Five-Point Correlation Numbers in Minimal Liouville Gravity

N-point correlation numbers in minimal Liouville gravity have been studied. It has been demonstrated how to use Al. Zamolodchikov’s higher equations of motion in Liouville field theory to explicitly calculate these numbers. An explicit expression has been obtained for the case of N = 5.

1. INTRODUCTION

There are several approaches to two-dimensional quantum gravity [1]. A “continuous” approach is based on the study of a functional integral over all Riemannian metrics in two dimensions and over matter fields. When the field theory of matter is conformal, after fixing a gauge, we arrive at the so-called Liouville gravity theory [2]; if the matter sector is described by minimal models of the conformal field theory [3], this is minimal Liouville gravity (MLG) [4–6]. Another “discrete” approach is based on the consideration of certain integrals over $L \times L$ matrices in the limit $L \to \infty$; this is the so-called approach of matrix models (MMs) (this approach is reviewed, e.g., in [7, 8]).

It is generally expected that these approaches should give equivalent results; the calculations of the gravity dimensions and correlation numbers performed in [4, 9–11] confirm this hypothesis. Since the exact calculation from the side of MLG is difficult, such verifications were carried out only up to four-point correlation numbers (four-point correlation function from the side of MLG was calculated in [12, 13]). At the same time, expressions for any N-point correlation function from the side of MMs in the single-matrix case can be obtained very easily [11]. In this work, we try to directly calculate higher correlation numbers in MLG (we consider only correlation functions on a sphere).

Correlation numbers in MLG are defined as vacuum averages of the product of physical (Becchi–Rouet–Stora–Tyutin (BRST)-closed) operators. Such operators are either local 0-dimensional fields or integrals of local densities ((1, 1)-dimensional fields) over the world surface. We study correlation functions of local operators ${{W}_{{m,n}}}$ with ghost number 1 and of integrals of the corresponding local densities ${{U}_{{m,n}}}(x)$ with ghost number 0 (the definitions and properties of all these fields are given in the next section). Because of the anomaly of the ghost number, such a correlation function should include three W operators for this correlation function to be nonzero on the sphere.

Consequently, the three-point function does not contain integrals over the world surface and to obtain it, it is sufficient to know the structure constants in minimal models [3] and in Liouville field theory [14–16]. The four-point correlation function includes one integral over the field position ${{U}_{{m,n}}}(x)$. In [13], it was proposed to calculate such a correlation function with the so-called “higher equations of motion” [17] in Liouville field theory, which allow one to reduce the integral $\int {{d}^{2}}x$ to computable boundary contributions from the vicinities of the points ${{x}_{i}},\;i = 1,2,3$ of insertion of the W operators and the point $x = \infty$.

An important circumstance used in this calculation is the BRST invariance of all other fields in the correlation function and, therefore, the possibility of rejecting the BRST-exact terms in the higher equations of motion. This is not the fact for correlation functions with more than one integral of the density $\int {{d}^{2}}{{y}_{i}}{\kern 1pt} {{U}_{{m,n}}}({{y}_{i}})$; such integrals are BRST-invariant only up to the boundary terms that cannot be omitted. In this work, we however demonstrate that the inclusion of $\mathcal{Q}$-exact terms in the higher equations of motion applied to one of the integrable densities $\int {{d}^{2}}{{y}_{1}}{\kern 1pt} {{U}_{{m,n}}}({{y}_{1}})$ reduces the integral with respect to ${{y}_{1}}$ to the boundary contributions in the form of a four-point correlation function; the contribution from the vicinity of the insertion of the second density ${{y}_{2}}$ has a form similar to the contributions from the vicinities of the points ${{x}_{i}}$.

This article is organized as follows. The properties of MLG necessary for the further calculations are briefly summarized in Section 2. Section 3 presents the proposed method for the calculation of the five-point correlation function and gives the result.

2. PRELIMINARIES

Minimal Liouville gravity is a conformal field theory with the total central charge 0, which consists of the Liouville field theory describing gravity, minimal model of conformal field theory as the matter sector, and the (B, C) system of reparameterizing BRST ghosts with a central charge of –26:

$${{A}_{{{\text{MLG}}}}} = {{A}_{{\text{L}}}} + {{A}_{{{{\mathcal{M}}_{{q,q'}}}}}} + \underbrace {\frac{1}{\pi }\int {{d}^{2}}x{\kern 1pt} \left( {C\bar {\partial }B + \bar {C}\partial \bar {B}} \right)}_{{{A}_{{{\text{ghost}}}}}}.$$

(1)

The central charge of the Liouville theory is determined by the relation c_L + c_M + c_gh = 0 and the numbers q and q', which are the characteristics of the minimal model.

Matter Sector ${{\mathcal{M}}_{{q,q'}}}$

The minimal models of conformal field theory is defined by a pair of coprime integers q and q'; its space of states consists of a finite set of irreducible representations of the Virasoro algebra with a degenerate highest weight, i.e., degenerate primary fields Φ_m,n with 1 ≤ m < q and 1 ≤ n < q' and their descendants. Let q/q' = b²; then the central charge of ${{\mathcal{M}}_{{q',q}}}$ is

$$c = 1 - 6({{b}^{{ - 1}}} - b{{)}^{2}},$$

(2)

and the dimensions of the degenerate fields Φ_m,n are

$$\begin{array}{*{20}{c}} {\Delta _{{m,n}}^{M} = - {{{({{b}^{{ - 1}}} - b)}}^{2}}{\text{/}}4 + \lambda _{{m, - n}}^{2},} \\ {{{\lambda }_{{m,n}}} = (m{{b}^{{ - 1}}} + nb){\text{/}}2.} \end{array}$$

(3)

The primary fields Φ_m,n will also be denoted as Φ_α, where the parameter α is related to the dimension as $\Delta _{\alpha }^{{(M)}} = \alpha (\alpha - {{b}^{{ - 1}}} + b)$. In addition, the following requirements are imposed on the minimal model.

(i) All degenerate fields Φ_m,n constitute the spectrum of the theory.

(ii) Factorization in the submodule generated by the singular mn-level descendant is performed in each Virasoro module generated by Φ_m,n:

$$D_{{m,n}}^{{{\text{(M)}}}}{{\Phi }_{{m,n}}} = \bar {D}_{{m,n}}^{{{\text{(M)}}}}{{\Phi }_{{m,n}}} = 0,$$

(4)

where $D_{{m,n}}^{{{\text{(M)}}}}$ ($\bar {D}_{{m,n}}^{{{\text{(M)}}}}$) are certain operators composed of holomorphic (antiholomorphic) Virasoro generators $L_{n}^{M}$ ($\bar {L}_{n}^{M}$) in the minimal model.

(iii) It is also assumed that ${{\Phi }_{{q - m,q' - n}}} = {{\Phi }_{{m,n}}}$.

These requirements unambiguously specify the theory and, in particular, make it possible to unambiguously determine the structure constants in the minimal model.

Liouville Field Theory

The gravity sector is described by the quantized version of the classical field theory based on the L-iouville action. It is a conformal field theory with central charge parameterized by the variable b or Q = b^–1 + b as

$${{c}_{{\text{L}}}} = 1 + 6{{Q}^{2}}.$$

(5)

According to the requirement of vanishing the total central charge in MLG, the parameter b in the Liouville theory should coincide with $\sqrt {q{\text{/}}q{\kern 1pt} '}$ defined in the preceding section. The parameter b enters the local Lagrangian of the theory

$${{\mathcal{L}}_{{\text{L}}}} = \frac{1}{{4\pi }}{{\left( {{{\partial }_{a}}\phi } \right)}^{2}} + \mu {{e}^{{2b\phi }}}.$$

(6)

Here, μ is an additional parameter of the theory called the cosmological constant, and ϕ is the dynamical variable remaining after fixing the gauge in the integral over metrics.

Primary fields in the theory are the exponential operators ${{V}_{a}} \equiv \exp \left( {2a\phi } \right)$, parameterized by a continuous (complex) parameter a; the corresponding conformal dimension is

$$\Delta _{a}^{{{\text{(L)}}}} = a(Q - a).$$

(7)

Two types of Liouville operators will be important below to construct the physical fields in MLG. Operators of the first type are degenerate fields ${{V}_{{m,n}}} \equiv {{V}_{{{{a}_{{m,n}}}}}}$ with

$${{a}_{{m,n}}} = - {{b}^{{ - 1}}}\frac{{(m - 1)}}{2} - b\frac{{(n - 1)}}{2}.$$

(8)

These fields satisfy the equations $D_{{m,n}}^{{({\text{L}})}}{{V}_{{m,n}}} = \bar {D}_{{m,n}}^{{({\text{L}})}}{{V}_{{m,n}}}$ = 0, which are analogous to those in the minimal models. The operators of the second type are the primary fields V_m,–n, whose role together with the ghost field C is to dress the primary fields V_m,–n used to construct the cohomology classes under study in MLG (see below).

The three-point correlation function ${{C}_{{\text{L}}}}({{a}_{1}},{{a}_{2}},{{a}_{3}})$ = ${{\left\langle {{{V}_{{{{a}_{1}}}}}(0){{V}_{{{{a}_{2}}}}}(1){{V}_{{{{a}_{3}}}}}(\infty )} \right\rangle }_{{\text{L}}}}$ in the Liouville theory [16] is known in the explicit form

$$\begin{gathered} {{C}_{{\text{L}}}}({{a}_{1}},{{a}_{2}},{{a}_{3}}) \\ = {{\left( {\pi \mu \gamma ({{b}^{2}}){{b}^{{2 - 2{{b}^{2}}}}}} \right)}^{{(Q - a)/b}}}\frac{{{{\Upsilon }_{b}}(b)}}{{{{\Upsilon }_{b}}(a - Q)}}\prod\limits_{i = 1}^3 \frac{{{{\Upsilon }_{b}}(2{{a}_{i}})}}{{{{\Upsilon }_{b}}(a - {{a}_{i}})}}, \\ \end{gathered} $$

(9)

where $a = {{a}_{1}} + {{a}_{2}} + {{a}_{3}}$ and ${{\Upsilon }_{b}}(x)$ is a certain special function [16]. The operator product expansion (OPE) in the Liouville theory has the form

$$\begin{gathered} {{V}_{{{{a}_{1}}}}}(x){{V}_{{{{a}_{2}}}}}(0) \\ = {{\int }^{'}}\frac{{dP}}{{4\pi }}C_{{{{a}_{1}},{{a}_{2}}}}^{{{\text{(L)}}Q/2 + iP}}{{(x\bar {x})}^{{\Delta _{{Q/2 + iP}}^{{{\text{(L)}}}} - \Delta _{{{{a}_{1}}}}^{{{\text{(L)}}}} - \Delta _{{{{a}_{2}}}}^{{{\text{(L)}}}}}}}\left[ {{{V}_{{Q/2 + iP}}}(0)} \right], \\ \end{gathered} $$

(10)

where the structure constant is related to Eq. (9) as $C_{{{{a}_{1}},{{a}_{2}}}}^{{{\text{(L)}}p}} = {{C}_{{\text{L}}}}(g,a,Q - p)$ and integration is performed over the imaginary axis if ${{a}_{1}}$ and ${{a}_{2}}$ are in the region

$$\left| {Q{\text{/}}2 - {\text{Re}}{{a}_{1}}} \right| + \left| {Q{\text{/}}2 - {\text{Re}}{{a}_{2}}} \right| < Q{\text{/}}2.$$

(11)

Analyticity in other regions of the parameters is required; i.e., the integral (10) should be supplemented by residues at the poles that cross the contour during the analytic continuation from region (11). These additional contributions (“discrete terms”) are particularly important for Eq. (10) to reproduce the operator product expansion with degenerate ${{V}_{{m,n}}}$ fields.

Ghosts and BRST Invariance

The ghost sector is the fermionic (B, C) system of with (2, –1)-dimensional fields

$${{A}_{{{\text{gh}}}}} = \frac{1}{\pi }\int (C\bar {\partial }B + \bar {C}\partial \bar {B}){{d}^{2}}x$$

(12)

and with a central charge of –26; this system appears after fixing the gauge by the Faddeev–Popov method. Odd BRST symmetry is generated in MLG by the (holomorphic) charge

$$\mathcal{Q} = \oint \left( {CT + C\partial CB} \right)\frac{{dz}}{{2\pi i}},$$

(13)

where T is the energy–momentum tensor of matter and Liouville theory. By definition the physical states in MLG belong to BRST chorology classes of the charge $\mathcal{Q}$ and the antiholomorphic charge $\bar {\mathcal{Q}}$.

Physical Fields and Their Correlation Functions

The simplest cohomology representatives of ghost number zero can be obtained by dressing the primary fields Φ_m,n with the Liouville operators V_m,–n; the total dimension ${{U}_{{m,n}}} \equiv {{V}_{{m, - n}}}{{\Phi }_{{m,n}}}$ is (1, 1). The BRST variation of U_m,n is

$$\mathcal{Q}{{U}_{{m,n}}} = \partial (C{{U}_{{m,n}}});$$

(14)

consequently, the integrals of U_m,n over the sphere are BRST invariant (with accuracy to possible boundary terms depending on other insertions).

To obtain physical fields of ghost number 1, the (0, 0) form ${{W}_{{m,n}}} \equiv C\bar {C}{{U}_{{m,n}}}$ can be considered instead of integration; it is BRST-closed, $\mathcal{Q}{{W}_{{m,n}}} = \bar {\mathcal{Q}}{{W}_{{m,n}}} = 0$. These fields will also be parameterized by the num-ber a as ${{W}_{a}} = {{V}_{a}}{{\Phi }_{{a - b}}}$.

In minimal gravity, there is an additional set of BRST-closed fields with ghost number zero that form the so-called “ground ring” [18]. They are constructed from descendants of degenerate fields in both sectors and have the general form

$${{O}_{{m,n}}}(x) = {{H}_{{m,n}}}{{\bar {H}}_{{m,n}}}{{\Theta }_{{m,n}}},\quad {{\Theta }_{{m,n}}} \equiv {{V}_{{m,n}}}{{\Phi }_{{m,n}}},$$

(15)

where H_m,n is the polynomial of the degree $mn - 1$ of Virasoro generators and ghosts B and C. The general form for H_m,n is unknown, but it can be found in each particular case by requiring $\mathcal{Q}$-closeness of O_m,n. These operators play an important role in the derivation of “the higher equations of motion.”

Ground ring operators have the following properties.

(i) Correlation functions are independent of their position; i.e.,

$$\partial {{O}_{{m,n}}} = {\text{BRST-exact}}.$$

(16)

(ii) Simple fusion rules in cohomology classes:

$${{O}_{{m,n}}}(x){{O}_{{m',n'}}}(0)$$

$$ = \sum\limits_{r = |m - m'| + 1:2}^{m + m' + 1} \sum\limits_{s = |n - n'| + 1:2}^{n + n' + 1} G_{{r,s}}^{{(m,n)|(m',n')}}{{O}_{{r,s}}}(0)$$

(17)

$$ + \;{\text{BRST-exact}}.$$

(iii) Simple fusion rules with operators W_a with ghost number 1:

$$\begin{gathered} {{O}_{{m,n}}}{{W}_{a}} = \sum\limits_{r = - m + 1:2}^{m - 1} \sum\limits_{s = - n + 1:2}^{n - 1} A_{{r,s}}^{{(m,n)}}(a){{W}_{{a + \frac{{r{{b}^{{ - 1}}} + sb}}{2}}}} \\ + \;{\text{BRST-exact}}. \\ \end{gathered} $$

(18)

The operators ${{O}_{{m,n}}}$ and ${{W}_{{m,n}}}$ can be renormalized to make $G_{{r,s}}^{{(m,n)|(m',n')}}$ and $A_{{r,s}}^{{(m,n)}}(a)$ identity matrices; more precisely,

$$G_{{r,s}}^{{(m,n)|(m',n')}} = \frac{{{{\Lambda }_{{m,n}}}{{\Lambda }_{{m',n'}}}}}{{{{\Lambda }_{{r,s}}}}};\quad {{\Lambda }_{{m,n}}} = \frac{{{{B}_{{m,n}}}}}{\pi }\mathcal{N}({{a}_{{m, - n}}}),$$

(19)

$$A_{{r,s}}^{{(m,n)}}(a) = \frac{{{{B}_{{m,n}}}}}{\pi }\frac{{\mathcal{N}(a)\mathcal{N}({{a}_{{m, - n}}})}}{{\mathcal{N}(a + {{\lambda }_{{r,s}}})}},$$

(20)

$$\mathcal{N}(a) = \frac{\pi }{{{{{(\pi \mu )}}^{{a/b}}}}}{{\left[ {\frac{{\gamma (2ab - {{b}^{2}})\gamma (2a{{b}^{{ - 1}}} - {{b}^{{ - 2}}})}}{{{{\gamma }^{{2a/b - 1}}}({{b}^{2}})\gamma (2 - {{b}^{{ - 2}}})}}} \right]}^{{1/2}}},$$

(21)

where ${{B}_{{m,n}}}$ is defined in Eq. (23). Thus, the necessary renormalization is specified as ${{\mathcal{O}}_{{m,n}}} = \Lambda _{{m,n}}^{{ - 1}}{{O}_{{m,n}}}$ and ${{\mathcal{W}}_{a}} = \mathcal{N}{{(a)}^{{ - 1}}}{{W}_{a}}$.

Higher Equations of Motion and MLG

The higher equations of motion in the Liouville theory involve the so-called logarithmic fields defined as

$$V_{a}^{'}(x) = \frac{1}{2}\frac{\partial }{{\partial a}}{{V}_{a}}(x).$$

(22)

Such derivatives with respect to the parameter at the point $a = {{a}_{{m,n}}}$ will be denoted as $V_{{m,n}}^{'}$.

According to the higher equations of motion, the descendants of the logarithmic operators can be identified (up to a constant) with primary fields ${{V}_{{m, - n}}}$ [17]; i.e.,

$$\begin{gathered} D_{{m,n}}^{{(L)}}\bar {D}_{{m,n}}^{{(L)}}V_{{m,n}}^{'} \\ = {{B}_{{m,n}}}{{V}_{{m, - n}}},{\kern 1pt} {{B}_{{m,n}}} = (\pi \mu \gamma ({{b}^{2}}){{b}^{{2 - 2{{b}^{2}}}}}{{)}^{n}}\frac{{\Upsilon _{b}^{'}(2{{\alpha }_{{m,n}}})}}{{{{\Upsilon }_{b}}(2{{\alpha }_{{m, - n}}})}}. \\ \end{gathered} $$

(23)

Consequently, (see [13, 19])

$${{U}_{{m,n}}} = B_{{m,n}}^{{ - 1}}(\bar {\partial } - \bar {\mathcal{Q}}{{\bar {B}}_{{ - 1}}})(\partial - \mathcal{Q}{{B}_{{ - 1}}})O_{{m,n}}^{'},$$

(24)

where $O_{{m,n}}^{'}: = {{H}_{{m,n}}}{{\bar {H}}_{{m,n}}}\Theta _{{m,n}}^{'}$, $\Theta _{{m,n}}^{'}: = {{\Phi }_{{m,n}}}V_{{m,n}}^{'}$, and ${{B}_{{ - 1}}}$ is the mode of the ghost field B.

Four-Point Correlation Function

We now apply Eq. (24) to calculate the four-point correlation function following [12]. Separating the normalization constants $\prod\nolimits_{i = 1}^3 {\mathcal{N}({{a}_{i}})} \times \mathcal{N}({{a}_{{m, - n}}})$ and taking into account the renormalized operators, we consider

$$\begin{gathered} {{C}_{4}}({{a}_{1}},{{a}_{2}},{{a}_{3}}{\text{|}}m,n) \\ \equiv \frac{1}{{{{Z}_{{\text{L}}}}}}\left\langle {\int {{d}^{2}}x{\kern 1pt} \frac{{{{U}_{{m,n}}}(x)}}{{\mathcal{N}({{a}_{{m, - n}}})}}{{\mathcal{W}}_{{{{a}_{1}}}}}({{x}_{1}}){{\mathcal{W}}_{{{{a}_{2}}}}}({{x}_{2}}){{\mathcal{W}}_{{{{a}_{3}}}}}({{x}_{3}})} \right\rangle . \\ \end{gathered} $$

(25)

Since the W operators are $\mathcal{Q}$-closed, omitting the BRST-exact terms, we obtain

$${{Z}_{{\text{L}}}}{\kern 1pt} {{C}_{4}}({{a}_{1}},{{a}_{2}},{{a}_{3}}{\text{|}}m,n)$$

$$ = \left\langle {\int {{d}^{2}}x{\kern 1pt} \partial \bar {\partial }\frac{{O_{{m,n}}^{'}}}{{{{B}_{{m,n}}}\mathcal{N}({{a}_{{m, - n}}})}}{\kern 1pt} {{\mathcal{W}}_{{{{a}_{1}}}}}{{\mathcal{W}}_{{{{a}_{2}}}}}{{\mathcal{W}}_{{{{a}_{3}}}}}} \right\rangle $$

(26)

$$ = \frac{1}{\pi }\left\langle {\int {{d}^{2}}x{\kern 1pt} \partial \bar {\partial }\mathcal{O}_{{m,n}}^{'}{\kern 1pt} {{\mathcal{W}}_{{{{a}_{1}}}}}{{\mathcal{W}}_{{{{a}_{2}}}}}{{\mathcal{W}}_{{{{a}_{3}}}}}} \right\rangle .$$

Then, we can use the Stokes formula to calculate the integral with respect to x because it includes a total derivative. The boundary contributions from the vicinities of the $x = {{x}_{i}}$ and $x = \infty$ points $O_{{m,n}}^{'}$ are nonzero because the fusion of $O{\kern 1pt} '$ with W results in logarithmic terms, which provide the delta function after differentiation. These logarithmic terms have to be calculated.

The contribution from the infinity point is determined by the behavior of $V_{{m,n}}^{'}$ at $x \to \infty$ (see [13]):

$$\begin{gathered} V_{{1,2}}^{'}(x) \sim - \Delta _{{m,n}}^{'}\log (x\bar {x}){{V}_{{1,2}}}(0),{\kern 1pt} \\ \Delta _{{m,n}}^{'} \equiv 2{{\lambda }_{{m,n}}} = m{{b}^{{ - 1}}} + nb. \\ \end{gathered} $$

(27)

The behavior of $O_{{m,n}}^{'}$ is similar; at infinity, it can be replaced by ${{O}_{{m,n}}}$ with the same coefficient and logarithm. The corresponding boundary contribution has the form

$$ - 2{{\lambda }_{{m,n}}}\left\langle {{{\mathcal{O}}_{{m,n}}}(0){{\mathcal{W}}_{{{{a}_{1}}}}}({{x}_{1}}){{\mathcal{W}}_{{{{a}_{2}}}}}({{x}_{2}}){{\mathcal{W}}_{{{{a}_{3}}}}}({{x}_{3}})} \right\rangle .$$

(28)

This correlation function is independent of the position of ${{O}_{{m,n}}}$ (following from the above properties); consequently, it can be shifted toward any of the $\mathcal{W}$ fields and the OPE can be performed, resulting in, e.g.,

$$\begin{gathered} - 2{{\lambda }_{{m,n}}}\sum\limits_{r = - m + 1:2}^{m - 1} \\ \times \;\sum\limits_{s = - n + 1:2}^{n - 1} \left\langle {{{\mathcal{W}}_{{{{a}_{1}} + {{\lambda }_{{r,s}}}}}}({{x}_{1}}){{\mathcal{W}}_{{{{a}_{2}}}}}({{x}_{2}}){{\mathcal{W}}_{{{{a}_{3}}}}}({{x}_{3}})} \right\rangle . \\ \end{gathered} $$

(29)

The logarithmic factors in the OPE of $O{\kern 1pt} '$$W$ can appear only from the differentiation of the power factors $(x\bar {x})$ in the discrete terms in the Liouville part with respect to a. In particular, for the OPE of $V_{{1,2}}^{'}{{V}_{a}}$, we obtain

$$\begin{gathered} \log (x\bar {x})(q_{{0,1}}^{{(1,2)}}(a)(x\bar {x}{{)}^{{ab}}}C_{L}^{ + }(a)[{{V}_{{a - b/2}}}(0)] \\ + \;q_{{0, - 1}}^{{(1,2)}}(a)(x\bar {x}{{)}^{{1 - ab + {{b}^{2}}}}}C_{L}^{ - }(a)[{{V}_{{a + b/2}}}(0)]), \\ \end{gathered} $$

(30)

$$q_{{r,s}}^{{(m,n)}} \equiv \left| {a - {{\lambda }_{{r,s}}} - \frac{Q}{2}} \right| - {{\lambda }_{{m,n}}}.$$

(31)

In other words, the result is similar to the OPE with the conventional primary field ${{V}_{{1,2}}}$, besides the additional factors $q_{{r,s}}^{{(1,2)}}$. This is valid for the general case $V_{{m,n}}^{'}$. Multiplying the OPE in the Liouville theory and minimal models and applying ${{H}_{{m,n}}}{{\bar {H}}_{{m,n}}}$, we obtain the expression similar to Eq. (18) in the form

$$\begin{gathered} \mathcal{O}_{{m,n}}^{'}(x){{\mathcal{W}}_{a}}(0) \\ = \log (x\bar {x})\sum\limits_{r = - m + 1:2}^{m - 1} \sum\limits_{s = - n + 1:2}^{n - 1} q_{{r,s}}^{{(m,n)}}(a){{\mathcal{W}}_{{a - {{\lambda }_{{r,s}}}}}} + \ldots . \\ \end{gathered} $$

(32)

The corresponding contributions in the four-point correlation function (with an additional sign of – because of the opposite orientations of the boundary contours for the vicinities of ${{x}_{i}}$ and $\infty$) have the form

$$ - \sum\limits_{i = 1}^3 \sum\limits_{r = - m + 1:2}^{m - 1} \sum\limits_{s = - n + 1:2}^{n - 1} q_{{r,s}}^{{(m,n)}}({{a}_{i}})\langle {{\mathcal{W}}_{{{{a}_{i}} - {{\lambda }_{{r,s}}}}}} \ldots {\kern 1pt} \rangle .$$

(33)

Then, since the three-point correlation functions normalized to $\mathcal{N}(a)$ are independent of the parameters ${{a}_{i}}$ and are $- {{b}^{{ - 2}}}({{b}^{{ - 4}}} - 1)$ [13, 20], we obtain

$$\begin{gathered} {{C}_{4}}({{a}_{1}},{{a}_{2}},{{a}_{3}}{\text{|}}m,n) = - ({{b}^{{ - 6}}} - {{b}^{{ - 2}}}) \\ \times \;\left[ { - 2mn{{\lambda }_{{mn}}} - \sum\limits_{i = 1}^3 \sum\limits_{r = - m + 1:2}^{m - 1} \sum\limits_{s = - n + 1:2}^{n - 1} q_{{r,s}}^{{(m,n)}}({{a}_{i}})} \right]. \\ \end{gathered} $$

(34)

3. FIVE-POINT CORRELATION FUNCTION

In this section, we modify the above method to the case of five-point (and higher) correlation functions. For simplicity, we consider $(2,2p + 1)$ MLG, where the normalized correlation function has the form

$${{C}_{5}}({{a}_{1}},{{a}_{2}},{{a}_{3}}{\text{|}}{{k}_{1}},{{k}_{2}}) = Z_{L}^{{ - 1}}$$

$$ \times \;\left\langle {\int {{d}^{2}}x{\kern 1pt} \frac{{{{U}_{{1,{{k}_{1}} + 1}}}(x)}}{{\mathcal{N}({{a}_{{1, - 1 - {{k}_{1}}}}})}}} \right.$$

(35)

$$ \times \;\left. {\int {{d}^{2}}y{\kern 1pt} \frac{{{{U}_{{1,{{k}_{2}} + 1}}}(y)}}{{\mathcal{N}({{a}_{{1, - 1 - {{k}_{2}}}}})}}{{\mathcal{W}}_{{{{a}_{1}}}}}({{x}_{1}}){{\mathcal{W}}_{{{{a}_{2}}}}}({{x}_{2}}){{\mathcal{W}}_{{{{a}_{3}}}}}({{x}_{3}})} \right\rangle .$$

Without loss of generality, we assume that ${{k}_{1}} \leqslant {{k}_{2}}$. First, we apply the higher equations of motion for the field ${{U}_{{{{k}_{1}}}}}$; the term with the second derivative $\partial \bar {\partial }\mathcal{O}_{{m,n}}^{'}$ can be reduced to the boundary contributions near the ${{x}_{i}}$, $y$, and $\infty$ points.

Contributions at x ~ x _i

Here, we apply the OPE to $\mathcal{O}'$${{\mathcal{W}}_{a}}({{x}_{i}})$. As above, only the logarithmic terms are important, which give

$$\begin{gathered} - \sum\limits_{i = 1}^3 \sum\limits_{s = - {{k}_{1}}:2}^{{{k}_{1}}} q_{{0,s}}^{{(1,{{k}_{1}} + 1)}}({{a}_{i}}) \\ \times \;\left\langle {\int {{d}^{2}}y{\kern 1pt} \frac{{{{U}_{{1,{{k}_{2}} + 1}}}(y)}}{{\mathcal{N}({{a}_{{1, - 1 - {{k}_{2}}}}})}}{\kern 1pt} {{\mathcal{W}}_{{{{a}_{i}} - {{\lambda }_{{0,s}}}}}}({{x}_{i}}) \ldots } \right\rangle ; \\ \end{gathered} $$

(36)

i.e., they are expressed in terms of the previously calculated four-point correlation function.

Contributions at x → ∞

In view of the asymptotic behavior of $O{\kern 1pt} '(x)$ at $x \to \infty$, the corresponding boundary contribution is given by the expression

$$ - 2{{\lambda }_{{1,{{k}_{1}} + 1}}}\int {{d}^{2}}y\left\langle {{{\mathcal{O}}_{{1,{{k}_{1}} + 1}}}(0){\kern 1pt} \frac{{{{U}_{{1,{{k}_{2}} + 1}}}(y)}}{{\mathcal{N}({{a}_{{1, - 1 - {{k}_{2}})}}}}}{\kern 1pt} \prod\limits_{i = 1}^3 {{\mathcal{W}}_{{{{a}_{i}}}}}} \right\rangle .$$

(37)

To calculate it, we apply the higher equations of motion to ${{U}_{{1,{{k}_{2}} + 1}}}(y)$. $\mathcal{Q}$-exact terms can be omitted because all other operators are $\mathcal{Q}$-closed. As a result, we obtain

$$\begin{gathered} - \frac{{2{{\lambda }_{{1,{{k}_{1}} + 1}}}}}{\pi } \\ \times \;\int {{d}^{2}}y{\kern 1pt} \partial \bar {\partial }\left\langle {\left( {\mathcal{O}_{{1,1 + {{k}_{2}}}}^{'}(y)} \right){{\mathcal{O}}_{{1,{{k}_{1}} + 1}}}(0)\prod\limits_{i = 1}^3 {{\mathcal{W}}_{{{{a}_{i}}}}}} \right\rangle . \\ \end{gathered} $$

(38)

The integral with respect to y is now reduced to the boundary terms near the x_i, 0, and ∞ points. A new term appearing here is the logarithmic contribution in the OPE of  $O_{{1,1 + {{k}_{2}}}}^{'}(y){{O}_{{1,{{k}_{1}} + 1}}}(0)$. Since the OPE with $V_{{1,k}}^{'}$ and ${{V}_{a}}$ in the logarithmic terms is the same as the OPE with the conventional primary field ${{V}_{{1,k}}}$ (except for the factors $q_{{r,s}}^{{(m,n)}}$), it is sufficient to add the same factors on the right-hand side of Eq. (17):

$$\begin{gathered} \mathcal{O}_{{1,{{k}_{2}} + 1}}^{'}(y){{\mathcal{O}}_{{1,{{k}_{1}} + 1}}}(x) = \log {\text{|}}y - x{{{\text{|}}}^{2}} \\ \times \;\sum\limits_{s = {{k}_{2}} - {{k}_{1}}}^{{{k}_{2}} + {{k}_{1}}} q_{{0,s - {{k}_{1}}}}^{{(1,{{k}_{2}} + 1)}}({{a}_{{1,{{k}_{1}} + 1}}}){{\mathcal{O}}_{{1,1 + s}}} + \ldots . \\ \end{gathered} $$

(39)

The four-point correlation function $\langle OWWW\rangle$ can now be calculated as above.

Contribution at x ~ y

The calculation of this contribution is the most difficult for two reasons. First, such a boundary term comes not only from $\partial \bar {\partial }O_{{1,1 + {{k}_{1}}}}^{'}$ but also from $\mathcal{Q}$-exact terms in the higher equations of motion. Second, because of the absence of additional C ghosts, the logarithmic terms in the OPE of $O_{{1,1 + {{k}_{1}}}}^{'}(x)$${{U}_{{1,1 + {{k}_{2}}}}}$ are not so simple as in Eq. (32). However, we demonstrate below that these two problems compensate each other. Indeed, using Eq. (24), we represent the operator product ${{U}_{{1,1 + {{k}_{1}}}}}(x){{U}_{{1,1 + {{k}_{2}}}}}(y)$ in the form

$${{B}_{{1,1 + {{k}_{1}}}}}{{U}_{{1,1 + {{k}_{1}}}}}(x){{U}_{{1,1 + {{k}_{2}}}}}(y)$$

$$ = (\bar {\partial }\partial - \bar {\mathcal{Q}}{{\bar {B}}_{{ - 1}}}\partial - \bar {\partial }\mathcal{Q}{{B}_{{ - 1}}} + \bar {\mathcal{Q}}{{\bar {B}}_{{ - 1}}}\mathcal{Q}{{B}_{{ - 1}}})$$

(40)

$$ \times \;O_{{m,n}}^{'}(x){{U}_{{1,1 + {{k}_{2}}}}}(y).$$

Transferring the action of the operators $\mathcal{Q}$ and $\bar {\mathcal{Q}}$ from $O_{{m,n}}^{'}(x)$ to ${{U}_{{1,1 + {{k}_{2}}}}}(y)$, we obtain the right-hand side of Eq. (40) in the form

$$\begin{gathered} (\bar {\partial }\partial O_{{m,n}}^{'}){{U}_{{1,1 + {{k}_{2}}}}} - \bar {\partial }{{B}_{{ - 1}}}O_{{m,n}}^{'}\mathcal{Q}{{U}_{{1,1 + {{k}_{2}}}}} \\ - \;\partial {{{\bar {B}}}_{{ - 1}}}O_{{m,n}}^{'}\bar {\mathcal{Q}}{{U}_{{1,1 + {{k}_{2}}}}} + {{{\bar {B}}}_{{ - 1}}}{{B}_{{ - 1}}}O_{{m,n}}^{'}\bar {\mathcal{Q}}\mathcal{Q}{{U}_{{1,1 + {{k}_{2}}}}}. \\ \end{gathered} $$

(41)

Finally, using the relation $\mathcal{Q}{{U}_{{1,1 + {{k}_{2}}}}} = {{\partial }_{y}}(C{{U}_{{1,1 + {{k}_{2}}}}})$, we represent the five-point correlation function in the form

$$\begin{gathered} \int {{d}^{2}}y{\kern 1pt} \int {{d}^{2}}x{\kern 1pt} {{\partial }_{x}}{{{\bar {\partial }}}_{x}}\left( {{{H}_{{1,1 + {{k}_{1}}}}}{{{\bar {H}}}_{{1,1 + {{k}_{1}}}}}\Theta _{{1,1 + {{k}_{1}}}}^{'}} \right) \\ \times \;{{U}_{{1,1 + {{k}_{2}}}}}(y){{W}_{{{{a}_{1}}}}}({{x}_{1}}){{W}_{{{{a}_{2}}}}}({{x}_{2}}){{W}_{{{{a}_{3}}}}}({{x}_{3}}) \\ \end{gathered} $$

(42)

$$\begin{gathered} - \;\int {{d}^{2}}y{\kern 1pt} \int {{d}^{2}}x{\kern 1pt} {{{\bar {\partial }}}_{x}}\left( {{{R}_{{1,1 + {{k}_{1}}}}}{{{\bar {H}}}_{{1,1 + {{k}_{1}}}}}\Theta _{{1,1 + {{k}_{1}}}}^{'}} \right) \\ \times \;{{\partial }_{y}}(C{{U}_{{1,1 + {{k}_{2}}}}}(y)){{W}_{{{{a}_{1}}}}}({{x}_{1}}){{W}_{{{{a}_{2}}}}}({{x}_{2}}){{W}_{{{{a}_{3}}}}}({{x}_{3}}) \\ \end{gathered} $$

(43)

$$\begin{gathered} - \;\int {{d}^{2}}y{\kern 1pt} \int {{d}^{2}}x{\kern 1pt} {{\partial }_{x}}\left( {{{{\bar {R}}}_{{1,1 + {{k}_{1}}}}}{{H}_{{1,1 + {{k}_{1}}}}}\Theta _{{1,1 + {{k}_{1}}}}^{'}} \right) \\ \times \;{{{\bar {\partial }}}_{y}}(\bar {C}{{U}_{{1,1 + {{k}_{2}}}}}(y)){{W}_{{{{a}_{1}}}}}({{x}_{1}}){{W}_{{{{a}_{2}}}}}({{x}_{2}}){{W}_{{{{a}_{3}}}}}({{x}_{3}}) \\ \end{gathered} $$

(44)

$$\begin{gathered} + \;\int {{d}^{2}}y{\kern 1pt} \int {{d}^{2}}x{\kern 1pt} {{R}_{{1,1 + {{k}_{1}}}}}{{{\bar {R}}}_{{1,1 + {{k}_{1}}}}}\Theta _{{1,1 + {{k}_{1}}}}^{'}{{\partial }_{y}}{{{\bar {\partial }}}_{y}} \\ \times \;\left( {C\bar {C}{{U}_{{1,1 + {{k}_{2}}}}}(y)} \right){{W}_{{{{a}_{1}}}}}({{x}_{1}}){{W}_{{{{a}_{2}}}}}({{x}_{2}}){{W}_{{{{a}_{3}}}}}({{x}_{3}}), \\ \end{gathered} $$

(45)

where ${{R}_{{1,1 + k}}}: = {{B}_{{ - 1}}}{{H}_{{1,1 + k}}}$. The Stokes theorem was applied to the integral with respect to x in Eq. (42) and all boundary terms except for x ~ y were calculated. After reduction to the integral over the boundary $\int {{d}^{2}}y$ in Eqs. (43)–(45), nonzero contributions remain only at x → y because sufficiently singular terms in the OPE of $CU{\text{/}}\bar {C}U$$W$ are absent and $CU{\text{/}}\bar {C}U$ decreases at infinity more rapidly than ${{\bar {z}}^{{ - 1}}}{\text{/}}{{z}^{{ - 1}}}$. To calculate these contributions, the logarithmic terms in the OPE of $R\bar {H}\Theta '(x)CU(y)$ or $R\bar {R}\Theta '(x)C\bar {C}U(y)$ are again necessary. The first several examples indicate that the sum with the boundary terms from the vicinity of x ~ y from Eq. (42) gives the contribution similar to Eq. (36):

$$ - \sum\limits_{s = - {{k}_{1}}:2}^{{{k}_{1}}} \frac{{q_{{0,s}}^{{(1,{{k}_{1}} + 1)}}({{a}_{{1, - {{k}_{2}} - 1}}})}}{{\mathcal{N}({{a}_{{1, - 1 - ({{k}_{2}} - s)}}})}}\left\langle {\int {{d}^{2}}y{\kern 1pt} {{U}_{{1,({{k}_{2}} - s) + 1}}}(y){{\mathcal{W}}_{{{{a}_{i}}}}}({{x}_{i}}) \ldots } \right\rangle .$$

(46)

We illustrate this by the example of ${{k}_{1}} = 1$. We use the explicit form

$${{H}_{{1,2}}} = L_{{ - 1}}^{M} - {{L}_{{ - 1}}} + {{b}^{2}}CB,\quad {{R}_{{1,2}}} = {{b}^{2}}B$$

(47)

and the OPE

$$V_{{1,2}}^{'}(x){{V}_{a}}(y) = \log ({\text{|}}x - y{{{\text{|}}}^{2}})\left[ {\overbrace {{\text{|}}x - y{{{\text{|}}}^{{2ab}}}\tilde {C}_{{\text{L}}}^{ + }(a)[{{V}_{{a - b/2}}}(y)]}^{(1)} + \overbrace {{\text{|}}x - y{{{\text{|}}}^{{2(1 - ab + {{b}^{2}})}}}\tilde {C}_{{\text{L}}}^{ - }(a)[{{V}_{{a + b/2}}}(y)]}^{(2)}} \right],$$

(48)

$${{\Phi }_{{1,2}}}(x){{\Phi }_{{a - b}}}(y) = \underbrace {{\text{|}}x - y{{{\text{|}}}^{{2(ab - {{b}^{2}})}}}C_{{\text{M}}}^{ + }(a - b)[{{\Phi }_{{a - b/2}}}(y)]}_{(3)} + \underbrace {{\text{|}}x - y{{{\text{|}}}^{{2(1 - ab)}}}C_{{\text{M}}}^{ - }(a - b)[{{\Phi }_{{a - 3b/2}}}(y)]}_{(4)}.$$

(49)

Here, $a = b + {{\alpha }_{{1,{{k}_{2}} + 1}}}$ and ${{\tilde {C}}_{{\text{L}}}}$ are the structure constants of the Liouville theory multiplied by the q factors as in Eq. (30). Multiplying the above expressions and applying the operator ${{H}_{{1,2}}}{{\bar {H}}_{{1,2}}}$, we obtain the OPE of $O_{{1,1 + {{k}_{1}}}}^{'}$${{U}_{{1,1 + {{k}_{2}}}}}$. Retaining only the logarithmic contributions, for the primary fields, we arrive at the expression

$$\log {\text{|}}x - y{{{\text{|}}}^{2}}\left[ {\tilde {C}_{{\text{L}}}^{ + }C_{{\text{M}}}^{ + }\left( { - \frac{{{{b}^{2}}}}{{x - y}} + {{b}^{2}}CB} \right)\left( { - \frac{{{{b}^{2}}}}{{\overline {x - y} }} + {{b}^{2}}\overline {CB} } \right){\text{|}}x - y{{{\text{|}}}^{{2(2ab - {{b}^{2}})}}}{{V}_{{a - b/2}}}{{\Phi }_{{a - b/2}}}(y)} \right.$$

$$\begin{gathered} + \;\tilde {C}_{{\text{L}}}^{ + }C_{{\text{M}}}^{ - }\left( {\frac{{1 - 2ab}}{{x - y}} + {{b}^{2}}CB} \right)\left( {\frac{{1 - 2ab}}{{\overline {x - y} }} + {{b}^{2}}\overline {CB} } \right){\text{|}}x - y{{{\text{|}}}^{2}}{{U}_{{a - b/2}}}(y) \\ + \;\tilde {C}_{{\text{L}}}^{ - }C_{{\text{M}}}^{ + }\left( {\frac{{2ab - 2{{b}^{2}} - 1}}{{x - y}} + {{b}^{2}}CB} \right)\left( {\frac{{2ab - 2{{b}^{2}} - 1}}{{\overline {x - y} }} + {{b}^{2}}\overline {CB} } \right){\text{|}}x - y{{{\text{|}}}^{2}}{{U}_{{a + b/2}}}(y) \\ \end{gathered} $$

(50)

$$ + \left. {\;\tilde {C}_{{\text{L}}}^{ - }C_{{\text{M}}}^{ - }\left( { - \frac{{{{b}^{2}}}}{{x - y}} + {{b}^{2}}CB} \right)\left( { - \frac{{{{b}^{2}}}}{{\overline {x - y} }} + {{b}^{2}}\overline {CB} } \right){\text{|}}x - y{{{\text{|}}}^{{2(2 - 2ab + {{b}^{2}})}}}{{V}_{{a + b/2}}}{{\Phi }_{{a - 3b/2}}}(y)} \right].$$

The second derivative ${{\partial }_{x}}{{\bar {\partial }}_{x}}$ of this expression gives the contribution from the vicinity of x ~ y in Eq. (42). Because of the absence of the C ghosts, terms different from ${{U}_{\# }} = {{V}_{\# }}{{\Phi }_{{\# - b}}}$ are not canceled and ghosts appear in the “correct” terms. However, the factors containing ghosts appear as the OPE

$${{R}_{{1,2}}}(x)C(y) = \frac{{{{b}^{2}}}}{{x - y}} + {{b}^{2}}BC + \ldots .$$

(51)

The boundary contributions given by Eqs. (43)–(45) have the same form. In particular, for Eq. (44), we have

$$ - {{\partial }_{x}}{{\bar {\partial }}_{y}}\log {\text{|}}x - y{{{\text{|}}}^{2}}\left[ {\tilde {C}_{{\text{L}}}^{ + }C_{{\text{M}}}^{ + }\left( {\frac{{{{b}^{2}}}}{{x - y}} - {{b}^{2}}CB} \right)\left( { - \frac{{{{b}^{2}}}}{{\overline {x - y} }} + {{b}^{2}}\overline {CB} } \right){\text{|}}x - y{{{\text{|}}}^{{2(2ab - {{b}^{2}})}}}{{V}_{{a - b/2}}}{{\Phi }_{{a - b/2}}}(y)} \right.$$

$$ + \;\tilde {C}_{{\text{L}}}^{ + }C_{{\text{M}}}^{ - }\left( {\frac{{{{b}^{2}}}}{{x - y}} - {{b}^{2}}CB} \right)\left( {\frac{{1 - 2ab}}{{\bar {x} - y}} + {{b}^{2}}\bar {C}B} \right){\text{|}}x - y{{{\text{|}}}^{2}}{{U}_{{a - b/2}}}(y)$$

$$ + \;\tilde {C}_{ - }^{{\text{L}}}C_{ + }^{{\text{M}}}\left( {\frac{{{{b}^{2}}}}{{x - y}} - {{b}^{2}}CB} \right)\left( {\frac{{2ab - 2{{b}^{2}} - 1}}{{\overline {x - y} }} + {{b}^{2}}\overline {CB} } \right){\text{|}}x - y{{{\text{|}}}^{2}}{{U}_{{a + b/2}}}(y)$$

$$ + \;\left. {\tilde {C}_{{\text{L}}}^{ - }C_{{\text{M}}}^{ - }\left( {\frac{{{{b}^{2}}}}{{x - y}} - {{b}^{2}}CB} \right)\left( { - \frac{{{{b}^{2}}}}{{\overline {x - y} }} + {{b}^{2}}\overline {CB} } \right){\text{|}}x - y{{{\text{|}}}^{{2(2 - 2ab + {{b}^{2}})}}}{{V}_{{a + b/2}}}{{\Phi }_{{a - 3b/2}}}(y)} \right].$$

For the primary fields, i.e., differentiating only the coefficients depending on $x - y$, we can replace ${{\bar {\partial }}_{y}}$ by $- {{\bar {\partial }}_{x}}$; after that, the contributions from Eqs. (42) and (44), which are proportional to ${{V}_{{a - b/2}}}{{\Phi }_{{a - b/2}}}$ and ${{V}_{{a + b/2}}}{{\Phi }_{{a - \frac{{3b}}{2}}}}$, are canceled in the derivative. The same cancellation also occurs for such contributions given by Eqs. (43) and (45). At the same time, summing the contributions to ${{U}_{{a - b/2}}}$ from all Eqs. (46)–(49), we obtain

$$\begin{gathered} \left( {\frac{{1 - 2ab}}{{x - y}} + {{b}^{2}}CB} \right)\left( {\frac{{1 - 2ab}}{{\overline {x - y} }} + {{b}^{2}}\overline {CB} } \right) + \left( {\frac{{1 - 2ab}}{{x - y}} + {{b}^{2}}CB} \right)\left( {\frac{{{{b}^{2}}}}{{\overline {x - y} }} - {{b}^{2}}\overline {CB} } \right) \\ + \;\left( {\frac{{{{b}^{2}}}}{{x - y}} - {{b}^{2}}CB} \right)\left( {\frac{{1 - 2ab}}{{\overline {x - y} }} + {{b}^{2}}\overline {CB} } \right) + \left( {\frac{{{{b}^{2}}}}{{x - y}} - {{b}^{2}}CB} \right)\left( {\frac{{{{b}^{2}}}}{{\overline {x - y} }} - {{b}^{2}}\overline {CB} } \right) \\ = \left( {\frac{{1 - 2ab}}{{x - y}} + {{b}^{2}}CB + \frac{{{{b}^{2}}}}{{x - y}} - {{b}^{2}}CB} \right)\left( {\frac{{1 - 2ab}}{{\overline {x - y} }} + {{b}^{2}}\overline {CB} + \frac{{{{b}^{2}}}}{{\overline {x - y} }} - {{b}^{2}}\overline {CB} } \right) = \frac{1}{{{\text{|}}x - y{{{\text{|}}}^{2}}}}{{(1 - 2ab + {{b}^{2}})}^{2}}. \\ \end{gathered} $$

(52)

It is seen that ghosts disappear and we obtain the same factor as for the OPE of $O{\kern 1pt} '$$W$ (see Eq. (20)).

All Together

Combining Eqs. (36), (38), and (46), we arrive the following expression for the five-point correlation function:

$${{C}_{5}}({{a}_{1}},{{a}_{2}},{{a}_{3}}{\text{|}}{{k}_{1}},{{k}_{2}}) = ({{b}^{{ - 6}}} - {{b}^{{ - 2}}})\left[ {{{\Sigma }_{1}} + {{\Sigma }_{2}} + {{\Sigma }_{3}}} \right];$$

(53)

$${{\Sigma }_{1}} = \sum\limits_{s = - {{k}_{1}}:2}^{{{k}_{1}}} q_{{0,s}}^{{(1,{{k}_{1}} + 1)}}({{a}_{{1, - {{k}_{2}} - 1}}})\left[ {2(1 + {{k}_{2}} - s){{\lambda }_{{1,1 + {{k}_{2}} - s}}} + \sum\limits_{i = 1}^3 \sum\limits_{l = - {{k}_{2}} + s:2}^{{{k}_{2}} - s} q_{{0,l}}^{{(1,1 + {{k}_{2}} - s)}}({{a}_{i}})} \right],$$

(54)

$${{\Sigma }_{2}} = \sum\limits_{i = 1}^3 \sum\limits_{s = - {{k}_{1}}:2}^{{{k}_{1}}} q_{{0,s}}^{{(1,{{k}_{1}} + 1)}}({{a}_{i}})\left[ {2(1 + {{k}_{2}}){{\lambda }_{{1,1 + {{k}_{2}}}}} + \sum\limits_{l = - {{k}_{2}}:2}^{{{k}_{2}}} \left( {q_{{0,l}}^{{(1,{{k}_{2}} + 1)}}({{a}_{i}} - {{\lambda }_{{0,s}}}) + \sum\limits_{j \ne i} q_{{0,l}}^{{(1,{{k}_{2}} + 1)}}({{a}_{j}})} \right)} \right],$$

(55)

$${{\Sigma }_{3}} = 2{{\lambda }_{{1,1 + {{k}_{1}}}}}\left[ {\sum\limits_{s = - {{k}_{1}}:2}^{{{k}_{1}}} \sum\limits_{l = - {{k}_{2}}:2}^{{{k}_{2}}} \left( {\sum\limits_{i = 1}^3 q_{{0,l}}^{{(1,1 + {{k}_{2}})}}({{a}_{i}}) + 2{{\lambda }_{{1,{{k}_{2}} + 1}}}} \right) + \sum\limits_{s = {{k}_{2}} - {{k}_{1}}}^{{{k}_{2}} + {{k}_{1}}} q_{{0,s - {{k}_{1}}}}^{{(1,{{k}_{2}} + 1)}}({{a}_{{1,{{k}_{1}} + 1}}})(1 + s)} \right].$$

(56)