical equations (2.13) and (2.17) in this approach. We will then see that (2.17) is equivalent to the definition of the isomonodromic τ -function when (λ, p) are PVI solutions. We also make a digression about the monodromy group and the moduli space of flat connections, summarizing how the different objects introduced in this paper can be labeled by the two PVI integration constants (σ, s). Finally, we finish this section arguing that the monodromy data of solutions of both heavy and light BPZ equations can be encoded in a Fuchsian system. We show that this can be consistently done only if c = 1. This has a two-fold purpose: first, to argue that the isomonodromic τ -function can be understood as a c = 1 correlator and, second, to show that the monodromy data of c = ∞ conformal blocks and c = 1 correlators can be encoded in the same Fuchsian system.

Isomonodromic Deformations and the Garnier System
In this section, we recover the standard c = 1 setup for isomonodromic deformations of Fuchsian systems [30][31][32]34]. Let us start with the following Fuchsian system for the vector where the A i are gl(2, C) matrices with We call this choice of Tr A i the canonical gauge. The Fuchsian system above has n singular points a = (a 1 , . . . , a n ) and corresponding monodromy coefficients θ = (θ 1 , . . . , θ n ) (see section 3.3 for details on the monodromy group). The integrability conditions of (3.1) are the Schlesinger equations [34, 52] which is a closed 1-form provided that the Schlesinger equations are satisfied. This is the generating function of the isomonodromic Hamiltonians One can show from the Schlesinger equations (3.3) that these Hamiltonians generate the isomonodromic flow for λ = λ(a) [30,50]. Let us focus now on the n = 4 case. Applying a Möbius transformation, we fix the singular points to a = (a 0 , a t , a 1 , a ∞ ) = (0, t, 1, ∞) and monodromy parameters θ = (θ 0 , θ t , θ 1 , θ ∞ ). Then we have the Fuchsian system where 2θ ∞ = κ 1 − κ 2 − 1 and κ 1 + κ 2 = −2(θ 0 + θ 1 + θ t ). These last conditions can be solved to Notice that we used the SL(2, C) gauge freedom to fix A ∞ to be in diagonal form. The solution Ψ has monodromies on the complex z-plane given by the eigenvalues of

3). A convenient parameterization for the A i was given by [31]
where p i and q i are functions of (λ, t) and the fixed monodromy parameters θ. The diagonal form of A ∞ in (3.7) implies the constraints i=0,1,t The second equation above implies that A 12 (z) must have a simple zero in z and, for consistency with (3.6), it has to be at z = λ , k ∈ C. (3.11) We can then solve for the q i 's in (3.10) via We have only two equations left in (3.10) for the three p i 's, so we introduce the variable and solve the constraints for them in terms of (λ, µ, t). The expressions for the p i 's are not particularly enlightening to display here and can be found in [31]. Let us consider the second order ODE for the first component of Writing (p i , q i ) in terms of (λ, µ, t) and using the parameterization (3.9) above, we get the deformed Heun equation in canonical form with the accessory parameter K = K(θ; λ, µ, t) given by Notice that (3.8) implies in As we saw above, the integrability conditions for the Fuchsian system are equivalent to the isomonodromic equations (3.3). Using the parameterization above in terms of (λ, µ, t), the isomonodromic equations reduce to the Garnier system [50,53] We denote a solution of the isomonodromic flow as (λ(t), µ(t)). The second order equation for λ(t) is the Painlevé VI equation [31]. The PVI solutions are, in general, transcendental, i.e., cannot be reduced to simple algebraic or special functions. Jimbo has used the isomonodromic technique [33] to find the asymptotics of the τ -function (3.5) and, consequently, of the PVI transcendents, near its critical points t = 0, 1, ∞. The full expansion of the PVI τ -function was found only recently in [34]. We will review this formula in section 4.

Semiclassical BPZ Equations from the Fuchsian System
We claimed in section 2 that a certain heavy-light 6-point correlator naturally encodes isomonodromic equations. To show this explicitly, we obtain the semiclassical Fuchsian equation (2.13) from the Fuchsian system (3.6). Applying the transformation to (3.15), we find the semiclassical Fuchsian equation where the monodromy parameters are encoded by and the accessory parameters are Now comparing (3.20) and (3.23) with (2.13) and (2.16), we find and thus we have recovered the semiclassical BPZ equations of section 2 from the isomonodromic Fuchsian system. At this point, probably it is not clear to the reader what is the relation between the classical intermediate momentum σ in the semiclassical action and the Fuchsian system parameters (λ, µ, t) (or (λ, p, t) in the Fuchsian equation (3.20)). We will clarify this point in the rest of this section by discussing the relationship between the monodromy group, the moduli space of flat connections and the semiclassical action.

Monodromy Group, Flat Connections and the Semiclassical Action
In this section, we will consider the Fuchsian system (3.6) in the SL(2, C) gauge, where Tr A i = 0. This can be obtained by the gauge transformation (3.39) discussed below. Assuming that λ and t are fixed, the formal solution of (3.6) is given by where P represents a path-ordered exponential and z 0 is an arbitrary base point. A consequence of this formula is that the poles of the gauge connection A(z) correspond to branch points of Ψ(z). If we do the analytic continuation of Ψ(z) around a closed path γ, enclosing one or more singular points, the solution will change by a monodromy matrix M γ , i.e., Ψ γ = M γ Ψ. Elementary paths enclosing only one singular point a i have monodromy matrix M i and we label those matrices by their trace Tr M i = 2 cos(2πθ i ). The four-point monodromy group is then generated by three out of four SL(2, C) matrices obeying the monodromy identity As Ψ(z) is analytic everywhere except at the branch cuts, the knowledge of the related monodromy data essentially determines the solution next to these points. The global information on how to connect different local solutions is encoded in the composite monodromies, obtained when a path encloses two singular points (two or more points for n > 4). For example, Ψ γ 0t = M 0 M t Ψ has composite monodromy parameter defined by Tr M 0 M t = −2 cos(2πσ 0t ) (see figure 2). All representations of the 4-point monodromy group are labeled by 4 elementary monodromies θ = (θ 0 , θ t , θ 1 , θ ∞ ) and 3 composite monodromies σ = (σ 0t , σ 1t , σ 01 ), as this is the sufficient data to generate all possible loops around singular points in the 4-punctured sphere. We call (θ; σ) the monodromy data associated to Ψ(z). Let us define the monodromy parameters by Assuming that the p i 's are fixed, irreducible representations of the monodromy group are labeled by three composite monodromies (p 0t , p 1t , p 01 ). For SL(2, C) matrices, the monodromy parameters also obey the Fricke-Jimbo relation [33] p 0t p 1t p 01 (3.28) and thus only two composite monodromies are independent of each other. Following [33,40], if we fix the θ and σ 0t , the Fricke-Jimbo relation can be parametrized in terms of s 0t as with coefficients given by (3.30d) Solving the system (3.29) for s 0t when we get s ± 0t (cos 2π(θ t ∓ σ 0t ) − cos 2πθ 0 )(cos 2π(θ 1 ∓ σ 0t ) − cos πθ ∞ ) = (cos 2πθ t cos 2πθ 1 + cos 2πθ 0 cos 2πθ ∞ ± i sin 2πσ 0t cos 2πσ 01 ) − (cos 2πθ 0 cos 2πθ 1 + cos 2πθ t cos 2πθ ∞ ∓ i sin 2πσ 0t cos 2πσ 1t )e ±2πiσ 0t . (3.32) The special cases when (3.31) is not true correspond to reducible representations, which are all listed in the context of PVI solutions in [28]. In those cases, to find s we should go back to the Fricke-Jimbo relation (3.28). In conclusion, we can label irreducible representations of the 4-point monodromy group by two parameters (σ 0t , s 0t ). This parametrization is essentially the same under the permutation of the composite monodromies and its related to the number of independent ways to slice the 4-punctured sphere into two pairs of pants [40].

Summary of Parameters
In section 3.1, we parametrized the gauge connection A(z) in terms of elementary monodromies θ and two extra parameters (λ, µ) (or (λ, p)). Therefore, for fixed t, the moduli space of flat connections A(z) can be labeled by (θ; λ, µ). From (3.25), it is clear that there should be a map between the moduli parameters (λ, µ) and the monodromy parameters (σ 0t , s 0t ). This is the Riemann-Hilbert map for Fuchsian systems, the map between irreducible representations of the monodromy group and the moduli space of flat connections [40,50]. As explicitly shown in [33], isomonodromic deformations define such map via the integration constants (σ 0t , s 0t ). Jimbo obtained the asymptotics of A(z; λ(t), µ(t), t) for small t and showed that the formulas only depend on (θ; σ 0t , s 0t ). This can then be used to find the asymptotics of the τ -function and the PVI solutions (λ(t), µ(t)) in terms of (θ; σ 0t , s 0t ). This will become clear in section 4 when we discuss the isomonodromic τ -function expansion. The semiclassical equation (3.20) is parametrized by λ and the accessory parameters H and p, for fixed t and θ. As H is a function of (λ, p, t), these parameters label the possible equations. If we assume that (λ(t), p(t)) are solutions of the isomonodromic equations, i.e.
then (λ(t), p(t)) are PVI solutions and can be labeled by the monodromy parameters (σ 0t , s 0t ). Therefore, for fixed (t, θ), we have that (λ(t), p(t)) ∼ (σ 0t , s 0t ). We also showed in (3.24) that the accessory parameters are given by derivatives of the semiclassical action S σ (λ(t), t). To complete the picture relating the semiclassical equation (3.20) and the monodromy parameters, we need to show that the classical intermediate momentum σ is the composite monodromy σ 0t . Let us assume that, for fixed λ = 0, 1, t, ∞, the solution ψ(z, λ, t) of (3.20) has a small t expansion as (3.34) Assuming that (3.34) is well-defined at both z = 0 and z = t, for t small enough, σ 0t represents the composite monodromy parameter. Substituting (3.34) in (3.20), we find at lowest order and thus The assumption that (λ(t), µ(t)) is a solution of the isomonodromic flow connects the different parameters, since all quantities of interest can be phrased in terms of (σ, s).

Fuchsian System and c = 1 BPZ equations
A natural question is whether it is possible to encode the level-2 heavy and light BPZ equations into a Fuchsian system for any value of c. This was proved for a single c = 1 BPZ equation in [54]. As we show below, we can recover each level-2 BPZ equation separately for arbitrary c from an appropriate Fuchsian system. However, we can only consistently recover both BPZ equations if c = 1. The relationship is true if the associated linear system allows for isomonodromic deformations. Let us first change the gauge of (3.1) by applying the transformation to (3.40), we get where 1 , 2 and α are three arbitrary constants and such that The choice of Tr B i sets the SL(2, C) gauge for the Fuchsian system. To derive a second order equation for Φ, we take the derivative of (3.40a) with respect to z Using the relation for sl(2, C) matrices we can easily show that where the accessory parameters are defined by From (3.40c) and (3.40b), we obtain Plugging (3.45) and (3.47) into (3.43) and dividing by 2 , we get Now we apply the transformation Φ(a|z, λ) = (z − λ)  Notice that (3.46) and (3.5) imply that h(a) is the τ -function up to a overall function of t. Then we get the following equation for χ where ∆ H = − 3 1 4 2 − 1 2 . Setting α = 2 and b 2 = 2 / 1 , this equation becomes the generalization of the BPZ equation (2.5a) for a correlator with n arbitrary insertions at z = a and two degenerate insertions, one at z and another at λ. However, if we repeat the same procedure for (3.40b), we only get the second BPZ equation (2.5b) if we choose α = − 1 . This means that α = 2 = − 1 is a sufficient condition for the Fuchsian system (3.40) consistently reproduce the two BPZ equations. Accordingly, b = i and thus c = 1 in the CFT interpretation. Using the parametrization (2.1) and (2.2), the c = 1 BPZ equations are thus where ∆ i = θ 2 i is the c = 1 conformal weight. This shows that the Fuchsian system (3.40) with 2 = − 1 simultaneously encodes the monodromy data of the two c = 1 BPZ equations above. Moreover, this fact is the starting point of the original argument of why the isomonodromic τ -function is equivalent to a c = 1 correlator [34]. For a proof of this fact, see [40].

Accessory parameter from the Isomonodromic τ -function
We previously discussed how the accessory parameter H(λ(t), p(t), t) in (3.20) can be written in terms of the monodromy data (θ; σ, s), given that (λ(t), p(t)) are solutions of the isomonodromic flow. Now we want to use this fact to write the accessory parameter expansion H x of Heun's equation (1.3) in terms of the monodromy data. The key point is to impose a special initial condition for the isomonodromic flow, which we discuss next. Then we will present our algorithm on how to solve this initial condition and obtain H x using the τ -function expansion of [34]. We review the τ -function expansion in section 4.2, solve the initial condition in section 4.3 and finally present the accessory parameter expansion in section 4.4.

Accessory Parameter from Initial Conditions
Let us focus on the n = 4 isomonodromic system (3.6). Here we discuss how to recover the accessory parameter of Heun's equation (1.3) from the deformed Heun's equation (3.15) by an appropriate initial condition on (λ(t), µ(t)) at t = x. The canonical form of (1.3), obtained by the transformation ψ = i=0,1,x (z − a i ) (1−2θ i )/2 y, is given by y(z) = 0, (4.1) where the accessory parameters of both equations are related by Consider the deformed Heun equation (3.15), which we repeat here for convenience to the reader, We recover (4.1) from (4.3) by applying the following initial condition for the isomonodromic flow to (4.3) [55,56] We show in Appendix A that this condition is well-posed with respect to the isomonodromic equations. As we discussed before, (λ(t), µ(t)) have (σ, s) as integration constants, so imposing (4.5) seems equivalent to fix the values of (σ, s) separately. However, as we are going to see next, due to the special structure of the isomonodromic flow, the condition λ(x) = x implies a non-trivial relation s = s(σ, x) between the composite monodromy parameters. This means that Heun's equation (4.1) lives in a proper subspace of all possible Fuchsian opers with 4 singulars points. Litvinov et al [3] found the accessory parameter expansion of H x by regularizing the PVI action via the solution of PVI equation with λ(x) = ∞. Here we present an alternative way to calculate the accessory parameter using the isomonodromic τ -function expansion [34] with the condition λ(x) = x. This gives a clearer implementation of the proposal [3] and, in addition, suggests a deeper connection between c = ∞ and c = 1 conformal blocks. Notice that the condition λ(x) = ∞ is equivalent to our choice by a bi-rational transformation [28].
Our algorithm to find the accessory parameter expansion consists of three steps 1. Write the accessory parameter H x in terms of the isomonodromic τ -function; 2. Solve the initial condition λ(x) = x to obtain a monodromy constraint s = s(σ, x); 3. Substitute the constraint s = s(σ, x) in the τ -function to obtain H x .
We will show below that the condition λ(x) = x can be phrased in terms of the τ -function, which is essential to find s(σ, x) as a series expansion in x. We remind the reader that the well-posedness of the initial condition is discussed in appendix A. After obtaining s(σ, x), to obtain the accessory parameter expansion is just a matter of straightforward calculation. This calculation is cumbersome at higher orders, so we show it explicitly only up to order x 0 in this section. In Appendix C, we reproduce the analytic CFT formulas up to order x 2 and present numerical evidence up to order x 5 , compared to the direct CFT approach.
The key to understand the relation s = s(σ, x) is to phrase the initial conditions in terms of the isomonodromic τ -function. We will do this in the next subsection. Here we express H x in terms of the τ -function. For convenience we define and θ 0,0 ≡ θ. The definition of the isomonodromic Hamiltonian (3.5) for our particular 4-point case is given by Using the parameterization (3.9) for the A i 's in (4.7), we find that H S,t (θ 0,+ ; λ, µ, t) = K(θ +,+ ; λ, µ, t) + 4θ Imposing the initial conditions (4.5) in (4.8), we obtain where we used the τ -function definition Notice that (4.9) substitutes the initial condition for µ(x) in (4.5), as it gives K x in terms of the monodromy data. Together with (4.2), we can then find H x in terms of the τ -function.

CFT Expansion of the τ -function
In order to proceed, we need to introduce the τ -function expansion of [34]. For later convenience, we define a slightly different τ -function This new τ -function changes the Hamiltonian, but does not change the equations of motion, as it is multiplied by a pure function of t. The complete expansion of the Painlevé VI τ -function, adapted to our definition (4.11), is given by τ (θ; t) = n∈Z C(θ, σ + n)s n t n(n+2σ) B(θ, σ + n; t), (4.12) where we assume that the real part of sigma σ obeys The structure constants are given in terms of Barnes functions 3 , (4.14) and the B's are the c = 1 conformal blocks, given by the AGT combinatorial series summing over pairs of Young diagrams λ, µ with where (i, j) denotes the box in the Young diagram λ, λ i the number of boxes in row i, λ j the number of boxes in column j and h λ (i, j) = λ i + λ j − i − j + 1 its hook length. A proof for this expansion is available in [40] and alternatively in [57].

Solving the condition λ(x) = x
Inspired by [3], our approach to find H x is to solve the condition λ(x) = x by assuming a series expansion for s = s(σ, x) in x. The Painlevé property implies that the PVI solution is regular at t = x for any x = 0, 1, ∞ [3,28,50]. As we are going to see below, this fact is consistent with the τ -function also being regular at t = x. Okamoto's formula [58] gives λ(t) in terms of a ratio of τ -functions where τ (θ, t) is given by (4.12). Evaluated at t = x, the left hand side vanishes because of our initial condition. Assuming that θ ∞ = 0 and neither of the τ -functions above vanishes or blows up at λ(x) = x, we have that In the following, we rewrite the condition (4.18) as a double series expansion. To do this, let us introduce some definitions. The isomonodromic Hamiltonian (4.10) and the accessory parameter expansion (4.9) do not depend on the normalization of the τ -function, so we normalize the structure constants as C n (θ, σ) . . = C(θ, σ + n) C(θ, σ) .

(4.19)
As we prove in Appendix B, these ratios can be factorized as C n (θ, σ) = C n (θ, σ)A(θ, σ) n , (n > 0) (4.20) where C n (θ, σ) is given by (B.7) and A(θ, σ) is given by (B.4). Another important formula isC For n < 0, we change σ → −σ in the formulas above. Plugging (4.19) and (4.20) into (4.12), we get τ (θ; t) = n∈Z C n (θ, σ)B(θ, σ + n; t)t n 2 [X(θ, σ, s; t)] n , (4.23) where X(θ, σ, s; t) . . = A(θ, σ)s t 2σ . (4.24) We can rewrite the c = 1 blocks (4.15) in terms of levels L is a restricted sum at level L of the coefficients (4.16). We will suppress the monodromy arguments in what follows, for simplicity, except the ones that are shifted. Using the definitions (4.23) and (4.25), we get where we used dX(t) dt = 2σ t X(t). Given that λ(t) admits a regular expansion near t = x, so does the right hand side of (4.17). This implies that X(x) also admits a Taylor expansion for x small where we assume that s = s(σ, x) is the source of the series expansion. Our goal is to extract the coefficients X k = X k (θ, σ) by solving (4.30) order by order in x. Therefore, this procedure gives us the coefficients of the s = s(σ, x) series solution.
We rewrite (4.30) as a double series condition where the means that the summation is over all n, p ∈ Z; L, M = 0, 1, 2, . . . such that n 2 + p 2 + L + M − 1 = α and n + p = β. From (4.31), we see that This means that the α = −1 term does not contribute to (4.30). For fixed α, we have that −n α ≤ β ≤ n α , where n α = √ α + 1 , with x being the floor function of x. To explain the β constraint, let us consider both n and p to be positive (or both negative). For fixed α, the upper bound on β can be reached when L = M = 0 so that α + 1 = n 2 + p 2 ≥ (n + p) 2 = β 2 .
Let us now go back to (4.30). For α = 0, we have −1 ≤ β ≤ 1 and thus which, using (4.31), (4.26), (4.22) and (B.7), we can show it is equivalent to This is essentially the same equation presented in [3] obtained from the PVI solution 4 . It gives two solutions for X 0 and we choose X 0 = 1 at this order, for consistency with the asymptotics of the PVI solution, as discussed in [3]. Plugging back X(x) = 1 + X 1 x + . . . to (4.33), the higher order equations in x will give only one solution for the coefficients X k . The next two coefficients are given in Appendix C in terms of σ and δ's.

Accessory Parameter Expansion
Substituting the definition (4.11) into (4.9) and then into (4.2), we get H x in terms of the τ -function (4.37) Notice that the first term in (4.37) is already the expected answer from CFT, as it comes from the leading behavior of the classical conformal block. The final step in our algorithm is to take the series solution for X(x) obtained above and plug it back into the τ -function in (4.37), with the substitution θ t = θ x − 1 2 . This computation is straightforward but demanding, so we will not discuss it here in general form. We will show below how the computation works for the first non-trivial term. This will make it clear how to proceed to obtain higher-order terms.
Remembering that X(t) = Ast 2σ , under the assumption 0 < σ < 1 2 , we have the small t expansion of (4.23) The τ -function has no well-defined Taylor expansion around t = 0 because its first derivative diverges as t −2σ . On the other hand, it is possible to do such expansion near a regular point t = x, for x sufficiently close to zero. In this sense, we are allowed to write Using that X(t = x) = 1 + X 1 x + . . . and B (1) to leading order in x. From (B.7), we see that and from (4.16) If we carefully sum (4.41) and (4.42), substituting θ 2 i = 1/4 − δ i and the shifted monodromy The second term above cancels with the second term of (4.37) and the first term above matches the standard CFT result [3]. The next order calculation is much more complicated, so we show here the result for (4.37) only up to first order in x We present the next two terms in the expansion above in appendix C. We implemented the algorithm of this section in a computer algebra program, as we do not have the explicit series solution for the constraint (4.33). We tested the accessory parameter expansion up to order x 2 analytically by comparing with the semiclassical limit of the CFT conformal block [3,25] using the inverse gram matrix approach [43,59]. We also checked this algorithm numerically, substituting the θ's with some fixed numbers from the beginning, up to order x 5 . More details are given in appendix C. This gives strong evidence that the isomonodromic approach presented here reproduces the classical conformal block expansion. Although we do not have a proof to all orders in x, we believe that our discussion above on the semiclassical limit and isomonodromic deformations, given the exponentiation hypothesis and the assumptions on the conformal weights, is enough mathematical evidence for the τ -function expansion.
In addition, according to our tests, it is numerically faster to calculate the accessory parameter expansion with our approach than inverting the Gram matrix and taking the semiclassical limit. We did not compare our approach with the Zamolodchikov recurrence formula, but, then again, the exact large c limit is also demanding in this case, while our algorithm already gives the analytic coefficients in the semiclassical limit.

Isomonodromic Approach to Classical Conformal Blocks
Now that we found the accessory parameter using the isomonodromic τ -function, an interesting question is if we can find an analogous formula for classical conformal blocks. One way to approach this problem is to use the symplectic structure of isomonodromic deformations [3,55]. As we discussed above, S σ (λ, t) is the action of the isomonodromic Hamiltonian systemλ determined by the Hamilton-Jacobi equation (2.17). The associated symplectic structure is given by Ω = dp ∧ dλ − dH ∧ dt. Here σ = σ 0t and ν is the canonically conjugate variable to σ [3,26]. These coordinates parameterize the moduli space of SL(2, C) flat-connections, similarly to (σ, s). The transformation

5)
H(θ; λ, p(µ), t) = K(θ; λ, µ, t)+ between the normal form (3.20) and the canonical form (3.15) is canonical with respect to Ω [50]. This induces a transformation by the function and thus dS c σ = µdλ − Kdt + νdσ. (5.9) Integrating (5.9) over a solution of the isomonodromic system, as the action-angle variables are constant on the orbits, we get Let us try to simplify the PVI Lagrangian as much as possible. First, we notice thaṫ and therefore Using Okamoto's formula (4.17), we can integrate the second term in (5.12) As both τ -functions have the same leading behaviour as t goes to zero, only the upper limit contributes to the second term above. Another equation proved by Okamoto in [58] is where we used (4.17) in the second line and τ + . . = τ (θ 0,+ ; t), τ − . . = τ (θ 0,− ; t), τ 0 . . = τ (θ −,0 ; t). (5.15) It also follows from (4.17) that We then use (5.14) and (5.16) to express the action only in terms of τ -functions where Although (5.17) is still complicated, substitution of (4.23) into (5.18) should give a formula for it only in terms of known functions of the monodromies. However, to get a closed form for the 4-point classical conformal block (1.1), we still need to impose the condition λ(x) = x into the resulting formula, following the procedure of the previous section. As we can only solve this condition order-by-order in x, a closed form is still out of reach. But, as we are going to see below, some special cases can be tractable. We leave the detailed study of this formula for future work.

Recovering the 4-point Classical Conformal Block
The 5-point semiclassical block (2.12) is given by where we used (5.7). We label the fields by their classical weights δ i and Π σ corresponds to the projection operator onto the intermediate state with momentum P = iσ/b. Notice that because λ(t) is a solution of isomonodromic deformations, we can change the position of the heavy degenerate field without changing the other monodromies. This has a nice AdS 3 interpretation, as degenerate insertions are conical defects in the bulk [60][61][62]. The PVI action governs the evolution of this conical defect in a way that the monodromies do not change. Therefore, we impose the boundary condition for the isomonodromic flow at t = x to be λ(x) = x. This entails to taking the fusion (5.19) clearly corresponds to taking C − = 0 so that the leading term in the OPE cancels with the appropriate term in right hand side of (5.19). Therefore, we end up with where we defined θ t = θ x − 1 2 and ν = σ ± 1 2 . The identification of the classical conformal block with the τ -function in (5.17) is a new result with important technical consequences, as the τ -function is a linear combination of c = 1 conformal blocks, as described in [34].

Linear Dilaton Case
In some special cases, the PVI solutions dramatically simplify. The simplest example is obtained by assuming i=0,1,t and σ ij = θ i + θ j , i, j = 0, 1, t. This implies that κ 2 + 1 = 0 (see (3.8)) and thus µ(t) = 0 is consistent with the equations of motion (3.18) [50]. Therefore, (5.13) gives S c σ = 0 and the PVI action is given by leading to the 4-point conformal block (5.21) where ν = θ 0 + θ x and according to the alternative Liouville definition ∆ i = α i (Q − α i ). This corresponds to the semiclassical limit of the linear dilaton correlator obeying the screening condition [48] i=0,1,x,∞ Notice that, in the c = 1 interpretation, the τ -function is a hypergeometric function [43,63], showing the non-triviality of this case in comparison to the c = ∞ result.

Conclusions
In this work, we discussed the deep mathematical relation between the large central charge limit of conformal blocks and the isomonodromic τ -function. We recovered the accessory parameter (4.37) of the 4-point Fuchsian equation (1.3) using the isomonodromic τ -function, with the additional fusion constraint λ(x) = x. We believe our approach gives a more straightforward algorithm than the one presented in [3].
The isomonodromic approach is relevant to applications of the theory of differential equations, as there is no need to take any semiclassical limit, in comparison to the CFT approach to calculate the accessory parameters. Thus, it can be applied to any particular problem that is governed by a Heun's equation. Recently, Piatek and Pietrykowski [64] recovered Floquet solutions of Heun's equation using CFT. This result nicely complements our discussion of accessory parameters. These techniques might be used in many concrete applications. In particular, the isomonodromic method has already proved useful for scattering problems in black hole physics [55,56,65].
We can also use it to calculate accessory parameters of confluent cases of Heun's equation, using the corresponding τ -functions described in [43,66]. Those are connected to the socalled irregular conformal blocks [67][68][69], which deserve to be better understood in CFT applications. Finally, we notice that it is straightforward to generalize the isomonodromic method for Fuchsian equations with any number of singular points by using the appropriate τ -function [40,70]. Our algorithm can then be generalized to find the accessory parameters and classical conformal blocks of n-point correlators.
The integral formula (5.17) for the PVI action is the first step to a closed expression for the 4-point classical conformal block (1.1). Using the τ -function c = 1 expansion, it should be possible to fully integrate the PVI action and impose the fusion constraint to obtain (1.1). This has many potential applications, in particular, related to the AdS/CFT correspondence: the emergence of AdS 3 gravity backgrounds [6,8], calculations of entanglement entropy [5] and the bulk computation of classical blocks from the geodesic approach [10][11][12][13][14]. In fact, the isomonodromic approach generalizes the monodromy method of [5,71]. The main obstruction is solving the PVI boundary condition in closed form. We gave a simple example where the PVI solution simplifies and recovers the c = ∞ limit of a linear dilaton conformal block [48]. Many other special limits remain to be explored, for example, the classical, algebraic and Riccati solutions of PVI [34,43]. Moreover, the connection between the isomonodromic and the AGT approaches [25,72] can give further insights on all these applications.
Although the relationship between classical conformal blocks, Painlevé VI equation and isomonodromic deformations has been discussed before [3,41,42], the importance of the τ -function has been fully appreciated only here. This relation is relevant not only from the technical point of view, but also highlights the intriguing map between the c = 1 and c = ∞ conformal blocks. Coulomb Gas (Dotsenko-Fateev) integral [59] and Fredholm determinant [70,73,74] representations of conformal blocks might also be useful to understand this map.
Another important question is what is the quantum counterpart of the c = 1 isomonodromic structure described here. A related development is that the canonical quantization of the isomonodromic equations is equivalent to the Knizhnik-Zamolodchikov (KZ) equations (equivalently the isomonodromic equations are the classical limit of the KZ equa-tions) [75,76]. Solutions of BPZ equations can be related to solutions of KZ equations [77][78][79], connecting Liouville theory to WZNW models. This web of relations, in connection to AGT correspondence and Hitchin systems, was reviewed in [41,42]. These papers also briefly mention the role of the isomonodromic τ -function in the semiclassical CFT limit. More recently, the authors of [80] have proposed a generalization of the isomonodromy/CFT correspondence to arbitrary central charge using q-Painlevé conformal blocks [81] and cluster algebras. It would be interesting to study how the classical limit studied here emerges from this quantum description.
The physical meaning of the c = 1 and c = ∞ relationship, if any, deserves further exploration in the future and we hope our work will be helpful in this direction too.

Acknowledgments
FN is grateful to Thomas Hartman, Oleg Lisovyy, Sergei Lukyanov and Rubik Poghossian for important discussions on aspects of this work. FN also thanks Bruno Carneiro da Cunha for his influential guidance on the results presented here. FN and ML are supported by the Brazilian ministries MCTI and MEC. The authors acknowledge support by the Simons Foundation, through the Simons Collaboration on the Non-perturbative Bootstrap, during the Bootstrap School at ICTP-SAIFR, São Paulo, where part of this work was done.

C Analytical and Numerical Checks of Accessory Parameter Expansion
As discussed in section 4.3, the solution of the condition λ(x) = x gives a series expansion X(x) = 1 + X 1 x + X 2 x 2 . . . for the PVI integration constant s(x) encoded in X(x). We present analytically the first permute δ 0 ↔ δ x , δ 1 ↔ δ ∞ . The conformal block is not invariant under this transformation, as we can see in rows 7 and 8. However, in the lines 3 and 4, the weights remain the same and this becomes a symmetry. Finally, in rows 5 and 6, we test the symmetry δ 0 ↔ δ 1 , δ x ↔ δ ∞ .