Xi Chen (陳曦) ,Wen-Li Yang (楊文力) ,Xiang-Mao Ding (丁祥茂) and Yao-Zhong Zhang (張耀中)

1 College of Intelligent Systems Science and Engineering,Hubei Minzu University 445000,China

2 Institute of Modern Physics,Northwest University,Xian 710069,China

3 Institute of Applied Mathematics,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100080,China

4 School of Mathematics and Physics,The University of Queensland,Brisbane,QLD 4072,Australia

Abstract This paper focuses on the finite dimensional irreducible representations of Lie superalgebra D(2,1;α).We explicitly construct the finite dimensional representations of the superalgebra D(2,1;α) by using the shift operator and differential operator representations.Unlike ordinary Lie algebra representation,there are typical and atypical representations for most superalgebras.Therefore,its typical and atypical representation conditions are also given.Our results are expected to be useful for the construction of primary fields of the corresponding current superalgebra of D(2,1;α).

Keywords: superalgebra,representations,shift operator,conformal field theory

1.Introduction

Affine Lie algebras and their corresponding conformal field theories(CFTs)have essential applications in many subfields of physics [1].Supersymmetry is the superalgebra associated with the symmetry generator.The concepts of supersymmetry relate to bosonic and fermionic degrees of freedom [2].Supersymmetry theory is a uniform framework for the systems of bosons and fermions.The conformal field theories are based on current algebras.Current superalgebras and their corresponding two-dimensional conformal field theory have played a fundamental role in the high-energy physics and statistical physics at critical point,such as logarithmic CFTs[3],topological field theory [4],disordered systems and integer quantum Hall effects [5–11].In most applications of conformal field theories,one needs to construct the finitedimensional representations of a superalgebra explicitly.

Unlike ordinary bosonic algebra representation,there are typical and atypical representations for most superalgebras.The typical representation is similar to the representation that appeared in bosonic algebra.The atypical representation can be irreducible or not fully reducible.There is no atypical representation?s counterpart in ordinary bosonic algebra representation [12,13].This makes the study of the representations of superalegbras extremely difficult.The superalgebras psl(n|n) and osp(2n+2|2n) stand out as a most interesting class due to the fact that the corresponding sigma models with their supergroups have a vanishing superdimension or vanishing dual Coexter number.The nonlinear sigma models based on the supergroups have a vanishing oneloop β function,which are expected to be conformal invariant without adding the Wess–Zumino terms [14].Finite-dimensional typical and atypical representations of osp(2|2) and gl(2|2) have been studied in several papers [15,16].

The sigma model associated with psl(4|4) (or su(2,2|4)) is related to the string theory on the AdS5×S5background.Recent studies show that the superalgebra D(2,1;α)is the one-parameter deformation of Lie superalgebra D(2,1)=osp(4|2) and has a vanishing dual Coexter number.It has played an important role in describing the origin of the Yangian symmetry of AdS/CFT[17,18] and the symmetry of string theory on AdS3×S3×S3×S1.There are two types of AdS3geometries which preserve superconformal symmetry;the finite-dimensional subalgebras of these superconformal algebras are psu(1,1|2)and D(2,1;α)[19].The parameter α is related to the relative size of the radius of geometry [20].Thus,the study of the D(2,1;α)model would provide essential insight into the quantization of the string theory on the AdS3×S3×S3×S1background.

This paper is organized as follows.In section 2,we review the definition of finite-dimensional exceptional superalgebra D(2,1;α) and its commutation relations.In section 3,we explicitly give the differential operator representations of all the generators.In section 4,we give the shift operators.In section 5,we construct the finite-dimensional representation of superalgebra D(2,1;α).In section 6,we give four atypical conditions.If none of the four atypical conditions are satisfied,then the representation is a typical representation.Section 7 is devoted to our conclusions.

2.D(2,1;α) superalgebra

The exceptional Lie superalgebra D(2,1;α) with α forms a continuous one-parameter family of superalgebras of rank 3 and dimension 17 [2].It is a deformation of the Lie superalgebra osp(4|2) with the parameter α ≠0,-1,∞.The bosonic(or even)part is a su(2)⊕su(2)⊕su(2)of dimension 9,and the fermionic(or odd)part is a spinor representation(2,2,2) of the bosonic part of dimension 8.In terms of the orthogonal basis vector ?1,?2,?3with the inner product

The even roots Δ0and the odd roots Δ1of D(2,1;α) are given by

and with each positive root δ,there are generators Eδ(raising operator),Fδ≡E-δ(lowering operator) and Hδ(Cartan generator).These operators have definite Z2-gradings:

For any two generators a,b ?D(2,1;α),the (anti)commutator is defined by

the commutation relations of D(2,1;α) are

and all the other commutators are zero.

3.Differential operator representation of D(2,1;α)

To obtain a shift operator [22] of D(2,1;α),one needs to construct the differential operator representations [23–31] of the Lie superalgebra D(2,1;α).Let〈Λ|be the highest weight vector in the representation of D(2,1;α) with the highest weights λi,satisfying the following conditions:

An arbitrary vector in the representation space is parametrized by the bosonic coordinate variablesand fermionic coordinate variables

We constructed the corresponding G+(x,θ) as follows:

and the associated Gδare given by (e is Euler?s number)

One can define a differential operator realization ρ(d)of the generators of Lie superalgebra D(2,1;α) by the following relation

Here,ρ(d)(g) is a differential operator of the bosonic and fermionic coordinate variablesassociated with the generator g.After some manipulations,we obtain the following differential operator representations of all generators of Lie superalgebra D(2,1;α):

One can directly check that the differential operator realizations satisfy the commutation relations of Lie superalgebra D(2,1;α) [21].

4.Shift operator of D(2,1;a)

The even part of Lie superalgebra D(2,1;α) is su(2)⊕su(2)⊕su(2),with the basis si,ti,ui(i=0,±),satisfying the relations

The odd part of Lie superalgebras D(2,1;α) is a spinor representation (2,2,2) of the even part,with components[22].In our assumption,the elements of D(2,1;α) are given by

The invariant scalars of the Lie subalgebra of D(2,1;α) are given by

Irreducible representations of Lie superalgebra can be reduced into the direct sum of a set of irreducible representations of subalgebra.The representation of su(2)⊕su(2)⊕su(2) can be labeled by(s,t,u),where s(s+1),t(t+1),u(u+1)are the eigenvalues of the subalgebra invariants S2,T2,U2.And the representations of D(2,1;α) are labeled by |s,ms;t,mt;u,mu;λ〉,where ms,mt,muare eigenvalues of the s0,t0,u0.The degeneracy representations can be labeled by λ.The operatoris defined by

The operators ?tand ?uare defined in the same way.Let(p,q,r)be the corresponding(s,t,u) values,and p be the maximum s value in the reduction of a D(2,1;α)representation.Therefore,the decomposition into su(2)⊕su(2)⊕su(2) is given by

The(s,t,u)=(p-1,q,r)is a twofold degeneracy.Therefore,the multiplicity of the (s,t,u) representation is denoted as |p-1,mp;q,mq;r,mr;λ〉(λ=1,2).

The shift operators Oi,j,kshift an eigenstate into one or two eigenstates (for the twofold degenerate case),

The normalized shift operator Ai,j,kis

5.Representations of D(2,1;α)

The exceptional Lie superalgebra D(2,1;α) (α ≠0,-1)forms

6.The typical and atypical representation of D(2,1;α)

The (s,t,u) components must satisfy

and the (p,q,r) also belongs to this set.If p ≥2,q ≥1,r ≥1,there are four atypical conditions [22] given by

If none of the four atypical conditions are satisfied,then the representation is a typical representation,which decomposes into 16 subalgebra irreducible representations.If one of the conditions is satisfied,the representation is reducible but indecomposable generally.The shift operator will separate the 16-dimensional lattice into two 8-dimensional lattices.Since

If p<2,q<1,r<1,only none-negative value elements appear in the decomposition of the (s,t,u) lattice.

7.Conclusions

First,we have reviewed the explicit differential operator representations for Lie superalgebra D(2,1;α).Based on the shift operator and differential operator representations,we have constructed the explicitly finite-dimensional representations of superalgebra D(2,1;α)by using bosonic and fermionic coordinates.Our results are expected to be useful for the construction of primary fields of the corresponding current superalgebra of D(2,1;α),which play an important role in the computation of quantization of the string theory on the AdS3×S3×S3×S1background.

Acknowledgments

This work received financial support from the National Natural Science Foundation of China (Grant No.11 405 051).Yao-Zhong Zhang was supported by the Australian Research Council Discovery Project DP190101529.Xiang-Mao Ding was supported by NSFC Grant 11 775 299.