Painlevé VI connection problem and monodromy of c = 1 conformal blocks

Generic c = 1 four-point conformal blocks on the Riemann sphere can be seen as the coefficients of Fourier expansion of the tau function of Painlevé VI equation with respect to one of its integration constants. Based on this relation, we show that c = 1 fusion matrix essentially coincides with the connection coefficient relating tau function asymptotics at different critical points. Explicit formulas for both quantities are obtained by solving certain functional relations which follow from the tau function expansions. The final result does not involve integration and is given by a ratio of two products of Barnes G-functions with arguments expressed in terms of conformal dimensions/monodromy data. It turns out to be closely related to the volume of hyperbolic tetrahedron.


Introduction
The two-dimensional conformal field theory (CFT) [4] has been intensively studied in the last three decades. A renewed interest to these studies is related to the recent discovery [1] of a relation between 2D CFTs and N = 2 4D supersymmetric gauge theories, commonly referred to as AGT correspondence.

JHEP12(2013)029
The infinite-dimensional conformal symmetry determines the structure of correlation functions and leads to the notion of conformal blocks: these are universal chiral parts of correlators corresponding to different choices of intermediate conformal families in the successive operator product expansions (OPE) of primary fields. From a mathematical standpoint, conformal blocks can be seen as new special functions arising in the representation theory of the Virasoro algebra. The AGT relation provides us with their explicit series representations.
Equivalence of different ways to decompose a correlation function into a sum over conformal families suggests the existence of duality transformations of conformal blocks, formalized by the concept of Moore-Seiberg groupoid [26]. In particular, there should exist an elementary invertible linear map connecting s-and t-channel four-point Virasoro conformal blocks on the Riemann sphere, AGT-related to weak/strong coupling S-duality on the gauge side. The integral kernel of this transformation is called the fusion matrix. Its explicit form was obtained in [30,31] by solving certain functional equations (which follow from the Moore-Seiberg formalism) with the help of representation theory of the modular double of U q (sl (2, R)). An alternative derivation, based on free-field representations of chiral vertex operators, was proposed later in [34,35].
The results of [30,31,34,35] hold for generic complex values of the Virasoro central charge c. Unfortunately, it is not clear whether/how they can be extended to the half-line c ∈ R ≤1 , including a particularly interesting point c = 1 [16,32,33] at the borderline between minimal models and Liouville theory.
The present work approaches the last problem by exploiting the relation of c = 1 conformal blocks and Painlevé VI equation [18,19]. It turns out to be mutually useful. We will show that c = 1 fusion matrix essentially coincides with a connection coefficient for Painlevé VI tau function expressed in terms of monodromy data of the auxiliary linear problem. Conformal expansions of the tau function imply that this coefficient satisfies certain recurrence relations. On the other hand, equivalence of different critical points of Painlevé VI can be seen as a kind of crossing symmetry condition. Connecting expansion parameters in different channels, it makes the recurrence relations highly nontrivial and restrictive. Their solution appears to be related to the Poisson geometry of the moduli space of monodromy data and complexified volume of generic hyperbolic tetrahedron.
It is worth mentioning that the connection problem for tau functions of Painlevé equations has a strong independent interest. Such questions arise, e.g. in the study of the large gap asymptotics of Fredholm determinants of integrable kernels arising in random matrix theory [2,11,14,24]. In this framework, the analogs of the connection coefficients are called Dyson constants. Their computation involves integrals of the classical Painlevé transcendents and so far seemed to be inaccessible with the existing tools of Painlevé theory. Most of the available exact results have been obtained on case by case basis by approximating the corresponding Fredholm determinants with Toeplitz and Hankel determinants [3,9,10,15,22]. Hopefully, our results will provide some new insight in this context. The paper is organized as follows. In section 2, we recall basic symmetry properties of conformal blocks, explain Ponsot-Teschner formula for the fusion kernel for generic c and JHEP12(2013)029 discuss a few explicit examples. In section 3, after a brief outline of the relation between c = 1 conformal blocks and Painlevé VI, we discuss monodromy data for the associated linear problem and their relation to hyperbolic tetrahedron. Connection problem for Painlevé VI tau function is solved in section 4. Its main result is the explicit formula (4.20) for the connection coefficient. The latter is related to c = 1 fusion matrix in section 5, see formula (5.4). The proofs of some technical results are relegated to appendix.

Symmetries
Let us start by fixing some notation. Throughout this paper, we use a Liouville-type parameterization of the central charge: To cover all possible complex values of c, it suffices to consider b from the first quadrant. The weak-coupling region c ≥ 25 then corresponds to b ∈ R ≥1 , the values c ≤ 1 to b ∈ iR ≥1 , and 1 ≤ c ≤ 25 to a quarter of the unit circle b = e iϕ , ϕ ∈ [0, π 2 ]. It is convenient to represent conformal weights of primary fields in the form where the parameters θ will be referred to as momenta. Four-point s-channel Virasoro conformal block on P 1 with external dimensions ∆ ν = c−1 24 + θ 2 ν attached to the points ν = 0, t, 1, ∞ and internal dimension ∆ σ = c−1 24 + σ 2 will be written in one of the following forms: It is a power series in t normalized as F c ({∆ ν }, ∆ σ ; 0) = 1. As a function of t, F c is believed to be analytically continuable to the universal cover of P 1 \{0, 1, ∞}. Some intuition about this analytic behavior may be gained by looking at the limit c → ∞, ∆'s finite, where conformal block reduces to Gauss hypergeometric series

JHEP12(2013)029
The present paper mainly deals with another special case c = 1, where conformal block function becomes a Fourier transform of the tau function of the sixth Painlevé equation with respect to one of its integration constants.
As a function of parameters, conformal block enjoys a number of symmetries, analogous to Euler-Pfaff fractional linear transformations of 2 F 1 (a, b; c; z): • Trivial symmetries. Changing the sign of any of θ 0,t,1,∞ or σ has no effect on conformal block as the latter depends on dimensions only.
• R-symmetries allow the exchange of columns and rows of external momenta: • T -symmetry enables one to exchange the dimensions in one column: • Regge-Okamoto symmetry. There is an identity where δ, δ 1t are defined by This is reminiscent of the unexpected Regge symmetry of Racah-Wigner 6j symbols and Okamoto symmetry of Painlevé VI [7]. The latter can be actually seen as a c = 1 specialization of the above. Though it is not easy to derive (2.4) from CFT first principles, this relation becomes almost obvious in the AGT representation where it corresponds to a permutation of masses of matter hypermultiplets. Being combined with trivial symmetries, it relates conformal blocks with three distinct (unordered) sets of external dimensions.

Linear transformations
Conformal blocks appear in the expansion of the four-point correlator of primary fields with additional prefactors. It is convenient to introduce the function

JHEP12(2013)029
It is useful to think of the variable t as being the cross-ratio t = (z 2 −z 1 )(z 4 −z 3 ) (z 3 −z 1 )(z 4 −z 2 ) of four points z 1 = 0, z 2 = t, z 3 = 1, z 4 = ∞. The mapping class group Γ = P SL 2 (Z) of the 4-punctured sphere is the quotient of the braid group on 3 strings by its center. It naturally acts on conformal blocks by braiding transformations of z 1,2,3,4 and appropriate permutations of dimensions. One of the generators of this action is given by the above T -transformation. The second generator acts as S : θ 0 ↔ θ 1 , t ↔ 1 − t. It can be checked that S and T satisfy the modular group defining relations S 2 = (ST ) 3 = 1.
It is expected that the linear span of conformal blocks (2.5) with different internal dimensions realizes an infinite-dimensional representation of Γ due to associativity of the operator product expansions. More precisely, there should be a linear "S-duality" relation between the conformal blocks calculated in different channels: The t-independent kernel F c is the fusion matrix. It may be assumed to be even function of parameters θ ν , σ, ρ and has a number of symmetries similar to (2.1), (2.2) and (2.4).
The explicit form of the fusion kernel was found by Ponsot and Teschner who identified it with the Racah-Wigner matrix for a class of infinite-dimensional representations of the quantum group U q (sl (2, R)) [30,31]. Their result reads (2.7) where we use the standard convention f α 1 , . . . , α n β 1 , . . . , are closely related to the Barnes double gamma function and quantum dilogarithm. They can be defined by analytic continuation of the integral representations The functionŜ b (x) has an infinite number of zeros and poles in the complex x-plane: • poles: x = ib m + 1 2 + ib −1 n + 1 2 with m, n ∈ Z ≥0 . This implies that, for instance, for real b ≥ 1 the integrand in (2.7) has eight infinite halflines of poles shown in figure 1a. As b = e iϕ , ϕ ∈ 0, π 2 , the half-lines open to 2D lattice sectors, see figure 1b. Similar picture holds for any b with Re b > 0.  The integration contour C in (2.7) runs from −∞ to +∞ passing between the upper and lower pole sectors. With this prescription, Ponsot-Teschner formula gives the fusion kernel for any complex value of the central charge except for the half-line c ∈ R ≤1 corresponding to purely imaginary b. The present paper is mainly concerned with the edge point c = 1 of this excluded region.

Checking Ponsot-Teschner formula: an example with c = 25
Let us illustrate the formula (2.7) for the fusion kernel with an explicit example. It will be based on the evaluation of conformal block with c = 25, arbitrary internal dimension and all external dimensions equal to 15 16 , found by Al. B. Zamolodchikov [38, footnote (1)]. This corresponds to setting b = 1 and θ 0,t,1,∞ = i 4 in the above. The answer appears in [38] in a parameterization particularly suitable for the modular transformations, which may be explained as follows. Consider a complex torus C/ (Z + τ Z) and identify its points related by multiplication by −1. This yields a double cover of P 1 with 4 ramification points. Their cross-ratio (our variable t) remains invariant under the action of the subgroup Γ(2) corresponding to pure analytic continuation of conformal blocks. It is explicitly given by the elliptic lambda function where ϑ 2,3 (z|τ ) are the usual Jacobi theta functions The inverse map can be written as where K(t) denotes complete elliptic integral of the 1st kind: It is clear that in the limit t → +0 one has τ → i∞. Even more specifically, one can check that lim t→+0 t −1 e iπτ = 1 16 . Now, using (2.9), the aforementioned result of [38] can be stated as . (2.10) The modular transformation S exchanging t and 1 − t maps τ to −τ −1 . Then, applying Jacobi's imaginary transformation to (2.10), it is straightforward to verify that the Sduality relation (2.6) is satisfied by Fourier transform conjugated by simple diagonal factors Next let us try to derive this relation from the Ponsot-Teschner formula. The functionŝ Γ b (x) andŜ b (x) in the limit b → 1 are expressed by means of the Barnes G-function: Thanks to the doubling identity the prefactor in the 1st line of (2.7) reduces to Similarly simplifying the integrand in the 2nd line, it is possible to show that the fusion kernel (2.11) will follow from (2.7) provided that The contour C passes between the half-lines of zeros of the Barnes functions in the denominator, as discussed above. In particular, for σ, ρ ∈ R it can be chosen as horizontal line with 0 < Im x < 1 2 . The intriguing integral identity (2.12) looks rather nontrivial and we will not attempt to rigorously prove it here. Instead, we contented ourselves with its numerical verification for several randomly chosen values of σ and ρ.

Probing c = 1 fusion
As c approaches the interval (−∞, 1], Ponsot-Teschner formula becomes singular. The sectors in figure 1b transform into overlapping half-planes containing an infinite number of dense lines of poles. The functionΓ b (x) has a natural boundary of analyticity at purely imaginary b. It is therefore legitimate to ask whether fusion transformations merely exist. Of course, they do for degenerate dimensions [12]. The present work suggests that this is also true for arbitrary dimensions at c = 1.
As an illustration, consider the Ashkin-Teller conformal block, characterized by c = 1 and all external dimensions equal to 1 16 . This is the second solvable case where conformal block function is known in a closed form for arbitrary internal dimension [38,Eq. (2.28)]. In the elliptic parameterization (2.8)-(2.9), one has ; .
It is very easy to check by evaluating Gaussian integrals that (2.6) is again satisfied by a Fourier-type fusion kernel ; (2.14) 3 Painlevé VI and hyperbolic tetrahedron

Conformal expansions
The problem of determining the c = 1 fusion matrix will be reformulated in subsection 5.1 as connection problem for the tau function τ (t) of the sixth Painlevé equation defined by its logarithmic derivative The relation of τ (t) to generic four-point c = 1 conformal blocks was observed in [18,19]. Painlevé VI parameters θ = (θ 0 , θ t , θ 1 , θ ∞ ) correspond to the external momenta, one of the constants of integration encodes the intermediate dimension spectrum and the other one is a generating parameter. Specifically, the tau function can be written as

JHEP12(2013)029
The first representation is particularly suitable in the vicinity of t = 0, and the second one gives the expansion of τ (t) around t = 1. The structure constants are given by .

(3.4)
Any of the two pairs of integration constants (σ 0t , s 0t ) and (σ 1t , s 1t ) specifies the initial conditions for Painlevé VI in the form of solution asymptotics near a given critical point.
The relation between the two pairs is most conveniently formulated in terms of monodromy data for the associated rank 2 linear problem which we shall now briefly discuss.

Monodromy and initial conditions
The space of monodromy data consists of conjugacy classes of triples (M 0 , M t , M 1 ) of monodromy matrices from SL(2, C). To describe it efficiently, one needs to introduce in addition to σ 0t , σ 1t a third exponent σ 01 which appears in the expansion at ∞. These exponents and parameters θ are related to monodromy matrices as follows: We define Painlevé VI monodromy manifold M as the corresponding SL(2, C)character variety of π 1 P 1 \{0, t, 1, ∞} . It is described by the Jimbo-Fricke affine cubic surface W (p 0t , p 1t , p 01 ) = 0, where [20] W (p 0t , p 1t , p 01 ) = p 0t p 1t p 01 + p 2 0t The parameters ω = (ω 0t , ω 1t , ω 01 , ω 4 ) depend only on θ appearing in Painlevé VI and are considered as fixed. They are explicitly given by The triples σ = (σ 0t , σ 1t , σ 01 ) satisfying the constraint W (p 0t , p 1t , p 01 ) = 0 parameterize the two-dimensional space of Painlevé VI initial conditions. Fixing p 0t in this constraint gives a quadric which admits rational parameterization. The quantity s 0t in (3.2) can be seen as the corresponding uniformizing parameter. The quantity s 1t from (3.
where we have introduced the notation q µν = ∂W ∂p µν so that, for instance, It turns out that Jimbo-Fricke cubic may be rewritten in terms of these variables in a nice determinantal form, e.g., Appendix contains several useful relations involving first minors of the matrix (3.9). In particular, they ensure consistency of the different sign choices in (3.6), (3.7).

Connection problem
The definition (3.1) of the Painlevé VI tau function contains an obvious normalization ambiguity, which implies that the coefficients χ 0,1 in (3.2)-(3.3) are intrinsically indefinite. However, their ratio χ 01 ( θ; σ 0t , σ 1t ; p 01 ) = χ −1 0 χ 1 (3.10) is completely fixed by the differential equation and initial conditions for ζ(t). It determines relative normalization of the expansions of τ (t) near 0 and 1, and will be called connection coefficient. Sometimes it becomes convenient to include the structure constants into the definition of relative normalization by introducinḡ In particular, for − 1 2 < Re σ 0t , Re σ 1t < 1 2 one can writē .

Canonical coordinates
There is a natural Poisson bracket {, } on monodromy manifold M inherited from the Atiyah-Bott symplectic structure on the moduli space of flat logarithmic SL(2, C)connections on the 4-punctured sphere. This bracket is defined by with q µν the same as above. Parameterizing s 0t , s 1t from (3.6), (3.7) as it can be easily verified (we have learned this from a recent work [28] containing an equivalent observation) that the local coordinates η 0t , η 1t are conjugate to monodromy exponents σ 0t , σ 1t : Two pairs of Darboux coordinates (σ 0t , η 0t ) and (σ 1t , η 1t ) are well-adapted for characterizing the expansions of τ (t) near t = 0 and t = 1, respectively, cf (3.2)-(3.3). Observe that generic c = 1 four-point Virasoro conformal blocks, as functions of t and 1 − t, literally coincide with Fourier expansion coefficients of the appropriately normalized Painlevé VI tau functions with respect to the dual coordinates η 0t and η 1t .
The pairs (σ 0t , η 0t ) and (σ 1t , η 1t ) are related by a canonical transformation whose generating function S( θ; σ 0t , σ 1t ) is defined by the equations (3.14) Remarkably, S( θ; σ 0t , σ 1t ) can be found in explicit form. It essentially coincides [28] with the complexified volume of the hyperbolic tetrahedron T with dihedral angles 2π θ, 2πσ 0t + π, 2πσ 1t + π, whose mnemonic graphical representation can be obtained by gluing external legs of s-or t-channel conformal blocks (see figure 2). A tetrahedral signature shows up already in (3.9): the 4 × 4 matrix G is nothing but the Gram matrix of scalar products of length √ 2 vectors normal to faces of T and oriented outwards.

The volume of T
The explicit formula for the volume is most conveniently written in terms of the Lobachevsky function, which is essentially a half of the imaginary part of the Euler dilogarithm: This definition differs from the standard one by a factor of π in the argument of Λ. The dilogarithms are evaluated on their main sheets, which implies that Λ(σ) is continuous and periodic. Define the parameters then the volume of T is given by [8,27] Here ω ± represent two nontrivial solutions z ± = e 2πiω ± of the equation which can be expressed in terms of θ, σ as z ± = 4 sin 2πσ 0t sin 2πσ 1t + 4 sin 2πθ t sin 2πθ ∞ + 4 sin 2πθ 0 sin 2πθ 1 ± q 01 2 4 k=1 e 2πi(ν Σ −ν k ) − e 2πi(ν Σ −λ k ) .

(3.19)
Their product does not contain p 01 and can be written as . (3.20) Note that for the genuine hyperbolic tetrahedra θ's, σ 0t and σ 1t are real. Also, q 01 is purely imaginary since G should have the signature (−, +, +, +) of the ambient space The parameters z ± then lie on the unit circle, which makes (3.17) compatible with our earlier conventions for Λ(σ).
The precise relation between Vol (T ) and the generating function S( θ; σ 0t , σ 1t ) from the previous subsection is provided by

JHEP12(2013)029
Let us also mention that the formula (3.17) for Vol (T ) can be rewritten in terms of ω + or ω − only (instead of using both of these parameters) thanks to the following result, cf [27, Theorem 2]: can be alternatively expressed as In the case of complex angles the function Vol (T ) may be defined via continuation from an open set U ⊂ C 6 . However, in doing this the periodicity with respect to angles will be lost, just as if instead of taking σ ∈ R and fixing the principal branches of Li 2 (z) in (3.15) we tried to continue Λ(σ) analytically from a suitable open subset of C.

Functional relations
In this section, we compute the connection coefficient defined by (3.10). The idea is to consider Painlevé VI parameters θ as fixed and obtain χ 01 ( θ; σ 0t , σ 1t ; p 01 ) by solving certain difference equations with respect to σ 0t and σ 1t .

JHEP12(2013)029
Our notation means the following: if p 01 is one of the two roots of the equation W (p 0t , p 1t , p 01 ) = 0, then p 01 = ω 0t − p 01 − p 0t p 1t denotes the other root. Also, denote by q µν , s µν the appropriate modifications of q µν , s µν . In particular, one has Now write s 0t , s 1t , s 0t , s 1t in terms of q 01 . It turns out, expectedly, that the products s 0t s 0t and s 1t s 1t depend only on q 2 01 . Using (3.9) and simplifying the corresponding expressions with the help of (A.1)-(A.8), it can be deduced from (4.1)-(4.2) that The general solution of (4.4)-(4.5) may be constructed in terms of Barnes G-function already encountered in subsection 2.3. Indeed, since G (1 + σ) = Γ (σ) G (σ), the function One then easily derives Lemma 3. The general solution of (4.4)-(4.5) is given by where κ per ( θ; σ 0t , σ 1t ) is an arbitrary periodic function of both σ 0t , σ 1t with periods 1.
What about κ per ( θ; σ 0t , σ 1t )? The simplest guess is to assume that this quantity does not depend on σ 0t , σ 1t (some arguments in favor of this hypothesis will be given in the next subsection). The guess is readily confirmed by numerical experiments, but in fact the numerics reveals much more: κ per ( θ; σ 0t , σ 1t ) is simply equal to 1! The final formula can now be written as .
The right side of (4.9) coincides with a ratio of three-point functions Φ in the time-like c = 1 Liouville theory [39]. A conceptual explanation of this intriguing coincidence is yet to be found.

Minimal solution
Let us now come back to the relations (4.1)-(4.2). As their solution is much easier to check than to guess, the reader interested only in the final result may jump directly to Lemma 4. What follows is an attempt to elucidate the origins of this Lemma.
Taking the logarithmic derivatives of both sides of (4.1)-(4.2) and recalling the parameterization (3.13), we obtain ∂ ∂σ 0t ln where all other arguments of χ 01 are temporarily omitted to lighten the notation. A tentative solution of (4.10)-(4.13) can be written in the form where the integral is calculated along a path starting at some fixed point P on the (infinitesheeted covering of) Jimbo-Fricke surface. Indeed, the integrand is a closed 1-form; its differential dσ 1t ∧ dη 1t − dσ 0t ∧ dη 0t vanishes since the transformation (σ 0t , η 0t ) → (σ 1t , η 1t ) is canonical, see subsection 3.4. Hence the integral value depends only on the homotopy class of the path on the Jimbo-Fricke surface with excluded one-dimensional subspaces corresponding to the singularities of the integrand. In particular, restricting to σ 1t = const, one obtains a complex curve with punctures at the poles of the integrand. Forχ 01 defined by (4.14) one has, e.g., Differentiating the right side, it is straightforward to check thatχ 01 satisfies (4.10)-(4.11). The other two relations are verified analogously. Thus we have shown that, up to an additive constant independent of σ 0t , σ 1t but a priori depending on homotopy class of integration path, lnχ 01 satisfies the same functional relations as ln χ 01 . On the other hand, χ 01 is expected to be a single-valued function of σ 0t , σ 1t . If it were possible to present it in the form (4.14), the integrals corresponding to different paths could only differ by integer multiples of 2πi. This appears not to be the case: the residues (e.g. calculated at the poles of the integrand restricted to σ 1t = const) are not integers. Therefore, one may try to use the freedom in the choice of the additive constant to correct the integrand by a closed 1-form with periodic coefficients which would ensure the necessary analytic properties.
Proof. Direct verification based on the identities of type (B.5) used in the proof of Lemma 1 in the appendix. Observe that the right side of (4.18) is a periodic function of ω + thanks to (3.18) and (4.7), which enables one to choose the solution of z + = e 2πiω + arbitrarily.
The periodic prefactor χ per ( θ; σ 0t , σ 1t ) can be fixed from the following considerations. Let σ 0t = α be a point where the connection coefficient tends to infinity. Unless this singular behavior is compensated by the coefficients of (3.2) and (3.3), all terms in the t = 0 tau function expansion vanish whereas the t = 1 series produces a nontrivial solution to Painlevé VI. This contradiction suggests that χ per ( θ; σ 0t , σ 1t ) is a nowhere vanishing holomorphic function of both σ 0t and σ 1t . Making an additional assumption of nice behavior at infinity, one concludes that χ per ( θ; σ 0t , σ 1t ) is in fact independent of the last two parameters.
The remaining dependence on θ can be strongly constrained using Painlevé VI solutions known in closed form and depending on continuous parameters. For instance, such solutions JHEP12(2013)029 are known for an infinite number of affine hyperplanes in the θ-space. In CFT language, they correspond to conformal blocks involving degenerate fields or (by Regge-Okamoto symmetry (2.4)) to conformal blocks of free-field exponentials with screening insertions. The simplest nontrivial example of this type is given by where θ are subject to the constraint θ 0 + θ t + θ 1 + θ ∞ = 1 and σ = (θ 0 + θ t , θ 1 + θ t , θ 0 + θ 1 ). More general formulas can be found in subsection 4.3 of [19]. Some further examples come from the continuous algebraic families of Painlevé VI transcendents living on affine θ-subspaces of dimensions 1 and 2. It turns out that the connection coefficients computed directly in these particular cases are reproduced by the simplest possible ansatz χ per ( θ; σ 0t , σ 1t ) = 1. It is further supported by numerical computations with random values of θ, σ and analytic checks using exceptional algebraic Painlevé VI solutions [6,13,21,23], discussed in subsection 4.3. This transforms Lemma 4 into the following Claim 5. Connection coefficient (3.10), (3.11) for the generic Painlevé VI tau function has the following expression in terms of monodromy data: It is worth noting that the formula (4.20) possesses a non-obvious symmetry: its right side remains invariant upon sign reversal of any of the parameters θ 0,t,1,∞ , σ 0t and σ 1t .

Algebraic checks
Algebraic solutions of Painlevé VI provide an instructive way to test the general expression Expanding the tau function (4.21) near the endpoints and using (4.23)-(4.24), one also finds that , (4.25) where the phase φ is a non-rational multiple of π, explicitly given by (4.27)

JHEP12(2013)029
First observe that where C Λ denotes the circle |s 0t | = e −2πΛ in the complex s 0t -plane and the last equality is obtained by combining (3.2)-(3.3) with the functional relation (4.2). The next step is to transform the integral over s 0t into an integral over σ 1t . The latter was considered so far as a function of σ 0t , s 0t implicitly determined by (3.6) -recall that s 0t parameterizes the pairs (p 1t , p 01 ) at fixed p 0t . It is not difficult to show that with α ± ( θ, σ 0t ) given by Observe that, as Λ → ∞, p 1t becomes very large, which means that either (a) Im σ 1t ∼ Λ so that e 2πiσ 1t ∼ s 0t or (b) Im σ 1t ∼ −Λ, in which case e 2πiσ 1t ∼ s −1 0t . Since Λ may indeed be chosen sufficiently large, (5.2) implies that σ 1t -integration contour may be chosen as a segment [σ * 1t , σ * 1t + 1] with Im σ * 1t being sufficiently large for all singularities of the integrand in (5.1) to be located below the line R + iΛ. The sum over n in (5.1) then produces an integral over the whole line so that and µ( θ, σ) = 1 2πi ∂ ln s 0t ∂σ 1t . The root p 01 of the Jimbo-Fricke relation is chosen as to reproduce the asymptotics s 0t → 0 as Im σ 1t → +∞.
Lemma 1 indicates that the prefactor µ( σ, θ) from the last relation can be rewritten in a symmetric form Vol(T ), (5.5) where Vol(T ) denotes the tetrahedral volume of section 3. One can also obtain an explicit trigonometric expression where G denotes the Gram matrix defined by (3.9) and the branch of the square root is chosen so that √ det G = q 01 . The formula (5.3) is a c = 1 analog of the fusion relation (2.6). One apparent difference is that here it becomes more convenient to integrate over a complex contour in the momentum space rather than R + . The integral kernel F θ 1 θt θ∞ θ 0 ; σ 1t σ 0t is the fusion matrix for c = 1 conformal blocks. We emphasize that, up to an elementary prefactor, it coincides with Painlevé VI connection constant explicitly given by (4.20).

Numerics
The expression (5.4) for the fusion kernel can in principle be checked numerically. Fix, for instance,  To specify the integration contour in (5.3), one needs to analyze the singularities of χ 01 with respect to σ 1t . The poles (and zeros) may only be located at θ 0 + θ ∞ + Z, θ t + θ 1 + Z, Z/2 with , = ±. One also has square root branch points corresponding to zeros of q 01 ; in our case they are given by ±(0.40 − 0.37i) + Z, ±(0.11 + 0.49i) + Z. This results into the singularity structure shown in figure 3. Hence in the integration contour R + iΛ on the right of (5.3) one may set Λ = 1.
In practice, the integrand decays very rapidly so that one can approximate the integral by the Riemann sum over uniform partition of the segment [−1.5 + i, 2.5 + i] into 400 subintervals. Numerical values of conformal block functions were obtained by taking several first terms in their series expansions (30 on the left and 15 on the right of (5.3)). Figure 4 shows the graphs of the left and right sides of (5.3), as functions of t ∈ (0, 1), calculated in this way. Discrepancies at the endpoints are related to the fact that efficient appoximation of conformal blockF 1 (t) by truncated series requires taking into account more and more terms as t approaches 1.
The origin of this simplification is that here µ( σ) = ∓1, cf e.g. (5.5) and (3.17). As explained in the previous subsections, the formulas (5.3)-(5.4) come with a prescription for the choice of p 01 . We should select from its two possible values the one characterized by vanishing of s 0t as Im σ 1t → ∞. This corresponds to picking out the lower sign in the above formulas, and finally yields the folowing expression for the fusion matrix: ; σ 1t σ 0t = 2 4σ 2 0t −4σ 2 1t e −2πiσ 0t σ 1t . (5.7) Since the integral in (5.3) is calculated over the whole line, the last expression is clearly equivalent to the fusion kernel (2.14) for Ashkin-Teller conformal blocks.

JHEP12(2013)029
Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.