Crossing invariant correlation functions at $c=1$ from isomonodromic $τ$ functions
Pavlo Gavrylenko Raoul Santachiara












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     M_0,4^SL(2,ℂ)(θ) M_0,4^SU(2)(θ) M_0,4^SL(2,ℂ)ττθ̃
    ^1 
    Center for Advanced Studies, Skolkovo Institute of Science and Technology,
     143026 Moscow, Russia
    ^2
     NRU HSE, International Laboratory of Representation Theory and Mathematical Physics,
     119048 Moscow, Russia^3 Bogolyubov Institute for Theoretical Physics,  03680 Kyiv, Ukraine^4 LPTMS, CNRS (UMR 8626), Univ Paris-Sud, Université Paris-Saclay, 91405 Orsay, 
    France
    pasha.145@gmail.com, 
    raoul.santachiara@gmail.com
    We present an approach that gives rigorous construction of a class of crossing invariant functions in c=1 CFTs from the weakly invariant distributions on the moduli space   of SL(2,ℂ) flat connections on the sphere with four punctures.
    By using  this approach we show how to obtain correlation functions in the Ashkin-Teller and the Runkel-Watts theory.
    Among the possible crossing-invariant theories, we obtain also the analytic Liouville theory,  whose consistence was assumed only  on the basis of numerical tests.Crossing invariant correlation functions at c=1 from isomonodromic τ functions
        Raoul Santachiara   ^4
        December 30, 2023
    ==============================================================================
    
    
    
    § INTRODUCTION
     
    Bootstrap solutions of two dimensional CFTs are known since longtime  <cit.>.
     The existence and the form of these solutions depend strongly on the value of the central charge c, that parametrizes the current conformal algebra, the Virasoro algebra. 
    When c takes rational values smaller than one, c∈ℚ and  c<1, one finds the  minimal  models whose spectrum contains a finite number of degenerate Virasoro representations <cit.>.
     For c irrational, c∉ℚ, we have the  generalized minimal models with a diagonal (i.e. with only spinless primary fields) spectrum formed by a discrete but infinite number of degenerate representations <cit.>. The correlation functions of the minimal and of the generalized minimal models are built from the solutions of Fuchsian differential equations and admit a (Coulomb gas) integral representation. This property makes the analytic proof of the crossing symmetry relatively simple. For central charge  c ≥ 25, one finds the celebrated  Liouville field theory that describes the two-dimensional quantum gravity <cit.> and that has been recently related to the random energy models <cit.>. The Liouville field theory has a continuous and diagonal spectrum <cit.> and its crossing symmetry has been proved in <cit.>. We notice also that the Liouville theory can be constructed by using a  (mathematically rigorous) probabilistic approach  <cit.>, thus setting the consistency of this theory on very solid ground. The Liouville theory can be analytically continued to complex values of the central charge, c∈ℂ-{-∞,1}. The general understanding  is that the analytic continuation of the Liouville theory is the unique bootstrap solution with continuous and diagonal spectrum in the region c∈ℂ-{-∞,1}<cit.>. 
    
    Despite the success of the above mentioned CFTs, there are many examples of critical points where a satisfying CFT description is lacking. We mention for instance the non-unitary critical systems with central charge c ∈ℝ_≤ 1 such as the critical percolation <cit.> or the random critical points <cit.>. These models point out the existence of bootstrap solutions still to uncover.   
    Motivated by this and by the success of  numerical bootstrap approaches in higher dimension CFT  <cit.>, the last few years have witnessed renewed efforts in the search of new CFTs. 
    The outcome of this research unveiled indeed new solutions: some presenting  a discrete and non-diagonal spectrum <cit.> and others with a continuous and diagonal spectrum <cit.>.  
    We refer to these latter solutions as the Liouville type solutions. In general, the crossing symmetry of these solutions is checked  only numerically,  mainly because the conformal blocks do not have integral representation nor satisfy differential equations.
    
    The central charge c=1 represents a particularly interesting case where many different known CFTs exist. In this paper we consider in particular the Gaussian free field, the Ashkin-Teller model and two different Liouville type solutions: one is the  Runkel-Watts theory <cit.> that can be found via a  c→ 1 limit  of the Liouville solution defined at c∈ℂ-{∞,1}<cit.>. The other Liouville type solution is the one proposed in <cit.> that is referred in this paper as the  analytic Liouville theory. 
    
    We will provide a derivation of the crossing symmetry of the c=1 theories by using the relation  between isomonodromic deformations and Virasoro  conformal blocks  <cit.>. In this work we will fully exploit this  connection to provide a unifying theoretical framework to find a class of bootstrap solutions in c=1 theories. Among these solutions, we find  the analytic Liouville theory. This proves for the first time the crossing symmetry of this latter theory. 
    
    
    
    
    § ISOMONODROMIC DEFORMATIONS – VIRASORO ALGEBRA RELATION
    
    We review here the connection between the theory of isomonodromic deformations of Fuchsian systems of differential equations and the c=1 Virasoro algebra conformal blocks.
    
    
     §.§ Rank 2 Fuchsian systems and the associated τ function
    
     A rank 2 Fuchsian system is defined by the system of the ordinary first order differential equations
    
        dY(z)/dz=( A_0/z+A_1/z-1+A_t/z-t) Y(z) ,
    
    where Y(z) is 2× 2 matrix of fundamental solution and  A_0,A_1,A_t are constant  2× 2-matrices that, for simplicity, we consider semi-simple (i.e. diagonalizable). Moreover we can assume,  without loss of generality, the matrices A_ν to be traceless. Indeed, the addition operation, sending (A_0,A_1,A_t)→(A_0-1/2Tr A_0 ·1,A_1-1/2Tr A_1 ·1,A_t-1/2Tr A_t ·1) , corresponds to the gauge transform Y(z)→ Y(z)·(z)^-1/2Tr A_0(z-1)^-1/2Tr A_1(z-t)^-1/2Tr A_t for the system (<ref>). We  can therefore assume that:
     
    
        A_ν∼diag(θ_ν, -θ_ν),  ν=0,1,t,
    
    where ∼ is used for similar matrices. The residue matrix at z=∞ is defined by 
    
        A_∞=-A_0-A_1-A_t     A_∞∼diag(θ_∞,-θ_∞).
    
    The monodromy group of the above system is characterized, in addition to the four parameters  θ=(θ_0,θ_t,θ_1,θ_∞), by the other two parameters that we denote as σ and s. The monodromy data (θ, σ, s) are considered as the coordinates on the monodromy manifold. We will usually  indicate a point of  this manifold by P,   P= (θ, σ, s). A precise definition of the monodromy manifold  as well as an explicit parametrization of all its elements, is provided in section <ref>. The rank 2 system (<ref>) with four singularities at z=0,1,t,∞ is not rigid <cit.>, i.e.  the monodromy data do not fix its form: there is one accessory parameter that one can deform by keeping the monodromy data invariant. Choosing the singularity position t as the deformation parameter, isomonodromic deformations of the system (<ref>) are governed by the equation
    Ỵ(z)/ṭ=-A_t/z-tY(z) .
    
    Compatibility condition of the latter formula with (<ref>) gives the Schlesinger equations that describe the non-linear evolution of {A_ν} under the isomonodromic transformation:
    Ạ_0/ṭ=[A_t,A_0]/t ,    Ạ_1/ṭ=[A_t,A_1]/t-1 ,    Ạ_t/ṭ=-[A_t,A_0]/t-
    [A_t,A_1]/t-1 .
    
    There is a statement that there exists tau function defined by formula
    /ṭlogτ(t)= A_0A_t/t+ A_1A_t/t-1 .
    
    The existence of a τ function is equivalent to the existence of a closed 1-form on the space of the deformation parameters <cit.>. In the 4-singularities case, where there is one single deformation parameter, this is a trivial statement. However, for more than four singularities, the existence of a τ function becomes non-trivial.
    Since isomonodromic deformations preserve monodromies, tau-function depends on t and on monodromy data,
    τ(t)=τ(P;t).
    
    
    
     §.§ Virasoro c=1 conformal blocks
    
    
    We refer the reader to <cit.> for the definition of the Virasoro algebra and its representation theory. 
    An important special function associated to the Virasoro representations is the four-point conformal block.  Once we fix the central charge to one, c=1, this is a function of six parameters: the value of the cross-ratio t, the dimensions Δ_ν, ν=0,1,t,∞ of the four primary fields sitting at positions z=0,1,t,∞, and the dimension Δ of the Virasoro representation that flows in the internal channel. 
    We recall that the internal field is the field produced in the fusion of the primary  at z=t with one of the other primaries sitting at z=0,1,∞. In the s-channel (t-channel) conformal block B^(s)({Δ_ν};Δ;t) (B^(t)({Δ_ν};Δ;t)), Δ is the dimension of the representation appearing in the fusion between the primaries with dimension Δ_t and Δ_0 (Δ_1): the corresponding conformal block is defined in a t expansion (1-t expansion). One has:
    
        B^(t)(Δ_0,Δ_t,Δ_1,Δ_∞;Δ;t)=(1-t)^Δ_0+Δ_∞-Δ_t-Δ_1 B^(s)(Δ_1,Δ_t,Δ_0,Δ_∞;Δ;1-t).
     
    Henceforth, we will drop the index of channel (s) or (t) because we will always imply  the conformal blocks to be in the s-channel, B^(s)({Δ_ν};Δ;t)→ B({Δ_ν};Δ;t).
    To  make the link with the isomonodromic transformations, one  parametrizes the dimensions Δ of the primary fields with the momenta θ such that Δ=θ^2:
     B(Δ_0,Δ_t,Δ_1,Δ_∞;Δ;t) → B(θ;σ;t),
    with Δ_ν =θ_ν^2,   (ν=0,t,1,∞),   and Δ=σ^2,
    
    From its definition, the   B(θ;σ;t) is manifestly invariant under the change of sign of  θ_ν and of σ. 
    
    
    
     §.§ τ functions in terms of Virasoro c=1 conformal blocks
    
    The main result in <cit.> is that the isomonodromic
    tau-function (<ref>) is a linear combination of conformal blocks with appropriate coefficients. In particular, the τ(P,t) function can be expressed as a series in t via the function τ_0t(P,t): 
    τ(P,t)=τ_0t(θ,σ_0t,s_0t,t)=_n∈ℤs_0t^n
    C(θ;σ_0t+n) B(θ;σ_0t+n;t) ,
    
    where the constants C(θ;σ_0t) are expressed in terms of the Barnes G-functions:
    
    C(θ;σ_0t)=∏_ϵ,ϵ'=±G(1+θ_t+ϵθ_0+ϵ'σ_0t)
    G(1+θ_1+ϵθ_∞+ϵ'σ_0t)/G(1+2σ_0t)G(1-2σ_0t)G(1+2 θ_0)G(1+2 θ_t)G(1+2 θ_1)G(1+2 θ_∞) .
    
    Differently from the conformal blocks, the constants C(θ;σ_0t) are not symmetric under the change of sign, θ_ν→ -θ_ν: this is reminiscent of the reflection coefficient in Liouville theory <cit.>.
     
    The form of the Fuchsian system (<ref>) is preserved by global conformal (SL(2,ℂ)) map z↦ (az+b)/(cz+d), and in particular by the map z↦ 1-z, which permutes 0 with 1 and maps t to 1-t. Using this map one finds another representation for the tau function as a series in 1-t:
    τ_1t(P̃,t)=_n∈ℤs_1t^n
    C(θ̃;σ_1t+n) B(θ̃;σ_1t+n;1-t) .
    
    In the above equation the point P̃:
    P̃=(θ̃,σ_1t,s_1t),   θ̃=(θ_1,θ_t,θ_0,θ_∞) ,
     is the image of P under the map z↦ 1-z, P→P̃, see section <ref>. 
    Since (<ref>)
    defines only logarithmic derivative of the tau function, it can get some non-trivial multiplicative factors after analytic continuation. This is actually the case for τ_0t(P,t) and τ_1t(P̃,t) that are tau functions of the same system. The ratio
    τ_0t(P,t)/τ_1t(P̃,t)=χ_01
    is called (up to some
    normalization) connection constant. Explicit formula for this constant has been conjectured in <cit.> and then rigorously proven in <cit.>.
    
    In principle, functions τ_0t(P,t) for different values of monodromies contain all possible c=1 conformal blocks — for example, they can
    be extracted by the inverse Fourier transformation. This procedure was used in <cit.> to obtain the fusion kernel for arbitrary c=1 conformal blocks from connection constant (<ref>). In principle, this formula for the fusion kernel could be useful for the study of crossing invariance in c=1 CFT. As a matter of fact, this turns out to be an extremely difficult approach because the fusion kernels are in general continous with a very subtle structure of singularities. 
    
    Before entering into details, let us first give the general sketch of the approach we will use to solve this problem.  We advocate a different point of view:  the good basis in the space of c=1 conformal blocks are not the 
    conformal blocks B(θ,σ_0t,t) themselves, but isomonodromic tau functions τ_0t(P,t) or τ_1t(P,t). The main reason for this choice is the much simpler formulas for the tau function crossing transformation (<ref>). 
    Another reason is based on the fact that the tau functions are common eigenvectors of the Verlinde loop operators that 
    act on the space of conformal blocks and that commute in c=1 case <cit.>. The CFT correlation function will be expressed in the tau function basis and the associated bootstrap relations will be solved by exploiting the formulas for the connection constant  (actually even simpler consequence of it). We propose the four-point correlation function  F_θ(t,t̅)  to have this form:
    
        F_θ(t,t̅) → F(t,t̅)[d μ_θ(P)]∝∫_P dμ_θ(P)τ_0t(P;t)τ_0t(ι(P),t̅),
    
    where ι(P) is the image of P under conjugation, z↦z̅, as explained in section <ref>. Therefore, given a distribution [We call by distribution some measure without
    positivity property and without normalization condition. One can also consider dμ_θ(P) as the element of the dual space to the (sub-)space of meromorphic functions of P, and the integral as pairing between function and distribution.]dμ_θ(P) on the monodromy data manifold, the correlation functions are averages of the product τ_0t(P;t)τ_0t(ι(P),t̅) of holomorphic and anti-holomorphic tau functions. The analytic continuations in the cross ratio t can be translated into the transformations of the monodromy manifold under the action of the braid group 𝔹_4, which exchanges the positions of the primary fields. This is the case for instance for the transformation (<ref>), see section (<ref>),  which is associated to the continuation t→ 1-t. The invariance of the four-point correlation function depends therefore on the transformation properties of the  distribution dμ_θ(P) under  the braid group 𝔹_4. As we will argue below, the  c=1 crossing symmetry solutions correspond to  distributions that are  weakly invariant. The precise definition of weakly invariant distributions, as well as the weak equivalence between distributions,  are given  in (<ref>) and in (<ref>). 
    
    
    In this paper we present several examples of weakly invariant distributions: one of them corresponds to  the  Runkel-Watts theory and another one to the  analytic Liouville theory <cit.>, proving then conjecture about their crossing invariance for c=1. One
    more example recovers particular correlation functions in Ashkin-Teller model, whose crossing symmetry was already proven in <cit.> by using the properties of elliptic functions.
    
    
    
    § CFTS AT C=1
    
    In this section we briefly review some CFT solutions at c=1.
    
    
    
     §.§ Gaussian free field
    
    The Gaussian free field (GFF) is described by a scalar field φ(x) and the action S[φ(x)]∼∫ d^2 x  (∇φ(x))^2.  It is the simplest example of CFT as it is a free theory and the correlation functions can be computed via the Wick theorem. The primary fields are the exponential  fields V_θ(x)=e^i θφ(x) with U(1) charge θ and conformal dimension Δ=Δ̅=θ^2.
    One can demand unitarity and consider therefore the representations with positive dimension. The spectrum 𝒮^GFF of this CFT is:
    
        𝒮^GFF:      {Δ, Δ},   Δ= θ^2 
           θ∈ℝ.
    
     For the four-point function, one obtains:
    ℱ^GFF_θ(t, t̅)=<V_θ_0(0)V_θ_t(t)V_θ_1(1)V_θ_∞(∞)> = δ_θ_0+θ_t+θ_1+θ_∞,0|t|^4 θ_0θ_t|1-t|^4 θ_1θ_t,
     
    where the Kronecker symbol δ_θ_0+θ_t+θ_1+θ_∞,0 comes from the U(1) charge symmetry. Using the fact that:
     
        B(θ_0,θ_t,θ_1,-θ_0-θ_t-θ_1;θ_0+θ_1;t)= t^2 θ_0θ_t(1-t)^2 θ_1 θ_t,
    
    the correlation (<ref>) can be factorized in the following way:
    ℱ^GFF_θ(t, t̅)=δ_θ_0+θ_t+θ_1+θ_0,0  B(θ;θ_0+θ_t;t) B(θ;θ_0+θ_t;t̅).
    
    
     §.§ The Ashkin-Teller model
    
    The Ashkin-Teller model <cit.> is a lattice spin model that can be defined as two coupled Ising models. The spin variables,  s_1(j⃗) and s_2(j⃗) are defined at each lattice site j⃗=(j_x,j_y) and  can take each  two possible values s_1,2(j⃗)=0,1. In addition to the  Ising couplings  s_1,2(j⃗) s_1,2(j⃗+e⃗_a), where j⃗ and j⃗+e⃗_a are neighbouring sites,  there is a four spin (energy-energy) interaction  of type s_1(j⃗) s_1(j⃗+e⃗_a)s_2(j⃗)s_2(j⃗+e⃗_a). 
    This model has a (self-dual) line of critical points that are described by a one parameter family of CFTs with central charge  c=1.  We denote this parameter with N. The study of the correlation function in this model has provided Virasoro c=1 solution <cit.> that we now briefly review. 
    
    The (untwisted) spectrum contains the representations with dimension: 
    
        𝒮^AT:     {Δ_n,m,  Δ̅_n,m},    n,m ∈ℤ
           Δ_n,m=(n/2N +   m N ) ^2,  Δ̅_n,m = (n/2N - N ) ^2,
    
    This spectrum is therefore  discrete and  non-diagonal, with fields with scaling dimension Δ+Δ̅=n^2/(2N^2)+2m^2N^2 and spin Δ-Δ̅=2mn. The values of  N=1/√(2) and N=1  correspond respectively to two decoupled Ising model, and to the four-states Potts model <cit.>. The (twisted sector of the) spectrum contains also fields that have dimension Δ=1/16 for any value of N. The set of parameters:
    θ_Picard=1/4,1/4,1/4,1/4,
     
    corresponds to four  Δ=1/16 representations. These representations are used in the computation of the magnetic correlations. There are four independent magnetic correlations <cit.>, that can be classified according to their behavior  under operator position exchange: the invariant magnetic correlation ⟨ s_1 s_1s_1s_1⟩=⟨ s_2 s_2s_2s_2⟩  and three magnetic correlations of type ⟨ s_1 s_1 s_2 s_2⟩ which transform one into another under this operation. The building blocks of such correlation are the functions B(θ_Picard;σ_0t,t) that have been computed exactly <cit.>:
     B(θ_Picard;σ_0t,t)= t^-1/8(1-t)^-1/8(16 q(t) )^σ_0t^2/ϑ_3(0|η(t)),
    
    where η(t) is a period of elliptic curve with branch-points 0,t,1,∞,  q(t)=e^iπη(t) is  the elliptic nome.
    
    
    
    The four different functions labeled by (ϵ,ϵ')=(0,0), (0,1), (1,0), (1,1):
    
        F^AT,(ϵ,ϵ')_θ_Picard(t,t̅)= 
           =∑_n,m∈ℤ(-1)^ϵ'm (16)^-(n+ϵ/2)^2/2N^2-2m^2 N^2 B(θ_Picard;n+ϵ/2/2N +   m N,t) B(θ_Picard;n+ϵ/2/2N - Nm,t̅) =
           =|t|^-1/4|1-t|^-1/4/|ϑ_3(0|η(t))|^2∑_n,m ∈ℤ (-1)^ϵ'mq(t)^Δ_n+ϵ/2,mq̅(t)^Δ̅_n+ϵ/2,m,
    
    were proposed to describe the invariant magnetic correlator ⟨ s_1 s_1s_1s_1⟩ (corresponding to ϵ=ϵ'=0),
    and also three correlators that transform one to another under the crossing transformations: ⟨ s_1s_2s_2s_1 ⟩, ⟨ s_1s_1s_2s_2 ⟩, and ⟨ s_1s_2s_1s_2 ⟩, corresponding to (ϵ,ϵ')=(1,0), (0,1), and (1,1), respectively <cit.>.
    The crossing symmetry of these functions can be proven by using the transformation properties of the elliptic functions. Notice that a generalization to any value of c<1 of the cases (1,0), (0,1), and (1,1) has been proposed in <cit.>.
    
    
    
     §.§ The Runkel-Watts theory
    
    In <cit.>, a new c=1 solution was found by taking, via a subtle procedure, the limit p→∞ of the unitary Virasoro minimal series ℳ_p,p+1. The spectrum 𝒮^RW of this theory is  continuous and  diagonal: it contains all the non-degenerate fields with positive dimension Δ≥ 0. It can be parametrized via  θ such that: 
    
        𝒮^RW:      {Δ, Δ},   Δ= θ^2 
           θ∈ℝ-ℤ/2.
    
    The  three point structure constant C^RW(θ_1,θ_2,θ_3) factorizes into two terms:
    
    C^RW(θ_1,θ_2,θ_3)=RW(θ_1,θ_2,θ_3)Φ(θ_1,θ_2,θ_3) .
    
    The term
    Φ(θ_1,θ_2,θ_3)=_ϵ_i=±G(1+ϵ_1θ_1+ϵ_2θ_2+ϵ_3θ_3)/_i=1^3_ϵ=±G(1+2ϵθ_i)
    is analytic in θ_i, i=1,2,3 while the term
    
    RW(θ_1,θ_2,θ_3)=[[  0: sinπ(θ_2+θ_3-θ_1)sinπ(θ_2+θ_3+θ_1)/sinπ(θ_2-θ_3-θ_1)sinπ(θ_2-θ_3+θ_1)>0 ,; 1: sinπ(θ_2+θ_3-θ_1)sinπ(θ_2+θ_3+θ_1)/sinπ(θ_2-θ_3-θ_1)sinπ(θ_2-θ_3+θ_1)≤ 0 . ].
    
    is a step-function. 
    The Runkel-Watts correlation function is:
    ℱ^RW_θ(t,t̅)= ∫_-∞^∞ d σ_0t C^RW(θ_0,θ_t,σ_0t)C^RW(σ_0t,θ_1,θ_∞) B(θ;σ_0t,t) B(θ;σ_0t,t̅) .
    
    Notice that the integration domain is over all σ_0t∈ℝ. This is the same as doing the integration over the spectrum (<ref>) as the poles in
    σ_0t∈1/2ℤ are suppressed by the factor (<ref>). 
    
    
    
     §.§ The c=1  analytic  Liouville theory
    
    
    Motivated by the study of the geometrical properties in random Potts model, a Liouville type theory was proposed for c≤ 1 in <cit.>.  The spectrum 𝒮^AL is continuous and diagonal. For c=1 one has 
    
        𝒮_c=1^AL:      {Δ, Δ},   Δ= θ^2 
           θ∈ℝ.
     
    The structure constant C^c≤ 1(θ_1,θ_2,θ_3) is given in terms of product of Barnes double Gamma functions <cit.>. At c=1 their expression simplify to
    
    C^c=1(θ_1,θ_2,θ_3)=Φ(θ_1,θ_2,θ_3) .
    
    Notice that, differently from the Runkel-Watts theory, there is no step function that suppresses the (non-integrable) divergences coming from the conformal blocks. The four point correlation function has therefore being defined by moving the contour in the complex plane:
    ℱ^AL_θ(t,t̅)= ∫_i h+ ℝ d σ_0t Φ(θ_0,θ_t,σ_0t)Φ(σ_0t,θ_1,θ_∞)  B(θ;σ_0t,t) B(θ;σ_0t,t̅),
    
    where h is an arbitrary constant. The property of the above integral will be discussed in detail below.
    
    
    § MODULI SPACE OF FLAT CONNECTIONS
    
    
    In this section we give self-contained description of the moduli space of SL(2,ℂ) and SU(2) flat connections on sphere with four
    punctures. Almost all content of this section can be found as well in <cit.>, <cit.>, <cit.> and <cit.>.
    
    
    
     §.§ Definitions
    
    
    Given some group G, the central object of our approach is the moduli space of G-flat connections [Since we work only with
    monodromies of the Fuchsian system, we will use only G⊆ GL(N,ℂ).] on sphere with 4 punctures: this is the space of all maps from the fundamental group of Riemann sphere with four punctures π_1(ℂP^1∖{0,t,1,∞})
    to G, defined up to overall conjugation. Henceforth we will denote this space as M_0,4^G, 
     M_0,4^G= Homπ_1(ℂP^1∖{0,t,1,∞}),G/G .
    
    The above map can be seen as a monodromy map that sends each path γ∈π_1(ℂP^1∖{0,t,1,∞}) to the monodromy of the solution of the Fuchsian system (equivalently the flat connection)  over this path <cit.>.
    The definition of M_0,4^G can be formulated in an equivalent and more explicit way in  terms of the set of equivalence classes of  monodromy matrices:
     M_0,4^G={M_0,M_t,M_1,M_∞|M_ν∈ G, M_0M_tM_1M_∞=1}/G .
     
    Two sets of matrices belong to the same equivalence class if they are related by an element g of the group G, via an overall conjugation: {M_ν}↦{g^-1M_ν g}, g∈ G. 
    
    We consider here two particular spaces, one for G=SL(2,ℂ) and another one for G=SU(2). Moreover, it is convenient  to consider the submanifold 
    M_0,4^G(θ) ⊂ M_0,4^G by setting M_ν∼diag(e^2π i θ_ν,e^-2π i θ_ν), i.e. by fixing the conjugacy classes [θ_ν]—=
    
    ={M_0,M_t,M_1,M_∞|M_ν∈ SL(2,ℂ), M_ν=2cos 2πθ_ν,M_0M_tM_1M_∞=1}/SL(2,ℂ) .
    
    In this way we get symplectic leaves of the Poisson bracket <cit.> on the moduli space of flat connections. Notice that the condition det M_ν=1 follows from the tracelessness condition Tr A_ν=0 for the residue matrices in (<ref>).
    In the G=SU(2) the (special) unitarity condition M_ν∈ SU(2) should also be imposed, so  we have 
    
        ⊂⊂.
    
    
     §.§ Explicit construction of the  and  spaces
    
    
    We give here explicit coordinate description
    of (the open charts of)  and . We suppose that M_0t=M_0M_t is diagonalizable
    (by unitary matrix in the unitary case):
    
    M_0M_t= e^2π iσ_0t   0
    0   e^-2π iσ_0t=e^2π i S_0t .
    
    The matrix M_t and its inverse M_t^-1 can be represented as linear combinations of identity matrix and rank 1 matrix:
    
    M_t= e^-2π iθ_t   0
    0   e^-2π iθ_t + e^2π iθ_t-e^-2π iθ_t u_1  v_1    u_2v_1
    u_1 v_2     u_2 v_2 ,
    
    M_t^-1= e^2π iθ_t   0
    0   e^2π iθ_t + e^-2π iθ_t-e^2π iθ_t u_1  v_1    u_2 v_1
    u_1 v_2     u_2v_2
    where u_1, v_1,u_2, v_2 ∈ℂ are complex numbers satisfying u_1 v_1+u_2 v_2=1. Using a more compact notation, M_t=e^-2π iθ_t1+ e^2π iθ_t-e^-2π iθ_t u^T⊗ v, M_t^-1=e^2π iθ_t1+ e^-2π iθ_t-e^2π iθ_t u^T⊗ v, where u and v are row vectors, u=(u_1,u_2) and v=(v_1,v_2). We further constraint the four variables u_1,u_2,v_1,v_2 by imposing the condition  M_0=diag(e^2iπθ_0,e^-2iπθ_0).
    Using the formula M_0=M_0tM_t^-1 and (1+AB)=(1+BA) we get:
    λ-M_0=λ-e^2π iθ_0λ-e^-2π iθ_0=
    =λ-e^2π i( S_0t+θ_t)+
     e^2π iθ_t-e^-2π iθ_t e^2π i S_0tu^T⊗ v=
    =
    λ-e^2π i(θ_t+σ_0t)λ-e^2π i(θ_t-σ_0t) 1+
     e^2π iθ_t-e^-2π iθ_t ve^2π i S_0t/λ-e^2π i( S_0t+θ_t)u^T 
    Computing this expression near the fake singularities of the right side we get two relations:
    
    u_1v_1=e^-2π iσ_0t e^2π i(θ_t+σ_0t)-e^2π iθ_0 e^2π i(θ_t+σ_0t)-e^-2π iθ_0/ e^2π iθ_t-e^-2π iθ_t
    e^2π i(θ_t+σ_0t)-e^2π i(θ_t-σ_0t)=
    =
    sinπ(θ_t+σ_0t-θ_0)
    sinπ(θ_t+σ_0t+θ_0)/sin 2πθ_t sin 2πσ_0t ,
    
    u_2v_2=e^2π iσ_0t e^2π i(θ_t-σ_0t)-e^2π iθ_0 e^2π i(θ_t-σ_0t)-e^-2π iθ_0/ e^2π iθ_t-e^-2π iθ_t
    e^2π i(θ_t-σ_0t)-e^2π i(θ_t+σ_0t)=
    =-
    sinπ(θ_t-σ_0t-θ_0)
    sinπ(θ_t-σ_0t+θ_0)/sin 2πθ_t sin 2πσ_0t .
    
    The four variables u_1, u_2, v_1, v_2 are defined modulo the transformation
    u,v ↦λ u, λ^-1v. We can thus set one of these coefficients to one, say v_2=1. The remaining three coefficients satisfy two conditions: it suffices then 
    to parametrize these four variables by one parameter  s_0t^A:
    
    u_1=sinπ(θ_t+σ_0t-θ_0)sinπ(θ_t+σ_0t+θ_0)/sin2θ_tsin2πσ_0te^-2π i σ_0t(s^A_0t)^-1 ,  
    v_1=s_0t^A e^2π iσ_0t ,
    
    u_2=-sinπ(θ_t-σ_0t-θ_0)sinπ(θ_t-σ_0t+θ_0)/sin2πθ_tsin2πσ_0t ,  
    v_2=1 .
    
    Now we check the unitarity condition for matrix M_t: M_t^-1=M_t^†. This leads to the condition u_iv_j=u_j^*v_i^* [
    which actually implies that u=α v^* for some α∈ℝ_+], which gives nontrivial relation for i=1, j=2:
    
    |s_0t^A|^2=-sinπ(θ_t+σ_0t-θ_0)sinπ(θ_t+σ_0t+θ_0)/sinπ(θ_t-σ_0t-θ_0)sinπ(θ_t-σ_0t+θ_0)
    The above relation can give solutions for s_0t^A only when the r.h.s. is positive, which imposes non-trivial constraint on σ_0t, like spherical triangle
    inequality: the step function RW(θ_t,σ_0t,θ_0),  defined in (<ref>), should equal to one. Moreover, obviously all θ- and
    σ-variables
    should be real: θ_ν∈ℝ, σ_μν∈ℝ.
    The same conditions should hold for the other triple of matrices, so the manifold
     will be defined by reality constraints and two conditions:
    
    RW(θ_0,θ_t,σ_0t)=1   and    RW(θ_1,θ_∞,-σ_0t)=1 .
    
    The above conditions fix the fusion rules of the Runkel-Watts theory. In <cit.> these fusion rules have been derived from a subtle limit of the minimal model fusion rules. It is quite remarkable that the same conditions can be associated to the monodromy submanifold .
    
    So far we have constructed monodromy matrices (M_0, M_t) only for one half of the sphere with 4 punctures, which is sphere with 3 punctures. We see
    from this construction that M_0,3^SL(2,ℂ)(θ), as well as M_0,3^SU(2)(θ), are just points: all monodromy matrices
    are defined only by the eigenvalues of M_0, M_t, M_0t, and s_0t^i can be removed by the overall diagonal conjugation (in the SU(2) case its modulus
    is fixed, but phase can be removed). In other words, we see well-known fact that the second order Fuchsian system with three singulariries is  rigid, i.e. the local exponents θ_0,θ_t and σ_0t are enough to parametrize the monodromy manifold.  
    
    Coming back to our problem, we can construct the remaining monodromies, M_1,M_∞  using the same method and by  replacing  θ_0→θ_1,
    θ_t→θ_∞, σ_0t→ -σ_0t (remember that M_1∞=M_0t^-1), s_0t^A→ s_0t^B e^2π iσ_0tsinπ(θ_∞-θ_1-σ_0t)/sinπ(θ_∞-θ_1+σ_0t),  exactly as it was done in <cit.>:
    
    M_∞=e^-2π iθ_∞ 1+ e^2π iθ_∞-e^-2π iθ_∞ũ^T⊗ṽ,  
    M_1=e^-2π i  SM_∞^-1 ,
    
    where
    ũ_1=-sinπ(θ_∞+σ_0t-θ_1)sinπ(θ_∞-σ_0t+θ_1)/sin2θ_∞sin2πσ_0t(s^B_0t)^-1 ,  ṽ_1=s_0t^B sinπ(θ_∞-θ_1-σ_0t)/sinπ(θ_∞-θ_1+σ_0t) ,
    ũ_2=sinπ(θ_∞+σ_0t-θ_1)sinπ(θ_∞+σ_0t+θ_1)/sin2πθ_∞sin2πσ_0t ,  ṽ_2=1 .
    
     Computing matrix elements explicitly we recover (up to simultaneous transposition) formulas
    (3.32a-d) from <cit.>. All matrices are defined up to the overall conjugation, so the point on monodromy manifold depends only on
    
    s_0t=s_0t^A/s_0t^B .
    
    In this way we constructed explicit uniformization of  by the two coordinates, σ_0t and s_0t. It is important to observe that
     is a holomorphic symplectic manifold with holomorphic symplectic form dω obtained from Goldman bracket <cit.> (or from Atiyah-Bott form), and σ_0t, log s_0t are its Darboux coordinates <cit.>:
    
    dω=1/2π i dσ_0t∧d s_0t/s_0t
    
    From the (<ref>) and (<ref>), in the unitary case we have:
    
    |s_0t|^2=R^unitary(σ_0t,θ)=
    
    =sinπ(θ_t+σ_0t-θ_0)sinπ(θ_t+σ_0t+θ_0)/sinπ(θ_t-σ_0t-θ_0)sinπ(θ_t-σ_0t+θ_0)sinπ(θ_1+σ_0t-θ_∞)sinπ(θ_1+σ_0t+θ_∞)/sinπ(θ_1-σ_0t-θ_∞)sinπ(θ_1-σ_0t+θ_∞) .
    
    
     §.§ Algebro-geometric description
    
    
    We have seen that the manifold  is a six-dimensional space. This can be seen also by considering the space of invariant functions on it, that are given by traces of all possible products of matrices. Such  products are associated to  elements (paths) of
    π_1(ℂP^1∖{0,t,1,∞}). Working in this geometric representation we can use the relation in SL(2,ℂ)
    A+A^-1= 1· A
    
    to reduce any path to a combination of non self-intersecting paths. It is possible because this relation is actually a “skein relation” shown
    in Figure <ref>, so one application of it reduces the number of self-intersections by 1.
    
    
    As a basis of non self-intersecting cycles first we should take cycles around one point — they give four functions p_μ= M_ν. And second
    we should take cycles around pairs of points: there are three such inequivalent cycles encircling 0t, 1t and 01, and they give rise to
    three functions  p_μν= M_μ M_ν for μν = 0t, 1t, 01, which are nevertheless not independent. There is one non-trivial relation called 
    Jimbo-Fricke affine cubic W=0:
    
    W=p_0tp_1tp_01+p_0t^2+p_1t^2+p_01^2-p_0t p_0p_t+p_1p_∞-p_1t p_1p_t+p_0p_∞-
    -p_01 p_0p_1+p_tp_∞+
    p_0p_tp_1p_∞+p_0^2+p_t^2+p_1^2+p_∞^2-4
    
    
    So algebro-geometric description of our monodromy manifold looks as follows: 
    = Specℂ[p_0t,p_1t,p_01,p_0,p_t,p_1,p_∞]/(W)
    
    Let us express  the quantities p_μν in terms of the variables (σ_0t,s_0t) introduced in (<ref>) and in (<ref>). We have:
    
    p_0t= M_0M_t=2cos2πσ_0t ,
    
    
    p_1t= M_1M_t=1/sin^2 2πσ_0t1/2 p_tp_1+p_0p_∞-1/4 p_0p_1+p_tp_∞ p_0t-.
    .- 4 _ϵ'=±s_0t^ϵ'_ϵ=±sinπ(ϵθ_0+θ_t-ϵ'σ_0t)sinπ(ϵθ_∞+θ_1-ϵ'σ_0t) ,
    
    
    p_01= M_0M_1=1/sin^2 2πσ_0t1/2 p_0p_1+p_tp_∞-1/4 p_tp_1+p_0p_∞ p_0t+ .
    .+ 4 _ϵ'=±s_0t^-ϵ' e^2π iσ_0tϵ'_ϵ=±sinπ(ϵθ_0+θ_t+ϵ'σ_0t)sinπ(ϵθ_∞+θ_1+ϵ'σ_0t) .
    
    One can verify that (<ref>) and (<ref>) are invariant under
    θ_ν↦θ_ν+k_ν, σ_μν↦σ_μν+k_μν,
    k_ν, k_μν∈ℤ. These are quite obvious symmetries since the monodromy properties of a Fuchs solution depend on the local exponents (i.e. the eigenvalues of the residue matrices A_ν) modulo an integer. There are also simple symmetries θ_ν↦-θ_ν for ν=0,∞. Their
    analogs for ν=t,1 are less trivial. For example, θ_t↦-θ_t, s_0t↦ s_0t∏_ϵ=±sinπ(ϵθ_0-θ_t+σ_0t)/sinπ(ϵθ_0+θ_t+σ_0t). Combining all such symmetries together we may construct transformation ϰ:
    ϰ(θ,σ_0t,s_0t)=(-θ,σ_0t,s_0t·∏_ϵ=±sinπ(θ_1-σ_0t+ϵθ_∞)
    sinπ(θ_t-σ_0t+ϵθ_0)/sinπ(θ_1+σ_0t+ϵθ_∞)sinπ(θ_t+σ_0t+ϵθ_0)) ,
    
    which will be used in the next subsection.
    Other symmetries will be discussed below, in section (<ref>).
    
    
    
     §.§ Involutions ι and ι'
    
    
    The crossing symmetry of  correlation function depends on the transformation properties of the holomorphic and anti-holomorphic conformal blocks. Consider an analytic  function ϕ(z,P) whose monodromies are described by the point P∈. It is therefore  natural to ask what are the monodromies of anti-analytic function ϕ(z̅,P). 
    
    The involution of the loops under the map z↦z̅ is denoted by ι, γ→γ^ι. The involutions of the generating elements of π_1(ℂP^1 {0,t,1,∞}) are shown on
    Figure <ref>. One can see that the base point is shifted, so to compare the monodromies one has to move it to initial position
    over some path [
    Different paths will give monodromies that differ by overall conjugation, so correspond to the same point of the monodromy manifold.]
    drawn with dashed line in Figure <ref>.
    Moreover, consider for instance the loop γ_t that is encircling  the point t and  passes  under the point 0. The transformed loop γ^ι_t is inverted and  passes  over the same point 0. Therefore, taking into account the branch cut associated to the singularity in 0, the loops γ^ι_t and γ_t live on different sheets. The loop γ^ι_t is topologically equivalent to the loop product γ_0^-1γ_t^-1γ_0.
    
    
    
    
     Let us consider the action of the involution ι on the  matrices in . This action (defined up to conjugation)  will be denoted by ι(M_ν) or, equivalently, by  M_ν^ι. On the basis of the above considerations, one has:
    
    M_0^ι=M_0^-1 ,  
    M_t^ι=M_0M_t^-1M_0^-1 ,  
    M_1^ι=M_0M_tM_1^-1M_t^-1M_0^-1 ,
    
    M_∞^ι=M_0M_tM_1M_∞^-1M_1^-1M_t^-1M_0^-1 .
    
    
    Now using the following identity in SL(2,ℂ) [
    We use the conjugation by matrix with ≠1, but its determinant can be easily fixed by introducing normalization factor, which obviously cancels.
    ] a   b
    c   d^-1= 0   b
    -c   0^-1 a   b
    c   d 0   b
    -c   0
    and the fact that the matrices M^ι_ν are defined up to overall conjugation, we can introduce new involution ι', whose
    action is given by:
    
    M_0^ι'=N_0M_0^-1N_0^-1 ,  
    M_t^ι'=N_0M_0M_t^-1M_0^-1N_0^-1 ,  
    M_1^ι'=N_0M_0M_tM_1^-1M_t^-1M_0^-1N_0^-1 ,
    
    where we introduce
    
    N_0=0   (M_0)_12
    -(M_0)_21   0,   N_1=0   (M_1)_12
    -(M_1)_21   0 .
    
    Using (<ref>) we get M_0^ι=M_0 by definition, M_0^ι M_t^ι=N_0M_0t^-1N_0^-1=M_0t by simple computation. Therefore
    M_t^ι'=M_t, as well. It remains to compute the action on M_1:
    
    M_1^ι'=(N_0M_0tN_1^-1) M_1 (N_0M_0tN_1^-1)^-1 .
    
    Matrix N_0M_0tN_1^-1 is diagonal, so diagonal elements of M_1 remain unchanged. Compute now one non-diagonal element:
    ι'(M_1)_12=(M_1)_12(M_1)_21/(M_1)_12(M_0t)_22/(M_0t)_11(M_0)_12/(M_0)_21 .
    
    In other words it means that
    ι'(u̅_1 v̅_2)=u̅_1v̅_2· e^4π iσ_0tu̅_2v̅_1/u̅_1v̅_2u_1 v_2/u_2 v_1 ,
    
    or equivalently
    ι'(s_0t)=1/s_0t∏_ϵ=±sinπ(θ_1+σ_0t+ϵθ_∞)
    sinπ(θ_t+σ_0t+ϵθ_0)/sinπ(θ_1-σ_0t+ϵθ_∞)sinπ(θ_t-σ_0t+ϵθ_0) .
    
    Resuming, under an involution ι, a point P=(θ,σ_0t, s_0,t), P∈ is transformed like:
    
    P= (θ,σ_0t, s_0t)↦ι'(P)= (θ,σ_0t,ι'(s_0t))
    
    We may also check that ι'(p_ν)=ι(p_ν)=p_ν, as well as ι'(p_0t)=ι(p_0t)=p_0t and ι'(p_1t)=ι(p_1t)=p_1t,
    but p_01 changes:
    ι'(p_01)=ι(p_01)= M_0^-1M_tM_1^-1M_t=p_tp_∞+p_0p_1-p_0tp_1t-p_01 .
    
    This is noting but permutation of two sheets of Jimbo-Fricke surface (<ref>). The same results have been obtained in <cit.>. Up to small misprint [in <cit.> one has to replace ω_0t with ω_01 in this formula], the reader can compare the above formulas with the ones found in <cit.> (notice that in <cit.> the primed  quantities correspond to the ones transformed under involution. For instance,  ι(p_01) here  corresponds to p_01' in <cit.>).
    It is also interesting to notice that involution ι coincides with the y-generator of extended modular group from the paper <cit.>,
    and to get the whole modular group one has to add two very similar generators.
    
    Here during the construction of ι' we have used the fact that in SL(2,ℂ) matrix and its inverse lie in the same conjugacy class 
    (<ref>), so we could keep θ the same.
    In the general case one should have θ↦-θ, so it is much
    more natural to combine ι' with the transformation ϰ from (<ref>): ι=ϰ∘ι'. Its action is much simpler:
    ι(θ,σ_0t,s_0t)=(-θ,σ_0t,s_0t^-1) .
    
    We remind that both transformations have the same action on the invariant functions, the only difference is the action on the 
    ambiguously defined variables like θ_ν. We will use both involutions: in some sense ι is simpler and more natural, but ι' is needed
    for compatibility with <cit.>.
    
    
    
     §.§ Transformation from s to t channel
    
    We review here how to connect the tau-functions τ_01(t) and τ_1t(t),   introduced in (<ref>) and in (<ref>). In order to do that we study here  the transformation of the loop γ∈π_1(ℂP^1∖{0,t,1,∞}) under the map z↦ 1-z, γ→γ̃. Figure <ref> shows the transformation of the generating loops γ_ν, ν=0,t,1,∞ under this map.  The corresponding action on the monodromy matrices M_ν will be denoted by M̃_ν, M_ν→M̃_ν, ν=0,t,1,∞.
    Consider two monodromy points P=(θ,σ_0t,s_0t) and P̃=(θ,σ_1t,s_1t) (P,P̃∈) that describe respectively the monodromies  of  τ_01(t) and τ_1t(t). We show here how P and P̃ are related one to the other. 
    
    
    By evoking the same considerations as the the ones discussed previously for the involution ι, the transformed monodromy matrices can be written as
    M̃_0=M_tM_1M_t^-1 ,  M̃_t=M_t ,  M̃_1=M_0 ,  M̃_∞=M_0^-1M_∞ M_0 .
    
    
    As far the invariant quantities p̃_0t=M̃_0M̃_t, p̃_1t=M̃_1M̃_t and  p̃_01=M̃_1M̃_∞ are concerned, formulas strictly analogous to (<ref>) should hold. In particular one expects that the expressions for  p̃_0t,p̃_1t and p̃_01 are  obtained from (<ref>) by replacing θ_0→θ_1, θ_0→θ_1,
    σ_0t→σ_1t, s_0t→ s_1t. This is true except a sign subtlety: from (<ref>), p̃_0t=p_1t and p̃_1t=p_0t, p̃_01= M_0M_t^-1M_1M_t=ι(p_01), where  ι(p_01) is defined in (<ref>). Therefore p_01=ι(p̃_01), which effectively  changes sign in the exponents. Using these results, we can express the quantities in P in terms of the coordinates of P̃:
    
    p_1t=2cos2πσ_1t ,
    
    
    p_0t=M̃_1M̃_t=1/sin^2 2πσ_1t1/2 p_tp_0+p_1p_∞-1/4 p_0p_1+p_tp_∞ p_1t-.
    .- 4 _ϵ'=±s_1t^-ϵ'_ϵ=±sinπ(ϵθ_1+θ_t+ϵ'σ_1t)sinπ(ϵθ_∞+θ_0+ϵ'σ_1t) ,
    
    
    p_01=ιM̃_0M̃_1=
    1/sin^2 2πσ_1t1/2 p_0p_1+p_tp_∞-1/4 p_tp_0+p_1p_∞ p_1t+ .
    .+ 4 _ϵ'=±s_1t^-ϵ' e^-2π iσ_1tϵ'_ϵ=±sinπ(ϵθ_1+θ_t+ϵ'σ_1t)sinπ(ϵθ_∞+θ_0+ϵ'σ_1t) .
    
    Notice that (<ref>) and (<ref>) are symmetric under the sign inversion: (σ_0t,s_0t)↦ (-σ_0t,s_0t^-1), and (σ_1t,s_1t)↦ (-σ_1t,s_1t^-1).
    This means that each pair of coordinates, (σ_0 1,s_01) or (σ_1 1,s_1t) actually realizes the two-fold cover of :
    
    (σ_μν,s_μν)≃ (-σ_μν,s_μν^-1) .
    
    
    By comparing the (<ref>) and (<ref>), we can find the relation between the coordinates (σ_0t,s_0t) and  (σ_1t,s_1t), i.e. the relation between P and P̃. We proceed like follows: first we use p_1t in terms of σ_0t and s_0t from (<ref>), thus fixing the value of σ_1t=σ_1t(σ_0t,s_0t) using the expression for p_1t in  (<ref>). Finally, using the following formula
    obtained from (<ref>):
    
    s_1t^±1
    =
    ∓ 2isin 2πσ_1t(p_01+e^∓ 2π iσ_1tp_0t)
    -e^∓ 2π iσ_1t(p_0p_1+p_tp_∞)
    +(p_tp_0+p_1p_∞)
    /16_ϵ=±sinπ(ϵθ_1+θ_t∓σ_1t)sinπ(ϵθ_∞+θ_0∓σ_1t) 
    we can express  the dependence of s_1t on (σ_0t,s_0t).
    
    
      §.§.§ Examples
    
    As an example of application of these formulas we consider the case θ_0=θ_t=θ_1=θ_∞=1/4, which corresponds to
    quasi-permutational monodromy. We introduce for convenience
    
    s_μν=-e^2π iη_μν .
    
    One can check that the following equality follows from (<ref>): cos 2πσ_1t=cos 2πη_0t. As a consequence we have either
    σ_1t=η_0t or σ_1t=-η_0t. Finally we get
    
      (σ_1t,η_1t)=(η_0t,-σ_0t)    or     (σ_1t,η_1t)=(-η_0t,σ_0t).
      
    We easily see that transformations (<ref>) preserve symplectic form (<ref>). This property holds also in the general case:
    
    dω=dσ_0t∧ dη_0t=dσ_1t∧ dη_1t .
    
    
    
    In the general case transformation (σ_0t,η_0t)↦(σ_1t,η_1t) is quite complicated. However, for our purposes, see  Subsection <ref>, we need to solve a simpler problem, namely the one of finding how this transformation changes the topological type of the cycle
     C_h,r={σ_0t,η_0t|Imσ_0t=h,Imη_0t=r} ,
      
    where we consider the limit [For practical applications it is enough to take A≫  1 and A≫θ_ν.]|h|>A, |r|>A, |r-h|>A, A →∞. First take this limit in (<ref>):
       p_0t=e^-2π iσ_0t h+O(e^-2π A) ,
       p_1t=e^-2π i(η_0t+θ_t+θ_1+1/2) r+O(e^-2π A) ,
       p_01=e^-2π i(η_0t-σ_0t+θ_t+θ_1)(r-h)+O(e^-2π A) .
      
      Now one has to fix sign for σ_1t:
    σ_1t=-(η_0t+θ_t+θ_1+1/2) r + O(e^-2π A) .
      
    Then compute the limit of s_1t using (<ref>):
    
      s_1t^±1=   O(e^-2π A)+
     +∓ e^-2π i(η_0t+θ_t+θ_1+1/2) r(e^-2π i(η_0t-σ_0t+θ_t+θ_1)(r-h)+ e^± 2π i(η_0t+θ_t+θ_1+1/2) r -2π i σ_0t h)/e^-4π i(η_0t+θ_t+θ_0) r ∓ 2π i(θ_t+θ_0) .
    
    Now there are two possible situation, h· r>0 and h· r<0.
    In the first situation only the second term in (<ref>) dominates in the formula for s_1t^-1:
    
    s_1t^-1= e^-2π iσ_0t r +4π i(θ_0-θ_1) r-2π i(θ_t+θ_0)+O(e^-2π A) ,
    
    while in the second situation only the first term in (<ref>) dominates in the formula for s_1t^+1:
    
    s_1t^+1 = e^2π iσ_0t r +4π i(θ_0-θ_1) r+2π i(θ_t+θ_0)+O(e^-2π A) .
    
    One can write these expressions in a unified way:
    η_1t=σ_0t r - 2(θ_0-θ_1) h - (θ_0+θ_t-1/2)+O(e^-2π A) .
      
    Notice that these limiting transformations (<ref>) and (<ref>) are very similar to (<ref>), and they also transform contour C_h,r to contour C_h̃,r̃ up to small perturbations of order O(e^-2π A), where
       h̃=-|r|-(θ_t+θ_1) r ,
       r̃=h r -2(θ_0-θ_1) h-θ_0-θ_t .
    
    So at the level of homology classes we have transformation
    [C_h,r]↦[C_h̃,r̃] .
      
    
     §.§ Braid group 𝔹_4
    
    
    More in general we consider the action of the braid group 𝔹_4 on the monodromy manifold. Indeed, the action of this group 𝔹_4 on the tau-function performs the analytic continuation of the tau-function in the variable t and different re-expansions
    between different channels, like s- and t-channels.
      In Artin representation, and element b of the braid group 𝔹_4 is generated by a product of  three basic elements,  (b_0t,b_t1,b_1,∞). For instance the element b_0t braids path passing through  0  below the path passing through t, as in this figure:
    
    [scale=1,every node/.style=scale=1]
    [number of strands=4] (braid) a_1;
     (0,-1) node[left]b_0t:;
     (1.4,-2) node[left]0;
     (2.2,-2) node[left]t;
     (3.5,-2) node[left]1;
     (4.5,-2) node[left]∞;
    
    
    Studying the corresponding transformation on the loops, one can derive the action of the generators (b_0t,b_t1,b_1,∞) on the matrices M_ν:  
    
    b_0t: (M_0,M_t,M_1,M_∞)↦ (M_t,M_t^-1M_0M_t,M_1,M_∞)
    
    b_t1: (M_0,M_t,M_1,M_∞)↦ (M_0,M_1,M_1^-1M_tM_1,M_∞) 
    
    b_1∞: (M_0,M_t,M_1,M_∞)↦ (M_0,M_t,M_∞,M_∞^-1M_1M_∞) .
    
    
    Let us consider for instance the element b=b_0t^2:
    
    [scale=1,every node/.style=scale=1]
    [number of strands=4] (braid) a_1 a_1;
     (0,-1) node[left]b_0t^2:;
     (1.4,-3) node[left]0;
     (2.2,-3) node[left]t;
     (3.5,-3) node[left]1;
     (4.5,-3) node[left]∞;
    
    
    Under the action of b_0t^2, the monodromy matrices transform as:
    
    b_0t^2:(M_0,M_t,M_1,M_∞)↦ (M_t^-1M_0M_t,M_t^-1M_0^-1M_tM_0M_t,M_1,M_∞),
    
    that, from (<ref>) and (<ref>), implies:
    
    b_0t^2(θ, σ_0t, s_0t)= ( θ, σ_0t, s_0te^4 π i σ_0t)
    
    Finally, one can verify that the s-channel to t-channel transformation (<ref>) corresponds to the element b_st∈𝔹_4, 
    
    b_st= b_t1b_0tb_t1,
     associated to the following braid :
    
    [scale=1,every node/.style=scale=1]
    [number of strands=4] (braid) a_2 a_1 a_2;
     (0,-1) node[left]b_st=b_t1b_0tb_t1;
     (1.4,-4) node[left]0;
     (2.2,-4) node[left]t;
     (3.5,-4) node[left]1;
     (4.5,-4) node[left]∞;
    
    
    We have that:
    
    b_st((θ, σ_0t, s_0t))=(θ̃, σ_1t, s_1t)
    
    where θ̃=(θ_1,θ_t,θ_0,θ_∞).
    
    Let us now explain the relation between our description of crossing (fusion) and braiding transformations and the Moore–Seiberg formalism <cit.>.
    In the case of general multi-point conformal blocks one has the action of the Moore-Seiberg groupoid generated by the two transformations: braiding, which moves two colliding fields around each other, and fusion, which performs local s–t channel transformation of conformal blocks, or changes pants decomposition.
    In general, fusion changes topological type of the pants decomposition.
    However, it turns out that for 4 and 5 points all pants decompositions are the same up to permutation of the external states (for 6 points we already have two inequivalent pictures).
    This means that one can fix some particular pants decomposition, for which t collides with 0 and 1 collides with ∞ [At the level of braids this means that we add some extra structure describing pants decomposition. Here we may say that strands are divided into two groups, (0, t) and (1, ∞).].
    Then we relate all other decompositions to this one by the action of the braid group permuting external states and acting on the complex modulus t by analytic continuation.
    So for the 4-point case we can consider 𝔹_4 instead of the Moore-Seiberg groupoid: for example, see Figure <ref> for the action of b_st on 4-point conformal block.
    
    
    
    The most important and non-trivial element is b_st, but there are also many elements which do not change the channel of expansion.
    They give rise to nice non-trivial relations between conformal blocks that can be found in <cit.>. For example, there is an identity for b_1∞:
     B(θ_0,θ_t,θ_1,θ_∞;σ_0t;t/t-1)=
    (-1)^σ_0t^2-θ_0^2-θ_t^2(1-t)^2θ_t^2 B(θ_0,θ_t,θ_∞,θ_1;σ_0t;t) .
    
    Here -1 should be treated either as e^π i or as e^-π i, dependently on the direction of analytic continuation, and comes from the power of t in the normalization factor.
    
    We also have the same simple formula for b_0tb_1∞^-1:
     B(θ_t,θ_0,θ_∞,θ_1;σ_0t;t)=
    (1-t)^θ_0^2+θ_∞^2-θ_1^2-θ_t^2 B(θ_0,θ_t,θ_1,θ_∞;σ_0t;t) .
    
    Together these three elements b_st, b_0t, b_1∞ generate the whole braid group. Notice also that since this braid group acts on sphere, we have extra relation b_0t^2=b_1∞^2, coming from the overall coordinate rotation z↦ z e^2π i, so effectively this group almost reduces to 𝔹_3.
    
    
    § CROSSING SYMMETRIC CORRELATION FUNCTION FROM TAU FUNCTIONS
    
    
    It is convenient to introduce the  tau-function τ̂_0t that differs from the (<ref>) only by a normalization:
    τ̂_0t(P;t)= ∑_n ∈ℤ s_0t^n C(θ;σ_0t+n)/C(θ;σ_0t) B(θ;σ_0t+n;t),   P=(θ,σ_0t,s_0t)
    
    
    A CFT correlation function  is obtained by gluing the holomorphic  and anti-holomorphic conformal block.  We have shown before the transformation of the point P under the conjugation z↦z̅. Using (<ref>) and (<ref>), the corresponding tau function can be written in two different forms:
    τ̂_0t(ι'(P),t̅)=∑_n ∈ℤ(ι'(s_0t))^n C(θ;σ_0t+n)/C(θ;σ_0t)  B(θ;σ_0t+n;t̅) ,
    τ̂_0t(ι(P),t̅)=∑_n ∈ℤ s_0t^-nC(-θ;σ_0t+n)/C(-θ;σ_0t)  B(θ;σ_0t+n;t) .
    
    then, using (<ref>), (<ref>) and the following property of the  Barnes functions
    G(1-ν+n)/G(1-ν)=(-1)^n(n-1)/2G(1+ν-n)/G(1+ν)π/sinπν^n ,
    
    we can show that these two functions coincide: τ̂_0t(ι'(P),t̅)=τ̂_0t(ι(P),t̅).
    
    Taking into account the identity:
    
    C(θ,σ_0t)C(-θ,σ_0t)=Φ(θ_0,θ_t,σ_0t)Φ(σ_0t,θ_1,θ_∞),
    
    where Φ has been introduced in (<ref>), we define the non-holomorphic function  as:
    (P,t,t̅)=τ_0t(P,t)τ_0t(ι(P),t̅)=Φ(θ_0,θ_t,σ_0t)Φ(σ_0t,θ_1,θ_∞)τ̂_0t(P,t)τ̂_0t(ι'(P),t̅) .
    
    One of the good properties of (P,t,t̅) is that it is an actual function on the monodromy manifold , while τ_0t(P,t) and 
    τ_0t(ι(P),t) are only sections of some line bundles.
    Namely, their periodicity properties are given by τ_0t(θ,σ_0t+1,s_0t;t)=
    s_0t^-1τ_0t(θ,σ_0t,s_0t;t) and τ_0t(-θ,σ_0t+1,s_0t^-1;t)=
    s_0tτ_0t(-θ,σ_0t,s_0t^-1;t), whereas functions on  are periodic under such shift.
    
    
    
    
    
    
     §.§ CFT correlation function in terms of  averages
    
    Using  the  (P,t,t̅) function, that contains a combination of holomorphic and anti-holomorphic conformal blocs, and the properties of the manifold ,  we want to define a function F_θ(t,t̅) that satisfies the following properties:
    
      *  To be single-valued in the variable t.
    
      *  To be crossing invariant, i.e. the two expressions, in terms of the s-channel and in terms of the  t-channel conformal blocks, are equivalent.
    
     
    
    Let consider a general  distribution dμ_θ(P) on  that is supported by , i.e.:
    
    dμ_θ(P) = 0   for  P=(θ',σ_0t,s_0t)  and θ'≠θ.
    
    We will focus on the function F(t,t̅)[dμ_θ], that is defined as the dμ_θ average of the function  (P,t,t̅)(<ref>) over the space :
     F(t,t̅)[dμ_θ]=_ dμ_θ(P) (P,t,t̅).
    
    The main point of this construction is that the crossing invariance of   F(t,t̅)[dμ_θ] can be assured by properly choosing the distribution dμ_θ. 
    
    
    
     §.§ Weakly equivalent and weakly invariant distributions
    
    To be more concrete, let us first introduce the notion of weak equivalence  between distributions on . We say that two distributions dμ_1(P) and dμ_2(P) are weakly equivalent if their integrals  with all reasonable [To be rigorous, one should say that by reasonable functions we mean the space of  functions at all possible t's.
    This definition seems to be not very useful for classificational problems, so probably there might be some simpler description: for example, functions whose singularities in the finite domain are only σ_0t=0,1/2. We don't actually know what is the best definition, since we are using these distributions only to average tau functions.] meromorphic test functions f(P) coincide:
    
    dμ_1≃ dμ_2⇔∀ f: ∫ dμ_1(P) f(P)=∫ dμ_2(P) f(P) .
    
    We can now define a “weakly invariant” distribution. Consider an element b of the four strands braid group 𝔹_4, b∈𝔹_4. A distribution μ_θ(P) is weakly  invariant if the transformed distribution, b dμ_θ(P) [By transformed distribution we denote the inverse of the pull-back:
        ∀ f: ∫ b(dμ(P)) f(b(P))=∫ dμ(P) f(P).
    
      Sometimes we also use another notation: dμ(b(P))≡ b(dμ(P)).]  is  weakly equivalent to the distribution with transformed θ:
    
    b dμ_θ(P)≃ dμ_b(θ)(P)
    
    The action of the element b on θ is simply a permutation of indexes. On the other hand, the action of the braid group on  the distribution is highly non-trivial as the transformation of the point P→ b(P) is in general quite complicated, as we have seen in the previous sections for some special braid action.
    
    
    
    
    
     §.§ Weakly invariant distribution and crossing invariance
    
    
    If the distribution dμ_θ(P) is weakly invariant under 𝔹_4 action, then the function F(t,t̅)[dμ_θ], defined by (<ref>), provides a crossing-invariant  CFT correlation function F_θ(t,t̅)=  F(t,t̅)[dμ_θ]. 
    In particular we show  that  the following theorem holds:
    
      Consider a weakly invariant distribution dμ_θ(P). The function F(t,t̅)[dμ_θ], defined by (<ref>), is a single-valued function of t and satisfies the properties:
    
        A)  F(e^2π i t,e^-2π it̅)[dμ_θ] =  F(t,t̅)[dμ_θ], 
           B)  F(t,t̅)[dμ_θ]= F(1-t,1-t̅)[dμ_θ̃] 
           with: θ= (θ_0,θ_t,θ_1,θ_∞),  θ̃=(θ_1,θ_t,θ_0,θ_∞)
    
    The above theorem implies the crossing invariance of the CFT correlation function [We might also claim the stronger invariance under the half-rotation t↦ e^π it/1-t that permutes θ_0 with θ_t or θ_1 with θ_∞, (<ref>), (<ref>), but for simplicity we do not focus on such finite symmetries.].
    
    
     Proof. We start by considering property A, see (<ref>). This property is related to the transformation of the correlation function under the double braid  action b_0t^2 that implements the rotation of  the position t around 0. The action of b_0t^2 on  is given by (<ref>). One verifies that 
    (θ;σ_0t,s_0t;e^2π it,e^-2π it)=(θ;σ_0t,s_0t· e^4π iσ_0t;t,t̅).
    This is the consequence of two identities:
    τ_0t(θ;σ_0t,s_0t;e^2π it)=e^2π i(σ_0t^2-θ_0^2-θ_t^2)τ_0t(θ;σ_0t,s_0t· e^4π iσ_0t;t) ,
    τ_0t(-θ;σ_0t,s_0t^-1;e^-2π it̅)=e^-2π i(σ_0t^2-θ_0^2-θ_t^2)τ_0t(-θ;σ_0t,s_0t^-1· e^-4π iσ_0t;t̅) .
    
     One can study the behavior of correlation function under such analytic continuation
     F(e^2π it, e^-2π it̅)[dμ_θ]=_ dμ_θ(P) (P,e^2π it,e^-2π it̅)=_ dμ_θ(P) (b_0t^2(P),t)=
    =
    _ dμ_θ(b_0t^2(P)) (b_0t^2(P),t)=_ dμ_θ(P̃) (P̃,t)= F(t, t̅)[dμ_θ]
    
    In the above series of identities we have used (<ref>) and the weak invariance of dμ_θ that allows  to replace dμ_θ(P)→ dμ_θ(b_0t^2(P)). Notice that if we  set  dμ_θ(P) to be given by the closed differential 2-form (in Darboux coordinates):
    
    dμ_θ(P) = dσ_0t∧d s_0t/s_0t,
    
    one can directly verify that the distribution is invariant under the change of coordinates (θ,σ_0t,s_0t)→(θ,σ_0t,s_0te^4π i σ_0t),  dμ_θ(P)= dμ_θ(b_0t^2(P)).
    
    We give a proof now of property B, see (<ref>). In this case we consider the braid element b_st defined by (<ref>). The action of b_st on  is given by (<ref>).
    We prove below that:
    (θ_1,θ_t,θ_0,θ_∞;σ_1t,s_1t;1-t)=
    (θ_0,θ_t,θ_1,θ_∞;σ_0t,s_0t;t) .
    
    The above identity, together with (<ref>), implies the property B:
     F(t, t̅)[dμ_θ]=_ dμ_θ(P) (P,t,t̅)=
    _ dμ_θ(P) (b_st(P),1-t,1-t̅)=
    =
    _ b_st(dμ_θ̃(P)) (b_st(P),1-t,1-t̅)=
    _ dμ_θ̃(b_st(P)) (b_st(P),1-t,1-t̅)=
    = F(1-t, 1-t̅)[dμ_θ̃]
    
    
    To show (<ref>), we use the definition (<ref>)(θ;σ_0t,s_0t;t,t̅)/(;σ_1t,s_1t;1-t,1-t̅)
    =
    Φ(θ_0,θ_t,σ_0t)Φ(σ_0t,θ_1,θ_∞)
    /Φ(θ_1,θ_t,σ_1t)Φ(σ_1t,θ_0,θ_∞)τ̂_0t(θ;σ_0t,s_0t;t)
    τ̂_0t(θ;σ_0t,ι'(s_0t);t̅)/τ̂_0t(;σ_1t,s_1t;1-t)
    τ̂_0t(;σ_1t,ι'(s_1t);1-t̅)
    =
    
    
    =
    Φ(θ_0,θ_t,σ_0t)Φ(σ_0t,θ_1,θ_∞)
    /Φ(θ_1,θ_t,σ_1t)Φ(σ_1t,θ_0,θ_∞)χ̅_01(θ;σ_0t,σ_1t;p_01)χ̅_01(θ;σ_0t,σ_1t;ι'(p_01))=1 .
    
    During this computation we first used <cit.> that defines connection constant:
    τ̂_0t(θ;σ_0t,s_0t;t)=χ̅_01(θ;σ_0t,σ_1t;p_01)τ̂_0t(;σ_1t,s_1t;1-t) ,
    
    and the most important relation <cit.>:
    χ̅_01(θ;σ_0t,σ_1t;p_01)χ̅_01(θ;σ_0t,σ_1t;ι'(p_01))
    =
    Φ(θ_1,θ_t,σ_1t)Φ(σ_1t,θ_0,θ_∞)/Φ(θ_0,θ_t,σ_0t)Φ(σ_0t,θ_1,θ_∞) .
    
    The latter relation is quite surprising, since connection constant for single tau function χ̅_0t is quite involved function of monodromy parameters,
    but it turns out that the product of two connection constants simplifies drastically, and we need only this simple part [
    It is witten in <cit.>“ A conceptual explanation of this intriguing coincidence is yet to be found ”, and we think that the application,
    which was found here, indicates that this coincidence is not accidental. We also expect it to hold in more general cases.].
    
    So now we can state that the problem of construction of the crossing-invariant functions with c=1 given by the ansatz (<ref>)
    is reduced to the problem of construction of weakly invariant distributions on .
    
    § EXAMPLES OF WEAKLY INVARIANT DISTRIBUTIONS
    
    
    We do not have yet a criteria for classifying completely neither the  weak equivalent  distributions, nor the weak invariant ones.  On the other hand we know some special cases that are shown below to provide non-trivial CFT solutions. These cases are:
    
      *  The distributions that are invariant,  b(dμ_θ(P))=μ_b(θ)(P). These distributions are obviously weakly invariant. As examples of invariant distribution, we may take either uniformly distributed measure on some invariant subset, or, more generally, invariant measures on invariant submanifolds.
    
      *  The distributions that are given by an invariant holomorphic 2-form concentrated on two-dimensional submanifolds M⊂, such that their homology classes [M] ∈ H_2,ℤ are invariant under
    the braid group action [b(M)]=[M].
    
    
    Below we show that, among the cases mentioned above, we can find important CFT solutions.
    
    
    
     §.§ The Gaussian free field
    
    In <cit.> it was observed that the Riccati solutions of the Painlevé VI were related to the conformal blocks appearing in the GFF.
    The parameters θ are given by:
    θ_Riccati = (θ_0,θ_t,θ_1,-θ_0-θ_t-θ_1) .
    
    The remaining two variables of the monodromy data, that we can choose to be (σ_0t,σ_1t) are also constrained to be:
    σ_0t= θ_0+θ_t,   σ_1t=θ_t+θ_1 .
    
    The corresponding tau function is simply given by <cit.>:
    τ_0t(P_Riccati,t) =  B(θ_Riccati,θ_0+θ_t, t) = t^2 θ_0 θ_t (1-t)^2 θ_1 θ_t .
    
    It is then clear that by taking as measure:
    
    d μ^Riccati_θ_Riccati(P)= δ(σ_0t-θ_0-θ_t) δ(σ_1t-θ_1-θ_t), 
    
    and using (<ref>) and (<ref>), one gets:
    ℱ_θ_Riccati(t,t̅)=ℱ(t,t̅)[d μ^Riccati_θ_Riccati] .
    
    
     §.§ The Runkel-Watts theory
    
    We reviewed this theory in section (<ref>). Now we show how this theory can be found in our approach.
    Let us consider the real two-dimensional submanifold
    ⊂
    which corresponds to unitary monodromies. This manifold is necessarily invariant with respect to any transformation since the  braid transformation (<ref>) maps unitary monodromies to unitary ones. Therefore we can construct invariant distribution by restriction of invariant symplectic form
    to  using (<ref>) and (<ref>)
    dμ^unitary_θ(P)=1/2π idσ_0t∧ds_0t/s_0t×
    ×δ(Imσ_0t)δ(|s_0t|^2-R^unitary(σ_0t,θ))RW(θ_0,θ_t,σ_0t) RW(θ_1,θ_∞,-σ_0t) .
    
    The Theorem <ref> assures the function F(t,t̅)[dμ^unitary_θ], defined in (<ref>), to be single-valued and crossing invariant. We show now that this function corresponds to the  correlation function F^RW_θ(t,t̅) defined in (<ref>):
     F(t,t̅)[dμ^unitary_θ]= F^RW_θ(t,t̅) .
    
    In order to compute the integral (<ref>) we introduce the short-hand notation
    R(σ_0t)=R^unitary(σ_0t,θ) ,    RW(σ_0t)=RW(θ_0,θ_t,σ_0t)RW(θ_∞,θ_1,σ_0t) .
    
    First we take the integral over s_0t:
     F(t,t̅)[dμ^unitary_θ]=∫_0^1 dσ_0t_|s_0t|^2=R(σ_0t)ds_0t/2π is_0tRW(σ_0t)τ_0t(P,t)τ_0t(ι(P),t̅)
    =
    =
    ∫_0^1 dσ_0tRW(σ_0t) _|s_0t|^2=R(σ_0t)ds_0t/2π is_0t_n,m∈ℤs_0t^n-mC(θ,σ_0t+n)C(-θ,σ_0t+m)×
    × B(θ,σ_0t+n,t) B(-θ,σ_0t+m,t̅)
    =
    =
    ∫_0^1 dσ_0t_n∈ℤRW(σ_0t+n)
    C(θ,σ_0t+n)C(-θ,σ_0t+n)| B(θ,σ_0t+n,t)|^2
    
    Now it remains to combine the sum into the integral over the real line and use (<ref>), (<ref>) and (<ref>):
     F(t,t̅)[dμ^unitary_θ]=∫_-∞^∞ dσ_0t RW(θ_0,θ_t,σ_0t)
    RW(θ_∞,θ_1,σ_0t)×
    ××Φ(θ_0,θ_t,σ_0t)Φ(θ_∞,θ_1,σ_0t) | B(θ,σ_0t,t)|^2 .
    
    Since the above expression reproduces the correlation function in the Runkel-Watts theory (<ref>), the identity (<ref>) is proven.
    
    
    
    
    
    
    
     §.§ Ashkin-Teller model
    
    
    Let us consider now the submanifold  ℳ_0,4^SU(2)(θ_Picard) generated by the following matrices
    
    M_0= 0   ie^2π iσ_0t
    ie^-2π iσ_0t   0,   M_t=0   -i
    -i   0,  
    M_1= 0   ie^-2π iσ_1t
    ie^2π iσ_1t   0
    The above matrices, that have all eigenvalues e^± 2iπ/4, are associated to the set of parameters (<ref>). We already considered this case at the end of Section <ref>. By conjugating all the matrices by M_t, one has
    (σ_0t,σ_1t)↦(-σ_0t,-σ_1t), which is the same as (<ref>).  
    
    It is convenient to represent a point P_Picard∈ℳ_0,4^SU(2)(θ_Picard) by using the variable σ_1t instead of s_0t,  
    
        P_Picard=(θ_Picard,σ_0t,σ_1t).
     
    The relation between s_0t and σ_1t greatly simplifies for ℳ_0,4^SU(2)(θ_Picard). From the expression of  p_1t(=2cos 2πσ_1t) as given in   (<ref>) and from  (<ref>), one obtains:
    for  θ= θ_Picard:    s_0t=-e^2π i σ_1t,   ι'(s_0t)=-e^-2π iσ_1t
    Accordingly, the closed 2–form (<ref>) in the new coordinates simply reads:
    1/2π i dσ_0t∧d s_0t/s_0t=dσ_0t∧ d σ_1t
    
    
    We focus first on the  function  (P_Picard,t,t̅), see the (<ref>). The associated  structure constants take the simple form:
    
    C(θ_Picard;σ_0t)=c^+/cos(πσ_0t)16^-σ_0t^2,   
    C(-θ_Picard;σ_0t)=c^- cos(πσ_0t) 16^-σ_0t^2.
    
    Using the above result, one can express  (P_Picard,t,t̅)  via the following double sum:
    (P_Picard,t,t̅)=c^+ c^-/|t^1/8(1-t)^1/8ϑ_3(0|η(t))|^2_(n,m)∈ℤ^2e^2π iσ_1t(n-m) q^(σ_0t+n)^2q̅^(σ_0t+m)^2,
    
    where the (<ref>) and (<ref>) have been used. Henceforth, we will neglect c^+c^- factor.
    
    We can study now the weakly invariant measures on ℳ_0,4^SU(2)(θ_Picard). The action  (<ref>) of the braid group on ℳ_0,4^SU(2)(θ_Picard) reads
    
    b_0t(M_0)∼ M_0,   b_0t(M_t)∼ M_t,  
    b_0t(M_1)∼ 0   ie^-2π i(σ_1t-σ_0t)
    ie^2π i(σ_1t-σ_0t)   0 .
    
    Notice that to obtain the above equations we applied  an appropriate simultaneous conjugation. A point P_Picard, under the action of b_0t and b_t1 transforms in the following way:
    
    b_0t(P_Picard)=(θ_Picard,σ_0t,σ_1t-σ_0t) ,
    
    b_t1(P_Picard)=(θ_Picard,σ_0t+σ_1t,σ_1t).
    
    These two transformations generate the SL(2,ℤ) group
    σ_0t
    σ_1t↦ a   b
    c   dσ_0t
    σ_1t ,    a   b
    c   d=1 .
    
    
    We know two possibilities to construct distribution, invariant under these transformations. 
    
    
    
      §.§.§ Continuous spectrum: back to the Runkel-Watts theory
     
    The first one is very universal: we just take
    dμ^unitary_θ_Picard given by (<ref>):
    
    dμ^unitary_θ_Picard=dσ_0t∧ dσ_1t·δ( Im σ_0t)δ( Im σ_1t)
    
    Using this distribution  we get
     F(t,t̅)[dμ^unitary_θ_Picard]=∫ dσ_0t∧ dσ_1t (P_Picard;t,t̅)=
    =
    _n∈ℤ∫_0^1   dσ_0t  e^-2π(σ_0t+n)^2  Im η(t)/|t^1/8(1-t)^1/8ϑ_3(0|η(t))|^2=
    ∫_-∞^∞  dσ e^-2πσ^2  Im η(t)/|t^1/8(1-t)^1/8ϑ_3(0|η(t))|^2=
    =
    |t^1/8(1-t)^1/8ϑ_3(0|η(t))|^-2(2  Im η(t))^-1/2
    The above correlation function, that can be found in <cit.>, is a   Runkel-Watts correlator, see Eq.45 in <cit.>.
    
    
    
      §.§.§ Discrete spectrum: Ashkin-Teller spin correlators
     
    
    Let us consider some vector (ω_1,ω_2)  and its SL(2,ℤ) orbit, which we denote
    
    O_(ω_1,ω_2)=SL(2,ℤ)· (ω_1,ω_2) .
    
    The second possibility to build an invariant measure is to define it as  supported by O_(ω_1,ω_2):
    
    dμ^O_(ω_1,ω_2)=_x⃗∈ O_(ω_1,ω_2)δ(σ_0t-x_1)δ(σ_1t-x_2)
    
    We have the following
    
    All finite orbits O_(ω_1,ω_2) coincide with some O_(0,1/p)=O_1/p for p∈ℤ, which, in its turn, can be obtained by
    additions and set-theoretic subtractions of the finite lattices 
    
    L_1/p=1/p(ℤ/pℤ)^2⊂ (ℚ/ℤ)^2⊂ (ℝ/ℤ)^2 ,
    
    so the basis of invariant measures is given by measures, supported by L_1/p.
    
    
    Proof of this proposition goes in the following way. First we check that any vector (ω_1,ω_2) is equivalent to (ω,0). Actually,
    if there is (ω_1,ω_2) with |ω_1|≤ |ω_2|, then we may transform it to (ω_1,ω_2±ω_1)— this procedure
    necessarily decreases min(|ω_1|,|ω_2|), and finally we come to (ω,0) [described procedure is nothing but the Euclid's algorithm]. Then ω should be rational number ω=p'/p [p and p' are not necessarily prime], otherwise
    we get infinite orbit.
    
    Obviously we have an inclusion O_p'/p⊂ L_1/p. Now we wish to describe the decomposition of L_1/p into the union of orbits.
    This is the same as to describe the orbits of SL(2,ℤ)⋉ (pℤ)^2 acting on ℤ^2: (A,x,y):1/p(a,c)↦ A·1/p(a+p x,c+p y).
    It is clear that the number d=(a,c,p) is invariant under such action.
    Now we show that it is the only invariant, so the point 1/p(a,c) belongs the orbit O_1/p(a,c,p). Moreover, we show that O_d̃'/p̃=O_1/p̃ if d̃'/p̃ is irreducible, so
    
    L_1/p=_p̃|pO_1/p̃ .
    
    
    Actually, first we can map point 1/p(a,c) to 1/p(0,d') using SL(2,ℤ) action to perform the Euclid's algorithm. Here d'=(a,c).
    Then we reduce fraction d'/p=d̃'/p̃. Since (d̃',p̃)=1, one can find such α, β∈ℤ that αd̃'-βp̃=1 and then construct  SL(2,ℤ) matrix A:
    
    A=d̃'   p̃
    β   α ,   A 0
    d̃' = p̃d̃'
    1+βp̃
    In this way we have shown that any orbit in L_1/p is equivalent to O_1/p̃, where p̃ is some divisor of p, so we proved (<ref>).
    
    Now we introduce the invariant distribution
    
    dμ^1/p=_a,c=0^p-1δ(σ_0t-a/p)δ(σ_0t-c/p) .
    
    Using (<ref>) we write
    
    dμ^1/p=_p̃|pdμ^O_1/p̃ ,
    
    and then using Möbius inversion formula we get
    
    dμ^O_1/p=_p̃|pμ(p̃/p)dμ^1/p̃ ,
    
    where μ(p̃/p) is the Möbius function. Therefore any invariant measure can be expressed in terms of dμ^1/p̃. □
    
    
    
    Now we start to study such measures, dμ^1/p, and first we study dμ^0, which gives 
     F(t,t̅)[dμ^0]=(θ_Picard;0,0;t,t̅)=|t|^-1/4|1-t|^-1/4 .
    
    So it is just the free field correlator.
    
    Moving to the general case we make two updates: first, for the technical reasons we can consider only even p=2N, and second, we can consider three more measures which are not SL(2,ℤ) invariant, but instead form its 3-element orbit:
    
    dμ^1/2N,ϵ,ϵ'=_a,c=0^2N-1δ(σ_0t-a+ϵ/2/2N)δ(σ_1t-c+ϵ'/2/2N) ,
    
    where ϵ,ϵ'∈ℤ/2ℤ. We see that the measure dμ^1/2N,0,0 is invariant, but three other measures dμ^1/2N,1,0=:dμ^1/2N,1, dμ^1/2N,0,1=:dμ^1/2N,2 and dμ^1/2N,1,1=:dμ^1/2N,3
    are permuted by SL(2,ℤ/2ℤ)=S_3. The two SL(2,ℤ) generators S= 0   1
    -1   0 and T= 1   1
    0   1 map to permutations as follows:
    
    S↦ (12) ,   T↦ (23) .
    
    To get correlation function first one has to compute the sum
    _m,n∈ℤ_a,c=0^2N-1e^π ic+ϵ'/2/N(n-m) q^(a+ϵ/2/p+n)^2q̅^(a+ϵ/2/2N+m)^2=
    
    
    =
    _n,k∈ℤ(-1)^ϵ'k_a=0^2N-1q^(a+ϵ/2/2N+n)^2q̅^(a+ϵ/2/2N+n+2Nk)^2=
    =
    _n,k∈ℤ(-1)^ϵ'kq^(n/2N)^2q̅^(n/2N+2Nk)^2=
    _n,k∈ℤ(-1)^ϵ'kq^(n+ϵ/2/2N+Nk)^2q̅^(n+ϵ/2/2N-Nk)^2/,.
    
     Using this sum the correlation function is rewritten as
      F(t,t̅)[dμ^1/2N,ϵ,ϵ']=|t|^-1/4|(1-t)|^-1/4/ϑ_3(0|η(t))|^2_n,k∈ℤ(-1)^ϵ'kq^(Nk+n+ϵ/2/2N)^2q̅^(Nk-n+ϵ/2/2N)^2
    that corresponds to ℱ_θ_Picard(t,t̅) defined previously in (<ref>).
    
     §.§ Analytic Liouville theory
    
    
    In the previous examples the weakly invariant distributions were actually just invariant.
    Here we consider the situation where the second example of weakly invariant distribution is realized: namely, in this case we take an integral over some 2-cycle, such that its homology class is invariant under the braiding transformations.
    It turns out that so constructed distribution provides the correlation functions of the analytic Liouville theory <cit.>.
    In this way we prove crossing invariance of this theory, conjectured in <cit.>.
    
    Consider the distribution concentrated on the contour C_h,r(<ref>):
    
    dμ^analytic(h,r)_θ=1/2π idσ_0t∧ds_0t/s_0t·δ(Imσ_0t-h)δ(|s_0t|-e^-2π r) ,
    
    where h≠0 and the associated function:
     F(t,t̅)[dμ^analytic(h,r)_θ]=∫_ih^ih+1
    dσ_0t_|s_0t|=e^-2π rds_0t/2π i s_0t (P,t,t̅)
    Using the fact that nothing depends on |s_0t|, the integration over s_0t follows strictly the one in (<ref>). We obtain:
       F(t,t̅)[dμ^analytic(h,r)_θ]=_n∈ℤ∫_ih^ih+1 dσ_0tΦ(θ_0,θ_t,σ_0t+n)Φ(θ_∞,θ_1,σ_0t+n)
    ×
    × B(θ,σ_0t+n,t)  B(θ,σ_0t+n,t̅)
    
    =∫_ih+ℝ dσ_0tΦ(θ_0,θ_t,σ_0t)Φ(θ_∞,θ_1,σ_0t) B(θ,σ_0t,t)  B(θ,σ_0t,t̅),
    
    that, comparing with (<ref>) shows that:
    
        F(t,t̅)[dμ^analytic(h,r)_θ]= ℱ^AL_θ(t,t̅).
    
    
    Note that the above formulas are very similar to the ones we saw in the case of dμ^unitary_θ(<ref>), since in the both cases we integrate the form dω over submanifold. However, as we already said, there is an important difference: in the unitary case the submanifold is invariant under the braid group action, while in the analytic case this is not true. Let's see this more in detail.
    
    We want to prove crossing invariance and single-valuedness of ℱ^AL_θ(t,t̅) by observing that the homology class of the submanifold of integration in (<ref>) is preserved by the braid group. We use the following strategy:
    
      * A: we check that the singularities of the integrand do not affect invariance of the integral.
      Namely, even when we move contour (<ref>) through the pole of the tau function, integral does not change because corresponding residue vanishes.
      In other words, one does not have to add extra cycles encircling singularities in the finite domain.
      So the only thing that we have to check is that integration cycle lies in the same homology class in H_2((ℂ^×)^2,ℤ)=ℤ after the transformation, where by (ℂ^×)^2 we denote a manifold with coordinates (e^2π iσ_0t,s_0t).
      
    
      * B: we check that homology class of the integration contour [ C_h,r] is invariant.
    
    
    Let's consider the point A. The integrand is analytic in σ_0t and s_0t everywhere except for σ_0t∈{0,1/2} (or, considering the  
    the real line, everywhere except for σ_0t∈1/2ℤ).
    We have to verify that moving the contour through these points does not change the value of the integral: therefore,  even if contour goes through the singular point, it can be correctly defined by shifting in one of the two directions, i.e. with h<0 or h>0.
    One observes that the two residues, at σ_0t=0 and  at σ_0t=1/2,  vanish due to the following relations:
    _n∈ℤRes_σ_0t=nΦ(θ_0,θ_t,σ_0t)Φ(θ_∞,θ_1,σ_0t)
     B(θ,σ_0t,t)  B(θ,σ_0t,t̅)dσ_0t=0 ,
    _n∈ℤRes_σ_0t=n+1/2Φ(θ_0,θ_t,σ_0t)Φ(θ_∞,θ_1,σ_0t)
     B(θ,σ_0t,t)  B(θ,σ_0t,t̅)dσ_0t=0 .
    
    To prove this we notice first that all the functions under the sum are even functions of σ_0t. This property is manifest in the definition of the constants (<ref>) and, concerning the conformal blocks B(θ,σ_0t,t) and B(θ,σ_0t,t̅), we recall that they depend only on the dimension σ_0t^2 of the intermediate channel. Now suppose that there is some even function
    f(σ)=f(-σ). Then the two functions
    f̃_0(σ)=_n∈ℤf(σ+n)    and    f̃_1/2(σ)=_n∈ℤf(σ+n+1/2)
    
    are ℤ-periodic and even: f̂_0(σ)=f̂_0(-σ), f̂_1/2(σ)=f̂_1/2(-σ). This implies automatically that
    Res_σ=0f̂_0(σ)dσ=Res_σ=0f̂_1/2(σ)dσ=0, and proves thus (<ref>).
    In this way we actually proved the point A above: singularities in the finite domain do not affect invariance of the integral. We also proved actually that the analytic distribution does not depend on h and r in a sense of weak equivalence:
    
    dμ^analytic(h,r)_θ≃ dμ^analytic(h',r')_θ≃ dμ^analytic_θ .
    
    One interprets this fact as effectively we do not have holes in the finite domain, so [ C_h,r]=[ C_-h,r]=[ C_h,-r]=[ C_-h,-r], and homology class does not depend on (h,r):
    [ C_h,r]=[ C_h',r']
    
    
    We can now pass to point B of the proof. We have already checked in Section <ref> that in the limit |h|→∞, |r|→∞, |h-r|→∞ homology class [ C_h,r] transforms to [ C_h̃,r̃].
    Because of (<ref>), first, it implies invariance of [ C_h,r] in the limit, and second, by the deformation argument [One can first pull initial contour to infinity, apply crossing transformation, and then pull it back. Notice that in the previous version of the paper, we computed the effects on the transformations in a region infinitesimal close to the Picard point θ_Picard and then we used an analytic continuation argument to extend the result to all values of θ. As the referee pointed out, this proof actually was not completed.] it implies invariance for all (h,r). 
    
    There is also simpler braiding transformation (<ref>) acting by b_0t^2: s_0t↦ s_0t e^4π i σ_0t.
    It maps C_h,r to C_h,r+2h, so also preserves [ C_h,r].
    The same also holds for half-braiding b_0t(<ref>). So all possible transformations preserve [ C_h,r], and this completes the proof.
    
    
    In this way we have proved  that correlation function in analytic Liouville theory (<ref>) is h-independent,
    crossing-invariant and single-valued.
    
    
     
    
    
    § CONCLUSIONS
    
    In this paper we discussed the construction of crossing invariant correlation functions in c=1 conformal field theories. Using the relation between isomonodromic deformations and Virasoro conformal blocks, we proposed the tau functions instead of the conformal blocks as a basis to construct crossing invariant correlation functions. The basic idea is that the transformation properties of the tau function under the braid group are simpler than the ones of the conformal blocks, in particular when a continuous spectrum is considered. We proposed the ansatz (<ref>) for crossing invariant correlation function according to which the problem of constructing of a class of CFT solutions reduces to the definition of weakly invariant distributions on the moduli space . We presented four examples of such distributions, given in (<ref>), (<ref>), (<ref>) and (<ref>). We showed that these four distributions provide respectively the correlation functions of the free Gaussian, of the Runkel-Watts, of the Ashkin-Teller and of the analytic Liouville theory.  
    
    We showed that our approach permits not only to recover very different theories by using a common framework. More important, we could prove crossing symmetry for the analytic Liouville theory, a case where other methods, based for instance on the Teschner-Ponsot formulas <cit.> seem too complicated for this value of the central charge. Nevertheless, there are crucial issues that need to be investigated further.
    We expect indeed that not all the possible c=1 CFT solutions are in the form (<ref>); for instance, this form seems not designed to describe correlations with non-diagonal external fields. And, even assuming the form (<ref>), we cannot characterize all its solutions as we do not know how to classify completely the weakly invariant distributions on . 
    
    There is also a series of questions and problems for future investigations:
    
    
      *  One obvious way to construct invariant measures on  is to take sums of delta-functions over the finite orbits of the action of the braid group: dμ(P)=_P'∈ Oδ(P,P').
    All such orbits were classified in <cit.>, corresponding solutions of Painlevé  VI are also known.
    Most of these finite orbits arise from monodromies corresponding to finite subgroups G⊂ SU(2)— 
    conjecturally, these cases should correspond to c=1 CFT's on orbifolds, and we should be able to construct closed formulas for correlation functions in such theories.
    More interesting question is what is the CFT meaning of the orbits that do not come from the finite subgroups.
    
    
      *  Despite the fact that the Ashkin-Teller correlation functions are known, this case needs better understanding from the methodological point of view.
    At the moment we are able to reproduce only the theories with integer parameter N by taking average over the finite orbit:
    the question is how this procedure has to be modified in the case of arbitrary N.
    
    
      *  As in the Virasoro case, recent efforts have been focused in finding new boostrap solutions in Toda theories <cit.>. Definitely, our construction should work for the N× N case related to W_N algebras with c=N-1, since relation between higher-rank isomonodromic deformations and W-algebras is also known <cit.>. 
    We conjecture that in this case there should be the analog of (<ref>), stating that the product of two connection constants equals to the ratio of 3-point functions.
    As a consequence, one should get four types of c=N-1 theories with W_N symmetry: 1) analog of the Runkel-Watts theory — the limit of W_N unitary minimal models with the fusion rules governed by existence of solutions of equation M_1M_3M_3=1 with fixed conjugacy classes of M_ν; 2) analog of the analytic Liouville theory; 3) analog of the Ashkin-Teller model; 4) analogs of the orbifold CFT's.
    Theory of the first type already appeared, for instance, in <cit.>.
    Some steps towards the study of the orbits of the braid group were already done in <cit.>, where the action of three involutions on the monodromy data is written.
    
    
      *  An approach worth to explore is the construction of  weakly invariant distributions from the study of the (linear) action of the braid group on cohomologies of  along the lines of <cit.> and <cit.>.
    
    
      *   The singularities of the tau functions at σ_0t∈1/2ℤ should be considered more in detail. There might be some construction in which the integral is “concentrated” around these submanifolds: we expect this mechanism to produce some correlation functions in the logarithmic c=1 CFT.
    
    
      *  There are several fundamental questions that arise concerning the general development of our approach. At the moment we are able to give some functions that are crossing invariant almost by construction. To guarantee that such functions are actual correlation functions in CFT, one should be able to represent them in a form of bilinear combinations of conformal blocks with factors decomposed into products of 3-point functions, like (<ref>), (<ref>), (<ref>).
    As a matter of fact, the functions we constructed in this paper turned out to fulfill this requirement.  However it would be very interesting to reformulate this factorization property as some additional property on dμ_θ(P).
    
    There is also the question whether the ansatz (<ref>) is the most general expression that gives crossing invariant functions, or there can be some more complicated “non-local” integrals. 
    Finally, we stress that, in our approach, only diagonal fields are allowed in the external channels (but arbitrary fields in the internal channel).
    So the question is how one can modify this construction, and which properties of the connection constant should be used to allow  general fields in all channels.
    
    If we wish to reformulate the problem of classification of c=1 crossing invariant CFTs as some problem about weakly invariant distributions, all these questions have to be answered.
    
    
    
    
    We hope to return to these problems elsewhere.
    
    
    
    § ACKNOWLEDGEMENTS
    
    
    We are grateful to M. Bershtein, B.  Estienne, N. Iorgov, O. Lisovyy, A. Litvinov, A. Marshakov for their interest to this work and for discussions, and especially to S. Ribault, for careful reading of the text and for many useful comments and questions. We thank Sylvain Ribault for pointing out the issue of the factorization property of our ansatz. We also thank the referee for pointing out issues in the first version of the proof in Section <ref>, see footnote in this Section. We are also grateful to the organizers of the conference “Random Geometry and Physics”, which was held in Paris in 2016, where this work was initiated. We are grateful to the University of Tours, and especially to O. Lisovyy, for his hospitality during the workshop in 2017 where the first results of this work were reported.
    
    The research of PG was carried out within the HSE University Basic Research Program and funded jointly by the Russian Academic Excellence Project '5-100', by RFBR grant mol_a_ved 18-31-20062, and by Young Russian Mathematics award.
    
    
    
    JHEP ]
