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FOCK SPACE RESOLUTIONS OF THE VIRASORO

arXiv:hep-th/9108023v1 27 Aug 1991CERN-TH.6196/91FOCK SPACE RESOLUTIONS OF THE VIRASOROHIGHEST WEIGHT MODULES WITH c ≤1Peter BouwknegtCERN - Theory DivisionCH-1211 Geneva 23SwitzerlandJim McCarthy†Department of PhysicsBrandeis UniversityWaltham, MA 02254Krzysztof Pilch‡Department of PhysicsUniversity of Southern CaliforniaLos Angeles, CA 90089-0484AbstractWe extend Felder’s construction of Fock space resolutions for the Virasoro minimalmodels to all irreducible modules with c ≤1. In particular, we provide resolutions for therepresentations corresponding to the boundary and exterior of the Kac table.Submitted to: Letters in Mathematical PhysicsCERN-TH.6196/91August 1991BRX TH-323revised August 1991USC-91/21† Supported by the NSF Grant #PHY-88-04561.‡ Supported in part by the Department of Energy Contract #DE-FG03-84ER-40168and by the USC Faculty Research and Innovation Fund.

1.IntroductionThe problem we address in this letter arises quite naturally in the so-called free fieldapproach to conformal field theories. The mathematical question one wants to answerwithin this framework is whether a given irreducible module L of the chiral algebra A hasa resolution in terms of Fock spaces.

More precisely, one wants to construct a family of freefield Fock spaces F(i), which are A-modules, and a set of A-homomorphisms (intertwiners)d(i) : F(i) →F(i+1), such that these spaces with the maps between them form a complexwhose (co)homology is isomorphic to L. The first example of such a construction wasgiven by Felder [1] for the class of representations of the Virasoro algebra corresponding tominimal models [2]. Later, this was extended to other conformal field theories, in particularWZNW and their coset models (for review see [3] and references therein).Felder’s construction relies on the complete classification of submodules of Fock spacesgiven by Feigin and Fuchs [4,5], and the explicit form of the intertwiners.

The latter aredefined, following Thorn [6], as multiple integrals of products of the screening currents. Onemay wonder whether the restriction to the representations in the fundamental range of theminimal series is important.

The answer turns out to be negative, and already examplesof similar resolutions outside this series have been discussed in [7]. In the following we willargue that a Fock space resolution can be explicitly constructed for any irreducible highestweight module of the Virasoro algebra provided one introduces additional intertwinersbesides those considered in [1,7].

The existence of such intertwiners, and their propertiesneeded in the computation of the cohomology, were demonstrated by Tsuchiya and Kanie[8]. We will discuss some of their results within the Dotsenko-Fateev formalism [9], asused in [1,3], which may be more familiar.

Not to make our presentation too long we willrestrict to modules with c ≤1.Our three main results, which cover the cases not analyzed previously, are summarizedin Theorems 5.2, 6.1 and 7.1.In particular, the first gives the resolution for modulescorresponding to the so-called boundary of the Kac table [10], whilst the last extendsFelder’s construction to modules outside the fundamental range.This letter is organized as follows: In Section 2, as well as introducing some definitionsand notation, we summarize the results of Feigin and Fuchs which will be used later.Then, in Sections 3 and 4, we introduce intertwiners and recall the construction of Felder’scomplex. After this review, we discuss the boundary case in detail in Sections 5 and 6, andin Section 7 describe the extension of the resolution for the irreducible modules beyond thefundamental range.

We conclude in Section 8 with some remarks on possible applicationsof these results.1

2. Feigin-Fuchs modulesThe generators of the Virasoro algebra, Vir, satisfy∗[Lm, Ln] = (m −n)Lm+n + c12m(m2 −1)δm+n,0 ,[Ln, c] = 0 ,m, n ∈ZZ .

(2.1)In this letter we will consider three classes of highest weight modules of Vir: the irreduciblemodules Lh,c, the Verma modules Mh,c, and the Feigin-Fuchs modules Fp,Q. We recallthat the central charge, c, and the conformal dimension, h, determine Lh,c and Mh,ccompletely, and that the latter is generated freely by L−n, n ≥1, acting on the vacuumvh,c, L0vh,c = hvh,c.The Feigin-Fuchs module Fp,Q is just the Fock space of a scalar field, φ(z), with abackground charge Q and momentum p such thatc = 1 −12Q2 ,h = 12p(p −2Q) .

(2.2)We follow the convention that the two-point function of φ(z) is⟨φ(z)φ(w)⟩= −ln(z −w) ,(2.3)and the stress energy tensor, T(z) = Pn∈ZZ Lnz−n−2, has the formT(z) = −12 : ∂φ(z)∂φ(z) : +iQ∂2φ(z) . (2.4)In terms of modes i∂φ(z) = Pn∈ZZ αnz−n−1,Ln = 12Xm∈ZZ: αmαn−m : −(n + 1)Qαn ,(2.5)where[αm, αn] = mδm+n,0 .

(2.6)The Fock space Fp,Q is generated from the vacuum vp, α0vp = pvp, by the free action ofthe creation operators α−m, m ≥1.As can easily be seen from (2.5), Fp,Q and F−p,−Q are isomorphic as Virasoro modules.Choosing one solution for Q in (2.2) leaves us, for a given h and c, with two Fock spacesFp,Q and F2Q−p,Q, dual to each other, i.e. F∗p,Q ≃F2Q−p,Q.∗See,e.g.

[11] for basic definitions and [2,12] for a review of conformal field theorytechniques used throughout this letter.2

The detailed structure of submodules of all Verma and Feigin-Fuchs modules of theVirasoro algebra has been obtained in [4,5]. There are three main types of modules, I, IIand III, depending on whether the equationxα+ + yα−+√2(p −Q) = 0 ,(2.7)has respectively zero, one or infinitely many integral solutions for x and y.

Here we haveintroduced α+ and α−, α+α−= −1, which also parametrize the background charge,Q =q12(α+ + α−) . (2.8)In the first case (type I) the Verma module and the Fock space are isomorphic andirreducible [5], and obviously no resolution is needed.

Thus we will assume that (2.7) hasat least one integral solution, or, equivalently, that the momentum p can be parametrizedby a pair of integers n and n′ (i.e. there is exactly one such pair for type II, and formodules of type III infinitely many such),p = pn,n′ =q12 ((1 −n)α+ + (1 −n′)α−) .

(2.9)For pn,n′ as in (2.9) we denote the corresponding conformal dimension, computed from(2.2), by hn,n′, and, for a fixed Q (and c) write Fn,n′ and Mn,n′ instead of Fpn,n′,Q andMhn,n′,c. Virasoro modules with the momentum given in (2.9) arise in the generalizedDotsenko-Fateev minimal models [9,7].In case III, in order to have infinitely many integral solutions to (2.7), there mustexist (relatively prime) integers p, p′, pp′ ̸= 0, such that pα+ + p′α−= 0.

We can thentake α+ =pp′/p and α−= −pp/p′. Clearly the central charge must be rational, and,depending on whether pp′ > 0 or pp′ < 0 we have c ≤1 or c ≥25, respectively.

In thefollowing we will restrict ourselves to c ≤1 and take p′ ≥p ≥1. In particular c = 1corresponds to p′ = p = 1, and for c < 1 we must have p′ > p .

Using the freedom inthe parametrization of the momentum, pn,n′ = pn+jp,n′+jp′, j ∈ZZ, we can always set0 ≤n′ ≤p′ −1 (or, equivalently, 0 ≤n ≤p −1). If, in addition, p′ > p > 1 and1 ≤n ≤p −1 ,1 ≤n′ ≤p′ −1 ,(2.10)we will say that the module belongs to the fundamental range of the minimal series [2],or to the interior of the Kac table [10].

The exterior of the table is defined by lettingn ̸= 0 mod p to be outside the range (2.10).Finally, n = 0 mod p or n′ = 0 mod p′correspond to the boundary of the table.In cases II and III, the detailed structure of Fn,n′ as a Virasoro module depends onthe momentum pn,n′ (2.9), and all possible subcases (in case III with c ≤1) are listed inthe following theorem which summarizes the results of Feigin and Fuchs.3

Theorem 2.1. ([5]) In case II there are four subcases:• Case II−(respectively II0): If nn′ < 0 (respectively nn′ = 0) then Fn,n′ and Mn,n′are isomorphic and irreducible.• Case II+(−): If n, n′ < 0 then Fn,n′ ≃Mn,n′ are reducible.

The maximal submoduleis isomorphic with F−n,n′ ≃M−n,n′.• Case II+(+): This case, where n, n′ > 0, is dual to II+(−), e.g. Fn,n′ ≃M∗−n,−n′.In case III for c ≤1 there are four subcases, in all of which the Fock space is a reduciblemodule.

In the first three, III−and III0−(±) below, p′ > p > 1 (i.e. c < 1), and m, m′denote labels in the fundamental range, 1 ≤m ≤p −1, 1 ≤m′ ≤p′ −1.

In III00−we havep′ ≥p ≥1 (i.e. c ≤1).

In all cases j ∈ZZ.• Case III−: The submodules of Fm+jp,m′ are generated by the vectors ui, vi, and wigiven by the following diagramv0←w0→v1←w1→v2←w2→· · ·ցւցւցւցւցւց· · ·u1←v−1→u2←v−2→u3←· · ·,(2.11)whose conformal weights are (see [1])h(v0) = hm+jp,m′ ,h(ui) = h−m+(|j|+2i)p,m′ , i ≥1 ,h(v−i) = hm−(|j|+2i)p,m′ , i ≥1 ,h(wi) = h−m−(|j|+2i)p,m′ , i ≥0 ,h(vi) = hm+(|j|+2i)p,m′ , i ≥1 . (2.12)• Case III0−(−): The submodules of i) Fm+jp,0, j ≥0, and ii) F0,m′+jp′, j ≥0, aregenerated by the vectors ui and vi,v0→u1←v1→u2←v2→u3←· · · ,(2.13)wherei)ii)h(vi) = hm+(j+2i)p,0 , i ≥0 ,h(vi) = h0,m′+(j+2i)p′ , i ≥0 ,h(ui) = hm−(j+2i)p,0 , i ≥1 ;h(ui) = h0,m′−(j+2i)p′ , i ≥1 .

(2.14)• Case III0−(+): The submodules of i) Fm+jp,0, j < 0, and ii) F0,m′+jp′, j < 0, aregenerated by the vectors ui and vi,u1←v1→u2←v2→u3←v3→· · · . (2.15)wherei)ii)h(vi) = hm−(j−2(i−1))p,0 , i ≥1 ,h(vi) = h0,m′−(j−2(i−1))p′ , i ≥1 ,h(ui) = hm+(j−2(i−1))p,0 , i ≥1 ;h(ui) = h0,m′+(j−2(i−1))p′ , i ≥1 .

(2.16)4

• Case III00−: The Fock space Fjp,0 ≃F−jp,0 is the direct sum of irreducible modules,Fjp,0 = L∞k=0 L(|j|+2k)p,0.In the diagrams (2.11), (2.13), and (2.15) vectors ui correspond to singular vectors inthe Fock space F⋆,⋆and generate the submodule, F′⋆,⋆, which is a direct sum of irreduciblehighest weight modules. In the quotient F⋆,⋆/F′⋆,⋆vectors vi become singular, and generateF′′⋆,⋆which is a direct sum of irreducible highest weight modules.

Note that v0 is special,since it is singular. Finally, wi are singular in F′⋆,⋆/F′′⋆,⋆, and generate a direct sum ofirreducible highest weight modules.

The arrows, 1 →2, indicate that the second vector isin the submodule generated by the first one.There are also similar diagrams for the composition series of singular vectors of theVerma modules, in which the singular vectors occur at precisely the same conformal weightsas above. Moreover, some arrows must be reversed, so that all of them describe embeddingsof the Verma submodules.3.

The intertwinersLet us first introduce a class of intertwiners between Fock spaces. They are constructedas products of the screening currentss+(z) =: exp(i√2α+φ)(z) :ands−(z) =: exp(i√2α−φ)(z) : ,(3.1)integrated over suitable multiple contours [6,5,8,1].

For positive integers r and r′ let usconsider operatorsQ(+)r [Ω] = [[ (s+)r ]]Ωr ,Q(−)r′ [Ω] = [[ (s−)r′ ]]Ωr′ ,(3.2)where [[ · · ·]]Ωk is defined by[[ si1 . .

. sik ]]Ωk =ZΩkdz1 .

. .

dzk si1(z1) . .

.sik(zk) ,i1, . .

., ik ∈{+, −} . (3.3)and Ωk is a set of contours for z1, .

. ., zk.

When acting on Fn,n′, the integrand in (3.3) isanalytic in Mk = ICk \ {zi = 0, zi = zj; i, j = 1, . .

., k} and has nontrivial monodromies dueto the presence of the factorYi

it is an element of H(Mk, Sα2±), the homology of Mk with coefficients in the localsystem Sα2± corresponding to (3.4) (this system depends only on α2±).5

In explicit computations one needs to have convenient representatives of these ho-mology classes. We will construct them using the following two classes of multicontours(singular at a point) that have been used in [1,13,3]: In the first class, which we denoteΓk, the integration variables z1, .

. ., zk are taken counterclockwise from 1 to 1 around 0,and nested according to |z1| > .

. .

> |zk|. In the second, bΓk, the z1 integration is along acontour surrounding 0, while z2, .

. .zk are integrated counterclockwise from the base pointz1 to z1 around 0, and the nesting is the same as in Γk.

The ambiguity in the phase of theintegrand (3.3) is fixed by analytic continuation from the positive real half-line (see [1,13]).Let us denote the resulting operators (3.2) by Q(+)r , Q(−)r′and bQ(+)r , bQ(−)r′ , respectively.We will also consider multiple contours obtained by putting several of those together.For example Γk∪Γk′ will denote a multiple contour z1, . .

.zk, . .

.zk+k′ in which the variablesof Γk′ are nested inside those of Γk.We may now state the main result about the intertwiners between Feigin-Fuchs mod-ules of type III due to Tsuchiya and Kanie [8].Theorem 3.1. 1) Consider Fn,n′, where n = m + jp, 1 ≤m ≤p, j ∈ZZ, and n′ isarbitrary.

Then for any non-negative integer k the operatorQ(+)m+kp[Ω] : Fn,n′ −→Fn−2(m+kp),n′ ,(3.5)is a well defined intertwiner provided Ωm+kp ∈H(Mm+kp, Sα2+).2) These intertwiners are nontrivial in the sense that such Ωm+kp exists for any k ≥0 andi) if n′ + (k −j)p′ ≤0 then Q(+)m+kp[Ω]vn,n′ ̸= 0 ;ii) if n′ + (k −j)p′ > 0 then Q(+)m+kp[Ω]χ = vn−2(m+kp),n′ for some χ ∈Fn,n′ .Here vn,n′ and vn−2(m+kp),n′ denote vacua of Fn,n′ and Fn−2(m+kp),n′, respectively.The analogous result holds for the operators Q(−)⋆[⋆].4. The Fock space resolution for type II+ and III−modulesWe begin by constructing explicitly intertwiners using contours Γr, Γr′, bΓr and bΓr′,1 ≤r ≤p and 1 ≤r′ ≤p′, introduced in the previous section.Lemma 4.1.

Operators Q(+)r, bQ(+)r: Fn,n′ −→Fn−2r,n′ and Q(−)r′ , bQ(−)r′: Fn,n′ −→Fn,n′−2r′ are well defined intertwiners between Virasoro modules if r = n mod p and r′ =n′ mod p′, respectively.Proof: Operators bQ(+)rand bQ(−)r′are well defined and commute with the action of theVirasoro algebra iffbΓ is a closed cycle. This amounts to the z1-contour being closed.

Forthe values of r and r′ as in the lemma this is easily verified using standard methods [1]. ForQ(+)rand Q(−)r′we use the following lemma which can be proven by standard manipulationswith the contours [1,13].⊔⊓6

Lemma 4.2. For 1 ≤r ≤p and 1 ≤r′ ≤p′,Q(+)r= 1r1 −q2r+1 −q2+bQ(+)r,Q(−)r′ = 1r′1 −q2r′−1 −q2−bQ(−)r′ ,(4.1)where q± = exp(iπα2±).Note that for positive r1, r2, r1+r2 ≤p we have Q(+)r1 Q(+)r2 = Q(+)r1+r2, provided Q(+)r1 andQ(+)r2 act on spaces on which they are well-defined.

In particular – after further investigationof the integral in bQ(+)p– (4.1) impliesQ(+)m Q(+)p−m = Q(+)p−mQ(+)m = Q(+)p= 0 ,(4.2)The same also holds for Q(−)m′ .Lemma 4.1 together with identity (4.2) is the basis for the construction of the complexof Fock spaces when m, m′ lie in the fundamental range (2.10). This complex, (F, d) ≡{(F(i)m,m′, d(i)), i ∈ZZ}, is defined as followsF(2j)m,m′ = Fm−2jp,m′ ,d(2j) = Q(+)m ,F(2j+1)m,m′= F−m−2jp,m′ ,d(2j+1) = Q(+)p−m ,j ∈ZZ .

(4.3)Using the submodule structure of Fock spaces F(i)m,m′ as summarized in Theorem 2.1,Felder [1] was able to compute the kernels and the images of all the intertwiners d(i), andprove the following important resultTheorem 4.3. Let 1 ≤m ≤p −1, 1 ≤m′ ≤p′ −1.

Then the complex (F, d) defined in(4.3) is a (two-sided) resolution of the irreducible module Lm,m′, i.e.H(i)(F, d) ≃δi,0Lm,m′ . (4.4)In fact there are three more resolutions, one in terms of modules F(i)p−m,p′−m′ (recallthat F∗m,m′ ≃Fp−m,p′−m′ and Lp−m,p′−m′ ≃Lm,m′), and two others if we use operatorsQ(−)⋆instead of Q(+)⋆.It may be worth noting that Felder’s proof also shows that the differential in thecomplex is nilpotent without resorting to the computation in (4.2).A similar construction for the modules of type II+(±) was carried out in [7].Theorem 4.4.

In the notation of Theorem 2.1, let n, n′ < 0. Then the complex0−→F−n,n′Q(+)−n−→Fn,n′−→0 ,(4.5)is a resolution for Ln,n′.

For the case II+(+) the resolution can be obtained by dualizationof (4.5).7

5.Resolutions for the modules of type III0−Using the results reviewed in the previous sections we will now construct resolutionsfor irreducible modules Lh,c of type III0−, for which h = hm+jp,0, 1 ≤m ≤p −1 orh = h0,m′+jp′, 1 ≤m′ ≤p′ −1, j ∈ZZ. Since hn,n′ = h−n,−n′, we may take j ≥0.Let us consider the Fock space F0,m′+jp′.

According to Theorem 2.1 this space hastype III0−(−) and its maximal submodule, F′0,m′+jp′, is generated by vectors ui, vi, i ≥1.It also follows from (2.13) that the irreducible module is isomorphic to the quotient, namelyL0,m′+jp′ = F0,m′+jp′/F′0,m′+jp′ . (5.1)By comparing (2.14) and (2.15) we find that the weights of ui and vi in F′0,m′+jp′ areprecisely those of the vectors generating submodules of the Fock space F0,m′−(j+2)p′.

Letus denote the latter vectors by u′i, v′i, i ≥1. This observation and (5.1) reduce the problemof constructing a resolution to that of finding a homomorphism d from F0,m′−(j+2)p′ intoF0,m′+jp′, such that du′i = ui and dv′i = vi, i ≥1.

Note that then d must be injective, i.e.it is an embedding.In a sense the problem is similar to the one in case II+(−), except that the structureof submodules of Fock spaces is somewhat more complicated. Since F0,m′+jp′ ≃F−jp,m′and F0,m′−(j+2)p′ ≃F(j+2)p,m′, the intertwiner should involve (j + 1)p currents s+(z).By Theorem 3.1 we know it exists.

To construct it explicitly we observe that, in view ofLemma 4.1 and (4.2), an obvious building block for such operators is bQ(+)p . Its propertiesare summarized in the following technical lemma whose proof will be outlined at the endof the section.Lemma 5.1.

Consider bQ(+)p: F0,m′+kp′ −→F0,m′+(k+2)p′, k ∈ZZ. Depending on k thereare four cases described by the diagrams i)–iv) below.

Operator bQ(+)pmaps special vectorsu′i and v′i from the first Fock space onto ui and vi in the second space, as indicated by thedownward arrows, i.e. bQ(+)pis nonzero along these arrows.i)k ≤−3 :u′1←v′1→u′2←· · ·↓↓↓u1←v1→u2←v2→u3←· · ·ii)k = −2 :u′1←v′1→u′2←v′2→· · ·↓↓↓↓v0→u1←v1→u2←v2→· · ·iii)k = −1 :u′1←v′1→u′2←v′2→u′3←· · ·↓↓↓↓v0→u1←v1→u2←· · ·8

iv)k ≥0 :v′0→u′1←v′1→u′2←v′2→· · ·↓↓↓v0→u1←v1→· · ·The main result of this section isTheorem 5.2. The intertwiner d = ( bQ(+)p )j+1 is an embedding of F0,m′−(j+2)p′ intoF0,m′+jp′, i.e.

the complex0−→F0,m′−(j+2)p′d−→F0,m′+jp′−→0 ,(5.2)is a Fock space resolution of L0,m′+jp′. Another resolution is given by the dual complex0−→F0,−m′−jp′d∗−→F0,−m′+(j+2)p′−→0 ,(5.3)where d∗= ( bQ(+)p )j+1.Proof: Using Lemma 5.1 we can compute the image of F0,m′−(j+2)p′ when acting withsubsequent bQ(+)p .The result is d(F0,m′−(j+2)p′) = F′0,m′+jp′.The second part of thetheorem follows from L∗0,m′+jp′ ≃L0,m′+jp′ and bQ(+)p∗= bQ(+)p .⊔⊓Clearly, for modules Lm+jp,0 there are analogous resolutions in which the differentialis bQ(−)p′ .Proof of Lemma 5.1: The general idea of the proof is the same as that of parts of Theorem4.3 (see [1]).

Let us begin with case i). By Theorem 3.1.

2.i) bQ(+)p u′1 ̸= 0. To verify thiswe choose a covector χ ∈F∗0,m′+(k+2)p′ (see [6] and Appendix in [1]) such that⟨χ, s+(z1) .

. .s+(zp)u′1⟩=pYℓ=1z−(k+1)p′−m′ℓpYℓ,ℓ′=1ℓ<ℓ′(zℓ−zℓ′)2α2+pYℓ=1z√2α+p0,m′+kp′ℓ.

(5.4)Then a straightforward computation yields⟨χ, bQ(+)p u′1⟩=ZbΓdz1 . .

. dzppYℓ,ℓ′=1ℓ<ℓ′(zℓ−zℓ′)2α2+pYℓ=1zα2+−p′−1ℓ= 2πi(−1)p−1p−1Yℓ=1(1 −q2ℓ+ )21 −q2+J0,p−1(2α2+, α2+ −p′ −1, α2+)= (2πi)p(−1)pp′+p−1p′ !

(p −1) !eiπα2+Γ(1 + α2+)pp−1Yℓ=1sin πℓα2+sin πα2+̸= 0 . (5.5)9

We used here an explicit result for the Dotsenko-Fateev integral [9]J0,n(α, β; ρ) ≡1n !Z 10dt1 . .

. dtnnYℓ=1(1 −tℓ)αtβℓnYℓ,ℓ′=1ℓ<ℓ′|tℓ−tℓ′|2ρ=nYℓ=1Γ(ℓρ)Γ(ρ)Γ(1 + α + (ℓ−1)ρ)Γ(1 + β + (ℓ−1)ρ)Γ(2 + α + β + (n + ℓ−2)ρ).

(5.6)The rest of the lemma in this case follows entirely from the embedding patterns ofsubmodules in both Fock spaces and bQ(+)pbeing an intertwiner. Let us outline the firstfew steps.Normalize u2 such that u2 = bQ(+)p u′1.

Since u′1 is in the submodule generated by v′1 wemust have bQ(+)p v′1 ̸= 0. Note that the latter remains nonzero after we divide F0,m′+(k+2)p′by the submodule generated by u1 and u2.

Indeed, suppose the opposite, i.e. that bQ(+)p v′1 =P1u1 + P2u2, where P1,2 are some polynomials in L−n, n ≥1.Since the submodulegenerated by u1 and u2 is a direct sum of two irreducible ones, we also have u1 = P ′1P1u1where P ′1 is some element in the enveloping algebra of Vir.

But if P1u1 ̸= 0 then u1 =bQ(+)p P ′1(v′1−P2u′1), and a simple examinination of the weights in both Fock spaces shows ther.h.s. must vanish, which is a contradiction.

Thus we can at most have bQ(+)p v′1 = P2u2. Butthen bQ(+)p (v′1 −P2u′1) = 0, i.e.

v′1 −P2u′1 ̸= 0 is in the kernel of bQ(+)p . Since the submoduleker bQ(+)pcontains neither u′1 nor v′1, there can be no such vectors at this level, i.e.

wemust have bQ(+)p v′1 ̸= P2u1. That bQ(+)p v′1 is singular after we divide out the submodulegenerated by u1 follows from the corresponding property for v′1.

Thus, finally, we mayset bQ(+)p v′1 = v2. In the next step a similar reasoning shows that bQ(+)p u′2 ̸= 0, so, up tonormalization it yields bQ(+)p u′2 = u3.

And so on.Case ii) is proven by exactly the same method. Cases iii) and iv) are dual to ii) and i),respectively.

Since bQ(+)p∗= bQ(+)p , we deduce from coker bQ(+)p∗≃ker bQ(+)pthat bQ(+)pmustbe onto. Because bQ(+)pcommutes with Vir, u′i are mapped onto ui, and the maps indicatedby the arrows are clearly nonzero.

After we divide both sides by submodules generated byu′i and ui, respectively, bQ(+)pbecomes a homomorphism from a direct sum of irreduciblehighest weight modules onto a direct sum of a subset of these modules, and the arrowsbetween v′i and vi follow.⊔⊓6. Resolutions for the modules of type III00−The problem of obtaining a resolution for a module Ljp,0, j ≥0, is similar to theone discussed above.

Given the Fock space Fjp,0 we must construct an embedding fromF(j+2)p,0 into Fjp,0. The result is10

Theorem 6.1. Let p′ ≥p ≥1 and j ≥0.

The operator bQ(+)p: F(j+2)p,0 −→Fjp,0 is anembedding and Ljp,0 ≃Fjp,0/ bQ(+)p (F(j+2)p,0).Proof: Since Fjp,0 and F(j+2)p,0 are direct sums of irreducible highest weight modules,the proof essentially consists of verifying that bQ(+)pdoes not annihilate any of the singularvectors in F(j+2)p,0. For simplicity let us only consider the case p = p′ = 1, i.e.

c = 1.Then bQ(+)1=Hdz : exp(i√2φ(z)) : is just a vertex operator. Using explicit formulae (interms of Schur polynomials) for singular vectors in the Fock space Fkp,0, k ≥0 [14,15], itis easy to check that, up to normalization, they coincide with the vectors ( bQ(+)1 )ℓvkp+2ℓ,0,ℓ≥0, where vkp+2ℓ,0 denotes the vacuum of Fkp+2ℓ,0.

The theorem is then obvious. Thegeneral case is similar.⊔⊓7.

Generalizations of Felder’s constructionConsider an irreducible module, Lm+jp,m′, of type III−, where m, m′ lie in the fun-damental range (2.10) and j is an arbitrary integer. The Fock space Fm+jp,m′ appearsin the complex (F, d) corresponding to the resolution of Lm,m′ for j even, and Lp−m,m′for j odd, constructed as in Section 4.

An important feature of (F, d) is that the vac-uum of the 0-th Fock space has the highest weight with respect to all other states in thecomplex. In particular, this guarantees that the cohomology contains at least the irre-ducible module.

Let us now consider the collection of Fock spaces obtained by removingfrom (F, d) all Fock spaces whose weights are higher than hm+jp,m′; i.e. we delete thespaces F−m+2jp,m′, .

. ., Fm−2jp,m′ for j > 0, and the spaces Fm−jp,m′, .

. ., F−m+(j+2)p,m′for j < 0.

On both sides of the deleted segment the differential is defined as before interms of Q(+)m and Q(+)p−m. In the middle we must use a new intertwiner of the formQ(+)m+kp = [[ (s+)m+kp ]]Ωm+kporQ(+)(p−m)+kp = [[ (s+)(p−m)+kp ]]Ω(p−m)+kp ,k ≥1 ,(7.1)where Ωm+kp = Γm ∪bΓp ∪.

. .

∪bΓp and Ω(p−m)+kp = Γ(m−p) ∪bΓp ∪. .

. ∪bΓp.

The resultingextension of the Fock space resolution to modules outside the fundamental range can besummarized by the following theorem.Theorem 7.1. Let 1 ≤m ≤p−1, 1 ≤m′ ≤p′ −1, and j ≥0.

Then the complex of Fockspaces· · ·Q(+)m−→F−m+(j+2)p,m′Q(+)(p−m)+jp−→Fm−jp,m′Q(+)m−→F−m−jp,m′Q(+)p−m−→· · · ,(7.2)is a resolution of the irreducible module Lm−jp,m′, while· · ·Q(+)m−→F−m+(j+2)p,m′Q(+)p−m−→Fm+jp,m′Q(+)m+jp−→F−m−jp,m′Q(+)p−m−→· · · ,(7.3)11

is a resolution of Lm+jp,m′.Proof: Contours Ω(p−m)+kp and Ωm+kp are nontrivial cycles in local homology. This canbe verified by a computation similar to the one in the proof of Lemma 5.1.

Thus theintertwiners Q(+)(p−m)+jp and Q(+)m+jp are precisely those given in Theorem 3.1. In particularthey satisfy i) and ii), respectively, which is precisely what one needs to extend the proofof Theorem 4.3 to the present case.⊔⊓It is clear that the three other resolutions discussed in Section 4 – the dual, and thetwo others constructed with Q(−)⋆instead of Q(+)⋆– may be extended in the same way tomodules outside the fundamental range.8.

Concluding remarksFormally, the new intertwiners in Sections 5, 6 and 7 are proportional to operators ofthe form[[ (s±)k ]]Γ[k]q± !,(8.1)where [n]q± denotes the usual q±-number. As has been extensively discussed for free fieldrealizations during the past two years, the screening currents inside [[ · ]]Γ satisfy the definingrelations of generators of a quantum group Uq±(n±) (see e.g.

[3] and references therein).It is worth noting that in the discussion of the general class of resolutions in this letter oneis automatically led to the “rescaled quantum group” generators of Lusztig [16].An interesting application of these resolutions is to extend the computation of theBRST cohomology of minimal models coupled to 2D quantum gravity [17] to generalizedDotsenko-Fateev models using methods discussed in [18]. This is elaborated in more detailin [19].References[1]G. Felder, Nucl.

Phys. B317 (1989) 215; erratum, ibid.

B324 (1989) 548.[2]A.A. Belavin, A.M. Polyakov and A.B.

Zamolodchikov, Nucl. Phys.

B241 (1984) 333.[3]P. Bouwknegt, J. McCarthy, and K. Pilch, Prog.

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[9]Vl.S. Dotsenko and V.A.

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Notes Math. 94 (1979) 441.[11]M.B.

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Bouwknegt, J. McCarthy, and K. Pilch, preprint CERN-TH.6162/91.[19]P. Bouwknegt, J. McCarthy, and K. Pilch, in preparation.13

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