The integrals of motion for the elliptic deformation of the Virasoro and W_{N} algebra

T.KOJIMA

Department of Mathematics, College of Science and Technology, Nihon University, Surugadai, Chiyoda-ku, Tokyo 101-0062, JAPAN

J.SHIRAISHI

Graduate school of Mathematical Science, University of Tokyo, Komaba, Megro-ku, Tokyo 153-8914, JAPAN

Abstract

We review the free field realization of the deformed Virasoro algebra V ir_{q,t} and the deformed W albegra W_{q,t}(ĝl_{N}). We explicitly construct two classes of infinitly many commutative operators I_{m}, G_{m}, (m ∈ ℕ), in terms of these algebras. They can be regarded as the elliptic deformation of the local and nonlocal integrals of motion for the conformal field theory [1–5]. This review is based on the works [15–17].

Key words: Exactly Solved Model, Virasoro algebra, W-algebra, Quantum group, Conformal field theory, Elliptic quantum group, Deformed W -algebra

1 Introduction

The Korteweg-de Vries (KdV) equation occupies a central place in the modern theory of completely integrable systems. Because of its integrability, the KdV equation has infinitly many conservation laws. The Hamiltonian aspects of the KdV theory connected it to the conformal field theory. The quantization of the second Poisson bracket {, }_{P.B.} of the KdV gives rise to the Virasoro algebra : [L_{m}, L_{n}] = (m − n)L_{m+n} + \frac{1}{12}c_{CF T} m(m^{2} − 1)δ_{m+n,0}. The quantum field theory of the KdV theory becomes the conformal field theory associated with the Virasoro algebra [1–3]. V.Bazhanov, S.Lukyanov, Al.Zamolodchikov [1] constructed quantum field theoretical analogue of the commuting transfer matrix T(z) acting the highest weight module of the Virasoro algebra. The commuting transfer matrix T(z) is constructed as the trace of an image of the universal R-matrix associated with the quantum affine symmetry U_{q}(ŝl_{2}).

Preprint submitted to Elsevier 3 August 2008

Hence the commutativity [T(z), T(w)] = 0 is a direct consequence of the Yang-Baxter relation. We call the coefficients of the asymptotic expansion of logT(z), (z → ∞), the local integrals of motion for the Virasoro algebra. They recover the conservation laws of the KdV in the classical limit c_{CF T} → ∞. We call the coefficients of the Taylor expansion of T(z) the nonlocal integrals of motion for the Virasoro algebra. See also the generalization to the W_{N} algebra [4,5].

In this paper we construct the elliptic deformation of the integrals of motion for the conformal field theory [1,4,5]. In this paper we construct two classes of infinitly many commutative operators I_{m}, G_{m}, (m ∈ ℕ), associated with the deformed Virasoro algebra and the deformed W -algebra W_{q,t}(ĝl_{N}). Because it is not so easy to calculate the trace of the image of the universal R-matrix of the elliptic quantum group, we prefer the completly differnt method of the construction for the integrals of motion in the elliptic deformation of the conformal field theory. Instead of considering the transfer matrix T(z), we directly give the explicit formulae of the integrals of motion I_{n} and G_{n} for the deformed W -algebra W_{q,t}(ĝl_{N}). The commutativity of the intregrals of motion are not understood as direct consequence of the Yang-Baxter relation. They are understood as consequence of the commutative family of the Feigin-Odesskii algebra [14].

The organization of this paper is as follows. In section 2 we give reviews on the deformed Virasoro algebra and the deformed W -algebra W_{q,t}(ĝl_{N}). In section 3 we give explicit formulae of the integrals of motion for the deformed Virasoro algebra and the deformed W -algebra, and state the main theorem.

2 Elliptic deformation of the Virasoro algebra and the W_{N}-algebra

In this section we review the elliptic deformation of the Virasoro algebra and the W_{N}-algebra. We fix three parameters x, r, s such that 0 < x < 1, Re(r) > 0 and Re(s) > 0. Let us set r^{∗} = r − 1. We set the parameters τ by x = exp (−π\sqrt{−1}/rτ ) We relate two variables z and u by z = x^{2u}. The symbol [u]_{r} stands for the Jacobi theta function

[u]_{r} = x^{\frac{u^{2}}{r} −u} \frac{Θ_{x^{2r}}(z)}{(x^{2r}; x^{2r})_{∞}} , Θ_{q}(z) = (z; q)_{∞}(q/z; q)_{∞}(q; q)_{∞},  (1)

where (z; q)_{∞} = ∏_{j=0}^{∞}(1 − q^{j}z). The elliptic theta function satisfies the quasiperiodicities,

[u + r]_{r} = −[u]_{r}, [u + rτ]_{r} = −e^{−π\sqrt{−1}τ−2π\sqrt{−1}u/r}[u]_{r}.  (2)

The symbol [a] stands for q-integer [a] = \frac{x^{a}−x^{−a}}{x−x^{−1}} .

2.1 Bosons

For N = 2, 3, 4, ⋯, we introduce the bosons β_{m}^{j}, (m ∈ ℤ_{≠0}; 1 ≦ j ≦ N), which satisfy the commuttion relation,

[β_{n}^{i}, β_{m}^{j}] = n\frac{[(r − 1)n]}{[rn]}δ_{n+m,0} × \{ \begin{matrix}\frac{[(s−1)n]}{[sn]} & (1 ≦ i = j ≦ N) \\ −\frac{[n]}{[sn]}x^{sn sgn(i−j)} & (1 ≦ i ≠ j ≦ N) \end{matrix} (3)

For N = 2, 3, 4, ⋯, we introduce the zero-mode operators P_{λ} and Q_{λ}. Let ϵ_{j}, (1 ≦ j ≦ N) be an orthonormal basis in ℝ^{N} relative to the standard inner product ( | ). Let us set ϵ̄_{i} = ϵ_{i} − ϵ, ϵ = \frac{1}{N} ∑_{j=1}^{N} ϵ_{j}. Let us set α_{j} = ϵ̄_{j} − ϵ̄_{j+1}. Let P_{λ}, Q_{λ} be the zero mode operators defined by the commutation relation

[iP_{λ}, Q_{μ}] = (λ|μ), (λ, μ ∈ ∑_{j=1}^{N} ℤϵ̄_{j}).  (4)

The action of the Dynkin-diagram automorphism η on the bosons is given by

η(β_{m}^{1}) = x^{− \frac{2s}{N} m}β_{m}^{2}, ⋯, η(β_{m}^{N−1}) = x^{− \frac{2s}{N} m}β_{m}^{N}, η(β_{m}^{N}) = x^{\frac{2s}{N} (N−1)m}β_{m}^{1}. (5)

The action of the Dynkin-diagram automorphism η on the zero-mode operator is given by

η(P_{λ}) = P_{η(λ)}, η(Q_{λ}) = Q_{η(λ)}, η(ϵ̄_{j}) = ϵ̄_{j+1}, (1 ≦ j ≦ N),  (6)

where we understand ϵ̄_{1} = ϵ̄_{N+1}. Let us introduce the Fock space F_{l,k}, (l, k ∈ ∑_{j=1}^{N} ℤϵ̄_{j}), of the bosons, generated by β_{−m}^{j}, (m > 0) over the vacuum vector |l, k⟩, (l, k ∈ ∑_{j=1}^{N} ℤϵ̄_{j}),

β_{−m}^{j}|l, k⟩ = 0, (m > 0; j = 1, 2, ⋯, N),  (7)
P_{α}|l, k⟩ = (α |l\sqrt{\frac{r}{r − 1}} − k\sqrt{\frac{r − 1}{r}}) |l, k ⟩,  (8)

|l, k⟩ = exp (\sqrt{\frac{r}{r − 1}}Q_{l} − \sqrt{\frac{r − 1}{r}}Q_{k}) |0, 0⟩.  (9)

2.2 Deformed W -algebra

In this section, we review the deformed Virasoro algebra and the deformed W algebra W_{q,t}(ĝl_{N}), following [7–10,15–17].

Definition 1 For N = 2, 3, 4, ⋯, the deformed W -algebra W_{q,t}(ĝl_{N}) is generated by the generators T_{m}^{(j)}, (1 ≦ j ≦ N, m ∈ ℤ), with the defining relations (10) of the series T_{j}(z) = ∑_{m∈ℤ} T_{m}^{(j)}z^{−m}.

f_{i,j}(z_{2}/z_{1})T_{i}(z_{1})T_{j}(z_{2}) − f_{j,i}(z_{1}/z_{2})T_{j}(z_{2})T_{i}(z_{1})
= c ∑_{k=1}^{i} ∏_{l=1}^{k−1} Δ(x^{2l+1}) (δ (\frac{x^{j−i+2k}z_{2}}{z_{1}} ) f_{i−k,j+k}(x^{−j+i})T_{i−k}(x^{−k}z_{1})T_{j+k}(x^{k}z_{2})
− δ (\frac{x^{−j+i−2k}z_{2}}{z_{1}} ) f_{i−k,j+k}(x^{j−i})T_{i−k}(x^{k}z_{1})T_{j+k}(x^{−k}z_{2})) , (1 ≦ i ≦ j ≦ N),

(10)

where we used the delta-function δ(z) = ∑_{m∈ℤ} z^{m}. Here we have set the constant c and the structure functions Δ(z) and f_{i,j}(z), (1 ≦ i, j ≦ N) by

c = −\frac{(1 − x^{2r})(1 − x^{−2(r−1)})}{(1 − x^{2})}, Δ(z) = \frac{(1 − x^{2r−1}z)(1 − x^{1−2r}z)}{(1 − xz)(1 − x^{−1}z)},  (11)
f_{i,j}(z)  (12)
= exp ( ∑_{m=1}^{∞} \frac{(1 − x^{2rm})(1 − x^{−2(r−1)m})(1 − x^{2mMin(i,j)})(1 − x^{2m(s−Max(i,j))})}{m(1 − x^{2m})(1 − x^{2sm})}(x^{|i−j|}z)^{m}) .

Example Upon the specialization N = s = 2, we have the deformed Virasoro algebra V ir_{q,t}. Upon this specialization the generators T_{m}^{(2)} can be regarded as T_{m}^{(2)} = 1. The generators T_{m}^{(1)} = T_{m} satisfy the following defining relation.

∑_{l=0}^{∞} f_{l}(T_{n−l}T_{m+l} − T_{m−l}T_{n+l}) = c(c^{2n} − x^{−2n})δ_{n+m,0},  (13)

where the structure constant f_{l} is given by ∑_{l=0}^{∞} f_{l}z^{l} = f_{1,1}(z). In the CFT limit (x → 1), we get the Virasoro algebra with the central charge c_{CF T} = 1− \frac{6}{r(r−1)}.

[L_{m}, L_{n}] = (m − n)L_{m+n} + \frac{1}{12}c_{CF T} m(m^{2} − 1)δ_{m+n,0}.  (14)

Proposition 2 For N = 2 the deformed W -algebra W_{q,t}(ĝl_{2}) is realized by the bosons (3), (4) on the Fock space.

T_{1}(z) = Λ_{1}(z) + Λ_{2}(z), T_{2}(z) =: Λ_{1}(x^{−1}z)Λ_{2}(xz) :,  (15)

where we have set

Λ_{1}(z) = x^{−\sqrt{r(r−1)}P_{α_{1}}} : exp ( ∑_{m≠0} \frac{1}{m}(x^{rm} − x^{−rm})β_{m}^{1}z^{−m}) :,  (16)

Λ_{2}(z) = x^{−\sqrt{r(r−1)}P_{α_{2}}} : exp ( ∑_{m≠0} \frac{1}{m}(x^{rm} − x^{−rm})β_{m}^{2}z^{−m}) : .  (17)

Proposition 3 For N = 3, 4, ⋯ the deformed W -algebra W_{q,t}(ĝl_{N}) is realized by the bosons (3), (4) on the Fock space.

T_{j}(z) = ∑_{1≦s_{1}<s_{2}<···<s_{j}≦N} : Λ_{s_{1}}(x^{−j+1}z)Λ_{s_{2}}(x^{−j+3}z) ⋯ Λ_{s_{j}}(x^{j−1}z) :, (18)

where we have set

Λ_{j}(z) = x^{−\sqrt{r(r−1)}Pϵ̄_{j}} : exp ( ∑_{m≠0} \frac{x^{rm} − x^{−rm}}{m} β_{m}^{j}z^{−m}) :, (1 ≦ j ≦ N). (19)

Here the symbol : ∗ : stands for usual normal ordering of bosons, i.e. β_{m}^{i} with m > 0 should be moved to the right.

2.3 Screening current

In this section we review the screening currents for the deformed Virasoro algebra and the deformed W algebra, following [7–13,15–17].

Definition 4 For N = 2 we introduce the operator F_{j}(z), (j = 1, 2), called the screening current for the deformed W -algebra W_{q,t}(ĝl_{2}). We define

F_{1}(z) = e^{−i\sqrt{\frac{r^{∗}}{r}} Q_{α_{1}}}z^{−\sqrt{\frac{r^{∗}}{r}} P_{α_{1}}+ \frac{r^{∗}}{r}} : exp ( ∑_{m≠0} \frac{1}{m}(β_{m}^{1} − β_{m}^{2})z^{−m}) :,  (20)

F_{2}(z) = e^{−i\sqrt{\frac{r^{∗}}{r}} Q_{α_{2}}}z^{−\sqrt{\frac{r^{∗}}{r}} P_{α_{2}}+ \frac{r^{∗}}{r}} : exp ( ∑_{m≠0} \frac{1}{m}(−x^{sm}β_{m}^{1} + x^{−sm}β_{m}^{2})z^{−m}) : .

(21)

Definition 5 For N = 3, 4, ⋯ we introduce the operator F_{j}(z), (1 ≦ j ≦ N), called the screening current for the deformed W -algebra W_{q,t}(ĝl_{N}). Let us set

F_{j}(z) = e^{−i\sqrt{\frac{r}{r−1}} Q_{α_{j}}} (x^{( \frac{2s}{N} −1)j}z)^{−\sqrt{\frac{r}{r−1}} P_{α_{j}} + \frac{r}{r−1}}
×: exp ( ∑_{m≠0} \frac{1}{m}(β_{m}^{j} − β_{m}^{j+1})(x^{\frac{2s}{N}} z)^{−m}) :, (1 ≦ j ≦ N − 1), (22)
F_{N}(z) = e^{−i\sqrt{\frac{r}{r−1}} Q_{α_{N}}} (x^{2s−N}z)^{−\sqrt{\frac{r}{r−1}} Pϵ̄_{N} + \frac{r}{2(r−1)}} z^{\sqrt{\frac{r}{r−1}} Pϵ̄_{1}+ \frac{r}{2(r−1)}}
×: exp ( ∑_{m≠0} \frac{1}{m}(x^{−2sm}β_{m}^{N} − β_{m}^{1})z^{−m}) : .  (23)

Proposition 6 For N = 2 the screening currents F_{1}(z), F_{2}(z) satisfy the following commutation relations.

[u_{1} − u_{2}]_{r}[u_{1} − u_{2} + 1]_{r}F_{j}(z_{1})F_{j}(z_{2})

= [u_{2} − u_{1}]_{r}[u_{2} − u_{1} + 1]_{r}F_{j}(z_{2})F_{j}(z_{1}), (j = 1, 2),  (24)

[u_{1} − u_{2} + \frac{s}{2} − 1]_{r} [u_{1} − u_{2} − \frac{s}{2}]_{r} F_{2}(z_{1})F_{1}(z_{2})

= [u_{2} − u_{1} + \frac{s}{2} − 1]_{r} [u_{2} − u_{1} − \frac{s}{2}]_{r} F_{1}(z_{2})F_{2}(z_{1}).  (25)

Proposition 7 For N = 2 the commutation relations between Λ_{j}(z) and F_{j}(z) are given by

[Λ_{1}(z_{1}), F_{1}(z_{2})] = (x^{−r^{∗}} − x^{r^{∗}})δ(x^{r}z_{1}/z_{2})A(x^{−r}z_{2}),  (26)
[Λ_{2}(z_{1}), F_{1}(z_{2})] = (x^{r^{∗}} − x^{−r^{∗}})δ(x^{−r}z_{1}/z_{2})A(x^{r}z_{2}),  (27)
[Λ_{1}(z_{1}), F_{2}(z_{2})] = (x^{r^{∗}} − x^{−r^{∗}})δ(x^{−r+s}z_{1}/z_{2})η(A(x^{r}z_{2})),  (28)
[Λ_{2}(z_{1}), F_{2}(z_{2})] = (x^{−r^{∗}} − x^{r^{∗}})δ(x^{r−s}z_{1}/z_{2})η(A(x^{−r}z_{2})),  (29)

where we have set

A(z) = e^{i\sqrt{\frac{r^{∗}}{r}} Q_{α_{1}}}z^{\sqrt{\frac{r^{∗}}{r}} P_{α_{1}}+ \frac{r^{∗}}{r}} : exp ( ∑_{m≠0} \frac{1}{m}(x^{rm}β_{m}^{1} − x^{−rm}β_{m}^{2})z^{−m}) : .

(30)

Proposition 8 For N = 3, 4, 5, ⋯, the screening currents F_{j}(z) satisfy the following commutation relations.

[u_{1} − u_{2} − \frac{s}{N} ]_{r} F_{j}(z_{1})F_{j+1}(z_{2}) = [u_{2} − u_{1} + \frac{s}{N} − 1]_{r} F_{j+1}(z_{2})F_{j}(z_{1}),

(31)

[u_{1} − u_{2}]_{r} [u_{1} − u_{2} + 1]_{r} F_{j}(z_{1})F_{j}(z_{2}) = [u_{2} − u_{1}]_{r} [u_{2} − u_{1} + 1]_{r} F_{j}(z_{2})F_{j}(z_{1}), (32)

for 1 ≦ j ≦ N. We understand F_{N+1}(z) = F_{1}(z). We have

F_{i}(z_{1})F_{j}(z_{2}) = F_{j}(z_{2})F_{i}(z_{1}), otherwise.  (33)

Proposition 9 For N = 3, 4, 5, ⋯ the commutation relations between Λ_{j}(z) and F_{j}(z) are given by

[Λ_{j}(z_{1}), F_{j}(z_{2})] = (x^{−r^{∗}} − x^{r^{∗}})δ(x^{− \frac{2s}{N} j+r}z_{1}/z_{2})A_{j}(x^{\frac{2s}{N} j−r}z_{1}), (1 ≦ j ≦ N − 1), (34)

[Λ_{j+1}(z_{1}), F_{j}(z_{2})] = (x^{r^{∗}} − x^{−r^{∗}})δ(x^{− \frac{2s}{N} j−r}z_{1}/z_{2})A_{j}(x^{\frac{2s}{N} j+r}z_{2}), (1 ≦ j ≦ N − 1),

(35)

[Λ_{N}(z_{1}), F_{N}(z_{2})] = (x^{−r^{∗}} − x^{r^{∗}})δ(x^{r−2s}z_{1}/z_{2})A_{N}(x^{−r}z_{2}),  (36)
[Λ_{1}(z_{1}), F_{N}(z_{2})] = (x^{r^{∗}} − x^{−r^{∗}})δ(x^{−r}z_{1}/z_{2})A_{N}(x^{r}z_{2}),  (37)

where we have set

A_{j}(z) = e^{i\sqrt{\frac{r^{∗}}{r}} Q_{α_{j}}} x^{−\sqrt{rr^{∗}}(Pϵ̄_{j} +Pϵ̄_{j+1})}(x^{−j}z)^{\sqrt{\frac{r^{∗}}{r}} P_{α_{j}} + \frac{r^{∗}}{r}}
×: exp ( ∑_{m≠0} \frac{1}{m}(x^{rm}β_{m}^{j} − x^{−rm}β_{m}^{j+1})z^{−m}) :, (1 ≦ j ≦ N − 1),(38)
A_{N}(z) = e^{i\sqrt{\frac{r^{∗}}{r}} Q_{α_{N}}} x^{−\sqrt{rr^{∗}}(Pϵ̄_{N} +Pϵ̄_{1})}(x^{2s−N}z)^{\sqrt{\frac{r^{∗}}{r}} Pϵ̄_{N} + \frac{r^{∗}}{2r}} z^{−\sqrt{\frac{r^{∗}}{r}} Pϵ̄_{1}+ \frac{r^{∗}}{2r}}
×: exp ( ∑_{m≠0} \frac{1}{m}(x^{(r−2s)m}β_{m}^{N} − x^{−rm}β_{m}^{1})z^{−m}) : .  (39)

3 Integrals of Motion

In this section we review the integrals of motion for the deformed Virasoro algebra and the deformed W -algebra, following [15–17].

3.1 Local integrals of motion I_{n}

We define the operators I_{n}, (n = 1, 2, 3, ⋯), which we call the local integrals of motion for the W_{q,t}(ĝl_{N}), (N = 2, 3, 4, ⋯).

Definition 10 For the regime Re(s) > 2 and Re(r^{∗}) < 0, we define

I_{n} = ∫ ⋯ ∫_{C} ∏_{j=1}^{n} \frac{dz_{j}}{z_{j}} T_{1}(z_{1})T_{1}(z_{2})T_{1}(z_{3}) ⋯ T_{1}(z_{n})
× ∏_{1≦j<k≦n} \frac{[u_{k} − u_{j}]_{s}[u_{k} − u_{j} + r]_{s}}{[u_{k} − u_{j} + 1]_{s}[u_{k} − u_{j} + r^{∗}]_{s}} , (n = 1, 2, 3, ⋯).  (40)

Here the contour C encircles z_{j} = 0 in such a way that z_{j} = x^{−2+2sl}z_{k}, x^{−2r^{∗}+2sl}z_{k}, (l = 0, 1, 2, ⋯) is inside and z_{j} = x^{2−2sl}z_{k}, x^{2r^{∗}−2sl}z_{k}, (l ∈ N) is outside for 1 ≦ j < k ≦ n. We call I_{n} the local integrals of motion.

The definitions of the local integrals of motion I_{n} for generic Re(s) > 0 and Re(r) > 0 should be understood as analytic continuation.

3.2 Nonlocal integrals of motion G_{n}

We define the operators G_{m}, (m = 1, 2, 3, ⋯), which we call the nonlocal integrals of motion for the W_{q,t}(ĝl_{N}), (N = 2, 3, 4, ⋯).

Definition 11 For N = 2 and the regime 0 < Re(s) < 2 and Re(r) > 0, we define

G_{m} = ∫ ⋯ ∫_{I} ∏_{t=1,2} ∏_{j=1}^{m} \frac{dz_{j}^{(t)}}{z_{j}^{(t)}} F_{1}(z_{1}^{(1)})F_{1}(z_{2}^{(1)}) ⋯ F_{1}(z_{m}^{(1)})F_{2}(z_{1}^{(2)})F_{2}(z_{2}^{(2)}) ⋯ F_{2}(z_{m}^{(2)})

× \begin{matrix}∏_{t=1,2} & ∏_{1≦j<k≦m} & [u_{j}^{(t)} − u_{k}^{(t)}] & r & [u_{k}^{(t)} − u_{j}^{(t)} − 1] & r \\ ∏_{j,k=1}^{m} & [u_{j}^{(1)} − u_{k}^{(2)} + \frac{s}{2}] & r & [u_{k}^{(2)} − u_{j}^{(1)} + \frac{s}{2} − 1] & r \end{matrix}  (41)

× ∏_{t=1,2} [∑_{j=1}^{m}(u_{j}^{(t)} − u_{j}^{(t+1)}) − \sqrt{rr^{∗}}Pϵ̄_{t+1} ]_{r} , (m = 1, 2, ⋯).

Here the contour I encircles z_{j}^{(t)} = 0, (t = 1, 2; 1 ≦ j ≦ m) in such a way that

|x^{s+2lr}z_{k}^{(2)}|, |x^{2−s+2lr}z_{k}^{(2)}| < |z_{j}^{(1)}| < |x^{−s−2lr}z_{k}^{(2)}|, |x^{s−2−2lr}z_{k}^{(2)}|,  (42)

for 1 ≦ j, k ≦ m and l ∈ ℕ. We call G_{m} the nonlocal integrals of motion for the deformed W -algebra W_{q,t}(ĝl_{2}).

Definition 12 For N = 3, 4, 5, ⋯ and the regime 0 < Re(s) < N and Re(r) > 0, we define

G_{m} = ∫ ⋯ ∫_{I} ∏_{t=1}^{N} ∏_{j=1}^{m} \frac{dz_{j}^{(t)}}{z_{j}^{(t)}} F_{1}(z_{1}^{(1)})F_{1}(z_{2}^{(1)}) ⋯ F_{1}(z_{m}^{(1)})
× F_{2}(z_{1}^{(2)})F_{2}(z_{2}^{(2)}) ⋯ F_{2}(z_{m}^{(2)}) ⋯ F_{N}(z_{1}^{(N)})F_{N}(z_{2}^{(N)}) ⋯ F_{N}(z_{m}^{(N)})

× \begin{matrix}∏_{t=1}^{N} & ∏_{1≦j<k≦m} & [u_{j}^{(t)} − u_{k}^{(t)}] & r & [u_{k}^{(t)} − u_{j}^{(t)} − 1] & r \\ ∏_{t=1}^{N−1} & ∏_{j,k=1}^{m} & [u_{j}^{(t)} − u_{k}^{(t+1)} + 1 − \frac{s}{N} ] & r & ∏_{j,k=1}^{m} & [u_{j}^{(1)} − u_{j}^{(N)} + \frac{s}{N} ] & r \end{matrix}  (43)

× ∏_{t=1}^{N} [∑_{j=1}^{m}(u_{j}^{(t)} − u_{j}^{(t+1)}) − \sqrt{rr^{∗}}Pϵ̄_{t+1} ]_{r} , (m = 1, 2, ⋯).

Here the contour I encircles z_{j}^{(t)} = 0, (1 ≦ t ≦ N; 1 ≦ j ≦ m) in such a way that

|x^{\frac{2s}{N} +2lr}z_{k}^{(t+1)}| < |z_{j}^{(t)}| < |x^{−2+ \frac{2s}{N} −2lr}z_{k}^{(t+1)}|, (1 ≦ t ≦ N − 1),  (44)
|x^{2− \frac{2s}{N} +2lr}z_{k}^{(1)}| < |z_{j}^{(N)}| < |x^{− \frac{2s}{N} −2lr}z_{k}^{(1)}|,  (45)

for 1 ≦ j, k ≦ m and l ∈ ℕ. We call G_{m} the nonlocal integrals of motion for the deformed W -algebra W_{q,t}(ĝl_{N}).

The definitions of the nonlocal integrals of motion G_{n} for generic Re(s) > 0 and Re(r) > 0 should be understood as analytic continuation.

3.3 Main results

In this section we state the main results.

Theorem 13 For N = 2, 3, 4, ⋯, the local integrals of motion I_{n} and the nonlocal integrals of motion G_{m} commute with each other.

[I_{n}, I_{m}] = [I_{n}, G_{m}] = [G_{n}, G_{m}] = 0, (m, n = 1, 2, ⋯).  (46)

These commutativities are understood as consequence of commuting family of the Feigin-Odesskii algebra [14].

Theorem 14 For N = 2, 3, 4, ⋯, the local integrals of motion I_{n} and the nonlocal integrals of motion G_{n} are invariant under the action of the Dynkindiagram automorphism η.

η(I_{n}) = I_{n}, η(G_{n}) = G_{n}, (n = 1, 2, ⋯).  (47)

Acknowledgement

We would like to thank Prof.B.Feigin, Prof.M.Jimbo and Mr.H.Watanabe for useful communications. We would like to thank Professors V.Bazhanov, A.Belavin, P.Bouwknegt, A.Chervov, S.Duzhin, V.Gerdjikov, K.Hasegawa, P.Kulish, W-X.Ma, V.Mangazeev, K.Takemura and M.Wadati for their interests in this work. This work is partly supported by Grant-in Aid for Young Scientist B(18740092) and Grant-in Aid for Scientific Research C(16540183) from JSPS.

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