BRST Construction

In this section we describe the BRST construction of Borcherds-Kac-Moody algebras g (also called generalised Kac-Moody algebras) from certain vertex algebras M of central charge 26:

M ^BRST g.

To this end we let M be a weak vertex operator algebra of central charge 26 and consider the tensor productW =M⊗V_gh.of M with the bosonic ghost vertex operator superalgebraV_gh.of central charge−26 from Section 7.1 above. ThenW is a weak vertex operator superalgebra of central chargec= 26−26 = 0.

On W, it is possible to define a BRST current with a corresponding BRST operator Qsuch that Qincreases the ghost number by one and

Q² = 0

(Proposition 7.2.1). This yields the cochain complex (W^•, Q^•) = (W^p, Q^p)p∈Zwherepis the ghost number. We call the corresponding cohomological spaces H_BRST^p (M), p∈ Z.

The space

g:=H_BRST¹ (M)

is of particular interest since it naturally carries the structure of a Lie algebra [LZ93]

(see Corollary 7.2.5) and is in general infinite-dimensional.

We make additional assumptions such that in particular the weak vertex operator algebra M is graded by the dual L⁰ of an even Lorentzian latticeL, in addition to the weight Z-grading. One obtains cochain complexes (W^•(α), Q^•) = (W^p(α), Q^p)p∈Z for each α∈L⁰ and associated cohomological spacesH^p(α) so that

H_BRST^p (M) = ^M

α∈L⁰

H^p(α).

The Lie algebra g = H_BRST¹ (M) is then also graded by L⁰ and a vanishing theorem [FGZ86, Zuc89] (see Theorem 7.2.7) together with the Euler-Poincaré principle allows us to compute the (finite) dimensions of the graded components ofg(see Theorem 7.2.8). In many interesting examples the Lie algebrag=H_BRST¹ (M) turns out to be a Borcherds-Kac-Moody algebra (see Sections 7.3 and 7.4).

Since we are working over the base field C, all Lie algebras in this text are complex, unless otherwise stated.

Vertex Superalgebra Cohomology

In the following we describe the general setting, in which the cohomology of vertex (super)algebras occurs.

Assumption. Let W be a weak graded vertex superalgebra, i.e. a graded vertex su-peralgebra without the requirement that the graded components be finite-dimensional

or that the grading be bounded from below. Assume that in addition to this weight grading there is aZ-grading in the sense of Definition 7.0.1

W =^M

p∈Z

W^p on W, denoted by upper indices.

Furthermore, let j ∈ W be some vector, homogeneous of degree 1 with respect to the upper grading and homogeneous of some weight wt(j) with respect to the weight grading. Then the zeroth mode Q := j₀ ∈ End_C(W) is an operator of degree 1 with respect to the upper grading, i.e. it raises the degree of a homogeneous element with respect to the upper grading by 1, and homogeneous with respect to the weight grading.

Finally, assume that Qsatisfies

Q² = 0 ⇐⇒ im(Q)⊆ker(Q).

For the moment let us viewW as aZ-gradedC-vector space (with the upper grading).

By definition, Qrestricts to

Q^p :=Q|_W^p:W^p →W^p+1 so that

im(Q^p−1)⊆ker(Q^p)⊆W^p

for all p∈Z. This defines acochain complex in the abelian category of C-vector spaces . . .^Q

p−2

−→ W^{p−1 Q}

p−1

−→ W^{p Q}

p

−→W^p+1Q

p+1

−→ . . .

denoted by (W^•, Q^•) = (W^p, Q^p)p∈Z. We define the p-th cohomological space of this complex as the quotient space

H^p := ker(Q^p)/im(Q^p−1) = (W^p∩ker(Q))/(W^p∩im(Q)) forp∈Z, measuring the non-exactness of the above sequence at positionp.

One can even show that the direct sum of these spaces H:= ker(Q)/im(Q) =^M

p∈Z

H^p

naturally carries the structure of a weak graded vertex superalgebra³, which also inherits the upper grading from W (see e.g. Section 5.7.3 of [FBZ04]).

3In particular, the homogeneity ofQwith respect to the weight grading implies thatHis again graded by weights.

BRST Cohomology

We now present the BRST cohomology at central charge 26. The weak graded vertex superalgebraW from above will be the tensor product W =M⊗V_gh. of a weak vertex operator algebra M of central charge 26 and the bosonic ghost vertex operator super-algebra V_gh., the upper grading will be the ghost number and for the vector j ∈W we will take the BRST currentj^BRST.

For the following results we need a series of increasingly strong assumptions on the vertex algebra M in thematter sector, starting with the following one:

Assumption. Let M be a weak vertex operator algebra of central charge 26, i.e. a vertex operator algebra but we do not require the weight grading on M to be bounded from below nor that the graded components be finite-dimensional.

In the ghost sector let V_gh. denote the bosonic ghost vertex operator superalgebra of central chargec=−26 from Section 7.1. We consider the tensor product

W :=M⊗V_gh.,

which naturally admits the structure of a weak vertex operator superalgebra of central charge c = 26−26 = 0 (see Section C.1). We obtain the usual tensor-product weight grading onW =M⊗Vgh.via ω=ωM⊗1_gh.+1_M⊗ωgh. and the tensor-product parity withM⊗_CV_gh.^¯0 being the even subspace andM⊗_CV_gh.^¯1 the odd one. Moreover, there is the ghost number operator idM⊗U. All three gradings are compatible, i.e. each graded subspace with respect to one grading decomposes into a direct sum with respect to the other gradings. This is the case if and only if all the grading operators commute.

We define the BRST current

j^BRST:= (idM⊗c₋₁)(ω_M ⊗1_Gh.+ 1

21_M ⊗ω_gh.) =ω_M ⊗c+ 1

21_M ⊗c−1ω_gh.

and theBRST operator

Q:=j₀^BRST.

We use the following shorthand notation for tensor products: we write operatorsA⊗id or id⊗AasAand vectorsa⊗1or1⊗aasa. Note that the modes of the vertex operators obey (a⊗1)n=an⊗id and (1⊗a)n = id⊗a_n by the left vacuum axiom and because 1 is of even parity. We also writeAB for the operator A⊗B when it is clear on which space the operatorsA and B act.

Using this shorthand notation the BRST current reads j^BRST=c−1(ωM +ωgh./2).

Proposition 7.2.1 ([FGZ86], [Zuc89], Section 4). The operator Q on W =M ⊗V_gh.

fulfils

Q² = 0,

and has weight 0 (i.e. [Q, L₀] = 0), ghost number 1 (i.e. [U, Q] = Q) and odd parity.

More generally

[Q, Ln] = 0 for all n∈Z. Moreover

{Q, b_n+1}=L_n.

We can now consider the BRST complex . . .^Q

p−2

−→ W^p−1Q

p−1

−→ W^{p Q}

p

−→W^{p+1 Q}

p+1

−→ . . . ,

a cochain complex in the category ofC-vector spaces, where the upper index is the ghost number. The corresponding cohomological spaces are

H_BRST^p (M) :=H^p := (W^p∩ker(Q))/(W^p∩im(Q)).

EachW^p and eachH^p is graded by the L0-weights sinceU commutes withL0 and Qis homogeneous with respect to the L₀-grading (Qeven commutes with L₀, too).

From{Q, b_n+1}=L_nwe see that ifx∈ker(Q), then

wt(x)x=L₀x={Q, b_n+1}x=Qb_n+1x

forL₀-homogeneousxand hencex=Q(b_n+1x/wt(x)), i.e.x∈im(Q) if wt(x)6= 0. This shows that the cohomology is only supported in weight 0 and means that if we study the subcomplex

. . .^Q

p−2

−→ W₀^p−1Q

p−1

−→ W₀^{p Q}

p

−→W₀^{p+1 Q}

p+1

−→ . . . ,

then its cohomological spaces are identical to those of the BRST complex (W^•, Q^•), i.e.

H^p ∼= (W₀^p∩ker(Q))/(W₀^p∩im(Q)) =H₀^p and H_n^p={0} forn6= 0 whereH^p =^Ln∈ZH_n^p (L₀-decomposition).

We also define the relative BRST subcomplex on the elements of W annihilated byb1

and of weight 0, i.e. on

C =W₀∩ker(b₁).

The weight and ghost number gradings both restrict to C sinceb₁ is homogeneous with respect to both gradings (i.e. [L0, b1] = 0 and [U, b1] = −b₁). The operatorQ restricts toC because of{Q, b₁}=L₀ and hence we get a subcomplex

. . .^Q

p−2

−→ C^{p−1 Q}

p−1

−→ C^{p Q}

p

−→C^{p+1 Q}

p+1

−→ . . . and the cohomological spaces

H_rel.^p (M) = (W₀^p∩ker(b₁)∩ker(Q))/(W₀^p∩ker(b₁)∩im(Q)).

We note that the inclusion map of C^p intoW^p induces an injective map H_rel.^p (M) → H_BRST^p (M). A priori, the BRST cohomological spaces and the relative cohomological spaces need not be the same. We will see however that under some stronger assump-tions H_BRST¹ (M) ∼= H_rel.¹ (M). More precisely, we will demand that W carry certain representations of the Heisenberg vertex operator algebra.

Lie Algebra Structure

In the following we will concentrate on H_BRST¹ (M). It is shown in [LZ93] that this space naturally carries a Lie algebra structure. Indeed, consider the bilinear bracket [·,·]:W ×W →W on W =M⊗Vgh. defined by

[u, v] :=−(−1)^|u|(b₀u)0v foru, v∈W.⁴

Proposition 7.2.2 ([LZ93], Section 2). The map Qacts as a derivation on [·,·], i.e.

Q[u, v] = [Qu, v]−(−1)^|u|[u, Qv] for u, v∈W.

It follows by a simple calculation that [·,·] restricts to a map [·,·]: ker(Q)×ker(Q)→ ker(Q) and even induces a well-defined bracket [·,·]: H×H→H where

H= ker(Q)/im(Q) =^M

p∈Z

H^p.

Theorem 7.2.3 ([LZ93], Theorem 2.2). The bracket [·,·]: H×H → H defines a Lie superbracket on the BRST cohomological spaceH with the rôles of even and odd elements interchanged, i.e. super-antisymmetry

[u, v] =−(−1)(|u|−1)(|v|−1)[v, u] and the super Jacobi identity

(−1)(|u|−1)(|t|−1)[u,[v, t]] + (−1)(|t|−1)(|v|−1)[t,[u, v]] + (−1)(|v|−1)(|u|−1)[v,[t, u]] = 0 hold for all u, v, t∈H.

For the Lie superalgebra relations to hold, it is necessary to consider the quotient H= ker(Q)/im(Q). Note that in [LZ93] the authors additionally show the existence of an associative, supercommutative “dot product” onH such that in total H carries the structure of aGerstenhaber algebra.

Proposition 7.2.4 ([LZ93], Theorem 2.2). The Lie superbracket on H is of degree -1 with respect to the ghost number, i.e. it restricts to

[·,·]:H^p×H^q →H^p+q−1 for all p, q∈Z.

Then, clearly, H¹ is closed under the bracket. Since H¹ has odd ghost number, it is even with respect to the Lie superalgebra structure onH and we obtain:

Corollary 7.2.5. g:=H¹=H_BRST¹ (M) becomes a Lie algebra with the bracket [u, v] = (b₀u)0v

for u, v∈H¹.

4We define the bracket with an additional minus sign relative to the definition in [LZ93].

Relative BRST Subcomplex

We return to the relative BRST subcomplex . . .^Q

p−2

−→ C^{p−1 Q}

p−1

−→ C^{p Q}

p

−→C^{p+1 Q}

p+1

−→ . . . on

C =W₀∩ker(b₁).

We shall see that the BRST complex and the relative BRST complex are connected by a short exact sequence of cochain complexes (see proof of Theorem 5.1 in [LZ89]). Indeed, consider the map

ψ:W₀^p →C^p−1, w7→(−1)^|w|b₁w.

Then for each p∈Zwe obtain a short exact sequence 0→C^p ,→W₀^p −→^ψ C^p−1 →0.

The surjectivity of ψ follows from {b₁, c−2} = id. The exactness in the middle is clear from the definition ofψ.

The map ψ is even a cochain map ψ: W₀^• → C^•−1 since ψQ = Qψ on W0, which follows from {Q, b₁}=L₀, and so is the inclusion map C^• ,→W₀^•. This means that we obtain a short exact sequence of cochain complexes

0→C^• ,→W₀^• −→^ψ C^•−1 →0.

The zig-zag lemma, which holds in any abelian category, then asserts the existence of the long exact sequence

. . .→H_rel.^p →H_BRST^p →H_rel.^p−1 →H_rel.^p+1 →H_BRST^p+1 →H_rel.^p →H_rel.^p+2→. . . . We can also define a Lie algebra structure on the relative cohomology. Indeed, the identity

(b0u)0v=b1(u−1v)−(b1u)−1v−(−1)^|u|u−1(b1v) holds for all u, v∈W (see Lemma 2.1 in [LZ93]) so that

(b₀u)0v=b₁u−1v∈ker(b₁)

foru, v∈ker(b1). The bracket [·,·] can hence be restricted to ker(b1) and we can define a Lie algebra structure onH_rel.¹ (M) in the same manner as for g=H_BRST¹ (M) above.

Grading

To proceed further we need additional assumptions on the vertex algebraM. Assumption. Let the weak vertex operator algebraM have an additional grading

M =^M

α∈Γ

M(α)

by some (additive) abelian group Γ in the sense of Definition 7.0.1. In particular, this grading is compatible with the weight grading on M. Then also W = M ⊗V_gh. is naturally graded by Γ and this grading is compatible with the weight and ghost gradings onW, i.e. the corresponding grading operators commute. Let us also assume thatQdoes not change the Γ-degree of a Γ-homogeneous element, i.e. that the Γ-grading operator and Qcommute.⁵

The BRST complex is now also graded by Γ, in addition to L0, and we get cochain complexes

. . .^Q

p−2

−→ W_n^p−1(α)^Q−→^p−1 W_n^p(α)−→^Qp W_n^p+1(α)^Q−→^p+1. . .

for all n∈Zand α∈Γ with corresponding cohomological spaces H_n^p(α) so that

H = ^M

n,p∈Z,α∈Γ

H_n^p(α) = ^M

p∈Z,α∈Γ

H₀^p(α),

noting that as beforeH_n^p(α) ={0}forn6= 0. Also the relative BRST complex is graded by Γ sinceb1 does not change the Γ-grading and we get the subcomplexes

. . .^Q

p−2

−→ C^p−1(α)^Q−→^p−1 C^p(α)−→^Qp C^p+1(α)^Q−→^p+1. . .

for α ∈ Γ with corresponding cohomological spaces H_rel.^p (α). We also obtain the long exact sequence

. . .→H_rel.^p (α)→H_BRST^p (α)→H_rel.^p−1(α)→H_rel.^p+1(α)→H_BRST^p+1 (α)→. . . forα∈Γ.

It is part of the above assumption (see Definition 7.0.1) that the Γ-grading on the vertex algebra M is such that for v ∈ M(α), the modes v_n, n ∈ Z, are operators of degreeα, i.e. they change the degree of a homogeneous element byα. This implies:

Proposition 7.2.6. The Lie algebra g =H_BRST¹ (M) =^Lα∈ΓH₀¹(α) is a Γ-graded Lie algebra, i.e. [H_BRST¹ (α), H_BRST¹ (β)]⊆H_BRST¹ (α+β) for α, β∈Γ.

Vanishing Theorem

We now make an even stronger assumption and demand that the weak vertex operator algebraMcarry certain representations of the Heisenberg vertex operator algebra related to the additional Γ-grading introduced above.

Assumption.

• LetV_L be a lattice vertex algebra associated with some even Lorentzian latticeL of rank k ≥ 2 and signature (k−1,1). VL is a weak vertex operator algebra of central charge k whose isomorphism classes of irreducible modules are naturally indexed byL⁰/L.⁶

5It follows from the definition ofj^BRST and Qthat this is the case for example when the Γ-grading operator commutes with all modesL^M_n ofωM.

6Dong’s classification result (see Theorem 5.2.3) even holds for even lattices that are not positive-definite as long as the quadratic form is non-degenerate (e.g. Lorentzian).

• LetU be some vertex operator algebra of central charge 26−ksatisfying Assump-tion SN (group-like fusion) and U₁ = {0}.⁷ Assume furthermore that the fusion groupF_U of U is isomorphic as finite quadratic space toL⁰/L= (L⁰/L,−Q_L), say via the mapχ:L⁰/L→FU.

• Assume thatU⊗V_L is isomorphic to a full weak vertex operator subalgebra ofM such thatM decomposes as a U⊗VL-module according to

M ∼= ^M

γ+L∈L⁰/L

U(χ(γ+L))⊗V_γ+L (7.1) where Vγ+L,γ +L ∈L⁰/L, are the irreducibleVL-modules and χ(γ+L) indexes the irreducibleU-modules. Note that sinceL⁰/L=L⁰/Las groups,χ also accepts elements ofL⁰/Las input.

• Let the isomorphism in (7.1) not only be an isomorphism of U⊗V_L-modules but even of weak vertex operator algebras, where on the right-hand side we consider the vertex algebra structure obtained as abelian intertwining subalgebra of the tensor-product abelian intertwining algebra⁸



 M

γ+L∈L⁰/L

U(χ(γ+L))



⊗



 M

γ+L∈L⁰/L

V_γ+L



, analogously to Proposition 3.1.7.

We will refer to the above decomposition (7.1) as the L⁰/L-decomposition of M.

Recall that byM_ˆh(1,0) =M_ˆh(1) =V_ˆh(1,0) we denote the Heisenberg vertex operator algebra (of level 1) associated with theC-vector spacehequipped with a non-degenerate symmetric bilinear form h·,·i (viewed as abelian Lie algebra with a non-degenerate, symmetric, invariant bilinear form). It has central chargec= dim_C(h) and its irreducible modules are given up to isomorphism byMˆh(1, α) for eachα∈hwith conformal weight hα, αi/2 (see e.g. [LL04], Section 6.3). By construction, given an even lattice L, the associated lattice vertex algebraV_L=M_ˆh(1,0)⊗Cε[L] can be decomposed into a direct sum of modules for the full vertex operator subalgebra Mˆh(1,0)⊗Ce₀ ∼=Mˆh(1,0) as

VL=^M

α∈L

Mˆh(1,0)⊗Ce_α ∼= ^M

α∈L

Mˆh(1, α)

where h= L⊗_ZC (andh·,·i is extended C-linearly from L to h) and similarly for the irreducible modulesVγ+L,γ+L∈L⁰/L.

7The assumption thatU1={0}leads to a very simple form of the Cartan subalgebra ofg.

8Theorem 5.2.6, which asserts the existence of an abelian intertwining algebra onL

γ+L∈L⁰/LVγ+L, also holds for even, non-degenerate (rather than only positive-definite) lattices. The abelian intertwining algebra structure onL

γ+L∈L⁰/LU(χ(γ+L)) is due to Theorem 3.2.3, the main result of Chapter 3.

We return to the vertex algebra M and note that by the above the direct sum of modules^Lγ+L∈L⁰/LV_γ+L is naturally graded by Γ :=L⁰, namely

M

γ+L∈L⁰/L

Vγ+L= ^M

γ+L∈L⁰/L

M

α∈γ+L

M_ˆh(1, α) = ^M

α∈L⁰

M_ˆh(1, α) (asM_ˆh(1,0)-module) withh=L⊗_ZC. This induces a Γ =L⁰-grading onM:

M = ^M

α∈L⁰

M(α) with M(α) =U(χ(α+L))⊗Mˆh(1, α). (7.2) TheL⁰-grading onM is a grading in the sense of Definition 7.0.1. This follows from the last two items of the above assumption. In particular, L₀ leaves the L⁰-grading of M invariant.

Recall that the L⁰-grading on M naturally induces anL⁰-grading on W =M ⊗V_gh.. The ghost number operator,L₀ andQonW all leave theL⁰-grading ofW invariant, i.e.

they commute with theL⁰-grading operator. In total, the grading assumption on p. 191 is satisfied.

Consequently, we get cochain complexes (W^•(α), Q^•) = (W^p(α), Q^p)p∈Z for each α∈ L⁰ and associated cohomological spacesH^p(α) so that

H_BRST^p (M) = ^M

α∈L⁰

H^p(α)

for p ∈ Z and the cohomological spaces of the relative subcomplex (C^•(α), Q^•) = (C^p(α), Q^p)p∈Z,α∈L⁰,

H_rel.^p (M) = ^M

α∈L⁰

H_rel.^p (α) forp∈Z.

The following vanishing theorem is the central result of this section:

Theorem 7.2.7 (Vanishing Theorem). Assume thatα6= 0. Then H_rel.^p (α) ={0}

for p6= 1.

Sketch of Proof. This is essentially Theorem 4.9 in [Zuc89]. It generalises Theorem 1.12 in [FGZ86]. The proof requires the identification of the relative BRST cohomology groups with the relative semi-infinite cohomology groups for the Virasoro algebra as introduced by Feigin [Fei84]. Proposition 3.3.5 in [Car12] gives a mathematical precise formulation in the casek= 2. In essence, the vanishing theorem is a statement about the representation theory of the Virasoro algebra and the Heisenberg vertex operator algebra.

Relevant for computingH_rel.^p (α) =H_rel.^p (M(α)) is thatM(α) =U(χ(α+L))⊗M_ˆh(1, α) is the tensor product of a Virasoro representation of central charge 26−kand a Heisenberg representation attached to some α∈R^k−1,1.

Knowing thatH_rel.^p (α) ={0}forp6= 1 andα6= 0 lets collapse the long exact sequence . . .→H_rel.^p (α)→H_BRST^p (α)→H_rel.^p−1(α)→H_rel.^p+1(α)→H_BRST^p+1 (α)→. . . .

forα6= 0 and we get

H_BRST¹ (α)∼=H_rel.¹ (α)∼=H_BRST² (α) forα6= 0 and

H_BRST^p (α) ={0} forp6= 1,2 and α6= 0.

In particular,

H_BRST¹ (α) =W₀¹(α)∩ker(Q)/W₀¹(α)∩im(Q)∼=

H_rel.¹ (α) =W₀¹(α)∩ker(b1)∩ker(Q)/W₀¹(α)∩im(b1)∩ker(Q)

for α 6= 0. The vector-space isomorphism is exactly the map H_rel.¹ (α) → H_BRST¹ (α) induced from the inclusion mapC^p(α),→W^p(α) described above.

Euler-Poincaré Principle

In this step we determine the dimension of H_rel.¹ (α) forα 6= 0 using the Euler-Poincaré principle.

Note that the weight grading of each component of the tensor product W(α) = U(χ(α+L))⊗M_ˆh(1, α)⊗V_gh. is bounded from below so that

dimC(Wn(α))<∞

for all α ∈ L⁰ and n ∈ Z. Then in particular the spaces Bn(α), C(α), Hrel.(α) and H_BRST(α) are finite-dimensional.

The Euler-Poincaré characteristic of the relative BRST complex (C^•(α), Q^•) is given by

χ(C^•(α)) :=^X

p∈Z

(−1)^pdimC(C^p(α)).

By theEuler-Poincaré principle, this is the same as the Euler-Poincaré characteristic of the cohomology H_rel.^• (α) = (H_rel.^p (α))p∈Z of that complex, i.e.

χ(C^•(α)) =χ(H_rel.^• (α)) :=^X

p∈Z

(−1)^pdimC(H_rel.^p (α))

for all α∈L⁰. Finally, using the above vanishing theorem forH_rel.^p (α) we get

−dim_C(H_rel.¹ (α)) = ^X

p∈Z

(−1)^pdim_C(C^p(α)) forα6= 0.

Now consider the supercharacter ofW(α)∩ker(b1) =:B(α), a subspace of the vertex superalgebraW of central charge 0,

schB(α)(q) =^X

n∈Z

sdim(B_n(α))qⁿ= ^X

n,p∈Z

(−1)^pdim_C(B^p_n(α))qⁿ

where sdim is the superdimension, i.e. the dimension of the even part minus the di-mension of the odd part. Recall that for V_gh. and hence for W = M ⊗V_gh. the even and odd part have exactly even and odd ghost number, respectively. Hence sdim(Bn(α)) = ^Pp∈Z(−1)^pdimC(B_n^p(α)). For the constant coefficient of schB(α)(q) we obtain

hschB(α)(q)ⁱ(0) = ^X

p∈Z

(−1)^pdim_C(B₀^p(α)) =^X

p∈Z

(−1)^pdim_C(C^p(α)) =−dim_C(H_rel.¹ (α)) forα6= 0 sinceC(α) =B0(α).

To compute the supercharacter of B(α) =M(α)⊗(V_gh.∩ker(b₁)) consider schB(α)(q) = schM(α)(q)·schVgh.∩ker(b₁)(q) = chM(α)(q)·schVgh.∩ker(b₁)(q).

The character of the kernel of b₁ inV_gh. is computed in Section 7.1 to be schVgh.∩ker(b₁)(q) =−η(q)².

The characters of the Heisenberg vertex operator algebra modules are known to be chMˆh(1,α)(q) = q^hα,αi/2

η(q)^dimC(ˆh) = q^hα,αi/2 η(q)^k

and we obtain for the character ofM(α) =U(χ(α+L))⊗M_ˆh(1, α):

chM(α)(q) = chU(χ(α+L))(q)·chMˆh(1,α)(q) = chU(χ(α+L))(q)q^hα,αi/2 η(q)^k . Finally, we compute the dimension ofg(α) =H_BRST¹ (α)∼=H_rel.¹ (α) and obtain

dimC(g(α)) =−^hschB(α)(q)ⁱ(0) =^hchM(α)(q)η(q)²ⁱ(0)

=

"

chU(χ(α+L))(q)q^hα,αi/2 η(q)^k−2

# (0)

=chU(χ(α+L))(q) 1 η(q)^k−2

(−hα, αi/2) forα6= 0. We have just proved:

Theorem 7.2.8 (Dimension Formula). The dimension of g(α) is dim_C(g(α)) =^hchM(α)(q)η(q)²ⁱ(0)

=^hchU(χ(α+L))(q)/η(q)^k−2i(−hα, αi/2) for α∈L⁰\ {0}.

Remark 7.2.9. In the special case of k = 2 the above formula simplifies and g(α) is isomorphic toU(χ(α+L))1−hα,αi/2, the subspace ofU(χ(α+L)) of weight 1− hα, αi/2.

Zero-Component

The vanishing theorem does not make a statement about the cohomology forα= 0∈L⁰. We compute the componentg(0) directly usingU₁ ={0}and that U is of CFT-type:

Proposition 7.2.10. The component g(0) is spanned by vectors of the form 1U⊗h(−1)1⊗c

for h∈h=L⊗_ZCand hence

g(0)∼=h∼=C^k or dimC(g(0)) = rk(L) =k.

Proof. The goal is to compute

g(0) =H_BRST¹ (0) =W₀¹(0)∩ker(Q)/W₀¹(0)∩im(Q),

where the elements ofW₀¹(0) have weight 0, ghost number 1 and degree 0∈L⁰. Recall thatW(0) =M(0)⊗V_gh.=U⊗M_ˆh(1,0)⊗V_gh.. The subspaceW₀¹(0) is spanned by the vectors

1_M ⊗c−21_gh. and 1_U⊗h(−1)1⊗c

forh∈h=L⊗_ZC.⁹ A simple calculation shows that the latter are in ker(Q) while the former is not. Moreover, W₀¹(0)∩im(Q) ={0}.

It is easy to see thatW₀¹(0)∩ker(Q) is in the kernel of b₁ so that

H_BRST¹ (0) =H_rel.¹ (0) =W₀¹(0)∩ker(b₁)∩ker(Q)/W₀¹(0)∩ker(b₁)∩im(Q). Together with H_BRST¹ (α) ∼=H_rel.¹ (α) for α 6= 0, which followed from the vanishing the-orem, this shows that

g=H_BRST¹ (M)∼=H_rel.¹ (M).

Recall that we defined a Lie algebra structure ong=H_BRST¹ (M) and on H_rel.¹ (M). The vector-space isomorphism betweenH_BRST¹ (M) and H_rel.¹ (M), which is induced from the inclusion map C¹ ,→W¹, is clearly also an isomorphism of Lie algebras.

Im Dokument A Cyclic Orbifold Theory for Holomorphic Vertex Operator Algebras and Applications (Seite 186-197)