aa r X i v : . [ h e p - t h ] J u l Prepared for submission to JHEP
Conical defects and N = 2 higher spin holography Yasuaki Hikida
Department of Physics, and Research and Education Center for Natural Sciences, Keio University,Hiyoshi, Yokohama 223-8521, Japan
E-mail: [email protected]
Abstract:
We study conical geometry with the maximal number of fermionic symmetryin the higher spin supergravity described by sl( N + 1 | N ) ⊕ sl( N + 1 | N ) Chern-Simonsgauge theory. It was proposed that a three dimensional N = 2 higher spin supergravityis holographically dual to the N = (2 , CP N Kazama-Suzuki model. Based one theduality, we find a map between conical geometries and primary states in the dual CFT. Inparticular, we construct geometric solutions corresponding to primary states in the RR-sector. The proposal is checked by the comparison of a few charges and by the relationbetween null vectors and higher spin symmetry.
Keywords:
Conformal and W Symmetry, AdS-CFT correspondence, Supergravity Mod-els
ArXiv ePrint: ontents CP N model 13 W -algebra charges 154.3 Null vectors v.s. higher spin symmetry 18 ( N + 1 | N ) Lie superalgebra 22
A.1 Generators, roots and weights 22A.2 Generators 24
B Degenerate representations of N = 2 W N +1 algebra 26 Higher spin gauge theories include gauge fields with spin two and higher, and they can bethought as a kind of extensions of gravity theory with spin two gauge field. They attract alot of attention since it is believed that they are related to the tensionless limit of superstringtheory. Furthermore, higher spin gauge theories on anti-de Sitter (AdS) background areproposed to be dual to vector-like conformal models with one-less dimensions. In [1, 2] itwas proposed that a four dimensional higher spin gauge theory developed by Vasiliev [3] isdual to three dimensional O( N ) vector model. There is also a proposal in lower dimensionsthat a three dimensional higher spin gauge theory in [4] is dual to a large N minimal model[5, 6]. In lower dimensions, there is a possibility that we can understand the duality quitedeeply. This is because three dimensional gravity theory is known to be topological whiletwo dimensional conformal field theory (CFT) is much restricted due to the large amountof symmetry. In this paper, we investigate an aspect of N = 2 supersymmetric version ofthe duality with lower dimensions in [7]. Concretely, we study maximally supersymmetric– 1 –onical defect (surplus) geometry in higher spin supergravity, and compare it to the primarystates in the dual CFT.The gravity side of the duality of [5] is given by a bosonic truncation of N = 2higher spin supergravity proposed by Prokushkin and Vasiliev in [4]. The gravity theoryconsists of gauge fields with higher spins s = 2 , , . . . and massive scalar fields with mass M = − λ . The gauge sector can be described by the Chern-Simons theory based onhigher spin algebra hs[ λ ], which can be truncated to sl( N ) at λ = ± N . The asymptoticsymmetry near the boundary of AdS is found to be a large N limit of higher spin W N algebra called as W ∞ [ λ ] [8–12]. On the other hand, the CFT side is a minimal model withrespect to the W N algebra, which can be described by the cosetsu( N ) k ⊕ su( N ) su( N ) k +1 (1.1)with the central charge c = ( N − (cid:18) − N ( N + 1)( N + k )( N + k + 1) (cid:19) . (1.2)Original proposal is that the gravity theory is dual to the ’t Hooft limit of the CFT, wherelarge N, k limit is taken with keeping the ’t Hooft parameter λ = Nk + N (1.3)finite. Then the parameter is identified with the one in the algebra hs[ λ ] and the mass ofthe dual scalars. There are many works on this duality, and in particular, the agreementof the spectrum has been shown in [13]. Moreover, holography involving minimal modelwith so( N ) instead of su( N ) has been proposed in [14, 15] and further refined in [16].Supersymmetric extensions have been done in [7] for N = 2 holography and [17] for N = 1holography.Classical geometry in higher spin gravity has been studied as well. A higher spinblack hole was constructed in [18] (see [19] and references therein), and conical defects areexamined in [20]. There is a large amount of gauge symmetry in higher spin gravity, andnotions like horizon and singularity are not gauge invariant. In particular, it was shownthat conical defects with the trivial holonomy in sl( N ) ⊕ sl( N ) Chern-Simons theory aremapped by gauge transformation into geometry without any conical singularity. It wasclaimed in [20] and later refined in [21] that the smooth geometry is dual to a primary statein a limit of the W N minimal model (1.1). Other states in the minimal model correspondto perturbative scalar fields in the gravity theory or their bound states with the conicalgeometry. The central charge of the model satisfies c ≤ N −
1, but the limit is given by ananalytic continuation as c → ∞ with finite N . The limit may be called as “semi-classical”limit. A justification of the analytic continuation is given in [12].The gravity theory for the N = 2 higher spin holography by [7] is the full N = 2 higherspin supergravity by Prokushkin and Vasiliev [4]. This theory includes fermionic higherspin gauge fields in addition to bosonic higher spin gauge fields, and they are described– 2 –y the Chern-Simons theory with shs[ λ ] ⊕ shs[ λ ] superalgebra, where shs[ λ ] reduces tosl( N + 1 | N ) for λ = N + 1. The theory also includes massive scalars and fermions whosemasses are organized by the parameter λ . The asymptotic symmetry is found to be a large N limit of N = (2 ,
2) super W N +1 algebra, which may be called as SW ∞ [ λ ] [7, 22–25].The dual CFT is proposed to be the CP N Kazama-Suzuki model [26, 27]su( N + 1) k ⊕ so(2 N ) su( N ) k +1 ⊕ u(1) N ( N +1)( N + k +1) (1.4)with the central charge c = 3 N kk + N + 1 . (1.5)Original proposal involves the ’t Hooft limit, where N, k → ∞ with finite (1.3). It is claimedthat the theory is a minimal model with respect to the N = (2 ,
2) super W N +1 algebra[28]. The spectrum of the supergravity has been reproduced by the ’t Hooft limit of thedual CFT [7, 29]. Boundary correlation functions are studied as well in [30, 31]. Recently,some classical geometry in the higher spin supergravity by sl( N + 1 | N ) ⊕ sl( N + 1 | N )Chern-Simons theory has been investigated in [32–34]. In particular, conical defects inthe Chern-Simons theory have been constructed in [32, 33]. In this paper, we study theproperties of the conical defects in more detail, and we interpret them in terms of the dualCFT.The rest of this paper is organized as follows; in the next section, we find out theconical defects in sl( N + 1 | N ) ⊕ sl( N + 1 | N ) Chern-Simons theory which preserve themaximal number of fermionic higher spin symmetry. The conical geometry is classified bya SL( N + 1 | N ) holonomy matrix with eigenvalues parametrized by integer numbers, and itcan be mapped to a smooth geometry by a gauge transformation. In section 3, we extendthe class of smooth geometry with maximal supersymmetry such that the interpretationin the dual CFT is possible. In section 4 we identify the smooth solutions with primarystates in the dual CFT. We allow both anti-periodic and periodic boundary conditions forthe Killing spinors along the spatial cycle of conical geometry, and each case correspondsto NSNS-sector or RR-sector of the dual CFT. As a check we compare some charges ofthe W -algebra. Furthermore, we map the null vectors in the dual CFT to the residualhigher spin symmetry of the conical geometry by following the recent argument in [21].Conclusion and discussions are given in section 5. In appendix A, we summarize someuseful formulas on sl( N + 1 | N ) Lie superalgebra. In appendix B we examine degeneraterepresentations of N = 2 W N +1 algebra. Note added
While completing this work, the revised version of [33] appeared in the arXiv. The authorsincluded study on fermionic symmetry of conical defects for higher spin supergravity de-scribed by sl( N + 1 | N ) ⊕ sl( N + 1 | N ) Chern-Simons theory with N ≥
3, while they dealtwith only N = 2 case in the previous version. There is overlap with the section 2 of thispaper. – 3 – Conical defects in higher spin supergravity
In [20] conical defects in a higher spin theory described by sl( N ) ⊕ sl( N ) Chern-Simonstheory have been studied and applied to the duality proposed in [5]. The arguments arerefined in [21]. In this section, we would like to investigate on the conical defect geometryin a higher spin supergravity described by sl( N + 1 | N ) ⊕ sl( N + 1 | N ) Chern-Simons gaugetheory. We will apply the results to the N = 2 duality by [7] in later sections. We would like to consider sl( N + 1 | N ) ⊕ sl( N + 1 | N ) Chern-Simons gauge theory. Its actionis given by S = S CS [ A ] − S CS [ ˜ A ] , (2.1)where S CS [ A ] = ˆ k π Z str (cid:18) A ∧ dA + 23 A ∧ A ∧ A (cid:19) . (2.2)Here the gauge fields take values in sl( N + 1 | N ) Lie superalgebra. The action is invariantunder the following gauge transformation as δA = dλ + [ A, λ ] , δ ˜ A = d ˜ λ + [ ˜ A, ˜ λ ] . (2.3)We define generalized dreibein and spin connection as e = ℓ A − ˜ A ) , ω = 12 ( A + ˜ A ) . (2.4)We identify a sl(2) subsector { L , L ± } of sl( N + 1 | N ) as a gravitational sector. Then, theChern-Simons level ˆ k , AdS radius ℓ and Newton’s constant G are related asˆ k = ℓ Gǫ N , ǫ N = str ( L L ) . (2.5)With this notation the metric is g µν = 1 ǫ N str ( e µ e ν ) . (2.6)In this paper, we only use the superprincipal embedding of osp(1 |
2) into sl( N + 1 | N ), andin that case ǫ N = N ( N + 1)4 . (2.7)See appendix A (and also [35]) for the details of the embedding. Some basics on the superlagebra including the definition of “str” may be found in appendix A. A reviewon superalgebras is given by [35]. – 4 –e would like to study locally AdS space with a conical singularity in a certain chosengauge. First we consider the ansatz for the gauge field configuration as [20, 32] A = b − a + bdx + + L dρ , ˜ A = − ba − b − dx − − L dρ . (2.8)The radial coordinate is ρ and the light-like coordinates are x ± = φ ± t . Here b = exp( ρL )and a + = N X k =1 B k ( a k , b k ) + N X ¯ k = N +2 B ¯ k ( a ¯ k , b ¯ k ) , a − = − N X k =1 B k ( c k , d k ) − N X ¯ k = N +2 B ¯ k ( c ¯ k , d ¯ k )(2.9)with [ B K ( x, y )] IJ = xδ I,K δ J,K +1 − yδ I,K +1 δ J,K . (2.10)We use the Capital letters for K, I, L = 1 , , . . . , N + 1, small letters for k = 1 , , . . . , N + 1and barred ones for ¯ k = N +2 , N +2 , . . . , N +1. The arguments a L , b L , c L , d L are constant.Assuming the static geometry with g ++ = g −− and the locally AdS metric, we have to set a L = b L = c L = d L up to a similarity transformation. The metric is now ℓ − ds = dρ − ( e ρ + M N e − ρ ) dt + ( e ρ − M N e − ρ ) dφ (2.11)with M N = 12 ǫ N N X k =1 a k − N X ¯ k = N +2 a k . (2.12)Here we redefine ρ → ρ + ln √ M N . The geometry has a conical singularity at e ρ = M N with deficit angle 2 π (1 − √ M N ). We consider the gauge field configuration corresponding to a conical defect, where in par-ticular the fermionic components are set to be zero. The fermionic higher spin symmetry isgenerated by the gauge transformation (2.3) which does not generate any non-zero fermioniccomponents. This condition is equivalent to the Killing spinor equation D µ ǫ ≡ ∂ µ ǫ + [ A µ , ǫ ] = 0 , (2.13)where ǫ is an odd element of sl( N + 1 | N ) superalgebra. Assuming that the supermatrices a + , a − in (2.8) are diagonalizable, we can write the ansatz (2.9) in the following form upto a bosonic gauge transformation a + = M X l =1 B l − ( a l − , a l − ) + M X ¯ l = M +1 B l ( a l , a l ) (2.14)– 5 –or N = 2 M with M ∈ Z and a + = M X l =1 B l − ( a l − , a l − ) + M − X ¯ l = M +1 B l − ( a l − , a l − ) (2.15)for N = 2 M −
1. With the help of bosonic gauge transformation, we set a ≥ a ≥ · · · ≥ a M − , and a M +2 ≥ a M +4 ≥ · · · ≥ a M for N = 2 M and a M +1 ≥ a M +3 ≥ · · · ≥ a M − for N = 2 M − N + 1 | N ) Lie superalgebra can be represented in terms of super-matrix, see appendix A. Let us write basic (2 N + 1) × (2 N + 1) supermatrices as ( e IJ ) KL = δ IK δ JL . The fermionic generators are then given by e i, ¯ and e ¯ ı,j where i, j = 1 , . . . , N + 1and ¯ ı, ¯ = N + 2 , . . . , N + 1. Thus ǫ can be expanded as ǫ = X i, ¯ ǫ i, ¯ e i, ¯ + X ¯ ı,j ǫ ¯ ı,j e ¯ ı,j . (2.16)The bosonic gauge field configuration we are considering does not mix e i, ¯ and e ¯ ı,j , so wecan safely set ǫ ¯ ı,j = 0. From the expression of L in (A.25) and (A.26), we have[ L , e i, ¯ ] = ( − i + ¯ − N − ) e i, ¯ . (2.17)Thus the Killing spinor can be set as ǫ i, ¯ = R ( ρ )ˆ ǫ i, ¯ ( x + ) , R ( ρ ) = exp(( i − ¯ + N + ) ρ ) . (2.18)For the x + -dependence, we use the properties of generators as[ B k ( a k , a k ) , e i, ¯ ] = a k ( − δ i,k e i +1 , ¯ + δ i,k +1 e i − , ¯ ) , (2.19)[ B ¯ k ( a ¯ k , a ¯ k ) , e i, ¯ ] = a ¯ k ( − δ ¯ , ¯ k e i, ¯ +1 + δ ¯ , ¯ k +1 e i, ¯ − ) . (2.20)For N = 2 M , this yields[ B l − ( a l − , a l − ) , e p − , ¯ ± ie p, ¯ ] = ± ia l − δ l,p ( e p − , ¯ ± ie p, ¯ ) , (2.21)[ B l ( a l , a l ) , e i, p ± ie i, p +1 ] = ± ia l δ ¯ l, ¯ p ( e i, p ± ie i, p +1 ) (2.22)for l, p = 1 , , . . . , M and ¯ l, ¯ p = M + 1 , M + 2 , . . . , M . The eigenvectors can be constructedas E η p , ¯ η ¯ l p, ¯ l = e p − , l + iη p e p, l + i ¯ η ¯ l ( e p − , l +1 + iη p e p, l +1 ) (2.23)with η p , ¯ η ¯ l = ±
1, whose eigenvalue is[ a + , E η p , ¯ η ¯ l p, ¯ l ] = i ( η p a p − + ¯ η ¯ l a l ) E η p , ¯ η ¯ l p, ¯ l . (2.24)We have another set of eigenvectors as E ¯ η ¯ L ¯ l = e N +1 , l + i ¯ η ¯ l e N +1 , l +1 , [ a + , E ¯ η ¯ L ¯ l ] = i ¯ η ¯ l a l E ¯ η ¯ L ¯ l (2.25)– 6 –or ¯ l = M + 1 , M + 2 , . . . , M . Therefore, the solutions to the Killing spinor equation aregiven by R ( ρ ) − ǫ = M X p =1 2 M X ¯ l = M +1 X η p , ¯ η ¯ l = ± c η p , ¯ η ¯ l p, ¯ l e − i ( η p a p − +¯ η ¯ l a l ) x + E η p , ¯ η ¯ l p, ¯ l (2.26)+ M X ¯ l = M +1 X ¯ η ¯ l = ± c ¯ η ¯ l ¯ l e − i ¯ η ¯ l a l x + E ¯ η ¯ l ¯ l with constants c η p , ¯ η ¯ l p, ¯ l , c ¯ η ¯ l ¯ l . In the same way, we have for N odd R ( ρ ) − ǫ = M X p =1 2 M − X ¯ l = M +1 X η p , ¯ η ¯ l = ± c η p , ¯ η ¯ l p, ¯ l e − i ( η p a p − +¯ η ¯ l a l − ) x + E η p , ¯ η ¯ l p, ¯ l (2.27)+ M X p =1 X η p = ± c η p p e − iη p a p − x + E η p p , where E η p , ¯ η ¯ l p, ¯ l = e p − , l − + iη p e p, l − + i ¯ η ¯ l ( e p − , l + iη p e p, l ) , (2.28) E η p p = e p − , N +1 + iη p e p, N +1 (2.29)with constants c η p , ¯ η ¯ l p, ¯ l , c η p p .If a part of supersymmetry is preserved, then the corresponding Killing spinors have tosatisfy anti-periodic boundary condition around the φ -cycle. Here we would like to requirethe maximal number of supersymmetry, thus we should have N ( N + 1) Killing spinors ǫ i, ¯ for all i, ¯ . This leads to the condition that a l − = p l with p l ∈ Z and a = q ¯ + 1 / q ¯ ∈ Z for N = 2 M , and a l − = p l + 1 / p l ∈ Z and a − = q ¯ with q ¯ ∈ Z for N = 2 M −
1. Notice that this condition coincides with the requirement that the holonomymatrix along the φ -cycle isHol φ ( A ) = exp (cid:18)I A φ dφ (cid:19) = ( − N sl( N +1) ⊗ ( − N − sl( N ) ⊗ u(1) (2.30)up to a similarity transformation. It is a center of the bosonic subalgebra sl( N + 1) ⊕ sl( N ) ⊕ u(1). In general, the notion of singularity is not gauge invariant in higher spin gaugetheory. Since the holonomy matrix is an gauge invariant operator, the trivial holonomymatrix suggests that our geometry is actually singularity free. In fact, by applying a gaugetransformation as in (3.31) of [20], we can map the conical defect geometry in (2.11) to asmooth wormhole geometry.We can also think of geometry with Killing spinors satisfying periodic boundary con-dition around the φ -cycle. Assuming the maximal number of Killing spinors, we have toset that a l − = p l with p l ∈ Z and a = q ¯ for N = 2 M or a − = q ¯ for N = 2 M − q ¯ ∈ Z . The holonomy matrix along the φ -cycle isHol φ ( A ) = sl( N +1) ⊗ sl( N ) ⊗ u(1) (2.31)– 7 –p to a similarity transformation. This implies that the geometry is singularity free, andagain we can map the conical defect geometry into a smooth wormhole geometry by agauge transformation.The AdS space corresponds to a l − = M + 1 − l , a = 2 M − ¯ + (2.32)for N = 2 M and a l − = M + − l , a − = 2 M − ¯ (2.33)for N = 2 M −
1. Both lead to M N = 1 / < M N < . (2.34)Let us check whether there are configurations satisfying this condition for small N cases,see also [32, 33]. For N = 1 ,
2, we can see that there is no such a solution. For N = 3,non-trivial solutions are( a , a , a ) = ( , , , ( , , , ( , , M = , , (2.35)for anti-periodic case and( a , a , a ) = ( p , , p ) , (1 , , M = , (2.36)with p ∈ Z for periodic case. Possibly there are more. For larger N , we can easily findout more solutions. In the previous section, we have dealt with conical defects with 0 < M N < / A, ˜ A with taking complex valuesand A † = − ˜ A . In this section, we examine with this setup more generic solutions whichare not included in the ansatz (2.14) or (2.15). In the next section, we will see that themap from these solutions to CFT primary states works very nicely as in the bosonic case[20, 21]. Performing the Wick rotation to (2.8), we consider the gauge field configuration with A = b − a + bdw + L dρ , ˜ A = − ba − b − d ¯ w − L dρ , (3.1)– 8 –here w = φ + iτ and ¯ w = φ − iτ . Here we assume that a + and a − are elements ofsl( N + 1 | N ) superalgebra with complex values and they can be diagonalized by somesupermatrix. Moreover, we set A † = − ˜ A and b = exp( ρL ).As argued in [33], solutions to the killing spinor equation (2.13) may be written as ǫ ( x ) = P exp( − Z xx A µ dx µ )ˆ ǫ ( x ) P exp( Z xx A µ dx µ ) . (3.2)The problem is to find out the gauge field configuration such that the maximal numberof Killing spinors satisfy anti-periodic or periodic boundary condition around the φ -cycle.When going around the cycle, the factor becomes holonomy matrix asHol φ ( A ) = exp (cid:18)I A φ dφ (cid:19) = S − exp π N +1 X I =1 θ I e II ! S , (3.3)where S is a supermatrix depending on ρ . Change the basis of spinor as ǫ ( x ) = S ˆ ǫ ( x ) S − = N +1 X l =1 2 N +1 X ¯ = N +2 ǫ l, ¯ e l, ¯ , (3.4)we can see that when the spinor with only ǫ l, ¯ = 0 goes around the cycle the phase factorbecomesexp − π N +1 X I =1 θ I e II ! e l, ¯ exp π N +1 X I =1 θ I e II ! = exp ( − π ( θ l − θ ¯ )) e l, ¯ . (3.5)The phase factor should be − l, ¯ ) when all Killing spinors satisfythe anti-periodic boundary condition. In the same way, the phase factor should be +1 forall ( l, ¯ ) when all spinors satisfy the periodic boundary condition.The condition of anti-periodicity for all Killing spinors is thus θ l − θ ¯ ∈ i ( Z + ) (3.6)for all l, ¯ . Generic solutions are θ l = i ( p l + β ) , θ ¯ = i ( q ¯ + + β ) (3.7)with p l , q ¯ ∈ Z . However, the supertraceless condition of sl( N + 1 | N ) reads β = − N +1 X l =1 p l + N +1 X ¯ = N +2 ( q ¯ + ) , (3.8)which is integer for N = 2 M and half-integer for N = 2 M −
1. Thus we see θ l = i (cid:18) p ′ l + 1 − ( − N (cid:19) , θ ¯ = i (cid:18) q ′ ¯ + 1 + ( − N (cid:19) , (3.9)– 9 –here q ′ l , p ′ j ∈ Z . It is convenient to write down θ L in terms of bosonic subalgebra sl( N +1) ⊕ sl( N ) ⊕ u(1) as − iθ i = ˜ l (1) i + ρ (1) i + ˜ mN + 1 = ˜ r (1) i − | ˜Λ (1) | N + 1 + N + 22 − i + ˜ mN + 1 , (3.10) − iθ N +1+ j = ˜ l (2) j + ρ (2) j + ˜ mN = ˜ r (2) j − | ˜Λ (2) | N + N + 12 − j + ˜ mN . (3.11)Here P i ˜ l ( a ) i = 0, ˜ r ( a ) i ∈ Z and | ˜Λ ( a ) | = P i ˜ r ( a ) i . The Weyl vectors ρ ( a ) i are defined in (A.13)and (A.14). From the condition for β , we find that˜ m ∈ − N | ˜Λ (1) | + ( N + 1) | ˜Λ (2) | + N ( N + 1) Z . (3.12)Holonomy matrix is nowHol φ ( A ) = e πi ( N + ˜ mN +1 ) sl( N +1) ⊗ e πi ( N +12 − ˜ mN ) sl( N ) ⊗ e − πi ˜ mN ( N +1) u(1) (3.13)up to a similarity transformation. Notice that it is a center of bosonic subalgebra sl( N +1) ⊕ sl( N ) ⊕ u(1) with complex elements, thus the configurations considered should correspondto smooth geometries in some gauge choice.Similarly, for the periodic case we need to assign θ l − θ ¯ ∈ i Z (3.14)for all l, ¯ , and solutions are θ l = i ( p l + β ) , θ ¯ = i ( q ¯ + β ) (3.15)with p l , q ¯ ∈ Z . The supertraceless condition leads β = − N +1 X l =1 p l + N +1 X ¯ = N +2 q ¯ , (3.16)which is also an integer number. Thus we may define − iθ i = ˜ l (1) i + ρ (1) i + ˜ mN + 1 = ˜ r (1) i − | ˜Λ (1) | N + 1 + N + 22 − i + ˜ mN + 1 , (3.17) − iθ N +1+ j = ˜ l (2) j + ρ (2) j + ˜ mN = ˜ r (2) j − | ˜Λ (2) | N + N + 12 − j + ˜ mN , (3.18)with P i ˜ l ( a ) i = 0 and ˜ r ( a ) i ∈ Z . From the condition for β , we have˜ m ∈ − N | ˜Λ (1) | + ( N + 1) | ˜Λ (2) | + N ( N + 1)( Z + ) . (3.19)The holonomy matrix isHol φ ( A ) = e πi ˜ mN +1 sl( N +1) ⊗ e − πi ˜ mN sl( N ) ⊗ e − πi ˜ mN ( N +1) u(1) (3.20)up to a similarity transformation. – 10 – .2 Asymptotically AdS geometry In order to compare the supergravity with the dual CFT, we have to search solutionswhich approach to AdS space at ρ → ∞ . For higher spin gauge theory, we need to assignboundary condition also for higher spin fields, which can be expressed as [8]( A − A AdS ) | ρ →∞ ∼ O (1) . (3.21)The gauge field configuration A AdS corresponding to the AdS background is given by (3.1)with a + = L and a − = − L − . The condition is shown to be equivalent to the Drinfeld-Sokolov reduction in [8], and in our case, the classical asymptotic symmetry under thecondition is N = (2 ,
2) super W N +1 algebra [7].The conical geometry considered has higher spin charges associated with the W -algebra, and we would like to compute them in this subsection. In order to assign theasymptotic boundary condition to the gauge fields, it is convenient to decompose thesl( N + 1 | N ) elements by its sl(2) subalgebra. The decomposition depends on how weembed sl(2), and we have chosen the one in [7] such thatsl( N + 1 | N ) = sl(2) ⊕ ( ⊕ N +1 s =3 g ( s ) ) ⊕ ( ⊕ Ns =1 g ( s ) ) ⊕ · ( ⊕ Ns =1 g ( s +1 / ) , (3.22)where g ( s ) is the (2 s − Z -gradingof the superalgebra. With this decomposition, the generators may be given by V ( s )+ n ( s = 2 , , . . . , N + 1) , V ( s ) − n ( s = 1 , , . . . , N ) , F ( s ) ± r ( s = 1 , , . . . , N ) , (3.23)where | n | ≤ s − , | r | ≤ s − /
2. The embedded sl(2) is generated by L m = V (2)+ m with m = 0 , ±
1. Some of the commutation relations may be found in appendix A.In terms of these generators, the gauge field configuration satisfying the asymptoticAdS condition (3.21) can be set as a + ( t + θ ) = L + 1ˆ k X s ≥ N + s L + s ( t + θ ) V ( s )+ − s +1 + X s ≥ N − s L − s ( t + θ ) V ( s ) −− s +1 (3.24)+ X s ≥ M + s +1 / G + s +1 / ( t + θ ) F ( s )+ − s +1 / + X s ≥ M + s +1 / G − s +1 / ( t + θ ) F ( s ) −− s +1 / ! by utilizing residual gauge transformation. Here we have defined N ± s = str ( V ( s ) ± s − V ( s ) ±− s +1 ) , M ± s +1 / = str ( F ( s ) ± s − / F ( s ) ±− s +1 / ) . (3.25)At the boundary, the functions L ± s ( θ ) , G ± s +1 / ( θ ) act as generators of classical N = 2 super W N +1 algebra, see [7, 22, 23]. In particular, the energy momentum tensor comes from L +2 ( θ ) and the central charge is c = 12ˆ kǫ N = 3 ℓ G (3.26)– 11 –n terms of parameters in (2.5). Note that this value is the same as the one obtainedfor the pure gravity in [36]. The other generators are primary with respect to the energymomentum tensor.As for our geometry, we assume the form of the gauge field as (3.1), where a + takes avalue in constant sl( N + 1 | N ) superalgebra. It is useful to define T ( s ) ± n = 12 ( V ( s )+ n ± V ( s ) − n ) , T (1) − = V (1) − , T ( N +1)+ n = V ( N +1)+ n , (3.27)where s = 2 , , . . . , N . In this notation, T ( s )+ n , T ( s ) − n ( s ≥
2) and T (1) − generate sl( N + 1),sl( N ) and u(1) bosonic subalgebras. Assigning the asymptotic AdS condition (3.21), thegauge field takes the form of a + ( t + θ ) = L + X s ≥ ˆ k − s/ t ( s )+ v ( s )+ T ( s )+ − s +1 + X s ≥ ˆ k − s/ t ( s ) − v ( s ) − T ( s ) −− s +1 (3.28)with t ( s ) ± = str ( T ( s ) ± s − T ( s ) ±− s +1 ) . (3.29)The constant coefficients v ( s ) ± are related to eigenvalues θ L in (3.3) by a gauge transfor-mation, and they correspond to the charges of N = 2 super W N +1 algebra. Notice thatthe fermionic components are set to be zero in our gauge configurations. The above formwith the normalization is particularly useful since we can just apply the result of [20] tothe bosonic subalgebras. The u(1) charge can be easily read off as v (1) − = − i ˆ k / ˜ m (3.30)by using the notation in appendix A. The other first few charges are [20] v (2)+ = − ˆ kC +2 (˜ n (1) ) ,v (3)+ = − i ˆ k / C +3 (˜ n (1) ) , (3.31) v (4)+ = ˆ k C +4 (˜ n (1) ) − C +4 ( ρ (1) )( C +2 ( ρ (1) )) ( C +2 (˜ n (1) )) ! , and v (2) − = ˆ kC − (˜ n (2) ) ,v (3) − = i ˆ k / C − (˜ n (2) ) , (3.32) v (4) − = − ˆ k C − (˜ n (2) ) − C − ( ρ (2) )( C − ( ρ (2) )) ( C − (˜ n (2) )) ! , where C + s (˜ n (1) ) = 1 s N +1 X j =1 (˜ n (1) j ) s , C − s (˜ n (2) ) = 1 s N X j =1 (˜ n (2) j ) s (3.33)– 12 –ith ˜ n ( a ) j = ˜ l ( a ) j + ρ ( a ) j . (3.34)Notice that this expression holds both for anti-periodic and periodic cases.In the previous subsection, we assumed that the supermatrix a + is diagonalizable. Asdiscussed in [20], the supermatrix of the form (3.28) can de diagonalized by supermatrixcorresponding to sl( N + 1) ⊕ sl( N ) bosonic subalgebra only when all ˜ n ( a ) j are distinct forboth a = 1 ,
2. Thus we need to require˜ n (1)1 > ˜ n (1)2 > · · · > ˜ n (1) N +1 , ˜ n (2)1 > ˜ n (2)2 > · · · > ˜ n (2) N . (3.35)In this case the holonomy matrix can be labeled by two Young diagrams Λ ( a ) along withthe u(1) charge ˜ m . In terms of parameters in (3.10), (3.11), (3.17) and (3.18), the Youngdiagram Λ ( a ) has ˜ r ( a ) j boxes in the j -th row. CP N model In [7] it was proposed that the N = 2 higher spin supergravity in [4] is dual to the ’t Hooftlimit (1.3) of the CP N Kazama-Suzuki model (1.4)su( N + 1) k ⊕ so(2 N ) su( N ) k +1 ⊕ u(1) N ( N +1)( k + N +1) (4.1)with the central charge c = 3 N kk + N + 1 . (4.2)The massless sector of the supergravity is described by shs[ λ ] ⊕ shs[ λ ] Chern-Simons theory,and the higher spin superalgebra shs[ λ ] can be truncated to sl( N + 1 | N ) at λ = N + 1.Based on the duality, we identify the classical smooth geometry of sl( N +1 | N ) ⊕ sl( N +1 | N )Chern-Simons theory considered in the previous section as a primary state in its dual CFT.We perform several checks of this identification. For the bosonic case, see [20, 21]. In order to compare with the sl( N + 1 | N ) ⊕ sl( N + 1 | N ) Chern-Simons theory, we will findthat it is necessary to shift the central charge c of the Kazama-Suzuki model (4.1) into nota physically allowed region. This implies that we need to move to a more generic theorywith the same symmetry. However, in this subsection, we still study the Kazama-Suzukimodel since the difference appears from the next leading order of 1 /c with large c as wewill see below.The states of the Kazama-Suzuki model are labeled by (Λ (1) , ω ; Λ (2) , m ). Here Λ (1) , Λ (2) are highest weights of su( N + 1) , su( N ) and the u(1) charge takes a value in m ∈ Z κ with κ = N ( N + 1)( k + N + 1). There are four representations of affine so(2 N ) with ω = − , , ,
2. Here ω = 0 and ω = 2 correspond to identity and vector representations,– 13 –espectively, and the fermions are in the NS-sector with anti-periodic boundary condition.On the other hand, ω = − ω = 1 correspond to co-spinor and spinor representations,and the fermions are in the R-sector with periodic boundary condition. The states of thecoset are then obtained by the decompositionΛ (1) ⊗ ω = ⊕ Λ (2) ,m (Λ (1) , ω ; Λ (2) , m ) ⊗ Λ (2) ⊗ m . (4.3)From the condition that the decomposition is possible, we have a selection rule | Λ (1) | N + 1 − | Λ (2) | N + mN ( N + 1) + ω , (4.4)where | Λ ( a ) | is the number of boxes of Young diagram corresponding to Λ ( a ) . See appendixA for the notations. In general, we should take case of field identification [37] as well, butit is not relevant for our purpose. The conformal weight of the primary state is in the NS-sector h (Λ (1) , ω ; Λ (2) , m ) = n + ω k + N + 1 (cid:18) C (1) (Λ (1) ) − C (2) (Λ (2) ) − m N ( N + 1) (cid:19) , (4.5)where C ( a ) (Λ ( a ) ) is the second Casimir in the representation Λ ( a ) of su( N + 1) for a = 1and su( N ) for a = 2. Here n is the grade at which (Λ (2) , m ) appears as a descendant of(Λ (1) , ω ) (see, e.g., [38]). The u(1) charge is q (Λ (1) , ω ; Λ (2) , m ) = 2 n ′ + ω − mN + k + 1 (4.6)with an integer n ′ . In the R-sector, we have h (Λ (1) , ω ; Λ (2) , m ) = n + N k + N + 1 (cid:18) C (1) (Λ (1) ) − C (2) (Λ (2) ) − m N ( N + 1) (cid:19) ,q (Λ (1) , ω ; Λ (2) , m ) = 2 n ′ + N ω − − mN + k + 1 (4.7)with some integers n, n ′ .We would like to compare these primary states to the classical solutions of sl( N +1 | N ) ⊕ sl( N + 1 | N ) Chern-Simons theory, where the classical limit corresponds to the limitwith large Chern-Simons level ˆ k . It is also the same as the limit with large central charge c for the asymptotic symmetry algebra as in (3.26). Thus, we should take the large centralcharge limit with c → ∞ but with N kept finite. This implies that we need to consider ananalytic continuation of k to an unphysical value as k = − ( N + 1) − N ( N + 1) c + O ( c − ) . (4.8) We should take the following identification among the states as (Λ (1) , ω ; Λ (2) , m ) ≃ ( A N +1 Λ (1) , ω +2; A N Λ (2) , m + k + N + 1), where A M is an outer automorphism of su( M ). Later we consider an analyticcontinuation on k , and the field identification does not make sense with an irrational k . – 14 –he validity of the analytic continuation is discussed in [25], see also [12, 16] for bosoniccases. With this limit, the conformal weight and the u(1) charge become h (Λ (1) , ω ; Λ (2) , m ) = − c ǫ N (cid:18) C (1) (Λ (1) ) − C (2) (Λ (2) ) − m N ( N + 1) (cid:19) , (4.9) q (Λ (1) , ω ; Λ (2) , m ) = c ǫ N m for all choices of ω .In the previous section, we found a set of smooth geometry preserving the maximalnumber of fermionic symmetry. It is classified by the holonomy matrix with eigenvalues θ L ( L = 1 , , . . . , N + 1), which are parametrized by two Young diagrams ˜Λ ( a ) ( a = 1 ,
2) andone integer ˜ m as in (3.10), (3.11), (3.17) and (3.18). Thus we may identify the parametersas ˜Λ (1) = Λ (1) , ˜Λ (2) = Λ (2) , ˜ m = m . (4.10)Notice that the condition for ˜ m (3.12) and (3.19) reproduces the selection rule in (4.4).From the ADM mass of the geometry, we may read off the boundary conformal dimensionfrom the classical geometry as (see (3.16) of [20]) h = − c M N = h (Λ (1) , ω ; Λ (2) , m ) − c , (4.11)where we have used C ( a ) (Λ ( a ) ) = 12 X i h ( l ( a ) i + ρ ( a ) i ) − ( ρ ( a ) i ) i (4.12)for a = 1 ,
2. Here M N is define in (2.12) and in terms of the holonomy matrix it is writtenas M N = − ǫ N N +1 X l =1 θ l − N +1 X ¯ = N +2 θ . (4.13)In this way, we have seen that the boundary conformal dimension from the geometryreproduces the one of dual CFT in (4.9), where the shift − c/
24 comes from the change ofworldsheet geometry from the cylinder of boundary AdS to the complex plane. W -algebra charges In this subsection, we would like to examine the charges of W -algebras for states primary tothe N = 2 super W N +1 algebra and compare them to the charges for the smooth solutionsof the gravity theory. The charges for the classical geometry have been computed in theprevious section as (3.30), (3.31) and (3.32). In order to compute charges in the CFT side,we utilize the free field realization of the N = 2 super W N +1 algebra in [39] and constructstates primary to the symmetry algebra in terms of free fields. The descendants are thenobtained by the action of W -algebra generators to the primary states, see (4.32) and (4.33)below. The charges of W -algebra can be read off from the action of zero modes of the– 15 – -algebra generators to these primary states. Review articles on W -algebra may be foundin [40, 41].First we focus on the NS-sector and then move to the R-sector. We introduce the supercoordinate Z = ( z, θ ) with a Grassmanian variable θ and the super derivative D = ∂ θ + θ∂ z .Furthermore, 2 N superfields are written as Φ j ( Z ) = φ j ( z ) + iθψ j ( z ) ( j = 1 , , . . . N ) withoperator products φ i ( z ) φ j (0) ∼ − δ i,j ln z , ψ i ( z ) ψ j (0) ∼ δ i,j z − . (4.14)With the preparation we introduce a Lax operator by [42, 43] L ( Z ) = ( a D + i Θ N +1 ( Z ))( a D + i Θ N ( Z )) · · · ( a D + i Θ ( Z )) (4.15)= ( a D ) N +1 + N +1 X j =2 U j ( Z )( a D ) N +1 − j , where U j/ ( Z ) ( j = 2 , , . . . , N + 1) are the generators of N = 2 super W N +1 algebra.Here Θ j ( Z ) = ( − j − ( λ j − λ j − ) · D Φ( Z ) ( λ = λ N +1 = 0), and the normal ordering isimplicitly assumed when operators are inserted at the same position. Moreover, λ j is thefundamental weight of sl( N + 1 | N ), see appendix A. From this equation, we have U j ( Z ) = ζ X ≤ l < ··· 2. Therefore, we can see the match of the first three charges of W -algebra as a v (1) − = u ,v (2)+ + v (2) − = h − ( u ) N ( N + 1) a − c , (4.28) a v (2) − = u − (1 − N ) a u − N )( u ) N by properly choosing the relative normalizations.Let us turn to the R-sector. In order to discuss this sector, it is useful to utilize thespectral flow symmetry of N = 2 superconformal algebra introduced in [46]. The algebrais invariant under the following transformation as J η ( z ) = J ( z ) + cη z ,G ± ,η / ( z ) = z ± η G ± / ( z ) , (4.29) T η ( z ) = T ( z ) + ηJ ( z ) + cη z , where η is a continuous parameter. If we set η = 1 / 2, then the transformation maps theNS-sector to the R-sector. It is argued that the spectral flow is generated by the operator[47] U η ( z ) = exp (cid:18) − iη r c ϕ ( z ) (cid:19) , J ( z ) = i r c ∂ϕ ( z ) . (4.30)However, the bosonic subsector of W -algebra generators defined in (3.24) from the gravityside decouple with the u(1) sector by definition. Therefore, with the basis, the bosonicgenerators are invariant under the spectral flow, and this implies that the charges in theR-sector match once the correspondence of charges is shown in the NS-sector. As mentioned above, descendants in the theory based on the N = 2 super W N +1 -algebraare obtained by the action of the W -algebra generators to the primary states. Let us denote | Λ i as a primary state and the mode expansions of W -algebra generators as W ( s ) a ( z ) = X r W ( s ) ar z r + h ( s ) a , (4.31)where W ( s )0 ( z ) = J s ( z ), W ( s ) ± ( z ) = G ± s +1 / ( z ), W ( s )1 ( z ) = T s +1 ( z ) and h ( s ) a denote theirconformal weights. The sum runs over r ∈ Z for bosonic operators and fermionic operatorsin the R-sector. For fermionic operators in the NS-sector, it runs over r ∈ Z + 1 / 2. Thecondition to be primary can be then written as W ( s ) ar | Λ i = 0 , r > , (4.32)– 18 –nd the descendant states are generated as | X i = W ( s ) a − r W ( s ) a − r · · · W ( s l ) a l − r l | Λ i (4.33)with r i > 0. When a descendant satisfies the condition of primary, the state is null andshould be removed from the spectrum.These null states are constructed by the action of null vectors to primary states. Forour case, there are the maximally possible number of null vectors from each state associ-ated with the weight Λ of sl( N + 1 | N ) Lie superalgebra as in (4.19). In the NS-sector,“independent” null vectors have been investigated in [39] (see also appendix B), and thereare 2 N fermionic null vectors at level Λ I + 1 / I = 1 , , . . . N ). It will be useful to noticethat Λ j − = r (1) j − r (2) j − Λ N , Λ j − = − r (1) j + r (2) j − + Λ N (4.34)with Λ N = | Λ (1) | N + 1 − | Λ (2) | N + mN ( N + 1) (4.35)in terms of the bosonic subalgebra. As an example, let us see what happens for the chiralprimary states with h = q/ 2. These states are labeled as [48]Λ (1) j = Λ (2) j ( j = 1 , , . . . , N − , m = N X j =1 j Λ (1) j , (4.36)which implies Λ j − = 0 (4.37)for all j . Therefore, we have N independent null vectors at level 1 / 2. One of themis given by the action of ( G +3 / ) − / , which arises from the definition of chiral primary.The others are generated by N − G + s +1 / ) − / with s = 1 , , . . . , N .It was pointed out in [21] that these null vectors should be identified as the residualhigher spin symmetries of the smooth gauge field configuration dual to the primary state.When we perform the path integral of the gauge theory, we have to divide the directions ofgauge symmetry, which corresponds to removing the null vectors. For both cases with anti-periodic and periodic Killing spinors, we can read off from (3.5) that the Killing spinorscorresponding to fermionic higher spin symmetry have w -dependence as exp( − ( θ l − θ ¯ ) w ).If we consider the spinors associated with ǫ ¯ ı,j in (2.16), then we have similarly the Killingspinors behaving as exp(( θ j − θ ¯ ı ) w ). Utilizing the parameters in (3.10), (3.11) and (3.34),we see that − i ( θ l − θ N +1+ j ) = ˜ n (1) i − ˜ n (2) j − ˜ mN ( N + 1) . (4.38)– 19 –herefore, the negative mode number of the Killing spinor with ǫ j,N +1+ j or ǫ N +1+ j +1 ,j coincides with the level of fermionic null vector Λ I +1 / 2. The other negative mode numberscan be obtained by the shifted Weyl reflections of sl( N + 1 | N ), which are the same asthose of bosonic subalgebra sl( N + 1) ⊕ sl( N ). These additional Killing spinors shouldcorrespond to null vectors appearing as descendants of the independent null vectors as inthe bosonic case [21, 49]. The maximally supersymmetric geometry also preserves themaximal number of bosonic higher spin symmetry. This is because the bosonic subgroupof the holonomy matrix along the spatial cycle is given by the center of SL( N + 1) ⊗ SL( N )up to a similarity transformation, see (3.13) and (3.20) above. In particular, we can showas in [21] that the w -dependence of the Killing vector is exp( − i (˜ n ( a ) i − ˜ n ( a ) j ) w ) for a = 1 , n ( a ) i − n ( a ) j ) with i < j . These Killing vectors should correspond to bosonic null vectorsappearing as descendants of fermionic independent null vectors.From the relation between higher spin symmetry and null vectors, we can say thatthe geometry dual to the primary states should have the maximal number of higher spinsymmetry. Moreover, we may obtain the one-loop partition function of the gravity theoryfrom the relation to the CFT. The one-loop partition function of the CFT can be writtenas a sum of characters of representation Ξ = (Λ (1) ; Λ (2) , m ) as Z CFT1-loop ( q ) = X Ξ | ch NS,RΞ ( q ) | , ch NS,RΞ ( q ) = tr Ξ q L . (4.39)Here the trace is over the states obtained by the action of W -algebra generators to theprimary state with label Ξ modulo the null vectors in the NS-sector or in the R-sector.From the CFT expression in (4.39) we expect that the one-loop partition function of thegravity theory can be obtained following [20] as the sum over the contributions from eachsmooth geometry. See section 5 for some discussions. In this paper, we have studied purely bosonic conical defects in sl( N + 1 | N ) ⊕ sl( N + 1 | N )Chern-Simons gauge theory with the maximal number of fermionic higher spin symmetry,where both anti-periodic and periodic boundary conditions of Killing spinors can be chosen.The gauge field configuration then is parametrized by two sets of integer number. Theholonomy matrix of the gauge field configuration is given by a center of bosonic subgroupup to a similarity transformation, which implies that the conical defect can be mappedto a non-singular geometry. As in [20] we have extended the class of smooth geometry,which is now labeled by two Young diagrams ˜Λ ( a ) with a = 1 , m . Thesmooth geometry is proposed to be dual to a primary state in the CFT with N = 2 W N +1 symmetry at the limit of large central charge c → ∞ . The primary state is labeled bysl( N + 1 | N ) weight as in (4.19), which can be expressed by two Young diagrams Λ ( a ) andu(1) charge m . We identify the labels as ˜Λ ( a ) = Λ ( a ) and ˜ m = m . Moreover, the cases with In the bosonic W N minimal model, the Weyl invariance of the null vector structure was shown in [49].In our supersymmetric model, it is an open problem to proof (or disproof) this. – 20 –nti-periodic and periodic Killing spinors are mapped to NSNS-sector and RR-sector of theCFT, respectively. This proposal is checked by comparing some W -algebra charges. Thenull vectors in the CFT are identified as the residual higher spin symmetry of the smoothgeometry.Once we know the relation between the null vectors and the residual higher spin sym-metry, we can guess the gravity partition function by following [21]. For the bosonic partof the gravity partition function, we can just use the result of [21] as Z B1-loop (Ξ) = Q ≤ i 2) super W N +1 algebra.The gravity partition function may differ from the CFT one by an overall factor as in thebosonic case, see the end of section 6 in [21]. It is an important open problem to reproducethe above expression by directly computing one-loop determinants of the supergravity.Moreover, we should compare the gravity partition function with the CFT one, which maybe possible by utilizing the expression in [29] or by generalizing the null vector analysis in[49] to our supersymmetric case. In particular, it is interesting to understand the structureof null vectors in the R-sector along with the detailed analysis on the NS-sector.One of the motivation to study this semi-classical limit of the N = 2 duality is that wecould have a AdS/CFT correspondence involving purely three dimensional Chern-Simonstheory without any matter fields coupled unlike for the bosonic case. Without the matterfields, we may have a chance to proof the duality by the application of the Drinfeld-Sokolov– 21 –eduction. For further understanding, we would like to study the duality beyond the large c limit by examining 1 /c corrections. For instance, at the next order of 1 /c , we cansee the difference between the states in the Kazama-Suzuki model (1.4) and the statescorresponding to the vertex operator (4.19), see appendix B. From the experience on thebosonic case in [20], it is natural to identify the states with ˆΛ = 0 in (B.18) as the smoothgeometry, and those with ˆΛ = 0 as a geometry dressed by perturbative corrections. Sincethe Kazama-Suzuki model is an unitary model with N = 2 W N +1 symmetry, it wouldbe important to see what kind of corrections make the theory unitary. It is also worthto study conical defects for so( N ) holography [14–16] and for N = 1 holography [17]. Inparticular, it was argued in [16] that the finite N effects for the so( N ) holography are abit more complex than the su( N ) case. It would be also interesting to study black holesolutions in the higher spin supergravity and see the relation to the duality. See a review[19] for the bosonic case. Some higher spin black holes in the supergravity have beenalready constructed in [32, 33]. Acknowledgements We are grateful to T. Creutzig, R. Gopakumar, P. Rønne and T. Ugajin for useful discus-sions. The work of YH was supported in part by Grant-in-Aid for Young Scientists (B)from JSPS. A sl ( N + 1 | N ) Lie superalgebra We summarize here useful formulas on sl( N + 1 | N ) Lie superalgebra. A.1 Generators, roots and weights The generators of sl( N +1 | N ) Lie superalgebra can be described (( N +1)+ N ) × (( N +1)+ N )supermatrices of the form M = A BC D ! , str M = tr A − tr D = 0 , (A.1)where A, D are even elements and B, C are odd elements with respect to the Z gradingof superalgebra. The traceless part of A, D generate sl( N + 1) , sl( N ) bosonic subalgebra,respectively, and the centralizer of the bosonic subalgebra is u(1). The bosonic subalgebrais thus sl( N + 1) ⊕ sl( N ) ⊕ u(1).The sl( N + 1 | N ) Lie superalgebra has a special property that we can choose a com-pletely odd simple root system. We introduce two orthogonal bases ε i ( i = 1 , , . . . , N + 1)and δ i ( i = 1 , , . . . , N ), which satisfy ε i · ε j = δ i,j , δ i · δ j = − δ i,j . (A.2) Such Lie superalgebras are given by sl( N ± | N ), osp(2 N ± | N ), osp(2 N | N ), osp(2 N + 2 | N ) andD(2 , α ) with α = 0 , ± 1, see, e.g., [35]. We borrow the notations in [39]. – 22 –hen the odd simple roots can be expressed as α i − = ε i − δ i , α i = δ i − ε i +1 (A.3)for i = 1 , , . . . , N . Then positive roots are α i + α i +1 + · · · + α j (A.4)with i ≤ j , where the root is even (odd) when i − j is even (odd). The fundamental weightsare defined by α i · λ j = δ i,j , (A.5)which can be expressed by the simple roots as λ i = α + α + · · · + α i − , λ i − = α i + α i +2 + · · · + α N . (A.6)As mentioned above, the sl( N + 1 | N ) superalgebra has sl( N + 1) ⊕ sl( N ) ⊕ u(1) bosonicsubalgebra. The abelian factor u(1) is generated by ν = N X i =1 ( λ i − λ i − ) , (A.7)whose norm is ν · ν = − N ( N + 1). The simple roots α (1) i and the fundamental weights λ (1) i for sl( N + 1) subalgebra are α (1) i = α i − + α i = ε i − ε i +1 , (A.8) λ (1) i = i − X j =1 ( − j − λ j + iN + 1 ν = i X j =1 ε j − iN + 1 N +1 X j =1 ε j (A.9)with i = 1 , , . . . , N . In the same way, the simple roots α (2) i and the fundamental weights λ (2) i for sl( N ) subalgebra are α (2) i = α i + α i +1 = δ i − δ i +1 , (A.10) λ (2) i = i X j =1 ( − j λ j − iN ν = − i X j =1 δ j + iN N X j =1 δ j (A.11)with i = 1 , , . . . , N − 1. The Weyl vector is ρ = N X i =1 λ i = 2( ρ (1) + ρ (2) ) , (A.12)where the Weyl vectors ρ (1) , ρ (2) for sl( N + 1) , sl( N ) are ρ (1) = N X i =1 λ (1) i = N +1 X i =1 ρ (1) i ε i = N +1 X i =1 (cid:0) N +22 − i (cid:1) ε i , (A.13) ρ (2) = N − X i =1 λ (2) i = − N X i =1 ρ (2) i δ i = − N X i =1 (cid:0) N +12 − i (cid:1) δ i . (A.14)– 23 – weight for sl( N + 1 | N ) Lie superalgebra is expressed asΛ = N X i =1 Λ i λ i (A.15)with non-negative integers Λ i . In terms of bosonic subalgebra, the weight may be expressedas Λ = N X i =1 Λ (1) i λ (1) i + N − X i =1 Λ (2) i λ (2) i + mN ( N + 1) ν (A.16)with Λ (1) i = Λ i + Λ i − , Λ (2) i = Λ i + Λ i +1 (A.17)and m = N X i =1 ( i Λ i − ( N + 1 − i )Λ i − ) . (A.18)With the decomposition, the weight may be labeled by two Young diagrams correspondingto Λ ( a ) = P j Λ ( a ) j λ ( a ) j with a = 1 , m . The diagrams have r ( a ) j boxesin the j -th low with r (1) j = N X i = j Λ (1) j , r (2) j = N − X i = j Λ (2) j . (A.19)In the orthogonal basis, the weights are decomposed asΛ (1) = N +1 X j =1 l (1) j ε j , Λ (2) = − N X j =1 l (2) j δ j (A.20)with l (1) j = r (1) j − | Λ (1) | N + 1 , l (2) j = r (2) j − | Λ (2) | N . (A.21)Here r (1) N +1 = r (2) N = 0 and | Λ ( a ) | is the number of boxes in the corresponding Youngdiagram. A.2 Generators The generators of sl( N + 1 | N ) are given by V ( s )+ n ( s = 2 , , . . . , N + 1) , V ( s ) − n ( s = 1 , , . . . , N ) , F ( s ) ± r ( s = 1 , , . . . , N ) (A.22)with | n | ≤ s − , | r | ≤ s − / 2. We have utilized the principal embedding of osp(1 | 2) intosl( N + 1 | N ) superalgebra, see [35] for instance. The embedded osp(1 | 2) corresponds to L n = V (2)+ n and G r = F (1)+ r , which satisfy[ L m , L n ] = ( m − n ) L m + n , [ L m , G r ] = ( m − r ) G m + r , { G r , G s } = 2 L r + s . (A.23)– 24 –he (anti-)commutation relations to other generators are[ L m , V ( s ) ± n ] = ( − n + m ( s − V ( s ) ± m + n , [ L m , F ( s ) ± r ] = ( − r + m ( s − )) F ( s ) ± m + n , [ G / , V ( s )+ m ] = − ( m − s + 1) F ( s − m +1 / , [ G / , V ( s ) − m ] = − F ( s ) − m +1 / , (A.24) { G / , F ( s − r } = 2 V ( s )+ r +1 / , { G / , F ( s ) − r } = ( r − s + ) V ( s ) − r +1 / . Other (anti-)commutation relations can be found in [50].It might be useful to express the generators L m of sl(2) subalgebra in terms of super-matrix. We use L n = K N +1 n K Nn ! , (A.25)with K M = 12 M − M − − M 00 1 − M , (A.26) K M = − √ M − p M − 2) 0. . .0 p | i ( M − i ) | 0. . . 0 √ M − , (A.27) K M − = √ M − p M − 2) 0. . .0 p | i ( M − i ) | 0. . .0 √ M − . (A.28)In particular, we find ǫ N = str ( L L ) = tr ( K N +10 K N +10 ) − tr ( K N K N ) = N ( N + 1)4 . (A.29)– 25 –or J o = V (1) − , we use J = N ( N +1) × ( N +1) 00 ( N + 1) N × N ! , (A.30)where M × M is the M × M identity matrix. The normalization is t ( s ) − = str( J J ) = − N ( N + 1) . (A.31) B Degenerate representations of N = 2 W N +1 algebra In this appendix we review the analysis in section 3.3 of [39] on degenerate representationsof N = 2 W N +1 algebra and their null vector structures. We introduce 2 N free bosons φ j and fermions ψ j ( j = 1 , , . . . , N ) with the operator products in (4.14). We consider theprimary fields of the form V Λ ( z ) = exp( ia Λ · φ ( z )) . (B.1)Here we have used Λ = N X j =1 Λ j λ j (B.2)in terms of sl( N | N + 1) fundamental weights λ l . At this stage Λ l takes any real number.We may express it in terms of bosonic subalgebras asΛ = N X i =1 Λ (1) i λ (1) i + N − X i =1 Λ (2) i λ (2) i + mN ( N + 1) ν . (B.3)Descendants are generated by the action of negative modes of N = 2 W N +1 currents as in(4.33). As explained in section 4.3, some descendants may satisfy the condition for primaryfields (4.32). In that case, we can remove the corresponding states from the spectrum in aconsistent way, and this could happen only for restricted classes of Λ.Before going into detailed analysis on these degenerate representations, we remarkon a global symmetry. The charges of the W -algebra for the primary operators can becomputed as in section 4.2. Primary fields with different Λ can have same W -charges, andthe corresponding states should be identified. As obtained in section 3.1 of [39] (for thebosonic case, see, e.g., [40]), the condition of the identification is − m + h m · Λ = − m ′ + h m ′ · Λ , − n + h n +1 · Λ = − n ′ + h n ′ +1 · Λ (B.4)where m ′ , n ′ are obtained by permutations of m, n . Here we have defined h m = λ m − − λ m = λ (2) m − − λ (2) m − νN , (B.5) h m +1 = λ m +1 − λ m = λ (1) m +1 − λ (1) m − νN + 1 . – 26 –n particular, the states with Λ and − ρ − Λ are dual to each other. In the following, westudy the condition that null vectors appear, however we should remember that there areidentifications among states as above.In order to construct null fields, we utilize screening charges which commute with the N = 2 W N +1 generators. There are three types of screening charges. One of them isobtained by the Hamiltonian reduction of sl( N + 1 | N ) WZW models as [28] S j ( z ) = α j · ψe ia − α j · φ ( z ) . (B.6)The other two are related to the bosonic subalgebras as [39] S (1) i = [( α i − α i − ) · ∂φ − ia ( α i · ψ )( α i − · ψ )] e − ia α (1) φ (B.7)for i = 1 , , . . . , N and S (2) i = [( α i − α i +1 ) · ∂φ + 2 ia ( α i · ψ )( α i +1 · ψ )] e ia α (2) φ (B.8)for i = 1 , , . . . , N − 1. Since the screening charges commute with the N = 2 W N +1 generators, we can construct a null field from a primary field V Λ ′ ( z ) as χ Λ ( z ) = Z du · · · du r S ( u ) · · · S ( u r ) V Λ ′ ( z ) (B.9)if the integral exists non-trivially. The integral contours are taken as in, e.g., [51].Let us start from a fermionic null field. Utilizing a fermionic screening operator (B.6),we have χ Λ ( z ) = Z duS j ( u ) V Λ ′ ( z ) = Z du ( u − z ) α j · Λ ′ α j · ψe ia (Λ ′ + α j a − ) φ ( z ) . (B.10)The integral exists for 1 + α j · Λ ′ = − N j (B.11)with a non-negative integer N j . Setting Λ ′ = − ρ − Λ − α j a − , we see that a null vectorappears at the level N j + 1 / j = N j (B.12)in (B.2). We have thus 2 N independent conditions for the fermionic null vectors, andmaximally degenerate representations are given fromΛ = N X j =1 N j λ j (B.13)with N j ≥ 0. In terms of bosonic subalgebras (B.3), we find the left hand side of (4.35)should be an integer number. This condition should be related to the selection rule (4.4)in the CP N Kazama-Suzuki model (1.4). – 27 –e can construct a bosonic null field with a bosonic screening charge (B.7). A nullfield is given by χ Λ ( z ) = Z du · · · du r i r i Y j =1 S (1) i ( u j ) V Λ ′ ( z ) (B.14)when the integral leads to a non-zero value. The condition is r i − r i + r i ( r i − a − r i a α (1) · Λ ′ = − r i s i (B.15)with positive integers r i , s i . We set Λ ′ = − ρ − Λ + r i α (1) i . Then a bosonic null vectorappears at the level r i s i for the case withΛ (1) i = ( r i − − s i a − (B.16)in (B.3). In the similar manner, we can construct a null field using the other type ofscreening charge (B.8). We find that a bosonic null vector appears at the level r ′ i s ′ i withpositive integers r ′ i , s ′ i for the case withΛ (2) i = ( r ′ i − 1) + s ′ i a − (B.17)in (B.3).Notice that even if we want assign the maximal number of the bosonic null vectorconditions, we can do so only for 2 N − m as in the selection rule (4.4). Moreover, the vacuum representation with Λ = 0 doesnot have bosonic null vectors of these kinds contrary to the bosonic case. The represen-tation with N j ∈ Z (including N j = 0) for all j has independent fermionic null vectors asmentioned above, and bosonic null vectors may be generated by these independent ones.In order to relate to the CP N Kazama-Suzuki model (1.4), we should set a − = k + N + 1,which is an integer number. Therefore, in that case, we could have the both types of nullvectors simultaneously. In fact, as pointed out in [52], the states in the CP N Kazama-Suzukimodel (1.4) may correspond to states labeled byΛ = N X j =1 (Λ j + a − ˆΛ j ) λ j . 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