Torus shadow formalism and exact global conformal blocks

Using the shadow formalism we find global conformal blocks of torus CFT2. It is shown that n-point torus blocks in the “necklace” channel (a loop with n legs) are expressed in terms of a hypergeometric-type function which we refer to as the necklace function.

h n where The parameters a i and c i are linear functions of conformal dimensions; the variables ρ i are particular combinations of z-coordinates and the modular parameter q; the symbol ∼ means that there is a prefactor called the leg factor which defines the conformal transformation properties of the torus block; the function F N is the necklace function which defines the bare conformal block.The case of 1-point blocks is exceptional.The paper is organized as follows.In Section 2 we describe the shadow formalism: in Section 2.1 we review the shadow formalism in plane CFT 2 ; in Section 2.2, following [4] we demonstrate how to apply this formalism to calculate the 4-point conformal block in the comb channel focusing on all basic features needed further in torus CFT 2 ; the known results for multipoint conformal blocks in the comb channel expressed in terms of the comb function are summarized in Section 2.3.In Section 3 we formulate the shadow formalism for torus CFT: in Section 3.1 we set our notation and conventions for torus CFT 2 and review the n-point torus correlation functions; in Section 3.2 we define torus CFT 2 conformal partial waves.Here, using the (anti)chiral factorization we reduce the whole analysis to the one-dimensional case where the torus is replaced by a circle.The main is Section 4, where we calculate multipoint torus conformal blocks: in Section 4.1 we calculate both 0-point block (character) and 1-point block; Section 4.2 explicitly describes our basic example of the 2-point torus block which is shown to be defined by the forth Appell function; in Section 4.3 we extend this calculation scheme to the case of n points and introduce the necklace function.In Section 5 we discuss our results and possible further developments.Appendix A collects the main formulae for n-point conformal blocks in the comb channel of plane CFT 2 and reviews the comb function [4].Appendix B introduces the necklace function and describes its properties.
In CFT 2 the conformal block factorizes into (anti)holomorphic parts The same formula is valid for the shadow block G h 1 ,...h 4 ; h1 ,..., h4 is obtained from the holomorphic conformal block by ∆ → 1 − ∆.When all points z i ∈ R the holomorphic block can be seen as CFT 1 conformal block.Thus, the full CFT 2 CPW (2.9) can be represented by knowing only one-dimensional conformal blocks.Having this in mind, one can define the one-dimensional CPW [4]: where from now on z i ∈ R.This function is a sum of the conformal block and its shadow, with some coefficients a, b depending on external/intermediate conformal dimensions.In order to distinguish between conformal and shadow blocks one notes that the conformal block G ∆ has a particular asymptotic behaviour in the OPE limit [4] 3 Finally, substituting the 3-point functions (2.7) into the CPW (2.11) one finds the following integral representation for the one-dimensional CPW: 4 (2.15) where h kl ≡ h k − h l . 3In particular, this asymptotics follows from the known exact expression for the 4-point conformal block (2.13) where χ1 = (z12z34)/(z13z24) is the cross-ratio [24]. 4Note that when considering (2.7) in d = 1 one substitutes zij → |zij|.

4-point conformal block
By making a suitable change of the integration variable the integral (2.15) can be reduced to Here, the cross-ratio χ 1 = (z 12 z 34 )/(z 13 z 24 ) < 1 that corresponds to ordering points as The integral in (2.16) has five singular points w = {−∞, 0, χ 1 , 1, +∞} and can be evaluated by splitting the integration domain into four regions as (2.17) where the integrand is given by The integrals (2.17) are found to be: where the coefficients a i , b i depend on conformal dimensions h i , ∆ only, and In any integration domain the integrals (2.19) involve 2 functions I ∆ and I 1−∆ related by ∆ → 1 − ∆, they have different asymptotics at χ 1 → 0 (i.e.z 3 → z 4 ).From (2.12) we conclude that these two functions contribute either to the conformal block G ∆ or to the shadow block G 1−∆ .Assuming that the correct asymptotics is given by (2.14) one concludes that I ∆ contributes to the conformal block G ∆ .In particular, knowing the asymptotics allows relating the coefficients in (2.12) and (2.19) as a = i a i and b = i b i .As noted in [4], in order to find G ∆ there is no need to integrate over the whole R in (2.16).Instead, the above splitting of the integration domain and the formulas (2.19) allows one to single out the conformal block contribution by choosing a suitable integration interval.E.g. choosing the interval (0, χ 1 ) one directly finds the conformal block.On the other hand, choosing the interval (1, +∞) one finds the shadow block which can be converted to the conformal block by ∆ → 1 − ∆.For other two intervals the result is given by a linear combination of the conformal and shadow blocks which can be disentangled by checking their asymptotics.Whatever the integration interval is chosen the 4-point conformal block is calculated to be (2.21) Here, one introduces the leg factor [4]: which guarantees correct conformal transformation properties of the conformal block; the function g h 1 ,h 2 ,h 3 ,h 4 ∆ (χ 1 ) is the bare 4-point conformal block which, by definition, is a conformally invariant function, Note that the bare conformal block in this form differs from the common one (2.13).Their relation can be seen by using the Euler identity Of course, these two representations are equivalent.However, splitting the conformal block into the leg factor and bare conformal block is ambiguous.In general, they are related by a factor of a function of the cross-ratios.The point of making the Euler transform is to choose a particular leg factor that would allow to represent multipoint bare blocks in the simplest form.In this respect, the leg factor (2.22) is preferable than the leg factor in (2.13).

Multipoint conformal blocks
The (n + 2)-point CPW with external dimensions {h j , h 0 , h ∞ , j = 1, ..., n}, and intermediate dimensions where the measure is defined to be This CPW is a linear combination of 2 n−1 terms: a conformal block in the comb channel (see Fig. 2) plus shadow blocks obtained by dualizing the intermediate dimensions .., n in all possible ways.Since the comb diagrams with n + 1 and n + 2 endpoints are related by gluing additionally a 3-point function a remarkable observation was made in [4] that a given (n + 2)-point CPW for n ≥ 3 can be recursively represented in terms of (n + 1)-point CPWs as It is this form of CPWs which finally allows finding closed-form formulas for multipoint conformal blocks.Since the (n + 1)-point CPW in the integrand (the first factor) is a sum of a conformal block and its shadows (for ∆ i , i = 2, ..., n − 1), then one can consider only the conformal block contribution by choosing an appropriate integration domain.This restricts the above recursion relation to conformal blocks only.By a straightforward algebra one finds the (n + 2)-point global conformal block which is expressed in terms of the comb function [4] (see Appendix A). 6

Global torus conformal blocks
Let us begin by briefly reviewing the basic structures from torus CFT 2 (for a more complete discussion see [11]).A two-dimensional torus T 2 can be viewed as a cylinder of the height Im τ , its circumference equals one, the boundaries are identified with a proper twist Re τ .The modulus τ takes values in the fundamental domain ⊂ the upper half-plane H.In local coordinates on the real plane t, t ∈ R 2 the cylinder is then realised as t ∼ t + 1 and t ∼ t + τ .By z = e 2πit the cylinder can be mapped onto the complex plane z, z ∈ C with z ∼ qz, where q = e 2πiτ , q q < 1 is the modular parameter.Using the latter description all calculations in torus CFT 2 can be reduced to those in plane CFT 2 , keeping in mind that the coordinates are subject to the equivalence relation, z ∼ qz .
More practically, this identification can be resolved as a finite domain of z-coordinates, namely,7 q q ≤ z z ≤ 1 .
Its boundaries are identified since 1 ∼ q due to (3.1), see Fig. 3.Note that a coordinate transformation that moves z out of the domain (3.2) must be accompanied by some modular transformation of q which properly changes the inner boundary q q (and vice versa), thereby guaranteeing that we stay on the same torus.
Let us introduce the group-theoretic notation.Global conformal symmetries in CFT 2 are governed by sl(2, C) with commutation relations [L m , L n ] = (m − n)L m+n (plus the same for Lm ), where m, n = 0, 1, −1.The (chiral) Verma module V h is spanned by the basis vectors (at is the Pochhammer symbol.

Torus correlation functions
Torus correlation functions of n primary operators φ i (z i , zi ) of the (anti)holomorphic dimensions h i , hi can be written as Here, we denoted Y (z, z) ≡ φ 1 (z 1 , z1 )... φ n (z n , zn ) and z = {z 1 , ... , z n } are points on C, the trace is taken over the Hilbert space of states where the domain D of admissible dimensions is determined by choosing a particular CFT 2 .
Expanding the trace in (3.3) yields Thus, the n-point torus correlation function is expressed through the (n + 2)-point plane matrix elements.Since L 0 |m, h = (m + h)|m, h , then the torus correlation function (3.5) is a power series in q.At q → 0 (when the torus decompactifies onto a plane) all higher-level matrix elements in (3.5) are suppressed and the torus n-point correlation function becomes equal to the plane (n + 2)-point correlation function (see [11] for a detailed discussion).When considering the torus conformal block as a part of the torus CPW we will treat this behaviour as the asymptotic condition: where F is the n-point torus conformal block, G is the (n + 2)-point plane block and ∆ 1 ∈ D is the torus intermediate dimension coming from the trace (see the next section).The torus geometry breaks the global conformal symmetry of the plane such that the global torus symmetry is only translational, u(1) ⊕ u(1) ⊂ sl(2, C), generated by L 0 , L0 (in the plane parametrization these transformations become dilatations).It constrains the torus correlation functions by implementing the global Ward identities.The holomorphic Ward identity reads as where L 0 is an action of L 0 on primary operators (the superscript means acting on i-th coordinate) along with the same relations for Lm .

Torus shadow formalism
In this section we define a torus CPW within the shadow formalism which will be the first step towards the calculation of torus conformal blocks. 8To this end, consider first 1-point torus functions.The respective CPW can be obtained by inserting the projector (2.4) in the correlation function (3.5) between the left states and Y (z, z) = φ(z 1 , z1 ): Representing the primary operator in any point as we find that it creates from the vacuum an infinite linear combination of descendant states where we used the conformal invariance of the vacuum state, L m |0 = 0 and Lm |0 = 0.Then, (3.9) can be rewritten as (to keep things simple we suppress z-dependence for a while) Here, we used h, m|n, ∆ = δ h,∆ (B ∆ ) mn = δ h,∆ δ m,n m!(2∆) m and then by resolving the Kronecker symbols we singled out the terms with h = ∆ and m = n; after that, using (3.11) we summed up the series for O.One can go further and expand q = e 2πiτ in τ : where we again used (3.11) and L 0 |n, ∆ = (∆ + n)|n, ∆ .Now, substituting this relation into (3.12)one obtains an expression which will be considered as a definition of the 1-point torus CPW: with the 3-point functions V h i ,h j ,h k given by (2.7).
In order to define an n-point torus CPW one follows essentially the same procedure with one more step of inserting additional projectors (2.4) between each pair of primary operators in Y (z, z): Here, the antiholomorphic measure is obtained from (2.26) by substituting w → w, we denoted ∆ = {∆ 1 , . . .∆ n }, h = {h 1 , . . ., h n }, and introduced identifications ∆ n+1 ≡ ∆ 1 and w n+1 ≡ qw 1 that guarantees a non-trivial q-dependence of the RHS, cf.(3.14) (+ the same in the antiholomorphic sector).In torus CFT 2 , the procedure of inserting projectors and/or using OPEs leads to topologically inequivalent channels. 9A channel obtained by inserting the projectors only (as in (3.15)) is called the necklace channel.In the rest of the paper we consider torus conformal blocks only in the necklace channel.
Having in mind the (anti-)chiral factorization of CFT 2 torus conformal blocks, it is convenient to introduce a chiral CPW: where, as before, ∆ n+1 ≡ ∆ 1 and w n+1 ≡ qw 1 .From (2.25) one derives an equivalent form From now on we consider the coordinates and the modular parameter to be real, z i ∈ R, 0 < q < 1.For real coordinates the identification (3.1) keeps its form, z ∼ qz, which can be now resolved as Similarly to (3.2) the boundary points here are identified as 1 ∼ q that results in z ∈ S 1 .Indeed, the real modular parameter q (i.e.Re τ = 0) corresponds to a rectangular torus T 2 = S 1 × S 1 where the first and second circles have radii 1 and Im τ , respectively.Then, using the cylinder map z = exp(2πi Re t − 2π Im t) and imposing the reality condition on z one finds that the periodic coordinate identification t ∼ t + 1 is not valid anymore since Re t = 0.It follows that the first circle in T 2 = S 1 × S 1 shrinks to a point and one is left with the second circle of radius Im τ realized as (3.19).This justifies a somewhat sloppy term "a CFT 1 torus CPW" which will be used in the sequel to designate one-dimensional CPW on a circle.
Note that considering the global transformations u(1) one can show that the CFT 1 torus CPW (3.17) has a scaling dimension h 1 + . . .+ h n .It is obvious since the torus CPW is obtained from the torus correlation function (3.3) upon inserting dimensionless projectors (2.4).As always, the global invariance can be used to fix one of z i .

n-point torus CPW
As discussed in Section 2, the plane CPW Ψ can be represented as a sum of the conformal block and its shadows.Since the torus CPW Υ is expressed through the plane CPWs Ψ by the means of (3.16) one concludes that Υ can also be represented as a sum of the conformal block and its shadows.In what follows, we explicitly calculate 1-point and 2-point conformal blocks thereby showing in detail the procedure of extracting conformal blocks from CPWs.Then, we generalize this consideration to the n-point case.

1-point torus block
Let us consider first the 1-point torus CPW (3.17)10 By substituting 3-point function (2.7) and changing the integration variable w → zw this expression can be cast into the form The integral here is similar to that one which defines the 4-point CPW in planar CFT 2 (2.16).Likewise, the integral is singular, as the integrand has five singular points w = {−∞, 0, 1, q −1 , +∞}.It can be calculated by splitting the integration domain into subregions whose boundaries are the singular points, cf.(2.17).The integrals in each subregion are given by linear combinations of two hypergeometric functions 2 F 1 that can be seen from Indeed, the global transformation w → λw changes singularities of the integrand as w = (0, z/q, z) → (0, z/λq, z/λ).As a result, Υ h ∆ (q, z) = λ −h Υ h ∆ (q, z/λ).
(2.19).Thus, the 1-point torus CPW (4.2) is a sum of two terms related by ∆ → 1 − ∆.The term satisfying the required asymptotic condition (3.6) is the 1-point torus block given by where is the torus 1-point leg factor and is the global 1-point torus conformal block [5].Note that the hypergeometric series in (4.5) converges in the prescribed domain q ∈ (0, 1).The torus leg factor (4.4) depends only on external dimension h and provides the covariance of the conformal block (4.3) under conformal transformations.It has the same dimension as the torus 1-point function.
At h = 0 the 1-point torus block (4.1) becomes equal to F h=0 ∆ (q, z) = q ∆ /(1 − q) which is the sl(2, R) character of the Verma module V ∆ .The same can be directly obtained from evaluating the 0-point CFT 2 torus CPW11 Using the 2-point function for the equivalent representations (2.2) and the standard formula for the δ-function (see the footnote 2) one obtains which is the sl(2, C) character of the Verma module V ∆ ⊗ V ∆.Thus, we conclude that the definition of the torus CPW (3.17)-(3.18)works well for the known examples.

2-point torus block
Just as in the case of the plane blocks, we will only focus on the functional dependence and we will not be interested in the overall coefficients depending on external/intermediate dimensions.
For n = 2 the CFT 1 torus CPW (3.18) reads as Here, the integral is given by the sum of 4 terms: the torus conformal block plus its shadows obtained by The plane 4-point CPW Ψ in the integrand of (4.8) is a sum of the plane conformal block and its shadow (2.12).To avoid shadow block terms we consider only the 4-point conformal block contribution to the plane CPW12 where the 4-point plane block is given by (2.21) (expanded as the hypergeometric series) with the leg factor (2.22) Thus, we have the 2-point torus CPW in the form The integral in the second line is recognized as the 4-point CPW (2.15): As in plane CFT 2 there is no need to integrate over all R to find the 2-point necklace block.By choosing an appropriate integration domain in . In the present case, the integration domain is split into four intervals by five singular points {−∞, 0, ρ 1 , 1, +∞}, where is just the plane cross-ratio χ 1 evaluated with respect to four arguments of Ψ ∆ 2 +m,h 2 ,h 1 ,∆ 2 +m ∆ 1 in (4.13).Note that x 1 is the u(1) invariant.Then, choosing the second interval (0, ρ 1 ) one is left with13 where the leg factor which is (2.22) in the new points is given by ) and the bare block (2.23) expanded into a series reads Using the identities Γ(1 − s)Γ(s) = π/ sin(πs) and (a) m = Γ(a + m)/Γ(a) the prefactor in (4.15) can be written as . (4.18) Thus, substituting (4.18) and (4.17) back into (4.15)along with using (a+m) n = (a) m+n /(a) m yields where F is the 2-point torus conformal block in the necklace channel with the 2-point torus leg factor and the 2-point bare torus block where F 4 is the fourth Appell function (B.3) and where x 1 is defined in (4.14).Note that ρ 2 arises when one substitutes (4.19) into (4.13) and then assembles all contributions in z and q of m-power into ρ ∆ 2 +m 2 that defines a second argument in the double power series.The quantities ρ 1 ∈ (0, 1) (4.14) and ρ 2 ∈ (0, 1) (4.24) will be referred to as the torus cross-ratios.The Appell series F 4 converges when . This inequality is satisfied in the prescribed domain of the modular parameter q ∈ (0, 1) and external points x 1 ∈ (q, 1) (see Section 3).Two comments are in order.First, the small-q asymptotics of (4.21) is given by where is a particular 4-point plane block in the comb channel (2.21) that agrees with the requirement (3.6).Second, one can explicitly check that the 2-point torus block solves the torus Casimir equations [11] (see Appendix B.2).
Torus cross-ratios.These particular combinations of z i and q arise naturally when calculating conformal blocks within the torus shadow formalism.We saw that the torus crossratios provide a convenient parametrization which allows representing the conformal block in a closed-form.For the bare block we have ) that is the original set of variables x 1 = z 2 /z 1 and q is replaced by ρ 1 (x 1 , q) and ρ 2 (x 1 , q), where the z-dependence is now spread out over two variables in globally-invariant way.The standard approaches, e.g. by representing the torus block as a power series in q and calculating the expansion coefficients being the matrix elements of primary operators, leave no chance to see that z i and q can be organized into the rational functions because that will require multiple non-trivial resummations.
Although the geometrical meaning of the torus cross-ratios is yet to be understood, here we notice that they are invariant under the following transformations (∀λ ∈ R): A few comments are in order.First, in terms of the modulus τ the transformations of q in (4.26) and (4.27) correspond to τ → ±τ which is the identical modular transformation of P SL(2, Z).Second, in terms of the u(1)-invariant ratio x 1 = z 2 /z 1 the transformation (4.26) trivializes, i.e. x 1 → x 1 and q → q.In particular, the torus cross-ratios are u(1) invariants.Third, the transformation (4.26) with the parameter λ = q is actually the equivalence relation (3.1), while the transformation (4.27) is an inversion of z i which, therefore, requires a proper modular transformation of q that aims to map the domain of z i back to (3.19) (or, in two dimensions, to (3.2)).
Reduction to the 1-point torus block.Setting h 2 = 0 and ∆ 1 = ∆ 2 one expects that the 2-point necklace block (4.21) is reduced to the 1-point torus block (4.3).Indeed, the 2-point torus leg factor (4.22) takes form where Λ h 1 (q, z 1 ) is the 1-point torus leg factor (4.4), while the 2-point bare necklace block (4.23) is given by where we used the following identity for the fourth Appell function [30] (4.30) Thus, as expected, one has Note that z 2 was kept arbitrary, nevertheless, it drops out identically once the constraints h 2 = 0, ∆ 2 = ∆ 1 are imposed.

n-point torus block
Recall that the n-point plane CPW Ψ is recursively expressed through the (n − 1)-point plane CPW (2.27).However, one can see that the n-point torus CPW Υ is not expressed through the (n − 1)-point torus CPW, but, instead, through the (n + 2)-point plane CPW Ψ.On the other hand, the recursion relation solves the (n + 2)-point plane CPW in terms of the 4-point plane CPW which implies that the n-point torus CPW Υ for any n can be represented as a linear combination of the 4-point plane CPWs.In fact, this phenomenon is seen already in the 2-point case (4.13) and, therefore, the calculation of the n-point necklace block follows essentially the same steps.
Let us consider the (n + 2)-point plane conformal block contribution to the n-point torus CPW (3.18) where the conformal block is represented as a product of the leg factor and the bare conformal block (A.1), (w, z 1 , ... , z n , qw) = L 1−∆ 1 ,h 1 ,...,hn,∆ 1 (w, z 1 , ... , z n , qw) with the leg factor (A.2), and cross-ratios are read off from (A.4) by setting z ∞ = w and z 0 = qw: (4.37) Plugging the leg factor (4.34) and the bare block (4.35) into (4.32)one can single out the 4-point CPW (2.15), namely, As in the 2-point case, the respective integral can be taken by splitting the integration domain by five singular points as {−∞, 0, ρ 1 , 1, +∞}, where which is the plane cross-ratio χ 1 evaluated for the four points which are the arguments of Ψ ∆n+mn,hn,h 1 ,∆ 2 +m 2 ∆ 1 in (4.38).Choosing the interval (0, ρ 1 ) one gets with the leg factor (2.22) .41) and the bare block (2.23) expanded as Substituting this expression into (4.38)one singles out the necklace torus block contribution to the torus CPW, Υ h ∆ (q, z) ⊃ F h ∆ (q, z) .(4.44) Here, the n-point block is represented as a product of the torus leg factor and the bare necklace block where the leg factor is defined to be and the bare block is given by where we introduced a new special function F N which we call the n-point necklace function (B.1).Here, ρ ≡ {ρ 1 , ρ 2 , ... , ρ n } are the torus cross-ratios: Similarly to the 2-point case the torus cross-ratios ρ 2 , ρ n arise when plugging (4.43) into (4.38) as combinations of z i and q packed into ρ m 2 +∆ 2 2 ρ mn+∆n n .This is in contrast to ρ 1 and ρ i for 3 ≤ i ≤ n − 1 which are the plane cross-ratios coming as arguments of the plane conformal blocks.Moreover, ρ i here coincide with the plane cross-ratios χ i−2 , cf. (A.4).Since the total degree in z-coordinates of any ρ i is zero, then one can rewrite them as functions of u(1)-invariant ratios x i ≡ z i+1 /z i (e.g.see (4.14) and (4.24) in the n = 2 case).In this way, the bare block is manifestly u(1)-invariant.
Also note that to reproduce the n = 2 torus block (4.21) one identifies z 3 ≡ qz 1 and z 0 ≡ z 2 /q.The case of 1-point torus blocks falls out the general formula (4.45) because the necklace function at n = 1 can not be reduced to a hypergeometric function 2 F 1 with the required arguments. 14Thus, the necklace function defines the torus conformal blocks only when n = 2, 3, ... .In this respect, the situation is similar to that with the plane conformal blocks where the comb function works properly only when n = 4, 5, ... .In both cases (plane and torus) one can see that the respective cross-ratios can be built only having a minimal number of points not completely fixed by a global conformal symmetry which equals two in the torus case and four in the plane case.

Conclusion
In this paper we have found exact functions of the global torus CFT 1 n-point conformal blocks in the necklace channel.Up to the leg factor these blocks are given by the necklace functions which are hypergeometric-type functions of n torus cross-ratios defined as particular combinations of the locations of the insertion points of primary operators as well as the modular parameter.To this end, we have elaborated the torus shadow formalism which is a version of the standard shadow formalism.Given the (anti-)holomorphic factorization of the CFT 2 torus blocks, the necklace functions also define the CFT 2 global necklace blocks. 14Despite having the relation (B.2) there is no such a transformation that brings the n = 1 necklace function to the 1-point block function.Nonetheless, we saw that 1-point blocks are perfectly well calculated within the torus shadow formalism and, in particular, they can be obtained from 2-point torus block by imposing extra constraints on ∆i , hi, see the end of Section 4.2.
Our results can be naturally extended in a few directions.First of all, we note that both the comb functions in plane CFT 2 and the necklace functions in torus CFT 2 belong to the same family of the hypergeometric-type series.It is tempting to speculate that considering CFT 2 on higher genus-g Riemann surfaces the conformal block functions in the channels which generalize the comb and necklace channels are members of the same hypergeometrictype family of functions.The second natural direction is to use the elaborated machinery to calculate conformal blocks in thermal CFT d and their asymptotics, see e.g.[16,[31][32][33].Third, up to a truncated Virasoro character the global torus blocks are the large-c asymptotics of the Virasoro torus blocks (the so-called light blocks) [6,9].Then, the Zamolodchikov's c-recursion in torus CFT 2 should be rephrased in terms of the torus cross-ratios which proved to be an efficient parameterization of torus blocks.Finally, it would be interesting to find an integral representation for the necklace function along with various Kummer-type and Pfaff-type transformations that could be useful for applications.
where the cross-ratios are , 1 ≤ i ≤ n − 3 , (A.4) and F K is the comb function [4]  where n = 1, 2, ... .At n = 1 it is given by which is the hypergeometric function.At n = 2 it is given by At n = 2 the equation system (B.9)-(B.11)can be cast into the form 12) which is a system of PDEs on the fourth Appell function F 4 (B.3)[30].Its general solution is given by [30] where C i , i = 1, 2, 3, 4 are integration constants.The first line defines the necklace block (4.23), while the others define the shadow block (see below).
Connection with the torus Casimir equations.When considering the torus conformal blocks the system (B.9)-(B.11)has to be regarded as the torus Casimir equations.For