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The semi-infinite cohomology of affine Lie algebras

We study in detail the semi-infinite or BRST cohomology of general affine Lie algebras. This cohomology is relevant in the BRST approach to gauged WZNW models. We prove the existence of an infinite sequence of elements in the cohomology for non-zero ghost numbers. This will imply that the BRST approach to topological WZNW model admits many more states than a conventional coset construction. This conclusion also applies to some non-topological models. Our work will also contain results on the structure of Verma modules over affine Lie algebras. In particular, we generalize the results of Verma and Bernstein-Gel'fand-Gel'fand,for finite dimensional Lie algebras, on the structure and multiplicities of Verma modules.


Introduction and summary of results
The present work studies the semi-infinite or BRST cohomology of affine Lie algebras. The motivation comes from the quantization of Wess-Zumino-Novikov-Witten (WZNW) models. These models play an essential part in the understanding and classification of conformal field theories. The BRST symmetry arises as a consequence of the gauging of a WZNW model w.r.t. a subgroup [1]. The constraints associated with this BRST symmetry are the generators of an affine Lie algebra g ′ = g k ⊕gk.
Here g k andgk correspond to the same finite dimensional Lie algebra, but have different central elements k andk = −k − 2cḡ (see section 2 for notation). The latter affine Lie algebra corresponds to an auxiliary, and in general non-unitary, WZNW model that arose in the derivation in [1]. The physical states in the gauged WZNW model are now given by the non-trivial elements of the resulting BRST cohomology. In [2] it was proved that the BRST approach was equivalent to the conventional coset construction, so that the states were ghost-free and satisfied the usual highest weight conditions w.r.t. the subalgebra g k . The conditions for this proof was that one selected a specific range of representations for the auxiliary WZNW model. For the original ungauged WZNW model the range of representations were assumed to be the integrable ones.
In this work we will consider completely general highest weight representations (an analagous treatment may be given for lowest weight representions). The motivation for this is that it may be that a more general situation than in ref. [2] is the physically relevant one. Our analysis of the cohomology is most straightforwardly applied to the case when the gauged subgroup coincides with the original group i.e. when we have a topological WZNW model. But, as we will show, it also generalizes to the most important class of non-topological models, namely those in which the ungauged WZNW model is unitary.
In [3] the explicit construction of elements in the BRST cohomology was considered. The procedure presented there for obtaining these elements showed that they were intimitely related to certain null-vectors. The key to the construction was to make a selection of null-vectors that generated the states in the cohomology. It turned out that these null-vectors are the highest weight vectors. Then by using the explicit form of highest weight null-vectors given by Malikov, Feigin and Fuchs [4], the elements may be constructed. Our work here may be seen as the theoretical basis of this construction. We will here prove that the procedure in [3] will always generate non-trivial states in the cohomology. We will also prove that the ghost numbers that appeared in the construction are the only possible ones. The ghost numbers will be uniquely determined by the representations of the algebras involved, and for fixed representations only one value (and its negative) will occur. It is still an open question whether the construction provides all the possible states. We also lack a general result on the dimensionality of the cohomology.
The plan of the paper and its main results are the following. In section 2 we give basic definition and facts for affine algebras and associated modules. In section 3 we discuss the structure of Verma modules. This is important since our analysis of the cohomology relies very heavily on this structure, in particular, on the embeddings of Verma modules into Verma modules. We make extensive use of a technique due to Jantzen [5] to perturb the highest weight of a reducible Verma module to obtain an irreducible one. This perturbation gives also a filtration of modules in a given Verma module. Section 3 contains results on the structure of Verma modules, which we have been unable to find in the literature. The main results are Theorem 3.10 and Theorem 3.11. These are generalizations of results of Verma [15] and Bernstein, Gel'fand and Gel'fand [11], respectively, for finite dimensional Lie algebras and of Rocha-Caridi and Wallach for affine Lie algebras with highest weights on Weyl orbits through dominant weights. The proof of Theorem 3.11 is almost identical to the proof of the finite dimensional case given in [14], Theorem 7.7.7 (which is used also in [6]). The proof of Theorem 3.10 only partly coincides with [6], as the latter does not extend to the case of antidominant weights.
In section 4 we proceed to introduce the BRST formalism. Most of the material (except Lemma 4.2) is well-known. In particular, we recapitulate a theorem due to Kugo and Ojima [7]. This theorem will partly be used in the main section, section 5. It is also conceptually important in understanding the basic mechanism behind the appearance of elements in the BRST cohomology for non-zero ghost numbers, which we now explain. The theorem, which applies only to irreducible modules, states that elements in the cohomology form either singlet or doublet (singlet pair) representations w.r.t the BRST algebra. Furthermore, elements that are trivial or outside the cohomology form so-called quartets in the terminology of [7] i.e. sets four states, in which two of the elements are BRST exact. In order to obtain an irreducible module, we use a trick due to Jantzen, to perturb a reducible module into an irreducible one. In the irreducible case one may prove (Corollary 5.2), that only ghost-free highest weight states are BRST non-trivial. As the perturbation is taken to zero and the module becomes reducible, certain quartets will evolve into singlet pairs in the following way. Two of states of the quartet will remain in the irreducible module and will then form a singlet pair in this module. The two other states will become null-states. One of the main results in this paper (Theorem 5.12) is the determination of the relevant null-states. This theorem gives the necessary and sufficient conditions on the null-states to be part of a quartet, that will contain a singlet pair as the perturbation is set to zero. The implications of the theorem is exploited in Theorems 5.14 and 5.15, which give the necessary and sufficient conditions on the ghost-numbers and weights for which the cohomology is non-trivial. In particular in Theorem 5.15 a sequence of non-trivial BRST invariant states is proved to exist. This sequence is exactly the one for which the construction has been given in [3]. The ghost numbers appearing are ±p, where p = l(λ)−l(λ) and l(λ) is the length of a Weyl transformation associated with λ (see section 3). This means that for given highest weights λ andλ of the original and auxiliary sectors, |p| is fixed to exactly one value. By Theorem 5.14 these ghost numbers and weights are the only non-trivial ones.
Let us also address the question of how the embedding of g into a larger algebra may affect our results. As our approach relies on the use of null-vectors, the crucial question is what happens to the relevant null-vectors as g is embedded. If the nullvector w.r.t. g will cease to be null in the larger algebra, then the entire quartet, to which the vector belongs for non-zero perturbation, will remain a quartet as Jantzen's perturbation is set to zero. Thus the corresponding elements in the cohomology of g will now be exact. In addition, many more elements may disappear from the cohomology group. This is most evident from the construction in [3], where one used non-trivial states at ghost number p − 1 (p > 0) to construct a BRST non-trivial element of ghost number p. In the extreme case the module over the larger algebra is irreducible and all elements, except the one at zero ghost number, will disappear.
There is one case in which the embedding will be straightforward. This will happen when we select integrable representations of the larger algebra. In this case it is known [9] that the irreducible module over the larger algebra is completely reducible w.r.t. to any subalgebra. Hence, the results given here generalize directly. This was the situation analyzed in [2]. Corollary 5.11 proves that the solutions given in [2] for a selected range of representations of the auxiliary sector, are in fact the unique solutions for zero ghost number for any selection of representations of the auxiliary sector.
The existence of extra elements in the cohomology, which have non-zero ghost numbers, implies that the BRST approach to WZNW models is different from the conventional coset approach. This applies to the topological case, but also to the non-topological case, at least when we take integrable representations of the original algebra. The rôle of these extra states is at this point unclear. It may be that their appearance will lead to inconsistencies. One may avoid the states by selecting an appropriate range of representations for the auxiliary sector. Then only ghost free states will appear in the cohomology. This was the situation treated in ref [2]. It may on the other hand be that the extra states are a new and important part of the quantization of WZNW models. In the latter case one may expect that the extra states will be needed to ensure S-matrix unitarity and hence will appear as poles in scattering amplitudes.

Preliminaries
Letḡ be a simple finite dimensional Lie algebra of rank r. We denote by g k the corresponding affine Lie algebra of level k. The set of roots ofḡ and g areᾱ ∈∆ and α ∈ ∆, respectively. The highest root ofḡ is denotedψ and its length is taken to be one. The restriction to positive roots are denoted by∆ + , ∆ + and to simple roots bȳ ∆ s , ∆ s . The weight and root lattices ofḡ and g are denoted byΓ w ,Γ r , Γ w and Γ r . Γ + r is the lattice generated by positive roots. Let Γ + w be the set of dominant weights, (α j ) 2 = δ ij for α j ∈ ∆ s } be the set of fundamental weights. Here λ i · α j denotes the invariant scalar product on g and (α j ) 2 = α j · α j . Define ρ as twice the sum of fundamental weights of g.ρ is the corresponding sum forḡ. ρ satisfies ρ · α i = (α i ) 2 , α i ∈ ∆ s . We define the set of antidominant weights Γ − w = {λ ∈ Γ w |α i · (λ + ρ/2) ≤ 0 for α i ∈ ∆ s }. A weight µ ∈ Γ w is said to be singular if it is orthogonal to at least one positive root and is said to be regular otherwise.
The Weyl group W of g is the set of transformations on Γ w generated by the simple reflections The length l(w) of w ∈ W is the minimal number of simple reflections that give w. We also define the ρ−centered reflections σ ρ i (λ) = σ(λ + ρ/2) − ρ/2. Similarly we write w ρ (λ) for a general ρ-centered Weyl transformation. We define an ordering between weights. Let µ, ν ∈ Γ w be such that µ − ν ∈ Γ + r . We then write µ ≥ ν. If µ − ν ∈ Γ + r /{0}, then this is denoted by µ > ν. Two weights λ and µ are said to be on the same Weyl orbit if there exists w ∈ W such that µ = w(λ). Similarly, they are said to be on the same ρ-centered Weyl orbit if µ = w ρ (λ).
We make a triangular decomposition of g, g = n − ⊕ h ⊕ n + . We will use the notation e α for the generators of n + , f α for those of n − and h i , i = 1, . . . , r + 2 for the generators of the Cartan subalgebra h. h i , i = 2, . . . , r + 1 spanh, h 1 is a central element of g with eigenvalue k/2 and h 0 is a derivation. We have a corresponding decomposition of U (g), the universal enveloping algebra of g, as U (g) = U (n − ) ⊗ U (h) ⊗ U (n + ).
Let M (λ) denote the highest weight Verma module over g of highest weight λ. The module is generated by a highest weight primary vector v 0λ satisfying Vectors in M ν (λ) will be called weight vectors of degree ν and their weights differ from the highest weight by ν. We consider throughout only vectors v ∈ M η (λ) with dimM η (λ) < ∞. The dimension of M ν (λ) is P (ν), which is the number of ways ν may be written as a linear combination of positive roots with non-negative coefficients. Let M ′ (λ) be the proper maximal submodule of M (λ). Then M (λ)/M ′ (λ) is irreducible and isomorphic to the unique irreducible g−module L(λ). Define a Hermitean form ..|.. as the mapping from M (λ)× M (λ) to the complex numbers by v 0λ |v 0λ = 1 where u ∈ U (g) and ( ) † denotes the Hermite conjugation defined by For v η , w µ ∈ M η (λ) we clearly have w µ |v η = 0 for η = µ. If η = µ, then F (λ) η = w η |v η may be viewed as a P (η) × P (η) matrix, whose entries are polynomials in λ. The determinant of F (λ) η is given by the Kac-Kazhdan formula [10] det F (λ) η = const.
where roots α ∈ ∆ + are taken with their multiplicities and P (η) = 0 if η ∈ Γ + . The zeros of the determinant are associated with highest weight vectors v µ that occur in M (λ) (see the following section). From eq.(2.4) one may infer that µ = λ−nα, which implies that the Verma module M (µ) is a submodule of M (λ). M (λ) is irreducible if and only if there does not exist n ∈ Z and α ∈ ∆ + such that Notice that this equation will for any imaginary root α (i.e. α 2 = 0) be equivalent to the condition k = −cḡ, where cḡ is the quadratic Casimir of the adjoint representation ofḡ.
ǫ → 0 this induces a filtration of modules in M (λ) Note that Jantzen's filtration is hereditary: Any irreducible subquotient of a g-module M (λ) is isomorphic to an irreducible g-module L(µ), λ − µ ∈ Γ + r . Denote by (M (λ) : L(µ)) the multiplicity of L(µ) in M (λ). M (1) (λ) is the maximal proper submodule of M (λ) and hence M (λ)/M (1) (λ) is isomorphic to the irreducible module L(λ). We will call the vectors in M (1) nullvectors of M (λ). We define π L to be the projection M (λ) The submodules of a given Verma module are generally not all of Verma type. It is convinient to introduce the notion of primitive vectors. Let V be a g-module. A vector v λ ∈ V is said to be primitive if there exists a submodule U of V such that v ∈ U, xv ∈ U for any x ∈ n + . (3.4) λ is called a primitive weight. Highest weight vectors are clearly primitive, but in general they do not exhaust all primitive vectors, even in the case of finite dimensional algebras, as was first noted in [11]. In fact, there may be infinitely many more primitive vectors than highest weight vectors (see [12] for an example for finite dimensional algebras). Any module V is generated by its primitive weights as a g-module.
We will call a module which is generated by acting freely with U (n − ) on a primitive vector, which is not of highest weight type, a Bernstein-Gel'fand (BG) module. The corresponding primitive vector will be called a Bernstein-Gel'fand primitive vector. Although every zero in the determinant eq.(2.4), i.e. every (α, n) for which the Kac-Kazhdan eq.(2.5) is satisfied, corresponds to a highest weight vector in M (λ) (cf. Proposition 3.8), the converse is in general not true. For a given λ there are usually more highest weight vectors than solutions (α, n). Let Hom g (M (µ), M (λ)) = 0 for a pair (α, n) in eq.(2.5) with α real i.e. µ = λ − nα, n ≥ 1 and α ∈ ∆ + ∩ ∆ R , where ∆ R is the set of real roots. Then we may write The inequality ensures that a solution to eq.(2.5) exists. In the form eq.(3.5) it is clear that by iteration, we will find new highest weight vectors not given by solutions to the Kac-Kazhdan equation for λ. It also follows that M (λ) is irreducible if and only if λ is antidominant. Notice that this requires k < −cḡ.
Proof. The proof is by explicit construction. Consider e.g. k < −cḡ and µ p = σ ρ ip (µ p−1 ). We take the sl 2 subalgebra generated by e ip , f ip and h ip satisfying is a highest weight vector and it will generate a submodule isomorphic to M (µ p−1 ). Hence, Hom g (M (µ p−1 ), M (µ p )) = 0.
By Theorem 3.1, Lemma 3.2 and Lemma 3.3 and we have the following: Proof. The proof of (i) is identical to that of Lemma 7.7.2 (ii) in [14] (cf [6], Lemma 8.2). Note that in the proof of Lemma 7.7.2 in [14], λ ∈ Γ + w is assumed. The weaker condition on λ, assumed in our case, does not affect (i). We prove (ii) for k > −cḡ.
As a corollary to this proposition we can generalize results obtained by [15] and [11] for finite dimensional Lie algebras. Corollary 3.9. A necessary and sufficient condition for M (µ) to be a submodule of M (ν) is that the condition (*) in Theorem 3.1 is satisfied.
Note here the following. Firstly, Theorem 3.1 and Corollary 3.9 imply that a BG We are now ready to formulate one of the main results of this section namely the dimension of the g-homomorphisms M (µ) → M (ν). This result generalizes the result of Verma [15] for finite dimensional Lie algebras and Rocha-Caridi, Wallach [6] for representations with highest weights on Weyl orbits through dominant weights.
Proof. We consider the cases k > −cḡ and k < −cḡ separately.
It is convinient to introduce the concept of length of a weight. Let µ ∈ Γ w . Then we define the length l(µ) as the smallest integer l(w) such that µ = w ρ (λ), w ∈ W , λ + ρ/2 ∈ Γ + w or λ ∈ Γ − w . We now prove some useful results involving this concept. First we have a result similar to Lemma 3.5.

4
The BRST formalism Define the algebra g ′ = g k ⊕ g −k−2cḡ , where cḡ is the quadratic Casimir of the adjoint representation. This algebra is invariant under the exchange k → −k − 2cḡ and, consequently, we may restrict to k > c g . The singular case k = −cḡ will not be treated here. We will denote g −k−2cḡ byg and the Verma module overg will be denoted M (λ), whereλ is its highest weight. Let B n ′ + , B n ′ − , B h ′ , Bg and B g ′ be bases of n ′ + , n ′ − , h ′ ,g and g ′ , respectively. The generatorsẽ α ,f α andh i is a realization of Bg and e ′ α , f ′ α and h ′ i a realization of B g ′ . Define M ′ λλ = M (λ) ⊗M (λ) and similarly L ′ λλ = L(λ) ⊗L(λ). π L ′ denotes the projection M ′ −→ L ′ . We define the anticommuting ghost and antighost operators c(x) and b(x), respectively, where x ∈ B g ′ , with the following properties Here δ x,y = 1 if x = y and 0 otherwise. Introduce a normalordering Define the BRST operator where ρ(x) is the component of ρ corresponding to the element x ∈ B h ′ . The BRST operator has the following two fundamental properties: d 2 = 0 and d † = d. The first property implies that x tot = {d, b(x)} = x + ρ + y b([x, y])c(y † ) generates an algebra g 0 which is centerless. Define a ghost module F gh . It is generated by the ghost operators acting on a vacuum vector v gh 0 satisfying We also define a restricted moduleF The dual F * gh of F gh has a vacuum vector v * gh The restricted dual isF * gh = {v * gh ∈ F * gh | c(x)v * gh = 0 for x ∈ B h ′ }. Define a Hermitean form for the ghost sector by v * gh 0 |v gh for a polynomial u in the ghost operators and v gh ∈ F gh , v * gh ∈ F * gh . If v gh = uv gh 0 then we denote by v * gh the vector u † v * gh 0 . The ghost number N gh of any vector v gh ∈ F gh is defined by N gh (v gh 0 ) = 0 and The ghost numbers of vectors in the dual module is similarly defined with N gh (v * gh 0 ) = 0. It is easily seen that (4.9) We have dω =dω for ω ∈Ĉ(g ′ , V ) and consequently on the relative subcomplex we may analyze the cohomology ofd in place of d. The cohomology associated withd, the semi-infinite or BRST relative cohomology is sometimes denoted byĤ ∞/2+p (g ′ , V ) to distinguish it from the conventional Lie algebra cohomology. We will, however, here for simplicity writeĤ p (g ′ , V ), where p refers to elements ω with N gh (ω) = p. Our primary interest here will be for V = L ′ λ,λ . But in order to gain knowledge of this case we will also study V = M ′ λ,λ and its submodules. It will be convinient to make a classification of vectors in the complex C(g ′ , V ) using the BRST operator. A central result due to Kugo and Ojima [7] states the following.
(iii) Quartet case: ω ∈ H * (g ′ , V ). There will exist four elements ω 1 , ω 2 , ψ 1 , ψ 2 ∈ C(g ′ , V ), where either ω = ω 1 or ω = ω 2 , such that ψ * 1 |ω 1 = 0 and ψ * 2 |ω 2 = 0, There will exist an analagous classification on the relative subcomplexĈ(g ′ , V ) w.r.t. d. In this classification all non-trivial elements in the cohomology will be singlets or singlet pairs. It should be remarked that the condition of irreducibility is essential for the theorem. In the following section, we will find that for V being a reducible Verma module the above classification does not hold. In particular, the non-trivial elements of the cohomology for non-zero ghost numbers will for this case not be members of singlet pairs. Define Jantzen's filtration for ξ ∈Ĉ(g ′ , M ′ ) as follows. Let We denote by ξ ǫ the vector v ǫ ⊗ṽ ǫ ⊗ v gh . An element ξ ǫ is always assumed to be finite as ǫ → 0. We denote by f (ǫ) ∼ ǫ n the leading order of a function f (ǫ) in the limit ǫ → 0. Note that our definition of Jantzen's filtration for g ′ implies that λ ǫ +λ ǫ is independent of ǫ. This is required if the cohomology should have at least one non-trivial element for ǫ = 0, namely the vacuum solution ξ 0ǫ = v 0ǫ ⊗ṽ 0ǫ ⊗ v gh 0 . In the next section the following result will be needed.

The BRST cohomology
We will now in detail study the semi-infinite relative cohomology associated with the BRST operator. The notation follows that of previous sections. ω ǫ , ξ ǫ , . . . always denote elements ofĈ(g ′ , . . .) that are finite in the limit ǫ → 0. Our starting point is Proposition 5.1 concerning the cohomology of Verma modules. This proposition was to our knowledge first given in [20], Proposition 2.29.
Proposition 5.1. Let M ′ be a highest weight Verma module over g ′ . Then H p (g ′ , M ′ ) = 0 for p < 0.
The results obtained so far are of importance for negative ghost numbers. We now turn to results relevant for positive ghost numbers. We will connect the two cases by the use of Jantzen's perturbation, Theorem 4.1 and Lemma 4.2.
Proposition 5.9.Ĥ p (g ′ , L ′ ) for p ≥ 1 are represented by elements of the form v ⊗ṽ 0 ⊗ v gh , or equivalently of the form v 0 ⊗ṽ ⊗ v gh , where v ∈ L,ṽ ∈L, v 0 is a primary highest weight vector w.r.t. g,ṽ 0 is a primary highest weight vector w.r.t. g and v gh satisfies c(x)v gh = 0, x ∈ n + .
Proof. Let ψ ∈Ĥ 0 (g ′ , M ′ ). Thendψ = 0 and we can use the gradationÑ gr and the homotopy operator as in the proof of Proposition 5.9 to conclude that since N gh (ψ) = 0 we must have ψ = v ⊗ṽ 0 ⊗v gh 0 . By using the gradation N gr we get the al-ternative form. The condition on the weights is a consequence of h tot (v⊗ṽ ⊗v gh 0 ) = 0.
Proof. The theorem is true for p = 0 by Corollary 5.11 and for p = 1 by Corollary 5.13. Assume the theorem to be true forĤ ±q (g ′ , L ′ µμ ), 0 ≤ q ≤ p − 1 and consider q = p.
Assume the contrary, ξ =dη with η ∈ M Remark 1: Results similar to Theorem 5.15 may be obtained for weights µ + ρ/2 andμ + ρ/2 being singular, provided the corresponding Verma modules satisfy the multiplicity condition of Lemma 3.17. It is clear, however, that this generalization does not hold for all singular cases.
Remark 2: The proof of Theorem 5.15 provides also an explicit method for finding the elements of the cohomology for negative ghost numbers. It is the same method as was presented in ref [3].