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Unitary Representations of W Infinity Algebras

arXiv:hep-th/9111058v1 28 Nov 1991Yukawa Institute KyotoYITP/K-955November 1991Unitary Representations of W Infinity AlgebrasSatoru Odake∗Yukawa Institute for Theoretical PhysicsKyoto University, Kyoto 606, JapanAbstractWe study the irreducible unitary highest weight representations, whichare obtained from free field realizations, of W infinity algebras (W∞, W1+∞,W 1,1∞, W M∞, W N1+∞, W M,N∞) with central charges (2, 1, 3, 2M, N, 2M + N).The characters of these representations are computed.We construct a new extended superalgebra W M,N∞, whose bosonic sectoris W M∞⊕W N1+∞. Its representations obtained from a free field realizationwith central charge 2M +N, are classified into two classes: continuous seriesand discrete series.

For the former there exists a supersymmetry, but forthe latter a supersymmetry exists only for M = N.∗e-mail address: odake@jpnrifp.bitnet.1

1IntroductionThe conformal field theory (CFT) in two dimensional space-time has made greatprogress in close contact with both the string theory and various branches of mathe-matics. The Virasoro algebra plays a central role in CFT, and to construct models ofCFT and extend this theory, one needs an extension of the Virasoro algebra.

In view ofthis several extensions (superconformal algebras, W algebras, parafermions, etc.) havebeen studied.

The notable example is Zamolodchikov’s W3 algebra and its WN gener-alization (AN−1 type W algebra) containing fields of conformal weight (spin) 2, · · ·, Nas conserved currents [1]. All of extended Virasoro algebras containing currents of spin> 2 have a non-linear property.By taking an appropriate N →∞limit of the WN algebra, one can obtain a linearalgebra with infinite number of fields.

The first example is the w∞algebra [2], whichcan be interpreted as the algebra of area-preserving diffeomorphisms of two dimensionalphase space.But w∞admits a central extension only in the Virasoro sector.Bydeforming w∞, Pope, Romans and Shen constructed the W∞algebra [3], which admitscentral extension in all spin sectors. This is another large N limit of WN.

In addition,they constructed the W1+∞algebra [4], which contains a spin 1 field too, and theirsuper extension, super W∞algebra [5], whose bosonic sector is W∞⊕W1+∞. Soonafterward Bakas and Kiritsis constructed the W M∞algebra [6], which is a u(M) matrixversion of W∞, and Sano and the present author constructed the csu(N)-W1+∞algebra[7], which is an extension of W1+∞and contains the SU(N) current algebra.

We willchange the notation csu(N)-W1+∞to W N1+∞. W N1+∞is to W N∞what W1+∞is to W∞[8].

In this paper we will construct a new superalgebra W M,N∞, whose bosonic sector isW M∞⊕W N1+∞. In this notation, super W∞[5] and super csu(N)-W∞[7] are W 1,1∞andW 1,N∞respectively.Given an algebra which generates the symmetry of a model, it is important todevelop its representation theory to find what fields appear in the model.

The successof the representation theory of the Virasoro algebra is a good example [9]. In spite ofdifficulties due to the non-linearity, the minimal representations of the WN algebra havebeen studied in detail by using the free field realization of Feigin-Fuchs type or the coset2

model of affine Lie algebras [1, 10]. Although W infinity algebras are linear algebras andtheir structure constants are explicitly known, their representation theories have beenstudied very poorly [11, 8].

In this paper we will initiate the study of representationtheories of W infinity algebras and construct their unitary representations based on freefield realizations.Our methods to study representation theories are as follows. First we prepare a freefield realization of the W infinity algebra and its Fock space.

In the Fock space, we tryfinding all the highest weight states (HWS’s) of the subalgebra (an affine Lie algebraor the Virasoro algebra), whose generators have the lowest spin. Next we check thesestates are also the HWS’s of the whole W infinity algebra.

To compute the charactersof W1+∞and W N1+∞, we use the character formulas of the affine Lie algebras and thefact that the generators of W1+∞and W N1+∞do not change the U(1) charge. For W∞and W M∞, we use the facts that the generators of the W infinity algebras are representedin terms of bilinears of the free fields and they preserve a certain quantum number.

Forsuper cases, we use the results of bosonic cases and the spectral flow invariance. TheWitten index is also computed.The organization of this paper is as follows.

In §2 we construct a new superalgebraW M,N∞and present its free field realization. In §3-5 the representations of W infinityalgebras (W1+∞, W N1+∞, W∞, W M∞, W 1,1∞, W M,N∞) are studied, and their characters arecomputed.

In §6 we present a discussion.2Algebras and Free Field RealizationsWe will define a new extended superalgebra W M,N∞, whose bosonic sector is W M∞⊕W N1+∞. Other W infinity algebras are contained in it as subalgebras.

W M,N∞is generatedby1W i,(αβ)(z),(i ≥−1; α, β = 1, 2, · · ·, N),(1)˜W i,(ab)(z),(i ≥0; a, b = 1, 2, · · ·, M),(2)1 As usual, the mode expansion of a field A(z) of conformal weight h is A(z) = P Anz−n−h, wherethe sum is taken over n ∈Z−h for the Neveu-Schwarz (NS) sector, n ∈Z for the Ramond (R) sector.3

Gi,aα(z),(i ≥0; a = 1, · · ·, M; α = 1, · · ·, N),(3)¯Gi,aα(z),(i ≥0; a = 1, · · ·, M; α = 1, · · ·, N). (4)W i,(αβ)(z) and ˜W i,(ab)(z) are bosonic fields with conformal spin i + 2, and generateW N1+∞and W M∞respectively.

W N1+∞(W M∞) contains W1+∞(W∞) as a subalgebra, andtheir generators areV i(z) =NXα=1W i,(αα)(z),˜V i(z) =MXa=1˜W i,(aa)(z). (5)V 0(z) and ˜V 0(z) are the Virasoro generators with central charge c and ˜c respectively.W N1+∞contains the U(N) current algebra, which is generated bycsu(N)k:Hi(z) = J(ii)(z) −J(i+1,i+1)(z),(i = 1, · · ·, N −1) CartanJ(αβ)(z),(α < β) raising ;(α > β) lowering,(6)ˆu(1)K:J(z) =NXα=1J(αα)(z),(7)where J(αβ)(z) = −4qW −1,(βα)(z) 2. k is the level of csu(N) and K stands for a normal-ization of ˆu(1) ([Jm, Jn] = Kmδm+n,0).

Gi,aα(z) and ¯Gi,aα(z) are fermionic fields withconformal spin i + 32. W i,(αβ)(z) ( ˜W i,(ab)(z), Gi,aα(z), ¯Gi,aα(z)) transform according tothe adjoint (trivial, N, ¯N) representation of su(N), and have U(1) charge 0 (0, 1, −1),respectively.

(Anti-)commutation relations of W M,N∞are given by3[W i,(αβ)m, W j,(γδ)n] =12Xr≥−1qrgijr (m, n)(δγβW i+j−r,(αδ)m+n+ (−1)rδαδW i+j−r,(γβ)m+n)+δijδαδδγβδm+n,0q2iki(m),(8)[ ˜W i,(ab)m, ˜W j,(cd)n] =12Xr≥−1qr˜gijr (m, n)(δcb ˜W i+j−r,(ad)m+n+ (−1)rδad ˜W i+j−r,(cb)m+n)+δijδadδcbδm+n,0q2i˜ki(m),(9)[W i,(αβ)m, Gj,cγn] = δαγ Xr≥−1qraijr (m, n)Gi+j−r,cβm+n,(10)2This definition is slightly different from [7].3 q is a deformation parameter and we can take it to be an arbitrary non-zero constant(e.g., q = 14).This q has nothing to do with q = e2πiτ appearing in the characters.4

[W i,(αβ)m, ¯Gj,cγn] = δβγ Xr≥−1qr(−1)raijr (m, n) ¯Gi+j−r,cαm+n,(11)[ ˜W i,(ab)m, Gj,cγn] = δbc Xr≥−1qr˜aijr (m, n)Gi+j−r,aγm+n,(12)[ ˜W i,(ab)m, ¯Gj,cγn] = δac Xr≥−1qr(−1)r˜aijr (m, n) ¯Gi+j−r,bγm+n,(13){Gi,aαm , ¯Gj,bβn} =Xr≥0qr(δabbijr (m, n)W i+j−r,(βα)m+n+ δαβ˜bijr (m, n) ˜W i+j−r,(ab)m+n)+δijδabδαβδm+n,0q2iˇki(m),(14)[W i,(αβ)m, ˜W j,(ab)n] = {Gi,aαm , Gj,bβn} = { ¯Gi,aαm , ¯Gj,bβn} = 0. (15)The structure constants [3, 4, 5, 6, 7] aregijr (m, n) =12(r+1)!φijr (0, −12)Ni,jr (m, n),(16)˜gijr (m, n) =12(r+1)!φijr (0, 0)Ni,jr (m, n),(17)aijr (m, n) =(−1)r4(r+2)!

((i + 1)φijr+1(0, 0) −(i −r −1)φijr+1(0, −12))Ni,j−12r(m, n),(18)˜aijr (m, n) =−14(r+2)! ((i −r)φijr+1(0, 0) −(i + 2)φijr+1(0, −12))Ni,j−12r(m, n),(19)bijr (m, n) =(−1)r4r!

((i + j + 2 −r)φijr ( 12, −14)−(i + j + 32 −r)φijr+1( 12, −14))Ni−12,j−12r−1(m, n),(20)˜bijr (m, n) = −4r! ((i + j + 1 −r)φijr ( 12, −14)−(i + j + 32 −r)φijr+1( 12, −14))Ni−12,j−12r−1(m, n),(21)andNx,yr(m, n) =r+1Xℓ=0(−1)ℓ r + 1ℓ!

[x + 1 + m]r+1−ℓ[x + 1 −m]ℓ×[y + 1 + n]ℓ[y + 1 −n]r+1−ℓ,(22)φijr (x, y) = 4F3h −12 −x −2y, 32 −x + 2y, −r+12 + x, −r2 + x−i −12, −j −12, i + j −r + 52; 1i,(23)4F3h a1, a2, a3, a4b1, b2, b3; zi=∞Xn=0(a1)n(a2)n(a3)n(a4)n(b1)n(b2)n(b3)nznn! ,(24)where [x]n = x(x−1) · · · (x−n+1), [x]0 = 1 and (x)n = x(x+1) · · · (x+n−1), (x)0 = 1andxn= [x]n/n!.

Since gijr = bijr = 0 for i −j −r < −1 and ˜gijr = aijr = ˜aijr = ˜bijr = 05

for i −j −r < 0, the summations over r are finite sums and the algebra closes. Thecentral terms areki(m) = kii+1Yj=−i−1(m + j) ,ki =22i−2((i + 1)!

)2(2i + 1)!! (2i + 3)!

!k,(25)˜ki(m) = ˜kii+1Yj=−i−1(m + j) ,˜ki =22i−3i! (i + 2)!

(2i + 1)!! (2i + 3)!

!˜k,(26)ˇki(m) = ˇkiiYj=−i−1(m + j + 12) ,ˇki = 22ii! (i + 1)!3((2i + 1)!!

)2 ˇk. (27)In the case of W M,N∞, the Jacobi identity requiresK = Nk ,c = Nk ,˜c = M˜k ,˜k = 2k ,ˇk = 3k.

(28)Since the level k of csu(N) (N > 1) is a positive integer for unitary representations, cen-tral charges c and ˜c must be multiples of N and 2M respectively. (Anti-)commutationrelations of W M,N∞are consistent with the hermiticity properties of the generators:W i,(αβ)†n= W i,(βα)−n, ˜W i,(ab)†n= ˜W i,(ba)−n, Gi,aα†n= ¯Gi,aα−n .

(29)The Cartan subalgebra of W M,N∞is generated byW i,(αα)0,˜W i,(aa)0. (30)The HWS of W M,N∞in the NS sector is defined byAn|hws⟩= 0,(n > 0; A = W, ˜W, G, ¯G)W i,(αβ)0|hws⟩= 0,(α > β)˜W i,(ab)0|hws⟩= 0,(a > b),(31)and, in the R sector, we require one more condition:Gi,aα,R0|hws⟩R = 0.

(32)The HWS’s of other W infinity algebras are defined in a similar way.Since W M,N∞contains a current algebra, there exists an automorphism, so calledspectral flow [12]. Namely (anti-)commutation relations are invariant under the trans-formations of the generators.

Explicit forms of the transformation rules are essentially6

the same as W 1,N∞[7]. Due to this property, representations in the R sector and thosein the NS sector have one-to-one correspondence.

We define the representations in theR sector as those mapped from the NS sector by the spectral flow with η = 12. Thenwe can show |hws⟩R = |hws⟩NS, becauseW i′n = Pij=−1 (coeff.) · W jn + (coeff.) · δn0,Gi′n = Pij=0 (coeff.)

· Gjn+ 12˜W i′n = Pij=0 (coeff.) · ˜W jn + (coeff.) · δn0,¯Gi′n = Pij=0 (coeff.) · ¯Gjn−12.

(33)W M,N∞with level k=1 is realized by N complex free fermions ψα(z) = Pn ψαnz−n−12(α = 1, · · ·, N) and M complex free bosons i∂ϕa(z) = Pn αanz−n−1 (a = 1, · · ·, M).Operator product expansions of the free fields are¯ψα(z)ψβ(w) ∼δαβz −w,i∂¯ϕa(z)i∂ϕb(w) ∼δab(z −w)2. (34)Generators of W M,N∞are represented in terms of bilinears of the free fields [4, 5, 6, 7] :W j,(αβ)(z) = 2j−1(j + 1)!

(2j + 1)!! qjj+1Xr=0(−1)r j + 1r!2(∂j+1−r ¯ψα∂rψβ)(z),(35)˜W j,(ab)(z) = 2j−1(j + 2)!

(2j + 1)!! qjjXr=0(−1)rj + 1 j + 1r!

j + 1r + 1! (∂j−ri∂¯ϕa∂ri∂ϕb)(z), (36)Gj,aα(z) = 2j+ 12(j + 1)!

(2j + 1)!! qjjXr=0(−1)r j + 1r!

jr! (∂j−ri∂¯ϕa∂rψα)(z),(37)¯Gj,aα(z) = 2j+ 12(j + 1)!

(2j + 1)!! qjjXr=0(−1)j+r j + 1r!

jr! (∂j−ri∂ϕa∂r ¯ψα)(z),(38)where the normal ordered product of two fields A(z) and B(z) is defined by (AB)(z) =Hzdx2πi1x−zA(x)B(z).

One of the methods to obtain the general level k realization is toprepare k copies of the above realization, because W M,N∞is linear. The spectral flowtransformation rules of the generators with a parameter η are easily derived from thoseof the free fields:ψα′(z) = zηψα(z), ¯ψα′(z) = z−η ¯ψα(z), ϕa′(z) = ϕa(z), ¯ϕa′(z) = ¯ϕa(z),(39)because eq.

(34) are invariant under this transformation. The transformation rules thatwill be needed later areW −1,(αβ)′n= W −1,(αβ)n−δαβδn014qη,W 0,(αβ)′n= W 0,(αβ)n−4qηW −1,(αβ)n+ δαβδn012η2,G0,aα′n= G0,aαn+η,¯G0,aα′n= ¯G0,aαn−η,˜W 0,(ab)′n= ˜W 0,(ab)n.(40)7

The vacuum states of the fermion and boson Fock spaces, |0⟩and |⃗p,⃗¯p⟩, are definedas usual: ψαm|0⟩= ¯ψαn|0⟩= αan|⃗p,⃗¯p⟩= ¯αan|⃗p,⃗¯p⟩= 0, (m ≥0, n > 0), αa0|⃗p,⃗¯p⟩= pa|⃗p,⃗¯p⟩,¯αa0|⃗p,⃗¯p⟩= ¯pa|⃗p,⃗¯p⟩. Hermiticity properties of the generators eq.

(29) are satisfied bythose of the free fields (ψα†n = ¯ψα−n, αa†n = ¯αa−n). In the following we take p∗a = ¯pa, sothat the unitarity of the representations is manifest.3Representations of W1+∞and W N1+∞We first consider the representations of W1+∞with c = 1 realized by one complexfree fermion.

We remark that the Virasoro generator V 0(z) agrees with the Sugawaraform of ˆu(1) current. For each integer n, we can find the HWS of the subalgebra ˆu(1)contained in the fermion Fock space, and we denote them as|n⟩def=ψ−12ψ−32 · · · ψ−n+ 12|0⟩n ≥1|0⟩n = 0¯ψ−12 ¯ψ−32 · · · ¯ψn+ 12|0⟩n ≤−1.

(41)These states are well known in Sato theory [13]. We can check that |n⟩is not only theHWS of ˆu(1) but also the HWS of W1+∞.

The conformal weight hn and U(1) chargeQn of |n⟩arehn = 12n2,Qn = n.(42)Although the eigenvalues of the higher-spin generators are easily calculated, we omitthem here.Since the dependence on the eigenvalues of higher-spin generators are very com-plicated, we consider the characters which count conformal weight and U(1) chargeonly:chW1+∞(θ, τ) def= trqV 00 −124zJ0,(43)where q = e2πiτ (Imτ > 0) and z = eiθ. Since W1+∞contains ˆu(1) as a subalgebra,the representation of W1+∞has more states than one of ˆu(1).On the other hand,the representation of W1+∞has less states than the Fock space with the fixed U(1)charge, because generators of W1+∞do not change U(1) charge.

These statements are8

expressed in terms of characters as follows:χˆu(1)1n(θ, τ) ≤chW1+∞n(θ, τ) ≤znχF ockn(τ),(44)where A ≤B means B −A is a q-series with non-negative coefficients. In general, thecharacter formula of ˆu(1)K with U(1) charge Q isχˆu(1)KQ(θ, τ) def= trqL0−124zJ0 =1η(τ)q12K Q2zQ,(45)where η(τ) = q124 Q∞n=1(1−qn).

On the other hands, the generating function of χF ockn(τ)isXn∈ZznχF ockn(τ) = trF ockqV 00 −124zJ0 = q−124∞Yn=1(1 + zqn−12)(1 + z−1qn−12). (46)Due to the Jacobi’s triple product identity, we have χˆu(1)1n(θ, τ) = znχF ockn(τ).

Thereforewe obtain the character formula of W1+∞,chW1+∞n(θ, τ) = χˆu(1)1n(θ, τ). (47)Next we consider the representations of W N1+∞with c = N realized by N complexfree fermions.

We use the same techniques as W1+∞case. We remark that, in the caseof level k =1, the Virasoro generator V 0(z) agrees with the sum of the Sugawara formof ˆu(1)N and csu(N)1 [14].

For each integer n, there exists the HWS of csu(N)1, and wedenote them as|n⟩def=Qmj=1(QNα=1 ψα−j+ 12) · Qaα=1 ψα−m−12|0⟩n ≥1|0⟩n = 0Q−m−1j=1(QNα=1 ¯ψN+1−α−j+ 12 ) · QN−aα=1 ¯ψN+1−αm+ 12|0⟩n ≤−1,(48)where we express n as n = Nm + a, (m ∈Z; a = 0, 1, · · ·, N −1). |n⟩is the HWS ofthe a-th rank antisymmetric representation of csu(N)1.

The state |n⟩is also the HWSof W N1+∞. The conformal weight hn and U(1) charge Qn arehn =12N n2 + a(N −a)2N,Qn = n.(49)The first and second factors of hn are contributions from ˆu(1)N and csu(N)1 respectively.9

Neglecting the dependence on the eigenvalues of higher-spin generators, we considerthe characters which count conformal weight, U(1) charge and eigenvalues of SU(N),chW N1+∞(θ, ⃗θ, τ) def= trqV 00 −N24eiθJ0ei⃗θ· ⃗H0,(50)where ⃗θ = PN−1i=1 θi⃗αi and ⃗αi is a simple root of su(N). W N1+∞contains ˆu(1)N ⊕csu(N)1as a subalgebra, and generators of W N1+∞do not change U(1) charge.

Therefore, thesimilar argument as W1+∞case shows that the character formula of W N1+∞is given bychW N1+∞n(θ, ⃗θ, τ) = χˆu(1)Nn(θ, τ)χ bsu(N)1a(⃗θ, τ),(51)where a ≡n(mod N), 0 ≤a ≤N −1. The character of ˆu(1) is given by eq.

(45), andthe character formula of csu(N)1 is given by [15]χ bsu(N)1a(⃗θ, τ) def= trqL0−N−124 ei⃗θ· ⃗H0 =1η(τ)N−1X⃗M∈ΛRq12 ( ⃗M+⃗Λa)2ei⃗θ·( ⃗M+⃗Λa),(52)where ΛR is a root lattice of su(N), and ⃗Λi (1 ≤i ≤N −1) is a fundamental weight ofsu(N) and ⃗Λ0 = ⃗0. To show eq.

(51), we need the identity:trF ockqV 00 −N24eiθJ0ei⃗θ· ⃗H0=q−N24NYj=1∞Yn=1(1 + zz−1j−1zjqn−12)(1 + z−1zj−1z−1j qn−12)=Xn∈Zχˆu(1)Nn(θ, τ)χ bsu(N)1a(⃗θ, τ),(53)where zj = eiθj, z0 = zN = 1, and a ≡n(mod N). This identity is proved by theJacobi’s triple product identity.4Representations of W∞and W M∞We first consider the representations of W∞with ˜c = 2 realized by one complex freeboson.

In the boson Fock space, the HWS’s of the Virasoro generator ˜V 0(z) are classifiedinto two classes: continuous series, whose momentum can be changed continuously, anddiscrete series, whose state exists for each integer n,|p, ¯p⟩,(p ̸= 0),(54)|n⟩def=(α−1)n|0, 0⟩n ≥1|0, 0⟩n = 0(¯α−1)−n|0, 0⟩n ≤−1. (55)10

We can check that these states are the HWS’s of W∞. Their conformal weights arehp¯p = |p|2,hn = |n|.

(56)Neglecting the dependence on the eigenvalues of higher-spin generators, we considerthe characters which count conformal weight only,chW∞(τ) def= trq˜V 00 −224. (57)Since ˜V in contains the terms ¯α0αn and ¯αnα0, and the momentum p is non-zero for thecontinuous series, the set of generators ˜V in is identified with the set of oscillators αn,¯αn.

Therefore the character formula of the continuous series of W∞ischW∞p¯p (τ) = q|p|2η(τ)2. (58)This result was first derived by Bakas and Kiritsis, using the Z∞parafermion [11].For the discrete series, ¯α0αn and ¯αnα0 are acting on the state as 0.

So, the numberof the states of the discrete series is less than one of the continuous series. Let us definethe quantum number B as (number of oscillators without¯) −(number of oscillatorswith¯).

Then B|n⟩= n|n⟩, and generators of W∞do not change B on |n⟩. By takingthe appropriate linear combinations of ˜V in, we can obtain all the oscillators of the form¯αnαm.

From these two facts, the generating function of the characters of the discreteseries isXn∈ZtnchW∞n(τ) = trF ockq˜V 00 −224tB =q−224Q∞n=1(1 −tqn)(1 −t−1qn). (59)From this, the character formula of the discrete series of W∞is expressed as4chW∞n(τ)=12η(τ)2Xm∈Zsign(m)(−1)mqmn−18(q12 (m+ 12)2 −q12(m−12)2)=1η(τ)2∞Xm=1(−1)mq12 m(m−1)+mn(qm −1).

(60)The relation between continuous and discrete series islimp→0 chW∞p¯p (τ) =Xn∈ZchW∞n(τ). (61)4 Recently Bakas and Kiritsis have constructed the non-linear deformation of W∞based on theSL(2, R)k/U(1) coset model, and investigated its characters, which include the characters of W∞inthe large k limit [16].11

Next we consider the representations of W M∞with ˜c = 2M realized by M complexfree bosons. Since the argument is the same as W∞case, we present the results only.The HWS’s of W M∞are|⃗p,⃗¯p⟩,⃗p = (0, · · ·, 0, pa, 0, · · ·, 0),pa ̸= 0,(62)|n⟩def=(α1−1)n|⃗0,⃗0⟩n ≥1|⃗0,⃗0⟩n = 0(¯α1−1)−n|⃗0,⃗0⟩n ≤−1.

(63)The degeneracy of the ground states are 1 andM+|n|−1|n|respectively. The conformalweights areh⃗p⃗¯p = |⃗p|2,hn = |n|.

(64)The characters which count conformal weight only, are defined bychW M∞(τ) def= trq˜V 00 −2M24 . (65)The character formula of the continuous series of W M∞ischW M∞⃗p⃗¯p (τ) =q|⃗p|2η(τ)2M .

(66)The generating function of the character formulas of the discrete series isXn∈ZtnchW M∞n(τ) =q−224Q∞n=1(1 −tqn)(1 −t−1qn)M. (67)The relation between continuous and discrete series islim⃗p→⃗0 chW M∞⃗p⃗¯p (τ) =Xn∈ZchW M∞n(τ).

(68)5Representations of W 1,1∞and W M,N∞We first consider the representations of W 1,1∞with ˜c = 2, c = 1 realized by one pairof complex free boson and fermion. Each state in the Fock space is expressed as a linearcombination of |˜∗⟩⊗|∗⟩, where the first and second factors are states in the boson andfermion Fock spaces respectively.

W 1,1∞contains W∞and W1+∞as subalgebras, and12

the generators of W∞and W1+∞are expressed by a boson and a fermion respectively.Therefore |˜∗⟩and |∗⟩are the HWS’s of W∞and W1+∞respectively.By using theresults obtained in the previous sections, we find the HWS’s of W 1,1∞, continuous seriesand discrete series:|p, ¯p⟩⊗|0⟩,(69)|n⟩def=|n −1⟩⊗|1⟩n ≥1|0⟩⊗|0⟩n = 0|n + 1⟩⊗| −1⟩n ≤−1. (70)The first and second factors of the tensor product are eqs.

(54,55) and eq. (41) respec-tively.

Their conformal weight h and U(1) charge Q arehp¯p = |p|2,Qp¯p = 0,(71)(hn, Qn) =(n −12, 1)n ≥1(0, 0)n = 0(−n −12, −1)n ≤−1. (72)Neglecting the dependence on the eigenvalues of higher-spin generators, we considerthe characters which count conformal weight and U(1) charge only,chW 1,1∞(θ, τ) def= trq˜V 00 −224+V 00 −124eiθJ0.

(73)˜V in contains the terms ¯α0αn and ¯αnα0, and Gin and ¯Gin contain the terms ¯α0ψn andα0 ¯ψn. For the continuous series, the momentum p is non-zero, so the set of generatorsof W 1,1∞is identified with the set of oscillators ψn, ¯ψn, αn, ¯αn.

Therefore the characterof the continuous series ischW 1,1∞p¯p(θ, τ) =q|p|2η(τ)2q−124∞Yn=1(1 + zqn−12)(1 + z−1qn−12)(74)= chW∞p¯p (τ)Xn∈ZchW1+∞n(θ, τ)(75)= chW∞p¯p (τ)f1,0(θ, τ),(76)where we define fK,Q(θ, τ) asfK,Q(θ, τ) def=1η(τ)Xn∈ZqK2 (n+ QK )2zK(n+ QK ). (77)13

For the discrete series, B|n⟩= n|n⟩, and the generators of W 1,1∞do not change B on|n⟩. By taking appropriate linear combinations of generators of W 1,1∞, we obtain all theoscillators of the form ¯ψnψm, ¯αnψm, αn ¯ψm, ¯αnαm.

From these two facts, the generatingfunction of the characters of the discrete series isXn∈ZtnchW 1,1∞n(θ, τ) =q−224Q∞n=1(1 −tqn)(1 −t−1qn) · q−124∞Yn=1(1 + tzqn−12)(1 + t−1z−1qn−12). (78)From this equation, the character formula of the discrete series ischW 1,1∞n(θ, τ) =Xℓ∈ZchW∞n−ℓ(τ)chW1+∞ℓ(θ, τ).

(79)The relation between continuous and discrete series islimp→0 chW 1,1∞p¯p(θ, τ) =Xn∈ZchW 1,1∞n(θ, τ). (80)Since W 1,1∞is a superalgebra, we must consider the R sector also.

By using thespectral flow, the character of the R sector is expressed in terms of the character of theNS sector:chW 1,1∞,R(θ, τ) = q324z36chW 1,1∞(θ + πτ, τ). (81)Explicitly they arechW 1,1∞,Rp¯p(θ, τ) = chW∞p¯p (τ)f1, 12(θ, τ),(82)chW 1,1∞,Rn(θ, τ) =Xℓ∈ZchW∞n−ℓ(τ)χˆu(1)1ℓ+ 12 (θ, τ).

(83)The conformal weight hR and U(1) charge QR in the R sector arehRp¯p = 18 + |p|2,QRp¯p = 12(84)(hRn, QRn ) =( 18 + n, 32)n ≥1( 18, 12)n = 0( 18 −(n + 1), −12)n ≤−1. (85)The ground states are singlets for n = 0, −1, and doublets for others.In order to study whether a supersymmetry exist or not, and if it exists, whether itis broken or unbroken, we define the Witten index:Index def= trRq˜V 00 +V 00 −hR(−1)F,(86)14

where F is the fermion number, and trace is taken over the representation space ofthe R sector. By using the property of the spectral flow and the fact that the fermionnumbers of the generators of W 1,1∞agree with their U(1) charges, the Witten index isexpressed as follows:Index = q324 −h−12QchW 1,1∞(π + πτ, τ).

(87)For representations n = 0, −1, the Witten indices areIndex0 = 1,Index−1 = −1. (88)For other representations, the Witten index vanishes.

Therefore there exists a (N = 2)supersymmetry for all representations and it is broken for n = 0, −1.Next we consider the representations of W M,N∞with ˜c = 2M, c = N realized by Mcomplex free bosons and N complex free fermions. Since the argument is the same asW 1,1∞case, we present the results only.

The HWS’s of W M,N∞are|⃗p,⃗¯p⟩⊗|0⟩,(89)|n⟩def=|n −N⟩⊗|N⟩n ≥N|0⟩⊗|n⟩−N < n < N|n + N⟩⊗| −N⟩n ≤−N. (90)The first and second factors of the tensor product are given by eqs.

(62,63) and eq. (48)respectively.

Their conformal weight h and U(1) charge Q areh⃗p⃗¯p = |⃗p|2,Q⃗p⃗¯p = 0,(91)(hn, Qn) =(n −12N, N)n ≥N( 12|n|, n)−N < n < N(−n −12N, −N)n ≤−N. (92)Neglecting the dependence on the eigenvalues of higher-spin generators, we considerthe characters which count conformal weight, U(1) charge and eigenvalues of SU(N),chW M,N∞(θ, ⃗θ, τ) def= trq˜V 00 −2M24 +V 00 −N24eiθJ0ei⃗θ· ⃗H0.

(93)The character formula of the continuous series ischW M,N∞⃗p⃗¯p(θ, ⃗θ, τ) =q|⃗p|2η(τ)2M q−N24NYj=1∞Yn=1(1 + zz−1j−1zjqn−12)(1 + z−1zj−1z−1j qn−12) (94)15

= chW M∞⃗p⃗¯p (τ)Xn∈ZchW N1+∞n(θ, ⃗θ, τ)(95)= chW M∞⃗p⃗¯p (τ)N−1Xa=0fN,a(θ, τ)χ bsu(N)1a(⃗θ, τ). (96)The generating function of the characters of the discrete series isXn∈ZtnchW M,N∞n(θ, ⃗θ, τ) =q−224Q∞n=1(1 −tqn)(1 −t−1qn)M×q−N24NYj=1∞Yn=1(1 + tzz−1j−1zjqn−12)(1 + t−1z−1zj−1z−1j qn−12).

(97)From this, the character formula of the discrete series ischW M,N∞n(θ, ⃗θ, τ) =Xℓ∈ZchW M∞n−ℓ(τ)chW N1+∞ℓ(θ, ⃗θ, τ). (98)The relation between continuous and discrete series islim⃗p→⃗0 chW M,N∞⃗p⃗¯p(θ, ⃗θ, τ) =Xn∈ZchW M,N∞n(θ, ⃗θ, τ).

(99)By using the spectral flow, the character of the R sector is expressed in terms of thecharacter of the NS sector:chW M,N∞,R(θ, ⃗θ, τ) = q3N24 z3N6 chW M,N∞(θ + πτ, ⃗θ, τ). (100)Explicitly they arechW M,N∞,R⃗p,⃗¯p(θ, ⃗θ, τ) = chW M∞⃗p,⃗¯p (τ)N−1Xa=0fN,a+ N2 (θ, τ)χ bsu(N)1a(⃗θ, τ)(101)chW M,N∞,Rn(θ, ⃗θ, τ) =Xℓ∈ZchW M∞n−ℓ(τ)χˆu(1)Nℓ+ N2 (θ, τ)χ bsu(N)1a(⃗θ, τ),(102)where, in the second equation, a ≡ℓ(mod N).

The conformal weight hR, U(1) chargeQR and the degeneracy of the ground states in the R sector arehR⃗p⃗¯p = 18N + |⃗p|2,QR⃗p⃗¯p = 12N,degeneracy =NXa=0 Na!= 2N,(103)(hRn , QRn , degeneracy)16

=( 18N + n, 32N, PNa=0NaM+n−a−1n−a)n ≥N( 18N + n, 12N + n,Pna=0NaM+n−a−1n−a)0 < n < N( 18N, 12N + n, NN+n)−N ≤n ≤0( 18N −(N + n), −12N,PNa=2N+nNaM−n−2N+a−1−n−2N+a)−2N < n < −N( 18N −(N + n), −12N, PNa=0NaM−n−2N+a−1−n−2N+a)n ≤−2N. (104)By using the property of the spectral flow and the fact that the fermion numbersof the generators of W M,N∞agree with their U(1) charges, the Witten index eq.

(86) isexpressed asIndex = qN+2M24−h−12QchW M,N∞(π + πτ,⃗0, τ). (105)For the continuous series, the Witten index vanishes for all M, N. Therefore, for thecontinuous series, there exists a supersymmetry, and it is unbroken.In the case of the discrete series with M ̸= N, the Witten index is not a number buta q-series.

Namely, at excited state, the number of bosonic states does not agree withone of fermionic states. Therefore a supersymmetry does not exist in the discrete seriesof W M,N∞(M ̸= N).

In the case of the discrete series with M = N, the Witten indexis just a number. So a (2N2 extended) supersymmetry exists.

For representations n(−N ≤n ≤0), the Witten index isIndexn = (−1)n NN + n!,(106)and a supersymmetry is broken. For other representations, the Witten index vanishesand a supersymmetry is unbroken.6DiscussionIn this paper we have studied the irreducible unitary highest weight representationsof W infinity algebras, which are obtained from free field realizations, and derivedtheir character formulas.

We have also constructed a new superalgebra W M,N∞, whosebosonic sector is W M∞⊕W N1+∞. Its representations obtained from a free field realizationare classified into two classes, continuous and discrete.

There exists a supersymmetry inthe continuous series, whereas a supersymmetry exists only for M = N in the discrete17

series.This is expected from the counting of the bosonic and fermionic degrees offreedom of the generators:W i−1,(αβ)(z)˜W i,(ab)(z)Gi,aα(z)¯Gi,aα(z)N2+M2−MN−MN=(M −N)2. (107)Perhaps a supersymmetry in the continuous series for M ̸= N may be an accidentalone.The representations with higher central charge and the realization independent rep-resentations are future subjects.

There are two difficulties in developing the realizationindependent representation theories of W infinity algebras. One is that there are infi-nite number of fields.

The other is the complicated dependence on the eigenvalues ofhigher-spin generators. For example, no one has succeeded in computing even the level1 Kac determinant.Although our representation theory is a restricted one, we have obtained the char-acter formulas.

It is interesting to apply these character formulas to models with Winfinity symmetry, for example [17], integrable non-linear differential equation systemssuch as the KP hierarchy and the Toda hierarchy, four dimensional self-dual gravity[18], Virasoro (W) constraints on the partition function of the multi-matrix model [19],and W infinity gravity [20, 21].The modular properties of the characters are alsosubjects of future research.Finally, we mention the anomaly-free conditions.In refs. [22, 23], the anomaly-free conditions for W∞, W1+∞and W 1,1∞are considered by the BRS formalism andζ function regularization, and it is shown that their critical central charges are −2,0 and −3 respectively.

Similar calculations have been done for W N1+∞and W 1,N∞byconsidering the ghost realizations of them, and their critical central charges are 0 and−2 −N respectively [8]. For W M,N∞, we obtain kghost = M, and the critical centralcharge is(˜c + c)critical = −(˜c + c)ghost = −2M2 −MN.

(108)18

AcknowledgmentsThe author would like to thank T. Inami for discussions and comments on the manuscript.He would like to acknowledge useful discussions with H. Nohara and T. Sano.19

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