Three Point Functions in Higher Spin AdS₃ Holography with 1/N Corrections

Abstract

We examine three point functions with two scalar operators and a higher spin current in 2d W${}_N$ minimal model to the next non-trivial order in $1/N$ expansion. The minimal model was proposed to be dual to a 3d higher spin gauge theory, and $1/N$ corrections should be interpreted as quantum effects in the dual gravity theory. We develop a simple and systematic method to obtain three point functions by decomposing four point functions of scalar operators with Virasoro conformal blocks. Applying the method, we reproduce known results at the leading order in $1/N$ and obtain new ones at the next leading order. As confirmation, we check that our results satisfy relations among three point functions conjectured before.

1. Introduction

Holography is expected to offer a way to learn quantum corrections of gravity theory from $1/N$ corrections in dual conformal field theory. In this paper, we address this issue by utilizing one of the simplest holographies proposed in [1]1, where 2d W${}_N$ minimal model is dual to Prokushkin-Vasiliev theory on AdS${}_3$ given by [6]. We examine three point functions with two scalar operators and one higher spin current in the minimal model up to the next leading order in $1/N$ expansion. They should be interpreted as one-loop corrections to three point interactions between two bulk scalars and one higher spin gauge fields in the dual higher spin theory. We develop a simple and systematic method to compute the three point functions by decomposing four point functions of scalar operators with Virasoro conformal blocks. Among others, we expect that this way of computation makes the dual higher spin interpretation easier. Applying the method, we reproduce known results at the leading order in $1/N$ obtained by [7,8]. Exact results are available only up to correlators with spin 5 current [9,10,11], but a simple relation was conjectured for generic s in [11]. We obtain the $1/N$ corrections of correlators with spin $s\le 8$ current, and the results for $s=6,7,8$ should be new. We check that they satisfy the conjectured relation as confirmation of our results.

We would like to examine the W${}_N$ minimal model in $1/N$ expansion, but we should specify the expansion in more details. The minimal model has a coset description

$$\begin{array}{c}\hfill \frac{\mathrm{su}{\left(N\right)}_k\oplus \mathrm{su}{\left(N\right)}_1}{\mathrm{su}{\left(N\right)}_{k+1}}\end{array}$$

(1)

whose central charge is given by

$$\begin{array}{cc}\hfill c& =(N-1)\left(1-\frac{N(N+1)}{(N+k)(N+k+1)}\right)\hfill \end{array}$$

(2)

The model has two parameters $N,k$. For our purpose, it is convenient to define the ’t Hooft coupling

$$\begin{array}{c}\hfill \lambda =\frac{N}{N+k}\end{array}$$

(3)

and label the model by $N,\lambda$ instead of $N,k$. We then expand the model in $1/N$, where each order depends on the other parameter $\lambda$. The expansion is almost the same as $1/c$ expansion because of $c\sim N(1-{\lambda }^2)+\mathcal{O}\left(N^0\right)$, but details are different.

The minimal model is argued to be dual to the higher spin theory of [6], which includes higher spin gauge fields ${\phi }^{\left(s\right)}$$(s=2,3,4,\dots )$ and complex scalar fields ${\varphi }_{\pm }$ with mass $m^2=-1+{\lambda }^2$. The large N limit of minimal model with $\lambda$ in Equation (3) kept finite corresponds to the classical limit of higher spin theory, where $\lambda$ is identified with the parameter in bulk scalar mass. The higher spin gauge fields ${\phi }^{\left(s\right)}$ and bulk scalars ${\varphi }_{\pm }$ are dual to higher spin currents $J^{\left(s\right)}$ and scalar operators ${\mathcal{O}}_{\pm }$, respectively. Here different boundary conditions are assigned to the bulk scalars ${\varphi }_{\pm }$ and the dual conformal dimensions are given by ${\Delta }_{\pm }=2h_{\pm }=1\pm \lambda$ at the tree level.

Basic data of conformal field theory may be given by spectrum and three point functions of primary operators. Since higher spin symmetry of the minimal model is exact, spectrum does not receive any corrections in $1/N$. Namely, there is no anomalous dimension for higher spin current $J^{\left(s\right)}$. Therefore, as simple but non-trivial examples, we examine three point functions and specifically focus on those with two scalar operators and one higher spin current as

$$\begin{array}{c}\hfill \langle {\mathcal{O}}_{\pm }\left(z_1\right){\overline{\mathcal{O}}}_{\pm }\left(z_2\right)J^{\left(s\right)}\left(z_3\right)\rangle \end{array}$$

(4)

with $s=2,3,4,\dots$. Here ${\overline{\mathcal{O}}}_{\pm }$ are complex conjugate of ${\mathcal{O}}_{\pm }$. In [7,8], the three point functions in the large N limit of the minimal model have been computed from classical higher spin theory. They were reproduced with conformal field theory approach in [8,12,13]2, but these methods are applicable only to the leading order analysis in $1/N$. Since the W${}_N$ minimal model is solvable, for instance, by making use of the coset description in Equation (1), we can obtain the three point functions in Equation (4) with finite $N,k$ in principle. However, in practice, the computation would be quite complicated, and only explicit expressions are available only with spin $3,4,5$ currents [9,10,11] (see also [15] for an alternative algebraic method).

In this paper, we develop a different way to compute the three point functions in Equation (4) from the decomposition of scalar four point functions by Virasoro conformal blocks. Our method may be explained as follows; Let us consider a generic operator product expansion of scalar operators ${\mathcal{O}}_i$ with conformal weights $(h_i,h_i)$ as

$$\begin{array}{c}\hfill {\mathcal{O}}_1\left(z_1\right){\mathcal{O}}_2\left(z_2\right)=\sum _p\frac{C_{12p}}{z_{12}^{h_1+h_2-h_p}{\overline{z}}_{12}^{h_1+h_2-{\overline{h}}_p}}{\mathcal{A}}_p\left(z_2\right)+\cdots \end{array}$$

(5)

where the coefficient $C_{12p}$ includes the information of three point function. Moreover, ${\mathcal{A}}_p$ has conformal weights $(h_p,{\overline{h}}_p)$, and dots denote contributions from descendants. Using the expansion, we can decompose scalar four point function as

$$\begin{array}{c}\hfill \langle {\mathcal{O}}_1\left(\infty \right){\mathcal{O}}_2\left(1\right){\mathcal{O}}_3\left(z\right){\mathcal{O}}_4\left(0\right)\rangle =\sum _p\frac{C_{12p}{C}_{34p}}{{\left|z\right|}^{2(h_3+h_4)}}\mathcal{F}(c,h_i,h_p,z)\overline{\mathcal{F}}(c,h_i,{\overline{h}}_p,\overline{z})\end{array}$$

(6)

Here $\mathcal{F}(c,h_i,h_p,z)$ is Virasoro conformal block, which can be fixed only from the symmetry in principle. Once we know scalar four point functions and Virasoro conformal blocks, we can read off coefficients as $C_{12p}$ by solving constraint equations coming from Equation (6). For our case with ${\mathcal{O}}_i={\mathcal{O}}_{\pm }$ or ${\overline{\mathcal{O}}}_{\pm }$, four point functions can be computed exactly with finite $N,k$, for instance, by applying Coulomb gas approach as in [16]. On the other hand, Virasoro conformal blocks are quite complicated, but explicit forms may be obtained by applying Zamolodchikov’s recursion relation [17], see also [18,19]. We can find other works on the $1/c$ expansion of Virasoro conformal blocks in, e.g., [20,21,22,23]. Gathering these knowledges, we shall obtain the coefficients as $C_{12p}$ up to the next leading order in $1/N$ expansion.

The paper is organized as follows; In order to study the decomposition in Equation (6), we need to examine scalar four point functions and Virasoro conformal blocks. In the next section we decompose scalar four point functions in terms of cross ratio z, and in Section 3 we give the explicit expressions of Virasoro conformal blocks in expansions both in $1/N$ and z. After these preparations, we compute three point functions in Equation (4) by solving constraint equations coming from Equation (6) in Section 4. In Section 4.1 we reproduce known results at the leading order in $1/N$. In Section 4.2 we obtain the $1/N$ corrections of three point functions for $s=3,4,\dots ,8$, and check that they satisfy the relation conjectured in [11]. Section 5 is devoted to conclusion and discussions. In Appendix A we examine Virasoro conformal blocks in expansions of $1/c$ and z by analyzing Zamolodchikov’s recursion relation. In Appendix B we compute three point functions with higher spin currents of double trace type.

2. Expansions of Four Point Functions

We would like to obtain the coefficients as $C_{12p}$ by solving Equation (6). For the purpose, we need information on the both sides of the equation, i.e., scalar four point functions and Virasoro conformal blocks. In this section we examine scalar four point functions. We are interested in three point functions of two scalar operators ${\mathcal{O}}_{\pm }$ and a higher spin current $J^{\left(s\right)}$ as in Equation (4). We consider the following four point functions with scalar operators ${\mathcal{O}}_{\pm }$ as

$$\begin{array}{c}\hfill G_{++}\left(z\right)\equiv \langle {\mathcal{O}}_{+}\left(\infty \right){\overline{\mathcal{O}}}_{+}\left(1\right){\mathcal{O}}_{+}\left(z\right){\overline{\mathcal{O}}}_{+}\left(0\right)\rangle \end{array}$$

(7)

$$\begin{array}{c}\hfill G_{--}\left(z\right)\equiv \langle {\mathcal{O}}_{-}\left(\infty \right){\overline{\mathcal{O}}}_{-}\left(1\right){\mathcal{O}}_{-}\left(z\right){\overline{\mathcal{O}}}_{-}\left(0\right)\rangle \end{array}$$

(8)

$$\begin{array}{c}\hfill G_{-+}\left(z\right)\equiv \langle {\mathcal{O}}_{-}\left(\infty \right){\mathcal{O}}_{+}\left(1\right){\overline{\mathcal{O}}}_{+}\left(z\right){\overline{\mathcal{O}}}_{-}\left(0\right)\rangle \end{array}$$

(9)

Exact expressions with finite $N,k$ may be found in [16]. From the expansions in z, we can read off what kind of operators are involved in the decomposition by Virasoro conformal blocks. In the rest of this section, we obtain the explicit forms of four point functions in z expansion for parts relevant to later analysis.

Let us first examine the z expansion of $G_{++}\left(z\right)$ in Equation (7), and see generic properties of the four point functions. The expression with finite $N,k$ is [16]

$$\begin{array}{c}\hfill G_{++}\left(z\right)={\left|z(1-z)\right|}^{-2{\Delta }_{+}}\left[{\left|{(1-z)}^{1+\lambda }{}_2{F}_1\left(1+\frac{\lambda }{N},-\frac{\lambda }{N};-\lambda ;z\right)\right|}^2\right.\\ \hfill \left.+{\mathcal{N}}_1{\left|z^{1+\lambda }{}_2{F}_1\left(1+\frac{\lambda }{N},-\frac{\lambda }{N};2+\lambda ;z\right)\right|}^2\right]\end{array}$$

(10)

with

$$\begin{array}{c}\hfill {\mathcal{N}}_1=-\frac{\Gamma (1+\lambda -\frac{\lambda }{N})\Gamma {(-\lambda )}^2\Gamma (2+\lambda +\frac{\lambda }{N})}{\Gamma (-1-\lambda -\frac{\lambda }{N})\Gamma (\frac{\lambda }{N}-\lambda )\Gamma {(2+\lambda )}^2}\end{array}$$

(11)

Here the exact value of conformal dimensions ${\Delta }_{+}=2h_{+}$ is

$$\begin{array}{c}\hfill {\Delta }_{+}=\frac{(N-1)(2N+1+k)}{N(N+k)}=1+\lambda -\frac{1}{N}-\frac{1}{N^2}\lambda +\mathcal{O}(N^{-3})\end{array}$$

(12)

which is expanded in $1/N$ up to the $N^{-2}$ order.

In the expansion in z, we would like to pick up the terms corresponding to the three point function in Equation (4). The operator product of ${\mathcal{O}}_{+}$ may be expanded as

$$\begin{array}{cc}\hfill {\mathcal{O}}_{+}\left(z\right){\overline{\mathcal{O}}}_{+}\left(0\right)=& \frac{1}{{\left|z\right|}^{2{\Delta }_{+}}}+\sum _s\frac{C_{+}^{\left(s\right)}{z}^s}{{\left|z\right|}^{2{\Delta }_{+}}}{J}^{\left(s\right)}\left(0\right)\hfill \\ & +\sum _{(s_1,s_2;s^{\prime })}\frac{C_{+}^{(s_1,s_2;s^{\prime })}{z}^{s^{\prime }}}{{\left|z\right|}^{2{\Delta }_{+}}}{J}^{(s_1,s_2;s^{\prime })}\left(0\right)+\sum _{n,m}{C}_{+}^{(n,m)}{z}^n{\overline{z}}^m{\mathcal{A}}_{(n,m)}\left(0\right)\cdots \hfill \end{array}$$

(13)

Here $J^{(s_1,s_2;s^{\prime })}\left(z\right)$ are higher spin currents of double trace type as

$$\begin{array}{c}\hfill J^{(s_1,s_2;s^{\prime })}=J^{\left(s_1\right)}{\partial }^{s^{\prime }-s_1-s_2}{J}^{\left(s_2\right)}+\cdots \end{array}$$

(14)

with $s^{\prime }\ge 6$ as $s_1,s_2\ge 3$ and $s^{\prime }-s_1-s_2\ge 0$. If we use the normalization as $\langle J^{\left(s\right)}{J}^{\left(s\right)}\rangle \propto N$, then the two point function of this type of operator becomes $\langle J^{(s_1,s_2;s^{\prime })}{J}^{(s_1,s_2;s^{\prime })}\rangle \propto N^2$. This is related to the fact that $C_{+}^{\left(s\right)}\propto N^{-1/2}$, while $C_{+}^{(s_1,s_2;s)}\propto N^{-1}$. There could be currents of other multi-trace type, but the contributions are more suppressed in $1/N$. Furthermore, ${\mathcal{A}}_{(n,m)}\left(z\right)$ are double trace type operators of the form as

$$\begin{array}{c}\hfill {\mathcal{A}}_{(n,m)}={\mathcal{O}}_{+}{\partial }^n{\overline{\partial }}^m{\overline{\mathcal{O}}}_{+}+\cdots \end{array}$$

(15)

and the conformal weights are $(h_{n,m},{\overline{h}}_{n,m})=(2h_{+}+n,2h_{+}+m)$. The dots in Equation (13) include the operators dressed by higher spin currents $J^{\left(s\right)}\left(z\right),{\overline{J}}^{\left(s\right)}\left(\overline{z}\right)$ for instance.

The operator product expansion in Equation (13) suggests that the contributions from $J^{\left(s\right)}$ or its descendants are included in terms like $z^{s+l}/{\left|z\right|}^{2{\Delta }_{+}}$, where $l=0,1,2,\dots$ corresponds to the level of descendant. In Equation (10), such terms appear as

$$\begin{array}{c}\hfill G_{++}\left(z\right)={\left|z\right|}^{-2{\Delta }_{+}}{(1-z)}^{-{\Delta }_{+}+1+\lambda }{}_2{F}_1\left(1+\frac{\lambda }{N},-\frac{\lambda }{N};-\lambda ;z\right)+\cdots \end{array}$$

(16)

Note that they also include effects from higher spin currents of double trace type $J^{(s_1,s_2;s^{\prime })}\left(z\right)$ among others. For the first term in Equation (10), the other contributions involve at least one anti-holomorphic current ${\overline{J}}^{\left(s\right)}\left(\overline{z}\right)$. For the second term in Equation (10), the expansions become polynomials of z and $\overline{z}$ at the leading order in $1/N$, and this implies that double trace type operators ${\mathcal{A}}_{(n,m)}$ should appear as ${\mathcal{A}}_p$ in Equation (5). At the leading order in $1/N$, we can expand Equation (16) around $z\sim 0$ as

$$\begin{array}{c}\hfill G_{++}\left(z\right)\sim {\left|z\right|}^{-2(\lambda +1)}\end{array}$$

(17)

This corresponds to the expansion by the identity operator in Equation (13). Thus the non-trivial contributions to our three point functions come form the terms at least of order $1/N$.

At the next and next-to-next orders in $1/N$, there are two types of contributions in Equation (16). One comes from

$$\begin{array}{c}\hfill {(1-z)}^{-{\Delta }_{+}+1+\lambda }={(1-z)}^{\frac{1}{N}+\frac{\lambda }{N^2}}+\mathcal{O}(N^{-3})\end{array}$$

(18)

which becomes

$$\begin{array}{c}{(1-z)}^{\frac{1}{N}}{(1-z)}^{\frac{\lambda }{N^2}}\hfill \\ =1-\frac{1}{N}\sum _{k=1}\frac{1}{k}{z}^k+\frac{1}{N^2}\left[\sum _{k=1}^{\infty }\left(-\frac{\lambda }{k}{z}^k\right)+\sum _{k=2}^{\infty }\frac{1}{k}{H}_{k-1}{z}^k\right]+\mathcal{O}(N^{-3})\hfill \end{array}$$

(19)

Here we have used for $k\ge 2$

$$\begin{array}{cc}\left(\genfrac{}{}{0pt}{}{\frac{1}{N}}{k}\right)& =\frac{\Gamma (1+\frac{1}{N})}{k!\Gamma (1-k+\frac{1}{N})}={(-1)}^{k-1}\frac{1}{Nk!}\left(1-\frac{1}{N}\right)\left(2-\frac{1}{N}\right)\cdots \left(k-1-\frac{1}{N}\right)\hfill \\ & ={(-1)}^{k-1}\frac{1}{Nk}\left(1-\frac{1}{N}\sum _{i=1}^{k-1}\frac{1}{i}\right)+\mathcal{O}(N^{-3})\hfill \end{array}$$

(20)

and the definition of harmonic number

$$\begin{array}{c}\hfill H_n=\sum _{j=1}^n\frac{1}{j}\end{array}$$

(21)

The other comes from the hypergeometric function, which can be similarly expanded in $1/N$ as

$$\begin{array}{c}{}_2{F}_1\left(1+\frac{\lambda }{N},-\frac{\lambda }{N},-\lambda ;z\right)=\frac{\Gamma (-\lambda )}{\Gamma (1+\frac{\lambda }{N})\Gamma (-\frac{\lambda }{N})}\sum _{n=0}^{\infty }\frac{\Gamma (1+\frac{\lambda }{N}+n)\Gamma (-\frac{\lambda }{N}+n)}{\Gamma (-\lambda +n)}\frac{z^n}{n!}\hfill \\ =1+\frac{\Gamma (1-\lambda )}{N}\sum _{n=1}^{\infty }\frac{\Gamma \left(n\right)}{\Gamma (n-\lambda )}{z}^n+\frac{1}{N^2}\left(\lambda z+\lambda \Gamma (1-\lambda )\sum _{n=2}^{\infty }\frac{\Gamma \left(n\right)}{n\Gamma (n-\lambda )}{z}^n\right)+\mathcal{O}(N^{-3})\hfill \end{array}$$

(22)

In total, we have

$$\begin{array}{c}\hfill {\left|z\right|}^{2{\Delta }_{+}}{G}_{++}\left(z\right)\sim 1+\frac{1}{N}\sum _{n=1}^{\infty }\left(-\frac{1}{n}+\frac{\Gamma (1-\lambda )\Gamma \left(n\right)}{\Gamma (n-\lambda )}\right)z^n+\frac{1}{N^2}\sum _{n=2}^{\infty }{f}_{++}^{\left(n\right)}{z}^n\end{array}$$

(23)

where

$$\begin{array}{c}\hfill f_{++}^{\left(n\right)}=-\sum _{l=1}^{n-1}\frac{\Gamma (1-\lambda )\Gamma \left(l\right)}{(n-l)\Gamma (l-\lambda )}-\frac{\lambda }{n}+\frac{H_{n-1}}{n}+\frac{\lambda \Gamma (1-\lambda )\Gamma \left(n\right)}{n\Gamma (n-\lambda )}\end{array}$$

(24)

First few expressions are

$$\begin{array}{cc}\hfill f_{++}^{\left(2\right)}& =\frac{1}{2}\left(-2-\frac{1}{\lambda -1}-\lambda \right)\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill f_{++}^{\left(3\right)}& =\frac{1}{3}\left(\frac{4}{\lambda -2}+\frac{1}{\lambda -1}-\lambda \right)\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill f_{++}^{\left(4\right)}& =\frac{1}{8}\left(1-\frac{18}{\lambda -3}+\frac{8}{\lambda -2}+\frac{14}{\lambda -1}-2\lambda \right)\hfill \end{array}$$

(27)

We would like to move to another four point function $G_{--}\left(z\right)$ in Equation (8), whose expression with finite $N,k$ can be again found in [16]. We use the four point function in order to obtain the three point function in Equation (4) with the other type of scalar operator ${\mathcal{O}}_{-}$. As for $G_{++}\left(z\right)$, the relevant part is

$$\begin{array}{c}\hfill G_{--}\left(z\right)={\left|z\right|}^{-2{\Delta }_{-}}{(1-z)}^{\frac{1}{N}-\frac{\lambda }{N^2}}{}_2{F}_1\left(1-\frac{\lambda }{N},\frac{\lambda }{N}-\frac{{\lambda }^2}{N^2};\lambda -\frac{{\lambda }^2}{N};z\right)+\cdots \end{array}$$

(28)

Here we may need

$$\begin{array}{cc}{\Delta }_{-}& =2h_{-}=\frac{N-1}{N}\left(1-\frac{N+1}{N+k+1}\right)\hfill \\ & =1-\lambda -\frac{1}{N}(1-{\lambda }^2)+\frac{1}{N^2}\lambda (1-{\lambda }^2)+\mathcal{O}(N^{-3})\hfill \end{array}$$

(29)

Similarly to $G_{++}\left(z\right)$ we can expand $G_{--}\left(z\right)$ in z as

$${\left|z\right|}^{2{\Delta }_{-}}{G}_{--}\left(z\right)\sim 1+\frac{1}{N}\sum _{n=1}^{\infty }\left(-\frac{1}{n}+\frac{\Gamma (1+\lambda )\Gamma \left(n\right)}{\Gamma (n+\lambda )}\right)z^n+\frac{1}{N^2}\sum _{n=2}^{\infty }{f}_{--}^{\left(n\right)}{z}^n$$

(30)

where

$$f_{--}^{\left(n\right)}=-{\sum }_{l=1}^{n-1}\frac{\Gamma (1+\lambda )\Gamma \left(l\right)}{(n-l)\Gamma (l+\lambda )}+\frac{\lambda }{n}+\frac{H_{n-1}}{n}+\frac{\lambda \Gamma (1+\lambda )\Gamma \left(n\right)}{\Gamma (n+\lambda )}\left({\sum }_{k=1}^{n-1}\frac{\lambda }{k+\lambda }-\frac{1}{n}\right)$$

(31)

First few expressions are

$$\begin{array}{cc}\hfill f_{--}^{\left(2\right)}& =\frac{\lambda }{2}-\frac{3}{2(\lambda +1)}+\frac{1}{{(\lambda +1)}^2}\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill f_{--}^{\left(3\right)}& =\frac{\lambda }{3}-\frac{13}{3(\lambda +1)}+\frac{2}{{(\lambda +1)}^2}+\frac{20}{3(\lambda +2)}-\frac{8}{{(\lambda +2)}^2}\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill f_{--}^{\left(4\right)}& =\frac{\lambda }{4}-\frac{31}{4(\lambda +1)}+\frac{3}{{(\lambda +1)}^2}+\frac{23}{\lambda +2}-\frac{24}{{(\lambda +2)}^2}-\frac{63}{4(\lambda +3)}+\frac{27}{{(\lambda +3)}^2}+\frac{1}{8}\hfill \end{array}$$

(34)

From the four point functions $G_{\pm \pm }\left(z\right)$, we can read off the square root of coefficients ${\left(C_{\pm }^{\left(s\right)}\right)}^2$, but relative phase factor cannot be fixed. In order to determine it, we also need to examine $G_{-+}\left(z\right)$ in Equation (9), which can be computed as [16]

$$\begin{array}{c}\hfill G_{-+}\left(z\right)={|1-z|}^{-2{\Delta }_{+}}{\left|z\right|}^{\frac{2}{N}}{\left|1+\frac{1-z}{Nz}\right|}^2\end{array}$$

(35)

with finite $N,k$. For later arguments, we need

$$\begin{array}{c}\hfill {|1-z|}^{2{\Delta }_{+}}{G}_{-+}\left(z\right)\sim 1+\frac{1}{N}\sum _{n=2}^{\infty }\frac{n-1}{n}{(1-z)}^n\end{array}$$

(36)

which is expanded in $(1-z)$ up to the $1/N$ order.

3. Virasoro Conformal Blocks

In the previous section we analyzed the left hand side of Equation (6). In order to obtain three point functions by solving the equations in Equation (6), we further need information on $\mathcal{F}(c,h_i,h_p,z)$. In general, the forms of Virasoro conformal blocks are quite complicated. In practice, we actually do not need to know closed forms but expansions in z up to some orders. For this purpose, a standard approach may be solving Zamolodchikov’s recursion relation in [17]. Following the algorithm developed in [18] (see also [19]), we obtain the expressions of Virasoro conformal blocks to several orders in z and $1/c$ in Appendix A. Related works may be found in [19,20,21,22,23], and in particular, some closed form expressions were given, e.g., in [20]. Our findings agree with their results after minor modifications.

Let us consider the four point function in the decomposition of Equation (6) with $h_1=h_2$ and $h_3=h_4$. In the decomposition, intermediate operator ${\mathcal{A}}_p$ can be the identity or other. As observed in the examples of previous section, only the Virasoro conformal block with the identity operator (called as vacuum block) survives at the leading order in $1/N$. This simply means that the four point functions are factorized into the products of two point ones at the leading order in $1/N$. Virasoro conformal block with ${\mathcal{A}}_p$ of single trace type would appear at the next leading order in $1/N$. We would like to examine $1/N$ corrections to three point functions, so we need $1/N$ corrections to the Virasoro block of ${\mathcal{A}}_p$. This also implies that we need the expression of vacuum block up to the next-to-next leading order in $1/N$.

Let us first examine the vacuum block with $h_1=h_3=h_{\pm }$. As was explained in Appendix A, the $1/c$-expansion of vacuum block is given by

$$\begin{array}{cc}{\mathcal{V}}_0\left(x\right)=& 1+\frac{2h_1{h}_3}{c}{z}^2{}_2{F}_1(2,2;4;z)\hfill \\ & +\frac{1}{c^2}\left[h_1^2{h}_3^2{k}_a\left(z\right)+h_1{h}_3(h_1+h_3)k_b\left(z\right)+h_1{h}_3{k}_c\left(z\right)\right]+\mathcal{O}(c^{-3})\hfill \end{array}$$

(37)

with

$$\begin{array}{c}{k}_a\left(z\right)=2z^4+4z^5+\frac{28z^6}{5}+\frac{34z^7}{5}+\frac{2687z^8}{350}+\mathcal{O}(z^9)\hfill \\ k_b\left(z\right)=\frac{2z^4}{5}+\frac{4z^5}{5}+\frac{39z^6}{35}+\frac{47z^7}{35}+\frac{263z^8}{175}+\mathcal{O}(z^9)\hfill \\ k_c\left(z\right)=\frac{2z^4}{25}+\frac{4z^5}{25}+\frac{109z^6}{490}+\frac{131z^7}{490}+\frac{1879z^8}{6300}+\mathcal{O}(z^9)\hfill \end{array}$$

(38)

The $1/c$ order term corresponds to the exchange of spin 2 current (energy momentum tensor) in terms of global block. We need to rewrite the expansion in $1/c$ by that in $1/N$ as

$$\begin{array}{c}\hfill {\mathcal{V}}_0\left(z\right)={\mathcal{V}}_0^{\left(0\right)}\left(z\right)+{\mathcal{V}}_0^{\left(1\right)}\left(z\right)\frac{1}{N}+{\mathcal{V}}_0^{\left(2\right)}\left(z\right)\frac{1}{N^2}+\mathcal{O}(N^{-3})\end{array}$$

(39)

The first two terms can be easily read off as

$$\begin{array}{c}\hfill {\mathcal{V}}_0^{\left(0\right)}\left(z\right)=1,{\mathcal{V}}_0^{\left(1\right)}\left(z\right)=\frac{1}{2}\left(\frac{1\pm \lambda }{1\mp \lambda }\right)z^2{}_2{F}_1(2,2;4;z)\end{array}$$

(40)

Since there are two types of contributions to ${\mathcal{V}}_0^{\left(2\right)}$, we separate it into two parts as

$$\begin{array}{c}\hfill {\mathcal{V}}_0^{\left(2\right)}={\mathcal{V}}_0^{(2,1)}\left(z\right)+{\mathcal{V}}_0^{(2,2)}\left(z\right)\end{array}$$

(41)

One comes from the $1/c$ order term with the next leading contribution from $h_{\pm }^2/c$ as

$$\begin{array}{cc}{\mathcal{V}}_0^{(2,1)}\left(z\right)& =f_{\pm \pm }^{\left(2\right)}{z}^2{}_2{F}_1(2,2;4;z)\hfill \\ & =f_{\pm \pm }^{\left(2\right)}\left(z^2+z^3+\frac{9z^4}{10}+\frac{4z^5}{5}+\frac{5z^6}{7}+\frac{9z^7}{14}+\frac{7z^8}{12}\right)+\mathcal{O}(z^9)\hfill \end{array}$$

(42)

where $f_{\pm \pm }^{\left(2\right)}$ were given in Equations (25) and (32). Here we have used

$$\frac{2h_{\pm }^2}{c}=\frac{1}{2N}\left(\frac{1\pm \lambda }{1\mp \lambda }\right)+\frac{1}{N^2}{f}_{\pm \pm }^{\left(2\right)}+\mathcal{O}\left(N^{-3}\right)$$

(43)

which are obtained from the $1/N$ expansions of $h_{\pm }$ as in Equations (12) and (29) and c in Equation (2) as

$$c=N(1-{\lambda }^2)\left[1-\frac{1}{N}\left(\lambda +\frac{1}{1+\lambda }\right)\right]+\mathcal{O}\left(N^{-1}\right)$$

(44)

The other comes from the $1/c^2$ order terms in Equation (37) as

$$\begin{array}{c}\hfill {\mathcal{V}}_0^{(2,2)}\left(z\right)=\frac{{(1\pm \lambda )}^2}{16{(1\mp \lambda )}^2}{k}_a\left(z\right)+\frac{(1\pm \lambda )}{4{(1\mp \lambda )}^2}{k}_b\left(z\right)+\frac{1}{4{(1\mp \lambda )}^2}{k}_c\left(z\right)\end{array}$$

(45)

with $k_a\left(z\right)$, $k_b\left(z\right)$, and $k_c\left(z\right)$ in Equation (38).

We also need Virasoro blocks of ${\mathcal{A}}_p$ up to the next non-trivial order in $1/N$. It is known that the Virasoro block is expanded in $1/c$ as (see, e.g., [23])

$$\begin{array}{c}\hfill {\mathcal{V}}_p\left(z\right)=g(h_p,z)+\frac{1}{c}\left[h_1{h}_3{f}_a(h_p,z)+(h_1+h_3)f_b(h_p,z)+f_c(h_p,z)\right]+\mathcal{O}(c^{-2})\end{array}$$

(46)

Here $g(h_p,z)$ is the global block of ${\mathcal{A}}_p$ and the expressions of $f_a(h_p,z)$, $f_b(h_p,z)$, and $f_c(h_p,z)$ were obtained in [23]. See also Appendix A. For our application, we set $h_1=h_3=h_{\pm }$ and $h_p=s$. We need the expansion in $1/N$ instead of $1/c$ as

$$\begin{array}{c}\hfill {\mathcal{V}}_s\left(z\right)={\mathcal{V}}_s^{\left(0\right)}\left(z\right)+{\mathcal{V}}_s^{\left(1\right)}\left(z\right)\frac{1}{N}+\mathcal{O}(N^{-2})\end{array}$$

(47)

The leading term ${\mathcal{V}}_p^{\left(0\right)}\left(z\right)$ is given by the global block as

$$\begin{array}{c}\hfill {\mathcal{V}}_s^{\left(0\right)}\left(z\right)=g(s,z)=z^s{}_2{F}_1(s,s;2s;x)\end{array}$$

(48)

The next order contributions in $1/N$ are

$$\begin{array}{c}\hfill {\mathcal{V}}_s^{\left(1\right)}\left(z\right)=\frac{1}{4}\frac{1\pm \lambda }{1\mp \lambda }{f}_a(s,z)+\frac{1}{(1\mp \lambda )}{f}_b(s,z)+\frac{1}{(1-{\lambda }^2)}{f}_c(s,z)\end{array}$$

(49)

where the functions $f_a(s,z)$, $f_b(s,z)$, and $f_c(s,z)$ are given by

$$\begin{array}{cc}{f}_a(s,z)& =z^s\left[2z^2+(s+2)z^3+\frac{(s+3)(5s(s+3)+6)z^4}{20s+10}+\frac{(s+3)(s+4)(5s(s+4)+8)z^5}{60(2s+1)}\right.\hfill \\ & \left.+\mathcal{O}(z^6)\right]\hfill \\ f_b(s,z)& =s(s-1)z^s\left[\frac{z^2}{2s+1}+\frac{(s+2)z^3}{4s+2}+\frac{(s+3)(5s(s+4)+18)z^4}{20\left(4s\right(s+2)+3)}\right.\hfill \\ & \left.+\frac{(s+3)(s+4)(5s(s+5)+24)z^5}{120\left(4s\right(s+2)+3)}+\mathcal{O}(z^6)\right]\hfill \\ f_c(s,z)& =\frac{s^2{(s-1)}^2}{2{(2s+1)}^2}{z}^s\left[z^2+\frac{(s+2)z^3}{2}+\frac{(s+3)(s(10s(2s+11)+191)+108)z^4}{40{(2s+3)}^2}\right.\hfill \\ & +\left.\frac{(s+3)(s+4)(s(10s(2s+13)+243)+144)z^5}{240{(2s+3)}^2}+\mathcal{O}(z^6)\right]\hfill \end{array}$$

(50)

4. Three Point Functions

After the preparations in previous sections, we now work on the decompositions of four point functions by Virasoro conformal blocks as in Equation (6). In the current case, the decompositions are

$$\begin{array}{c}\hfill {\left|z\right|}^{2{\Delta }_{\pm }}{G}_{\pm \pm }\left(z\right)={\mathcal{V}}_0\left(z\right)+\sum _{s=3}^{\infty }{(C_{\pm }^{\left(s\right)})}^2{\mathcal{V}}_s\left(z\right)+\sum _{(s_1,s_2;s^{\prime })}{(C_{\pm }^{(s_1,s_2;s^{\prime })})}^2{\mathcal{V}}_{s^{\prime }}\left(z\right)+\cdots \end{array}$$

(51)

Here $G_{\pm \pm }\left(z\right)$ are four point functions defined in Equations (7) and (8), and the expansions in z were obtained as in Equations (23) and (30). Moreover, ${\mathcal{V}}_0\left(z\right)$ is the vacuum block and ${\mathcal{V}}_s\left(z\right)$ is the Virasoro block of higher spin current $J^{\left(s\right)}$ (or $J^{(s_1,s_2;s)}$). Their expansions in z can be found in the previous section.

Solving constraint equations from Equation (51), we read off the coefficients $C_{\pm }^{\left(s\right)}$, which are proportional to the three point functions in Equation (4). It is convenient to expand the coefficients in $1/N$ as

$$\begin{array}{c}\hfill C_{\pm }^{\left(s\right)}=\frac{1}{N^{1/2}}\left(C_{\pm ,0}^{\left(s\right)}+\frac{1}{N}{C}_{\pm ,1}^{\left(s\right)}+\mathcal{O}(N^{-2})\right)\end{array}$$

(52)

Then we can see that the constraint equations from Equation (51) at the order $N^0$ is trivially satisfied as $1=1$. The non-trivial conditions arise from order $1/N$ terms, and they determine the leading order expressions $C_{\pm ,0}^{\left(s\right)}$ as seen in the next subsection. The main purpose of this paper is to compute $C_{\pm ,1}^{\left(s\right)}$, which are $1/N$ corrections to the leading order expressions. We derive them by solving order $N^{-2}$ conditions up to $s=8$ in Section 4.2. Notice that we should take care of $C_{\pm }^{(s_1,s_2;s^{\prime })}$ in Equation (51) for $s\ge 6$, which may be expanded as

$$\begin{array}{c}\hfill C_{\pm }^{(s_1,s_2;s^{\prime })}=\frac{1}{N}{C}_{\pm ,0}^{(s_1,s_2;s^{\prime })}+\mathcal{O}(N^{-2})\end{array}$$

(53)

The coefficients $C_{\pm ,0}^{(s_1,s_2;s^{\prime })}$ are analyzed in Appendix B.

4.1. Leading Order Expressions in $1/N$

We start from three point functions at the leading order in $1/N$. We examine the constraint equations from Equation (51) up to $1/N$ order. Up to this order, the vacuum block is given by (see Equation (37))

$$\begin{array}{c}\hfill {\mathcal{V}}_0\left(z\right)=1+\frac{1}{N}{(C_{\pm ,0}^{\left(2\right)})}^2{z}^2{}_2{F}_1(2,2;4;z)+\mathcal{O}(N^{-2})\end{array}$$

(54)

where we have defined

$$\begin{array}{c}\hfill C_{\pm ,0}^{\left(2\right)}={\left.\sqrt{\frac{2{\left(h_{\pm }\right)}^2}{c}}\right|}_{\mathcal{O}\left(N^{-1/2}\right)}=\sqrt{\frac{1}{2}\frac{1\pm \lambda }{1\mp \lambda }}\end{array}$$

(55)

The Virasoro block of $J^{\left(s\right)}$ is

$$\begin{array}{c}\hfill {\mathcal{V}}_s\left(z\right)=z^s{}_2{F}_1(s,s;2s;z)+\mathcal{O}(N^{-1})\end{array}$$

(56)

as in Equation (47) with Equation (48). Therefore, the expansion in Equation (51) can be written as

$$\begin{array}{c}\hfill {\left|z\right|}^{2{\Delta }_{\pm }}{G}_{\pm \pm }\left(z\right)=1+\frac{1}{N}\sum _{s=2}^{\infty }{(C_{\pm ,0}^{\left(s\right)})}^2{z}^s{}_2{F}_1(s,s;2s;z)+\cdots \end{array}$$

(57)

up to the order of $1/N$. The four point functions $G_{\pm \pm }\left(z\right)$ can be expanded as

$$\begin{array}{c}\hfill {\left|z\right|}^{2{\Delta }_{\pm }}{G}_{\pm \pm }\left(z\right)\sim 1+\frac{1}{N}\sum _{n=1}^{\infty }{z}^n\left(-\frac{1}{n}+\frac{\Gamma (1\mp \lambda )\Gamma \left(n\right)}{\Gamma (n\mp \lambda )}\right)+\cdots \end{array}$$

(58)

as in Equations (23) and (30) up to the same order. On the other hand, the global blocks can be written as

$$\begin{array}{c}\hfill z^s{}_2{F}_1(s,s;2s;z)=\frac{\Gamma \left(2s\right)}{{(\Gamma \left(s\right))}^2}\sum _{n=0}^{\infty }\frac{{(\Gamma (s+n))}^2}{\Gamma (2s+n)}\frac{z^{n+s}}{n!}\end{array}$$

(59)

Comparing the coefficients in front of $z^n$, we obtain

$$\begin{array}{c}\hfill -\frac{1}{n}+\frac{\Gamma (1\mp \lambda )\Gamma \left(n\right)}{\Gamma (n\mp \lambda )}={(\Gamma \left(n\right))}^2\sum _{s=2}^n\frac{\Gamma \left(2s\right){(C_{\pm ,0}^{\left(s\right)})}^2}{{(\Gamma \left(s\right))}^2\Gamma (s+n)(n-s)!}\end{array}$$

(60)

They are the constraint equations for ${(C_{\pm ,0}^{\left(s\right)})}^2$ with $s=3,4,\dots$.

In order to fix relative phase factor, we examine $G_{-+}\left(z\right)$ in Equation (9) as well. The decomposition in Equation (6) become

$$\begin{array}{cc}\hfill {|1-z|}^{2{\Delta }_{+}}{G}_{-+}\left(z\right)& \sim 1+\frac{1}{N}\sum _{s=2}^{\infty }{(-1)}^s{C}_{-,0}^{\left(s\right)}{C}_{+,0}^{\left(s\right)}{(1-z)}^s{}_2{F}_1(s,s;2s;1-z)\hfill \end{array}$$

(61)

in this case. The extra phase factor ${(-1)}^s$ may require explanation; Now we need to use a slightly different expression of operator product expansion as

$$\begin{array}{c}\hfill {\mathcal{O}}_{+}\left(1\right){\overline{\mathcal{O}}}_{+}\left(z\right)=C_{+}^{\left(s\right)}\frac{{(1-z)}^s}{{|1-z|}^{2{\Delta }_{+}}}{J}^{\left(s\right)}\left(1\right)+\cdots \end{array}$$

(62)

Then the coefficients in front of global blocks are given by

$$\begin{array}{c}\hfill C_{+}^{\left(s\right)}\langle {\mathcal{O}}_{-}\left(\infty \right)J^{\left(s\right)}\left(1\right){\overline{\mathcal{O}}}_{-}\left(0\right)\rangle =C_{+}^{\left(s\right)}{(-1)}^s\langle {\mathcal{O}}_{-}\left(\infty \right){\overline{\mathcal{O}}}_{-}\left(1\right)J^{\left(s\right)}\left(0\right)\rangle \propto {(-1)}^s{C}_{-}^{\left(s\right)}{C}_{+}^{\left(s\right)}\end{array}$$

(63)

Here the factor ${(-1)}^s$ can be obtained from the coordinate dependence of three point function, which is completely fixed by conformal symmetry, see Equation (65) below. Therefore, we have constraint equations for three point functions as

$$\begin{array}{c}\hfill \frac{n-1}{n}={(\Gamma \left(n\right))}^2\sum _{s=2}^n\frac{{(-1)}^s\Gamma \left(2s\right)C_{-,0}^{\left(s\right)}{C}_{+,0}^{\left(s\right)}}{\Gamma {\left(s\right)}^2\Gamma (s+n)(n-s)!}\end{array}$$

(64)

by comparing the coefficients in front of $z^n$.

Now we have three types of constraint equation as in Equations (60) and (64), and we would like to show that the known results satisfy these equations. At the leading order in $1/N$, the three point functions have been computed as [8]

$$\begin{array}{c}\hfill \langle {\mathcal{O}}_{\pm }\left(z_1\right){\overline{\mathcal{O}}}_{\pm }\left(z_2\right)J^{\left(s\right)}\left(z_3\right)\rangle =\frac{{\eta }_{\pm }^{\left(s\right)}}{2\pi }\frac{\Gamma {\left(s\right)}^2}{\Gamma (2s-1)}\frac{\Gamma (s\pm \lambda )}{\Gamma (1\pm \lambda )}{\left(\frac{z_{12}}{z_{13}{z}_{23}}\right)}^s\langle {\mathcal{O}}_{\pm }\left(z_1\right){\overline{\mathcal{O}}}_{\pm }\left(z_2\right)\rangle \end{array}$$

(65)

The phase factors ${\eta }_{\pm }^{\left(s\right)}$ depends on the convention of higher spin currents, but we may set ${\eta }_{+}^{\left(s\right)}=1$ and ${\eta }_{-}^{\left(s\right)}={(-1)}^s$. The two point function of higher spin current $J^{\left(s\right)}$ in Equation (65) is (see (6.1) of [8])

$$\begin{array}{c}\hfill \langle J^{\left(s\right)}\left(z_1\right)J^{\left(s\right)}\left(z_2\right)\rangle =\frac{B^{\left(s\right)}}{z_{12}^{2s}},B^{\left(s\right)}=\frac{N}{2^{2s}{\pi }^{5/2}}\frac{sin\left(\pi \lambda \right)}{\lambda }\frac{\Gamma \left(s\right)\Gamma (s-\lambda )\Gamma (s+\lambda )}{\Gamma (s-\frac{1}{2})}\end{array}$$

(66)

at the leading order in $1/N$. The coefficients $C_{\pm ,0}^{\left(s\right)}$ are given by normalization independent ratios as

$$\begin{array}{c}\hfill C_{\pm ,0}^{\left(s\right)}={\left.\frac{\langle {\mathcal{O}}_{\pm }{\overline{\mathcal{O}}}_{\pm }{J}^{\left(s\right)}\rangle }{\langle {\mathcal{O}}_{\pm }{\overline{\mathcal{O}}}_{\pm }\rangle {\langle J^{\left(s\right)}{J}^{\left(s\right)}\rangle }^{1/2}}\right|}_{\mathcal{O}\left(N^{-1/2}\right)}\end{array}$$

(67)

which become

$$\begin{array}{c}\hfill C_{\pm ,0}^{\left(s\right)}={\eta }_{\pm }^{\left(s\right)}\sqrt{\frac{\Gamma {\left(s\right)}^2}{\Gamma (2s-1)}\frac{\Gamma (1\mp \lambda )}{\Gamma (1\pm \lambda )}\frac{\Gamma (s\pm \lambda )}{\Gamma (s\mp \lambda )}}\end{array}$$

(68)

The first few coefficients are

$$\begin{array}{cc}{C}_{\pm ,0}^{\left(3\right)}={\eta }_{\pm }^{\left(3\right)}\sqrt{\frac{1}{6}\frac{(2\pm \lambda )(1\pm \lambda )}{(2\mp \lambda )(1\mp \lambda )}},\hfill & \hfill C_{\pm ,0}^{\left(4\right)}=\sqrt{\frac{1}{20}\frac{(3\pm \lambda )(2\pm \lambda )(1\pm \lambda )}{(3\mp \lambda )(2\mp \lambda )(1\mp \lambda )}}\end{array}$$

(69)

along with Equation (55) for $s=2$. Using these explicit expressions, we can check that the constraint equations in Equations (60) and (64) are indeed satisfied3.

4.2. $1/N$ Corrections

We would like to move to the main part of this paper. In this subsection, we derive $1/N$ corrections to three point functions by examining the equations in Equation (51). With the help of analysis in previous sections, we have already ingredients necessary to the task. For examples, the expansions of $G_{\pm \pm }\left(z\right)$ were given in Equations (23) and (30) up to order $1/N^2$. Moreover, the vacuum block and the Virasoro block of $J^{\left(s\right)}$ are expanded as in Equations (39) and (47), respectively. Using these expansions, the equations in Equation (51) become

$$\begin{array}{cc}\sum _{m=2}^{\infty }{f}_{\pm \pm }^{\left(m\right)}{z}^m=& {\mathcal{V}}_0^{\left(2\right)}\left(z\right)+\sum _{s=3}^{\infty }2C_{\pm ,1}^{\left(s\right)}{C}_{\pm ,0}^{\left(s\right)}{\mathcal{V}}_s^{\left(0\right)}\left(z\right)+\sum _{s=3}^{\infty }{(C_{\pm ,0}^{\left(s\right)})}^2{\mathcal{V}}_s^{\left(1\right)}\left(z\right)\hfill \\ & +\sum _{(s_1,s_2;s^{\prime })}{(C_{\pm ,0}^{(s_1,s_2;s^{\prime })})}^2{\mathcal{V}}_{s^{\prime }}^{\left(0\right)}\left(z\right)\hfill \end{array}$$

(70)

at the order of $1/N^2$. Here $f_{\pm \pm }^{\left(m\right)}$ are defined in Equation (24) and Equation (31). At this order we should include the effects from higher spin currents of double trace type as ${(C_{\pm ,0}^{(s_1,s_2,s^{\prime })})}^2$ in Equation (70) with $s^{\prime }\ge 6$.

Let us examine the equations in Equation (70) from low order terms in z. There are no $z^0$ and $z^1$ order terms in the both sides. We can see that the equality in Equation (70) is satisfied at the order of $z^2$ from Equation (42). Non-trivial constraint equations appear at the $z^3$ order as

$$f_{\pm \pm }^{\left(3\right)}=f_{\pm \pm }^{\left(2\right)}+2C_{\pm ,0}^{\left(3\right)}{C}_{\pm ,1}^{\left(3\right)}$$

(71)

where $f_{\pm \pm }^{\left(2\right)}$ comes from ${\mathcal{V}}_0^{(2,1)}$ in Equation (42). Solving them we find

$$\frac{C_{+,1}^{\left(3\right)}}{C_{+,0}^{\left(3\right)}}=-\frac{1}{2}\left(-\lambda +\frac{1}{1+\lambda }+\frac{4}{2+\lambda }\right),\frac{C_{-,1}^{\left(3\right)}}{C_{-,0}^{\left(3\right)}}=\frac{1}{2}\left(-\lambda +\frac{1}{\lambda +1}+\frac{4}{\lambda +2}-6\right)$$

(72)

The $z^4$ order constraints are

$$\begin{array}{c}\hfill f_{\pm \pm }^{\left(4\right)}=f_{\pm \pm }^{\left(2\right)}\frac{9}{10}+\frac{{(1\pm \lambda )}^2}{8{(1\mp \lambda )}^2}+\frac{(1\pm \lambda )}{10{(1\mp \lambda )}^2}+\frac{1}{50{(1\mp \lambda )}^2}+2C_{\pm ,0}^{\left(4\right)}{C}_{\pm ,1}^{\left(4\right)}+2C_{\pm ,0}^{\left(3\right)}{C}_{\pm ,1}^{\left(3\right)}\frac{3}{2}\end{array}$$

where the contribution from Equation (45) starts to enter. The constraints lead to

$$\begin{array}{c}\frac{C_{+,1}^{\left(4\right)}}{C_{+,0}^{\left(4\right)}}=\frac{1}{10}\left(5\lambda +\frac{6}{\lambda -1}-\frac{11}{\lambda +1}-\frac{20}{\lambda +2}-\frac{45}{\lambda +3}\right)\hfill \\ \frac{C_{-,1}^{\left(4\right)}}{C_{-,0}^{\left(4\right)}}=\frac{1}{10}\left(-5\lambda +\frac{6}{\lambda -1}-\frac{1}{\lambda +1}+\frac{20}{\lambda +2}+\frac{45}{\lambda +3}-60\right)\hfill \end{array}$$

(73)

We would like to keep going to the cases with $s\ge 5$, where $f_a(s,z)$, $f_b(s,z)$, and $f_c(s,z)$ in Equation (50) contribute. For $s=5$, the conditions become

$$\begin{array}{c}\hfill f_{\pm \pm }^{\left(5\right)}=f_{\pm \pm }^{\left(2\right)}\frac{4}{5}+\frac{{(1\pm \lambda )}^2}{4{(1\mp \lambda )}^2}+\frac{(1\pm \lambda )}{5{(1\mp \lambda )}^2}+\frac{1}{25{(1\mp \lambda )}^2}+2C_{\pm ,0}^{\left(5\right)}{C}_{\pm ,1}^{\left(5\right)}+2C_{\pm ,0}^{\left(4\right)}{C}_{\pm ,1}^{\left(4\right)}·2\\ \hfill +2C_{\pm ,0}^{\left(3\right)}{C}_{\pm ,1}^{\left(3\right)}·\frac{12}{7}+{(C_{\pm ,0}^{\left(3\right)})}^2\left[\frac{1}{2}\frac{1\pm \lambda }{1\mp \lambda }+\frac{6}{7(1\mp \lambda )}+\frac{18}{49(1-{\lambda }^2)}\right]\end{array}$$

(74)

We then find

$$\begin{array}{c}\frac{C_{+,1}^{\left(5\right)}}{C_{+,0}^{\left(5\right)}}=\frac{\lambda }{2}+\frac{25}{7(\lambda -1)}-\frac{57}{14(\lambda +1)}-\frac{2}{\lambda +2}-\frac{9}{2(\lambda +3)}-\frac{8}{\lambda +4}\hfill \\ \frac{C_{-,1}^{\left(5\right)}}{C_{-,0}^{\left(5\right)}}=-\frac{\lambda }{2}+\frac{25}{7(\lambda -1)}-\frac{43}{14(\lambda +1)}+\frac{2}{\lambda +2}+\frac{9}{2(\lambda +3)}+\frac{8}{\lambda +4}-10\hfill \end{array}$$

(75)

by solving the constraints.

For $s\ge 6$, the contributions from higher spin currents of double trace type should be considered. They are given by

$$\begin{array}{c}\hfill J^{(3,3;6)}\sim :J^{\left(3\right)}{J}^{\left(3\right)}:,J^{(3,4;7)}\sim :J^{\left(3\right)}{J}^{\left(4\right)}:\end{array}$$

(76)

for $s=6,7$ and4

$$\begin{array}{c}\hfill J^{(4,4;8)}\sim :J^{\left(4\right)}{J}^{\left(4\right)}:,J^{(3,5;8)}\sim :J^{\left(3\right)}{J}^{\left(5\right)}:,J^{(3,3;8)}\sim :J^{\left(3\right)}{\partial }^2{J}^{\left(3\right)}:\end{array}$$

(77)

for $s=8$. Their precise forms are fixed such as to be primary in the sense of Virasoro algebra as derived in Appendix B5. Once we have the expressions of these currents, we can obtain the coefficients ${(C_{\pm ,0}^{(s_1,s_2;s^{\prime })})}^2$, which are defined as

$${(C_{\pm ,0}^{(s_1,s_2;s^{\prime })})}^2={\left.\frac{{\langle {\mathcal{O}}_{\pm }{\overline{\mathcal{O}}}_{\pm }{J}^{(s_1,s_2;s^{\prime })}\rangle }^2}{{\langle {\mathcal{O}}_{\pm }{\overline{\mathcal{O}}}_{\pm }\rangle }^2\langle J^{(s_1,s_2;s^{\prime })}{J}^{(s_1,s_2;s^{\prime })}\rangle }\right|}_{\mathcal{O}(N^{-2})}$$

(78)

In Appendix B we also compute the three point functions $\langle {\mathcal{O}}_{\pm }{\overline{\mathcal{O}}}_{\pm }{J}^{(s_1,s_2;s^{\prime })}\rangle$ and the two point functions $\langle J^{(s_1,s_2;s^{\prime })}{J}^{(s_1,s_2;s^{\prime })}\rangle$ for the currents in Equations (76) and (77) at the leading order in $1/N$.

Utilizing these results, we obtain $1/N$ corrections to three point functions with single trace currents of $s=6,7,8$. The constraint equations for $s=6$ are

$$\begin{array}{cc}{f}_{\pm \pm }^{\left(6\right)}& =f_{\pm \pm }^{\left(2\right)}\frac{5}{7}+\frac{7{(1\pm \lambda )}^2}{20{(1\mp \lambda )}^2}+\frac{39(1\pm \lambda )}{140{(1\mp \lambda )}^2}+\frac{109}{1960{(1\mp \lambda )}^2}+2C_{\pm ,0}^{\left(6\right)}{C}_{\pm ,1}^{\left(6\right)}+2C_{\pm ,0}^{\left(5\right)}{C}_{\pm ,1}^{\left(5\right)}·\frac{5}{2}\hfill \\ & +2C_{\pm ,0}^{\left(4\right)}{C}_{\pm ,1}^{\left(4\right)}·\frac{25}{9}+2C_{\pm ,0}^{\left(3\right)}{C}_{\pm ,1}^{\left(3\right)}·\frac{25}{14}+{(C_{\pm ,0}^{\left(3\right)})}^2\left[\frac{5}{4}\frac{1\pm \lambda }{1\mp \lambda }+\frac{15}{7(1\mp \lambda )}+\frac{90}{98(1-{\lambda }^2)}\right]\hfill \\ & +{(C_{\pm ,0}^{\left(4\right)})}^2\left[\frac{1}{2}\frac{1\pm \lambda }{1\mp \lambda }+\frac{4}{3(1\mp \lambda )}+\frac{8}{9(1-{\lambda }^2)}\right]+{(C_{\pm ,0}^{(3,3;6)})}^2\hfill \end{array}$$

(79)

where the effect of $J^{(3,3;6)}$ in Equation (76) enters. Solving these equations we find

$$\begin{array}{cc}\frac{C_{+,1}^{\left(6\right)}}{C_{+,0}^{\left(6\right)}}& =\frac{\lambda }{2}-\frac{5}{3(\lambda -2)}+\frac{1315}{84(\lambda -1)}-\frac{1357}{84(\lambda +1)}-\frac{1}{3(\lambda +2)}-\frac{9}{2(\lambda +3)}-\frac{8}{\lambda +4}\hfill \\ & -\frac{25}{2(\lambda +5)}\hfill \\ \frac{C_{-,1}^{\left(6\right)}}{C_{-,0}^{\left(6\right)}}& =-\frac{\lambda }{2}-\frac{5}{3(\lambda -2)}+\frac{1315}{84(\lambda -1)}-\frac{1273}{84(\lambda +1)}+\frac{11}{3(\lambda +2)}+\frac{9}{2(\lambda +3)}+\frac{8}{\lambda +4}\hfill \\ & +\frac{25}{2(\lambda +5)}-15\hfill \end{array}$$

(80)

For spin 7, another double trace operator $J^{(3,4;7)}$ in Equation (76) should be considered as

$$\begin{array}{cc}{f}_{\pm \pm }^{\left(7\right)}& =f_{\pm \pm }^{\left(2\right)}\frac{9}{14}+\frac{17{(1\pm \lambda )}^2}{40{(1\mp \lambda )}^2}+\frac{846(1\pm \lambda )}{2520{(1\mp \lambda )}^2}+\frac{131}{1960{(1\mp \lambda )}^2}+2C_{\pm ,0}^{\left(7\right)}{C}_{\pm ,1}^{\left(7\right)}+2C_{\pm ,0}^{\left(6\right)}{C}_{\pm ,1}^{\left(6\right)}·3\hfill \\ & +2C_{\pm ,0}^{\left(5\right)}{C}_{\pm ,1}^{\left(5\right)}·\frac{45}{11}+2C_{\pm ,0}^{\left(4\right)}{C}_{\pm ,1}^{\left(4\right)}·\frac{10}{3}+2C_{\pm ,0}^{\left(3\right)}{C}_{\pm ,1}^{\left(3\right)}·\frac{25}{14}+{(C_{\pm ,0}^{\left(3\right)})}^2\frac{(3467\pm 42\lambda (89\pm 24\lambda \left)\right)}{490(1-{\lambda }^2)}\hfill \\ & +{(C_{\pm ,0}^{\left(4\right)})}^2\frac{(7\pm 3\lambda )}{6(1-{\lambda }^2)}+{(C_{\pm ,0}^{\left(5\right)})}^2\frac{(31\pm 11\lambda )}{242(1-{\lambda }^2)}+{(C_{\pm ,0}^{(3,3;6)})}^2·3+{(C_{\pm ,0}^{(3,4;7)})}^2\hfill \end{array}$$

(81)

which lead to

$$\begin{array}{c}\frac{C_{+,1}^{\left(7\right)}}{C_{+,0}^{\left(7\right)}}=\frac{\lambda }{2}-\frac{490}{33(\lambda -2)}+\frac{8183}{132(\lambda -1)}-\frac{8249}{132(\lambda +1)}+\frac{424}{33(\lambda +2)}-\frac{9}{2(\lambda +3)}\hfill \\ -\frac{8}{\lambda +4}-\frac{25}{2(\lambda +5)}-\frac{18}{\lambda +6}\hfill \\ \frac{C_{-,1}^{\left(7\right)}}{C_{-,0}^{\left(7\right)}}=-\frac{\lambda }{2}-\frac{490}{33(\lambda -2)}+\frac{8183}{132(\lambda -1)}-\frac{8117}{132(\lambda +1)}+\frac{556}{33(\lambda +2)}+\frac{9}{2(\lambda +3)}\hfill \\ +\frac{8}{\lambda +4}+\frac{25}{2(\lambda +5)}+\frac{18}{\lambda +6}-21\hfill \end{array}$$

(82)

The constraint equations for $C_{\pm ,1}^{\left(8\right)}$ are

$$\begin{array}{c}{f}_{\pm \pm }^{\left(8\right)}=f_{\pm \pm }^{\left(2\right)}\frac{7}{12}+\frac{{(1\pm \lambda )}^2}{{(1\mp \lambda )}^2}\frac{2687}{5600}+\frac{(1\pm \lambda )}{{(1\mp \lambda )}^2}\frac{263}{700}+\frac{1}{{(1\mp \lambda )}^2}\frac{1879}{25200}+2C_{\pm ,0}^{\left(8\right)}{C}_{\pm ,1}^{\left(8\right)}\hfill \\ +2C_{\pm ,0}^{\left(7\right)}{C}_{\pm ,1}^{\left(7\right)}\frac{7}{2}+2C_{\pm ,0}^{\left(6\right)}{C}_{\pm ,1}^{\left(6\right)}\frac{147}{26}+2C_{\pm ,0}^{\left(5\right)}{C}_{\pm ,1}^{\left(5\right)}\frac{245}{44}++2C_{\pm ,0}^{\left(4\right)}{C}_{\pm ,1}^{\left(4\right)}\frac{245}{66}+2C_{\pm ,0}^{\left(3\right)}{C}_{\pm ,1}^{\left(3\right)}\frac{7}{4}\hfill \\ +{(C_{\pm ,0}^{\left(3\right)})}^2\left(\frac{387\pm 418\lambda +113{\lambda }^2}{40(1-{\lambda }^2)}\right)+{(C_{\pm ,0}^{\left(4\right)})}^2\left(\frac{7(47977\pm 41162\lambda +8833{\lambda }^2)}{21780(1-{\lambda }^2)}\right)\hfill \\ +{(C_{\pm ,0}^{\left(5\right)})}^2\left(\frac{7{(31\pm 11\lambda )}^2}{484(1-{\lambda }^2)}\right)+{(C_{\pm ,0}^{\left(6\right)})}^2\left(\frac{{(43\pm 13\lambda )}^2}{338(1-{\lambda }^2)}\right)+{(C_{\pm ,0}^{(3,3;6)})}^2\frac{147}{26}+{(C_{\pm ,0}^{(3,4;7)})}^2\frac{7}{2}\hfill \\ +{(C_{\pm ,0}^{(4,4;8)})}^2+{(C_{\pm ,0}^{(3,5;8)})}^2+{(C_{\pm ,0}^{(3,3;8)})}^2\hfill \end{array}$$

(83)

Here we have taken care of double trace operators $J^{(4,4;8)}$, $J^{(3,5;8)}$, and $J^{(3,3;8)}$ in Equation (77). We then have

$$\begin{array}{c}\frac{C_{+,1}^{\left(8\right)}}{C_{+,0}^{\left(8\right)}}=\frac{\lambda }{2}+\frac{525}{143(\lambda -3)}-\frac{12572}{143(\lambda -2)}+\frac{101311}{429(\lambda -1)}-\frac{203051}{858(\lambda +1)}+\frac{12286}{143(\lambda +2)}\hfill \\ -\frac{2337}{286(\lambda +3)}-\frac{8}{\lambda +4}-\frac{25}{2(\lambda +5)}-\frac{18}{\lambda +6}-\frac{49}{2(\lambda +7)}\hfill \\ \frac{C_{-,1}^{\left(8\right)}}{C_{-,0}^{\left(8\right)}}=-\frac{\lambda }{2}+\frac{525}{143(\lambda -3)}-\frac{12572}{143(\lambda -2)}+\frac{101311}{429(\lambda -1)}-\frac{202193}{858(\lambda +1)}+\frac{12858}{143(\lambda +2)}\hfill \\ +\frac{237}{286(\lambda +3)}+\frac{8}{\lambda +4}+\frac{25}{2(\lambda +5)}+\frac{18}{\lambda +6}+\frac{49}{2(\lambda +7)}-28\hfill \end{array}$$

(84)

as solutions to the constraint equations.

Since the three point functions were already obtained with finite $N,k$ in [9,10,11] for $s=3,4,5$, they can be compared to our results in principle. Instead of doing so, we utilize a simpler relation, which is on the ratio of three point functions (see (4.52) of [11])

$$\begin{array}{c}\hfill \frac{\langle {\mathcal{O}}_{+}{\overline{\mathcal{O}}}_{+}{J}^{\left(s\right)}\rangle }{\langle {\mathcal{O}}_{-}{\overline{\mathcal{O}}}_{-}{J}^{\left(s\right)}\rangle }={(-1)}^s\frac{(k+N+1)}{(k+N)}\prod _{n=1}^{s-1}\left[\frac{nk+(n+1)N+n}{nk+(n-1)N}\right]\end{array}$$

(85)

The relation was derived for $s=2,3,4,5$ by using the explicit results and conjectured for generic s based on them. The expression up to the $1/N$ order becomes

$$\begin{array}{cc}\frac{\langle {\mathcal{O}}_{+}{\overline{\mathcal{O}}}_{+}{J}^{\left(s\right)}\rangle }{\langle {\mathcal{O}}_{-}{\overline{\mathcal{O}}}_{-}{J}^{\left(s\right)}\rangle }& =\frac{C_{+,0}^{\left(s\right)}+\frac{1}{N}{C}_{+,1}^{\left(s\right)}}{C_{-,0}^{\left(s\right)}+\frac{1}{N}{C}_{-,1}^{\left(s\right)}}+\mathcal{O}(N^{-2})\hfill \\ & ={(-1)}^s\prod _{n=1}^{s-1}\left(\frac{n+\lambda }{n-\lambda }\right)\left[1+\frac{1}{N}\left(\lambda +\sum _{m=1}^{s-1}\frac{m\lambda }{m+\lambda }\right)+\mathcal{O}\left(N^{-2}\right)\right]\hfill \end{array}$$

(86)

Thus, at the leading order in $1/N$, we have

$$\begin{array}{c}\hfill \frac{C_{+,0}^{\left(s\right)}}{C_{-,0}^{\left(s\right)}}={(-1)}^s\prod _{n=1}^{s-1}\left(\frac{n+\lambda }{n-\lambda }\right)\end{array}$$

(87)

We can easily check that Equation (68) satisfy this condition. The relation in Equation (86) at the next leading order in $1/N$ implies

$$\begin{array}{c}\hfill \frac{C_{+,1}^{\left(s\right)}}{C_{+,0}^{\left(s\right)}}-\frac{C_{-,1}^{\left(s\right)}}{C_{-,0}^{\left(s\right)}}=\lambda +\sum _{m=1}^{s-1}\frac{m\lambda }{m+\lambda }\end{array}$$

(88)

We have confirmed our results (and the conjectured relation in Equation (86)) by showing that our results on $C_{\pm ,1}^{\left(s\right)}$ for $s=3,\dots ,8$ satisfy this equation.

Before ending this section, we would like to make comments on normalized three point functions

$$\begin{array}{c}\hfill C_{\pm }^{\left(2\right)}=\frac{\langle {\mathcal{O}}_{\pm }{\overline{\mathcal{O}}}_{\pm }{J}^{\left(2\right)}\rangle }{\langle {\mathcal{O}}_{\pm }{\overline{\mathcal{O}}}_{\pm }\rangle {\langle J^{\left(2\right)}{J}^{\left(2\right)}\rangle }^{1/2}}\end{array}$$

(89)

with the energy momentum tensor $T\propto J^{\left(2\right)}$. They do not appear in the decomposition of Virasoro conformal blocks but can be fixed by the conformal Ward identity as

$$\begin{array}{c}\hfill C_{\pm }^{\left(2\right)}=\frac{1}{N^{1/2}}\left(C_{\pm ,0}^{\left(2\right)}+\frac{1}{N}{C}_{\pm ,1}^{\left(2\right)}+\mathcal{O}(N^{-2})\right)=\sqrt{\frac{2h_{\pm }^2}{c}}\end{array}$$

(90)

In particular, they lead to Equation (55) and

$$\begin{array}{c}\hfill 2C_{\pm ,0}^{\left(2\right)}{C}_{\pm ,1}^{\left(2\right)}=f_{\pm \pm }^{\left(2\right)}\end{array}$$

(91)

with Equation (43), or equivalently

$$\begin{array}{c}\hfill \frac{C_{+,1}^{\left(2\right)}}{C_{+,0}}=\frac{\lambda }{2}-\frac{1}{2(\lambda +1)},\frac{C_{-,1}^{\left(2\right)}}{C_{-,0}}=-\frac{\lambda }{2}+\frac{1}{2(\lambda +1)}-1\end{array}$$

(92)

As a consistence check, we can show that they satisfy Equation (88) as well.

5. Conclusions and Open Problems

We have developed a new method to compute three point functions of two scalar operators and a higher spin current in Equation (4) in 2d W${}_N$ minimal model. This model can be described by the coset in Equation (1) with two parameters $N,k$, and we analyze it in $1/N$ expansion in terms of ’t Hooft parameter $\lambda =N/(N+k)$ in Equation (3). We decompose scalar four point functions $G_{\pm \pm }\left(z\right)$ in Equations (7) and (8) and $G_{-+}\left(z\right)$ in Equation (9) by Virasoro conformal blocks. The four point functions were computed exactly with finite $N,k$ in [16], and Virasoro conformal blocks can be obtained including $1/N$ corrections, say, by analyzing Zamolodchikov’s recursion relation [17]. Solving the constraint equations from the decomposition, we can obtain three point functions including $1/N$ corrections. At the leading order in $1/N$, we can easily reproduce the known results in [8] because Virasoro conformal blocks reduce to global blocks in this case. At the next leading order, we have obtained $1/N$ corrections to the three point functions up to spin 8. Previously exact results were known for $s=3,4,5$ in [9,10,11], and our findings for $s=6,7,8$ are new. We have confirmed our results by checking that the conjectured relation in Equation (88) is satisfied.

We have evaluated $1/N$ corrections only up to spin 8 case because of the following two obstacles. One comes from $1/c$ corrections to Virasoro conformal blocks. Up to the required order in $1/c$, closed forms can be obtained, for instance, by following the method in [23] except for $f_c(s,z)$ in Equation (49). In Equation (50) (or in [23]), the function $f_c(s,z)$ is given up to the order $z^{5+s}$, but we need the term at order $z^{6+s}$ with $s=3$ for spin 9 computation. We have not tried to do so, but it should be possible to obtain the terms at higher orders in z without a lot of efforts. Another is related to the contributions from higher spin currents of double trace type as analyzed in Appendix B. In order to obtain primary operators of this type, we have used commutation relations in Equation (A15), which are borrowed from [24]. For spin 9, a current of the form $J^{(3,6;9)}\sim :J^{\left(3\right)}{J}^{\left(6\right)}:$ would give some contributions. However, in order to find its primary form, we need the commutation relation between $W,Y$, which is currently not available. At the order in $1/c$ which do not vanish at $c\to \infty$, we can derive the commutation relations involving more higher spin currents, for instance, from dual Chern-Simons description as in [25,26,27,28]. The computation is straightforward but might be tedious. In any case, it is definitely possible to obtain the $1/N$ corrections of three point functions for $s\ge 9$, and it is desired to have expressions for generic s.

There are many open problems we would like to think about. Because of the simplicity of our method, it is expected to be applicable to more generic cases. For example, it is worth generalizing the current analysis to supersymmetric cases. Recently, it becomes possible to discuss relations between 3d higher spin theory and superstrings by introducing extended supersymmetry to the duality by [1]. Higher spin holography with $\mathcal{N}=3$ supersymmetry has been developed in a series of works [29,30,31], while large or small $\mathcal{N}=4$ supersymmetry has been utilized through the well-studied holography with symmetric orbifold in [32,33]. Previous works on the subject may be found in [34,35,36]. As mentioned in introduction, the main motivation to examine $1/N$ corrections in 2d W${}_N$ minimal model is to learn quantum effects in dual higher spin theory. We would like to report on our recent progress in a separate publication [37].