Spin-chain with PSU(2|2)xU(1)^3 and Non-linear Sigma-model with D(2,1;gamma)

We propose that the spin-chain with the PSU(2|2)xU(1)^3 symmetry is equivalent to the non-linear sigma-model on PSU(2|2)xU(1)^3/{HxU(1)} with a certain subgroup. To this end we show that the spin-variable of the former theory is identified as the Killing scalar of the latter and their correlation functions can have the same integrability. It is crucial to think that the respective theory gets the PSU(2|2)xU(1)^3 symmetry by a symmetry reduction the exceptional supergroup D(2,1;gamma), rather than by an extension of PSU(2|2).


Introduction
The study of the string/QCD duality has a long history going back to the late 1970s. During the decade the subject has been studied with a renewed interest. The most clear-cut assertion of the string/QCD duality was made by calculating the anomalous dimensions of a spin-chain system on one side and the N = 4 SUSY QCD on the other and showing a remarkable agreement between them [1]. A spin-chain system of variables ψ m (x) was defined by correlation functions taking the form [2] e ip 1 x 1 +ip 2 x 2 +······ < ψ m 1 (x 1 )ψ m 2 (x 2 ) · · · · · · > . (1) Here ψ m (x) is assumed to be the fundamental vector of the PSU(2|2)⊗U(1) 3 symmetry. It is also assumed that the correlation functions obey the exchange algebra with the R-matrix for two adjacent variables ψ m (x)s. If the symmetry of the spin-system were strictly PSU(2|2), the R-matrix would be universally given by the plug-in formula for PSU(2|2) [3,4], and the correlation functions would be position-independent as those of the topological theory [5]. The PSU(2|2) symmetry had to be centrally extended to PSU(2|2)⊗U(1) 3 in order to give an account of the duality to the N = 4 SUSY QCD. A position-dependent R-matrix was found in an explicit form [2], which is not of the difference form of the two spectral parameters. This unusual feature of the R-matrix attracted a vivid interest among the community of mathematical physics [6]. A keen insight into the matter was given by realizing PSU(2|2)⊗U(1) 3 as a symmetry reduction of the exceptional group D(2,1;γ) [2,7]. The above arguments are based on the integrability and the PSU(2|2)⊗U(1) 3 symmetry. They are just assumed and their origin is obscure. These assumptions get a firm base by considering a 2-d non-linear σ-model with the PSU(2|2)⊗U(1) 3 symmetry as an equivalent theory to the spin-system. Namely the non-linear σ-model is integrable admitting an infinite number of conserved currents. The PSU(2|2)⊗U(1) 3 symmetry is regarded as descending from the superconformal symmetry of the IIB superstring.
According to [8] a quantity corresponding to the spin-variable ψ m with the PSU(2|2) symmetry may be constructed as the Killing scalar on a coset space PSU(2|2)/H with a certain group. 1 The exchange algebra for the Killing scalar may be discussed by studying the Poisson structure in the non-linear σ-model on PSU(2|2)/H. The R-matrix of the exchange algebra is universal following the quantization of [8].
The aim of this letter is to show that when the spin-system has the centrally extended symmetry PSU(2|2)⊗U(1) 3 [2], the equivalent theory is the non-linear σ-model on an enlarged coset space PSU(2|2)⊗U(1) 3 /{H⊗U(1)}. That is, we show that the Killing scalar of this generalized non-linear σ-model has the same transformation property as the spinvariable ψ m with the PSU(2|2)⊗U(1) 3 symmetry. As the result the non-linear σ-model has the position dependent R-matrix of the spin-system and correlation functions taking the centrally extended form identical to (1). But the coset space PSU(2|2)⊗U(1) 3 /{H⊗U(1)} is meaningless as it is, because the ordinary non-linear realization is not applicable to a non-simple group such as PSU(2|2)⊗U(1) 3 . To give it a precise meaning we consider the non-linear σ-model on a further enlarged coset space D(2,1;γ)/{H⊗U(1)}. The nonlinear σ-model on PSU(2|2)⊗U(1) 3 /{H⊗U(1)} is defined by the symmetry reduction of D(2,1;γ) to PSU(2|2)⊗U(1) 3 in the model thus generalized. The symmetry reduction is undertaken in the same way as was done for the spin-system [2,7].
This letter is organized as follows. In section 2 we explain the Lie-algebra of D(2,1;γ) and a reducing process to go to the subgroups PSU(2|2)⊗U(1) 3 , SU(2|2), PSU(2|2) at an algebraic level. In section 3 the matrix representation for those subgroups are given. The non-linear representation of D(2,1;γ) is discussed in section 4. In section 5 we undertake the symmetry reduction, discussed at the algebraic level in section 2, in the coset space D(2,1;γ)/{H⊗U(1) 3 }. It is then shwon that the non-linear σ-model on the reduced coset space is equivalent to the spin-chain with the PSU(2|2) symmetry.

Non-linear representation of D(2,1;γ)
In this section we discuss the coset space D(2,1;γ)/{H⊗U(1)} which non-linearly realizes the exceptional supergroup D(2,1;γ), whose Lie-algebra was discussed in section 2. We choose the simplest case where H=SU(2)⊗SU (2). Then the generators of D(2,1;γ) are decomposed into the subsets denoted by For a left multiplication of an element e iǫ·T D ∈ D(2,1;γ) the coset element changes as with a compensator e −i[λ(ϕ,ǫ)·Ĥ+λ C (ϕ,ǫ)C] . Here use was made of (23) defines the transformation of the coordinates ϕ i → ϕ ′i (ϕ, ǫ). When ǫ is infinitesimally small this relation defines the Killing vectors R Ξi (ϕ) as and the parameter functions λ Ξ (ϕ) and λ Ξ C (ϕ) of the compensator as According to [12] they can be calculated in a purely algebraic way. We only outline the calculation. For the details the reader may refer to [13]. For infinitesimally small parameters ǫ Ξ , we may write the transformation (23) as This becomes by using the following formulae: for matrices E and X if E ≪ 1. Here α n are the constants , · · · · · · .
5 Symmetry reduction to PSU(2|2)⊗U(1) 3 Now we come to the main point of this letter. Let us reduce the D(2,1;γ) symmetry to PSU(2|2)⊗U(1) 3 in the non-linear representation. The Lie-algebra in this limit was given (14) and (15), while (11)∼(13) vanishing after the rescaling (C, P, K) → 1 γ (C, P, K). To find the Killing vectors (28) in the scaling limit, the calculation in the previous section should be redone with these algebra. We then find them to tend to It amounts to dropping the coupling terms of the parameters ǫ C , ǫ P , ǫ K and the coordinate x in (28). It is because the multiple commutators with C, P, K in the r.h.s. of (25) are vanishing when calculated in the limit. Correspondingly the coset space D(2,1;γ)/{H⊗U(1)} is reduced to PSU(2|2)⊗U(1) 3 / {H⊗U(1)}. The transformation (34) represents the PSU(2|2)⊗U(1) 3 symmetry on this reduced coset space. Thus a non-simple group symmetry such as PSU(2|2)⊗U(1) 3 has been non-linearly realized in a definite way. Without the above symmetry reduction it would have been found hardly. This is an important point in this letter. Thus the coset space PSU(2|2)⊗U(1) 3 /{H⊗U(1)} gets well-defined by the Killing vectors (34). For this coset space we may find the Killing scalar as well. Here also we had better recalculate it as has been done for the Killing vectors (34), rather than resort to a scaling argument of (33). It takes the form with Υ PSU (θ, ω) the Killing scalar for the coset space PSU(2|2)/H. The phase factor is due to the decoupling of C, P, K in the limit. This form of the Killing scalar may be alternatively understood by writing the coset space PSU(2|2)⊗U (1) by the non-linear transformation (34) by the construction. It is interesting to observe that the linear transformation (36) is obtained although the phase factor e ixP +iyK is subjected to the non-linear transformation by the Killing vector R Ξ P in (34). (36) is isomorphic to the transformation (21). Hence Υ PSU⊗U(1) 3 may be identified with the spin-variable ψ m of the spin-system with PSU(2|2)⊗U(1) 3 . Consequently we are led to claim that the non-linear σ model on PSU(2|2)⊗U(1) 3 /{H⊗U(1)} is equivalent to the spin-system with the centrally extended symmetry PSU(2|2)⊗U(1) 3 .
It corresponds to (35) in the previous argument, and has the same transformation property as (36) by the non-linear transformation realized on the PSU(2|2)⊗U(1) 3 /U(1) 3 . Either of the Killing scalars Υ(ϕ, σ) and Υ(x, y, X S 2 ⊗S 2 , Θ), given in a form such as (33), satisfies the exchange algebra with the universal R-matrix of D(2,1;γ), when nonlinear σ-models on those coset spaces are quantized following [8,12]. The universal R-matrix becomes position-dependent as the consequence of the symmetry reducing of these Killing scalars as (35) and (40) respectively. The real problem is to understand how the U(1) phase factors in (35) and (40) are braided in the correlation functions (1) so that the R-matrix satisfies the Yang-Baxter equation [2]. We hope that the non-linear representation presented in this letter would shed new light on such a study.