Braiding properties of the N=1 super-conformal blocks (Ramond sector)

Using a super scalar field representation of the chiral vertex operators we develop a general method of calculating braiding matrices for all types of N=1 super-conformal 4-point blocks involving Ramond external weights. We give explicit analytic formulae in a number of cases.


Introduction
One of the fundamental principles of the 2-dimensional conformal field theory (CFT) is the convergence of the operator product expansion [1]. It in particular implies that any 4-point function factorizes in three different ways corresponding to the scattering channels s, t, u. Equivalence of these decompositions is one of the basic consistency condition of the theory usually referred to as the crossing symmetry or the bootstrap equation.
Using the factorization and the conformal properties of 3-point functions one can express any 4-point correlator in terms of structure constants and holomorphic and antiholomorphic 4-point conformal blocks [1]. In the rational CFT the crossing symmetry implies monodromy relations between conformal blocks in different channels [2,3]. Monodromy matrices between s−t, and t−u channels, are called the fusion and the braiding matrices, respectively. As the spectrum of a rational CFT is finite they are finite-dimensional.
A well known example of the CFT with a continuous spectrum is the Liouville theory. In this case the fusion and braiding matrices were first calculated by Ponsot and Teschner using representations of U q (sl(2, R)) [4,5]. These matrices can be also obtained calculating the exchange relations for the chiral operators in the scalar field representation [6,7]. The explicite form of the integral kernels of the fusion and the braiding matrices was used in the analytic proof that the Liouville structure constants [8,9] satisfy the bootstrap equation [4]. Although derived in the context of the Liouville field theory the fusion and braiding integral kernels are universal for all CFT. For instance for the central charge and the spectrum of a minimal model the continuous integral over intermediate weights reduces to a finite sum by the "contour pinching" mechanizm [10,11].
The structure of conformal blocks in the N = 1 superconformal theory is considerably more complicated. It has been recently analyzed in the context of recursion relations in a number of papers [12,13,14,15,16,17,18,19]. The form of the fusion matrix for the Neveu-Schwarz (NS) superconformal blocks was first proposed in [20] on the basis of the properties of super-symmetric extensions of b-hypergeometric functions. This result has been confirmed in [21] where the fusion matrix was derived from the exchange relations of the chiral operators in the super scalar free field representation. As in the case of the bosonic Liouville theory the explicit form of the fusion matrix was used to check the bootstrap equation in the NS sector of the N = 1 super-symmetric Liouville theory with the structure constants proposed in [22,23].
The main aim of the present paper is to derive the integral kernels of the braiding matrices in the Ramond (R) sector of N = 1 SCFT by calculating the exchange relation for chiral vertex operators in the free super field representation. The extension of the theory by the Ramond sector leads to four types of chiral vertex operators. Their different compositions correspond to different 4-point blocks of the Neveu-Schwarz and Ramond fields [19]. We derive all technical ingredients necessary for calculating the braiding matrices for all types of N = 1 superconformal blocks involving external Ramond weights. Due to a proliferation of types of superconformal blocks we present detailed calculations only in a few cases. They were chosen to illustrate all the technicalities involved. The methods developed are general and can be easily applied to all other blocks.
Our first motivation was to complete the proof of the bootstrap equation in both sectors of the (GSO projected) N = 1 super Liouville theory. The matrices we are to calculate are however universal and can be used to check the bootstrap equation and to calculate 4-point correlation functions in any N = 1 SCFT. The second interesting problem is to find out if there is a supersymmetric counterpart of the relation between the fusion and the modular matrices recently found in the standard CFT [24]. Due to the technical complexity both problems are postponed to subsequent papers.
The paper is organized as follows. Following [12,18,19] we define in Section 2 the Neveu-Schwarz (NS) and the Ramond (R) chiral vertex operators and analyze some of their properties. In Section 3 the construction of chiral superscalar field space representation of the NS and the R vertex operators is described. In Subsection 3.1 we derive braiding relations for the chiral fermion fields. Our method is to decompose the chiral fermion Fock space into Virasoro Verma modules and then use known braiding properties of the Ising model chiral vertices. In Subsection 3.2 we introduce the chiral fields and clarify their relation to the chiral vertex operators of Section 2. In Subsection 3.3 the matrix elements of the chiral fields necessary for their normalization are calculated.
In Section 4 we calculate the braiding matrices. In Subsection 4.1 we derive the braiding kernel for the compositions of ordered exponentials and screening charges. This result along with the results of Section 3 are used in Subsection 4.2 to calculated the braiding kernel in several cases including pairs of NS-NS, R-NS and R-R chiral vertex operators.
The paper is supplemented by a number of appendixes. Appendix A collects the properties of the chiral vertex operators of the Ising model we need in our construction of the chiral fermion fields. In Appendix B we derive in some specific case the Ward identity for the fermionic current S in the presence of Ramond fields. Appendix C contains some relevant properties of the Barnes double gamma function. In Appendix D we derive the orthogonality relations we use in Subsection 4.1.
The paper is rather technical and some remarks concerning conventions and notations can be helpful. Let us first emphasize that the choice of chiral vertex operators in the Ramond sector is determined by the fact that the full theory is based on "small representations", i.e. irreducible representations of the left and the right N = 1 Ramond algebras extended only by the common parity operator [18,19].
We shall adopt the symmetric form of the OPE of the fermionic current with the Ramond fields [26]: and the standard normalization of the two-point function Formulae (1.1), (1.2) determine the braiding relations of the fermionic current with the Ramond fields up to a sign. This is in order related to the normalization of the two-point function of the R − β fields as can be easily verified analyzing analytic properties of the three-point functions 1 S(z)R ± β 2 (w 2 ,w 2 )R ∓ β 1 (w 1 ,w 1 ) .
1 See Appendix B for a similar analysis of chiral correlators.
In the present paper we chose (1. 3) The opposite convention is also possible. It is used for instance in [18,19]. For the chiral Ramond fields we assume: (1.6) Choosing the principal argument of a complex number ξ in the range −π ≤ Arg ξ < π one can write (1.6) as . (1.8) Braiding properties (1.6), (1.7) are crucial for most of the calculations in the present paper. Our notation for the chiral vertex operators and conformal blocks is organized as follows. In the NS sector the chiral vertex operators are denoted by where f = e, o is the parity index and the weights in the square brackets denote: ∆ 2 -the weight of the vertex itself, ∆ 1 -the weight of the source and ∆ 3 -the weight of the target NS Verma module. The "star" vertices are defined by In the other three sectors the rules are similar but the vertices acquire an additional ± index: This is related to the structure of Ward identities in these sectors. In contrast to the NS sector the 3-point conformal blocks are determined up to four rather then two structure constants. The ± values of the additional index correspond the choice of a basis of 3-point blocks required by the "small" representation mentioned above. The conformal weights of the Ramond modules are denoted by parameter β which emphasizes the sign dependence but also encodes information about the sectors. The notation of vertices is consistent with the notation of conformal blocks introduced in [18,19]. 3 One has for instance Let us note that the ± in front of β-s in the symbol of a conformal block is related to the ± index of the corresponding vertex operator rather then an actual sign of this parameter.
(When there are two β-s in a column we write the signs in front of the upper one.) According to these notational rules all braiding relations for the chiral vertex operators can be easily translated into analytic continuation formulae for corresponding 4-point blocks.
Although very economic for denoting vertices and blocks the ∆, β notation is not well suited for the analytic expressions for the braiding matrices. For this purposes we use in both sectors the α parametrization of conformal weights: The relation α to β in the Ramond sector is straightforward α = Q 2 − √ 2β.

Chiral vertex operators
The relations of N = 1 superconformal algebra extended by the fermion parity operator 3 The blocks themselves are different as in the present paper our conventions for braiding and hence the Ward identities are different.
where m, n ∈ Z and k, l ∈ Z + 1 2 in the Neveu-Schwarz algebra sector and k, l ∈ Z in the Ramond algebra one.
The NS supermodule V ∆ of the highest weight ∆ and the central charge c is defined as a free vector space generated by all vectors of the form where K = {k 1 , k 2 , . . . , k i } and M = {m 1 , m 2 , . . . , m j } are arbitrary ordered sets of indices and ν ∆ is the highest weight state with respect to the extended NS algebra: In the Ramond sector the highest weight state is defined in a similar way A novel property is that the zero level subspace of the R supermodule Hermitian forms . , . c,∆ on V ∆ and . , . c,β on W β are uniquely determined by the relations They are block-diagonal with respect to the L 0 -and (−1) F -gradings. Following [19] we introduce 3-point blocks as chiral 3-forms (anti-linear in the left argument and linear in the central and the right ones): satisfying the "bosonic" (with respect to L n ) and the "fermionic" (with respect to S k ) Ward identities. The "bosonic" identities are the same for 3-point blocks of all types. We shall not use them in the present discussion (see [19] for their explicit form).
The "fermionic" Ward identities for the NN type of 3-point block take the form [12]: The 3-form ̺ RR (η 3 , ξ 2 , η 1 |z) is determined up to four rather then two constants: As before the parity of the total number of fermionic excitations is preserved and therefore Using Ward identities (2.9) one can derive the relations for the even number of fermionic operators #M + #N ∈ 2N and for #M + #N ∈ 2N + 1. One also has (2.12) Identical relations hold for ν replaced by * ν.
An appropriate basis for the 3-point blocks takes the form: The corresponding chiral vertex operators are given by RR,f (η 3 , * ν, η 1 |z).
For states with definite parities some forms identically vanish: The decomposition of the 4-point functions of Ramond fields into conformal blocks suggests the following convenient choice of a basis of the 3-point blocks [18,19] (2.20) The chiral vertex operators are then defined by their matrix elements as follows (2.21)

Chiral fermion
In the NS sector the chiral fermion field decomposes into half-integer modes: The algebra of modes is realized in the Fock space F NS generated out of the vacuum |Ω F satisfying ψ r |Ω F = 0, r > 0, In the R sector ψ(z) has the integer mode decomposition: and the vacuum state of the Fock space F R is doubly degenerated Both Fock spaces carry the c = 1 2 Virasoro algebra representation where k ∈ Z + 1 2 in the NS sector and k ∈ Z in the R sector. Each Fock space can be decomposed into Virasoro Verma modules which leads to the decomposition of the total Hilbert space of the chiral fermion into Virasoro Verma modules with conformal weights 0, 1 2 , 1 16 , 1 16 : In this decomposition U 0 ⊕ U + 1 16 and U 1 2 ⊕ U − 1 16 are the positive and the negative eigenspaces of the partity operator (−1) F , respectively.
We introduce new chiral fields σ ± (z) satisfying: The operators σ ± are uniquely determined by the relations above. One could in principle calculate them in terms of modes. It is however more convenient to use decomposition (3.3) and to express all the fields in terms of the (Virasoro) chiral vertex operators: where we have applied the standard notation 1, ε, σ for the Ising chiral fields with the weights 0, 1 2 , 1 16 , respectively. The representation above can be easily verified using well known properties of the Ising chiral vertices summarized in Appendix A.
In a similar way one can calculate the braiding relation we shall need in the following: It follows from (3.5) that for even ξ ∈ V and for odd ξ ∈ V. One also has It is probably worth to mention that in the chiral fermion theory described above the algebra of chiral fields 1, ψ, σ + , σ − does not close with respect to OPE. For instance the OPE of σ + with itself contains a new local operator which is neither a conformal primary nor a descendent of a primary field. In this respect the chiral fermion with a partity operator in both sectors is not a complete chiral CFT. In consequence the corresponding superscalar model we shall describe in the next subsection is not a complete chiral N = 1 superconformal field theory. Nevertheless, it provides an appropriate representation of the chiral vertex operators in the N = 1 superconformal theory.

Chiral fields
The chiral boson field with periodic boundary can be defined in terms of the decomposition where the modes satisfy They are realized on the Hilbert space where F B is the Fock space with the vacuum state |Ω B defined by the conditions a n |Ω B = 0, n > 0, Ω B |Ω B = 1. The superscalar Hilbert space is defined as a tensor product and carries the representation of the N = 1 superconformal algebra with the central charge c = 3 2 + 3Q 2 : where k, l ∈ Z + 1 2 in the NS sector and k, l ∈ Z in the R sector. The superscalar Hilbert space H can be seen as a direct integral over the specturm of the operator p: and p|p = p|p . The representation (3.11) defines on H p NS the structure of the NS Verma module V ∆ with the conformal weight ∆ = 1 8 Q 2 + 1 2 p 2 , and the structure of the R Verma module W ∆ with the conformal weight ∆ = 1 8 Q 2 + 1 16 + 1 2 p 2 on H p R . For our purposes it is convenient to work with the boson and the fermion fields on the unit circle, transformed back to the zero time slice of the infinite cylinder: In terms of these fields we construct the ordered exponential where k ∈ Z + 1 2 in the NS sector and k ∈ Z in the R sector. For real b the screening charge Q(σ) is hermitian, its square is positive and Q(σ) 2 t may be uniquely defined for complex t. Following [21] we define the "even" and the "odd" complex powers of the screening charge 13) and the compositions (3.14) The chiral NS fields are diagonal with respect to the sector decomposition , The chiral R fields are off-diagonal: The fields can be extended to Euclidean fields on the whole cylinder by the analytic continuation to the imaginary time (3.17) They are related to the fields on the complex plane z = e w via The chiral fields satisfy a simple braiding relation with functions of p: An important feature of the fields V α f s (z), V * α f s (z) is that for the conformal weights It thus follows from definition (2.8) that one can use V α f s (z) and V * α f s (z) to represent the chiral vertex operators in the NS sector: A similar property holds in the R sector. For the conformal weights (3.20) Chiral vertices (2.13) can then be represented as follows and (3.22) One can repeat the above considerations for the R chiral fields. For the conformal weights In a similar way, for the conformal weights (3.24)

Matrix elements
In this subsection we shall calculate the matrix elements using a suitable modification of the procedure proposed in [7] and adapted to the NS sector of the N = 1 superconformal theory in [21]. The idea is to find an explicit form of an appropriate four-point, chiral correlator containing a degenerate Ramond field and then, by studying its different limits, to express the matrix elements we are after through the matrix elements of the chiral NS field computed in [21]. For β = β + ≡ 1 2 √ 2 b −1 + 2b the Ramond supermodule W β + is degenerate (with respect to scalar product (2.6)). The vector is orthogonal to all vectors in W β + . Using (2.5) we can rewrite the condition that χ + is null in the form of a pair of operator equations which should hold in arbitrary correlation function. Introducing we get in particular For the new function h(z): we obtain from (3.27) where A = bα 3 − 1 2 b 2 . The solution of (3.28) corresponding to the sum of correlators f + (z) + f − (z) can be singled out by its leading behavior at z → 0. It follows from the momentum conservation that all the states obtained by the action of W ± β + f (z) on the vector ν 1 have momenta equal to The small z behavior of f ± (z) can thus be calculated by inserting a projection on the highest weight states of W q with an appropriately chosen parity. This gives where we have used the identities It follows from (3.20) that the asymptotics in (3.30) are identical, hence where 2 F 1 denotes the hypergeometric function.
In a similar way one can deduce from (3.27) the differential equation satisfied by the We shall now determine ν 4 | W + β 3 e s 3 (1)|w + q by comparing the leading behavior of the left and the right hand side of (3.32) for z → 1. We start by calculating the leading term of we have for z 3 → z 2 : where the equality follows form the definition of the screening charge Q, its even power (3.13) and braiding property (3.4). Further, from the definition of the screening charge and braiding properties of the normal ordered exponentials (see the next section for details) one has, Finally, from (3.5) and (3.12): and we get .
In conclusion, for z → 1 : where the matrix element on the r.h.s. was calculated in [21] and reads: On the other hand, we can analyze the z → 1 behavior of the r.h.s. of (3.32). Using the analytic continuation formula for the hypergeometric function, we obtain a coefficient in front of (1 − z) Using (3.35) and the "shift identities" for the Barnes functions (see Appendix C) we finally get where the first relation of (3.23) has been added for completeness.
The same procedure can be applied to other matrix elements. For instance, in order to compute give the formulae for w + 3 | W + β 2 e s (1) |ν 1 and w − 3 | W + β 2 o s (1) |ν 1 , respectively, and the matrix elements w + 3 | V α 2 e s (1)|w + 1 and w − 3 | V α 2 o s (1)|w + 1 can be obtained from the correlators The result reads Matrix elements (3.35), (3.40), (3.42), supplemented by the matrix element calculated in [21] form a complete set of eight independent normalizations required for representation of all chiral vertices in both sectors.

Braiding of normal ordered exponentials and screening charges
In this subsection we shall calculate the braiding matrix for operators (3.14). We follow the procedure proposed in [7] and extended to the NS sector of the N=1 superconformal theory in [21]. Let us assume the existence of a braiding relation of the form The sum over the indices g is restricted by the parity conservation which holds in the chiral superscalar model and the integration measure dµ( t) is proportional (by the momentum conservation) to the Dirac delta δ(t 2 + t 3 − s 2 − s 3 ). The additional subscript ♮ = R, NS denotes the sector in which relation (4.1) is considered.
If the product of fields in (4.3) act on the NS state, the fermion field ψ(x) appearing in the definition of the screening charges is anti-periodic, ψ(x + 2π) = −ψ(x), and On the other hand, while acting on the Ramond state the fermion field is periodic, ψ(x + 2π) = ψ(x). In this case Q ′ I = e iπb 2 e 2πbp Q I . It follows from the definition of the screening charges and from braiding relation (4.2) that the operators Q I , Q c I and e πbp satisfy the Weyl-type algebra form a representation of (4.5). In this representatioñ Representation (4.7) allows to transform the powers of sums ofQ−s into the normal ordered form, with x operator to the left of p and t operators. Using the shift property of the Barnes functions: and one has e bx e − 1 2 iπbt + i e Introducing a matrix notation one can present the result of the calculations above in the compact form In a similar way one obtains Defining an even and an odd complex power of τ 1 we get: Evaluated on a common eigenstate of operators p and t, p |p, τ = p |p, τ , t |p, τ = τ |p, τ , τ ∈ iR, the right hand sides of formulae (4.8) take the form of analytic functions of p and τ, multiplied by the operator e (s 1 +s 2 )x = e (t 1 +t 2 )x .
Stripping off this factor from both sides of formula (4.4) one gets a relation between analytic functions of p and τ. In terms of the Barnes S functions 5 this relation takes the form and The Barnes functions in the integrand of (4.9) depend on the integration variable t 3 only via the parameter p u . Multiplying both sides of equation (4.9) by S NS (from the right) and choosing the integration measure to be dµ(t 3 ) = θ(p u ) dp u one gets In order to calculate the braiding kernel from (4.11) one can then use the orthogonality relation 6 : where This yields For further applications it is convenient to regard the braiding kernel as a function of new variables: In order to make it more readable we introduce the notation Repeating the steps above for ǫ = sgn(σ 3 − σ 2 ) < 0 one gets where The explicit form of the braiding matrix B can be read off directly from formula (4.14). For instance The other cases differ from the expressions above only by constant phases and NS/R indices of the Barnes functions.

Braiding of chiral vertex operators
In this subsection we shall derive the braiding properties of the chiral vertex operators. Rather than presenting the general formula (which would be quite clumsy due to a plethora of indices) we will discuss several examples choosing vertex operators from different sectors. All other cases can be easily obtained in a similar way.

NS-NS braiding
The braiding properties of the Neveu-Schwarz vertex operators in the NS sector were already discussed in [21]. In this subsection we shall calculate two examples of braiding matrices for the NS operators in the Ramond sector. Let us first consider the composition which in representation (3.21) takes the form The notation R indicates that the product of chiral fields V α 3 e s 3 (z 3 )V α 2 e s 2 (z 2 ) acts on the states from the Ramond sector. It follows from (3.15) that for this product one can apply braiding relation (4.1) derived in the previous subsection. In notation (4.16) we get Using (3.21) one can express the r.h.s. in terms of chiral vertex operators Thus, if we define the braiding matrix by the relation  The braiding of all other pairs of the NS chiral vertex operators can be calculated in an essentially the same way. For instance, for the composition the relevant braiding relation (4.1) reads As before, one can express the r.h.s. of (4.23) in terms of chiral vertex operators: Equations (4.23), (4.24) and (4.25) suggest the following definition of the braiding matrix Then the results above yield (4.26)

NS-R braiding
As an example of the braiding between the NS and the R fields consider the composition which can be represented, (3.21), (3.24), as: By definitions (3.15), (3.16) Then using braiding relation (4.1) and notation (4.16) we obtain By formulae (3.19) and (3.24) the r.h.s. can be expressed in terms of chiral vertex operators This leads to the braiding relation (4.27)

R-R braiding
We shall start from the composition of Ramond vertex operators in the NS sector In representation (3.21): From definition (3.16) and braiding properties (3.4), (3.6) one gets It the present case the braiding relation of g fields we need reads e e g α 2 g 2 t 2 (σ 2 )g α 3 g 3 t 3 (σ 3 ) NS .
Using braiding relations (3.4) and representation (3.21) one gets: We are thus lead to following form of the braiding relation As our last example we present the result for a composition of two Ramond vertex operators in the Ramond sector,

B. Conformal Ward Identities for the fermionic current S(ξ)
Correlation functions of the fermionic current S(ξ) in the presence of the Ramond fields are no longer single-valued and derivation of their form, even if standard, is subtle. As as example we shall discuss the correlator Due to OPE (1.1) it has square root branch cuts at ξ = 1 and ξ = z. The function should then be single-valued and analytic on the complex ξ plane save the simple poles at ξ = 0, z, 1. With the principal argument of a complex number ξ in the range −π ≤ Arg ξ < π one has: 1 − ξ = e − iπ 2 sgn Arg(ξ−1) ξ − 1, ξ − z = e This, together with the condition s(ξ) → 0 for ξ → ∞, completely determines the form of the function s(ξ) and yields

C. Some properties of special functions related to the Barnes double gamma
For ℜ x > 0 the Barnes double function Γ b (x) has an integral representation of the form: With a help of relations Using relations (C.1) and definitions (C.2), (C.3) one can easily establish some basic properties of these functions.

D. Orthogonality relations
For ξ ∈ iR + we define etc.
The orthogonality relations where p 1 , p 2 ∈ R + , were derived in [20]. They followed from the "Saalschutz summation formulae" which were also derived in that paper. For our present purposes we need another two Saalschutz summation formulae, which may be derived following the steps that lead to is finite. This gives Using the reflection formula for the Barnes S−functions, Eq. (C.6), the fact that S ♮ (x) are real analytic and that p i ∈ R, τ ∈ iR, we can rewrite (D.1), (D.6) and (D.7) in the form were for brevity we have not written the "−0 + " prescription explicitly. It is now easy to check that for every choice of the parity indices h 2 , h 3 , and g 2 , g 3 , the orthogonality formula (4.12) reduces to one of the equations (D.8), (D.9).