We review the free field realization of the deformed Virasoro algebra $Vir_{q,t}$ and the deformed $W$ algebra $W_{q,t}(\hat{gl_N})$. We explicitly construct two classes of infinitly many commutative operators ${\cal I}_m$, ${\cal G}_m$, $(m \in {\mathbb N})$, 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.
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 + 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 [1,2,3]. 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 ( sl 2 ).
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 ( gl 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 ( gl 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 ( gl 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 √ -1/rτ We relate two variables z and u by z = x 2u . The symbol [u] r stands for the Jacobi theta function
where (z; q) ∞ = ∞ j=0 (1 -q j z). The elliptic theta function satisfies the quasiperiodicities,
The symbol [a] stands for q-integer [a] = x a -x -a x-x -1 .
For N = 2, 3, 4, • • • , we introduce the bosons β j m , (m ∈ Z =0 ; 1 ≦ j ≦ N), which satisfy the commuttion relation,
(1
For N = 2, 3, 4, • • • , we introduce the zero-mode operators P λ and
Let us set α j = ǭj -ǭj+1 . Let P λ , Q λ be the zero mode operators defined by the commutation relation
The action of the Dynkin-diagram automorphism η on the bosons is given by
The action of the Dynkin-diagram automorphism η on the zero-mode operator is given by
where we understand ǭ1 = ǭN+1 . Let us introduce the Fock space
In this section, we review the deformed Virasoro algebra and the deformed W algebra W q,t ( gl N ), following [7,8,9,10,15,16,17].
For N = 2, 3, 4, • • • , the deformed W -algebra W q,t ( gl N ) is generated by the generators T (j) m , (1 ≦ j ≦ N, m ∈ Z), with the defining relations (10) of the series
where we used the delta-function δ(z) = m∈Z z m . Here we have set the constant c and the structure functions ∆(z) and
Example Upon the specialization N = s = 2, we have the deformed Virasoro algebra V ir q,t . Upon this specialization the generators T (2) m can be regarded as T (2) m = 1. The generators T (1) m = T m satisfy the following defining relation.
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 = 1-6 r(r-1) .
[
For N = 2 the deformed W -algebra W q,t ( gl 2 ) is realized by the bosons (3), (4) on the Fock space.
where we have set
Proposition 3
) is realized by the bosons (3), (4) on the Fock space.
where we have set
Here the symbol : * : stands for usual normal ordering of bosons, i.e. β i m with m > 0 should be moved to the right.
In this section we review the screening cur
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