aa r X i v : . [ h e p - t h ] N ov BOSONIC GHOSTS AT c = AS A LOGARITHMIC CFT
DAVID RIDOUT AND SIMON WOODNovember 24, 2014A
BSTRACT . Motivated by Wakimoto free field realisations, the bosonic ghost system of central charge c = EYWORDS : Logarithmic conformal field theory, vertex algebras, modular transformations, fusion.
1. I
NTRODUCTION
Ghost systems have a long history in conformal field theory, particularly with regard to Faddeev-Popov gauge fix-ing of superstrings, see [1] for example, but also as ingredients for constructing more complicated theories, Wakimotorealisations of Wess-Zumino-Witten models [2] and quantum hamiltonian reductions [3] being notable examples. Theintrinsic appeal of ghost systems is that they are examples of free field theories. On the other hand, these theories arestrongly non-unitary and, in the case of bosonic ghost systems, the spectrum of conformal weights is well known tobe unbounded below.The fermionic ghost system with central charge c = − c = − c = c = and modular invariant partition functions.Because this formalism is tailored to studying the modular properties of the theory’s characters, we could have cho-sen any bosonic system in which the ghost fields have integer conformal weight (to facilitate the T-transformationof characters). The choice c = c .Of course, the c = − Z -orbifold theory that coincides with the fractional level Wess-Zumino-Witten model b sl ( R ) − / . Being non-free, one has to work fairly hard to establish the modular transformation properties [10]and fusion rules [11] for this affine algebra. Here, it is very important to realise that category O is not sufficient —the physically relevant module category is far larger. It is, unfortunately, not clear how to determine this physicallyrelevant category. We can only insist that it be closed under fusion and conjugation, as well as have the property thatone can construct a modular invariant partition function from the characters. Granting these results for b sl ( R ) − / ,the fusion rules may then be lifted to the c = − We will denote the fusion product of vertex operator algebra modules by × , reserving the symbol ⊗ , and the term “tensor product”, for the tensorproduct of complex vector spaces. While fusion is expected to have the properties of an abstract tensor product, this has only been proven undercertain assumptions on the algebra and the category of modules, see [9] for example, that do not seem to be met in this paper. extensions [12] and we summarise them, for ease of comparison, in Appendix A. The analysis reported here for c = c = For each Borel subalgebra, we choose a certain parabolic subalgebra and find a continuum ofparabolic highest weight modules parametrised by R / Z . We refer to [17] for definitions and basic properties ofparabolic subalgebras and modules (in the context of semisimple Lie algebras).Characters are then computed in Section 4, where we quickly detail the contradictions inherent in regarding themas meromorphic functions (on the product of the Riemann sphere and an open disc), see also [18]. We would like tostrongly emphasise that writing bosonic ghost characters in terms of modular forms can therefore lead to incorrectconclusions. Proceeding instead in a distributional setting [10], we determine expressions for the characters of theparabolic modules and construct resolutions to deduce formulae for the highest weight characters. The core of theanalysis now follows in Section 5. There, we apply the modular S-transformation to the parabolic characters inTheorem 2 and show that the result defines a unitary integral operator on the space spanned by the characters thatis symmetric with respect to the canonical basis and squares to the conjugation map. This turns out to involve asurprisingly non-trivial automorphy factor which is dealt with by a judicious extension of the characters and theirtransformation properties. S-transformation formulae for the highest weight characters follow.We then apply the Verlinde formula in Section 6, showing explicitly that the resulting “Verlinde product rules”,for decomposing products of characters, have non-negative integer coefficients (Theorem 6). Conjecturing that theserules coincide with the image of the fusion rules in the Grothendieck ring, we effortlessly arrive at almost all ofthe fusion rules involving simple modules. The remaining simple-simple fusion rules are then calculated explicitly inSection 7. This is a technical matter utilising the Nahm-Gaberdiel-Kausch algorithm [19,20]. The actual computationsare relatively simple, but there is a conceptual problem to overcome in that the modules that we would like to fuse allhave trivial “special subspaces”. Nevertheless, a careful analysis shows (Theorem 8) that the resulting fusion productsare staggered modules in the sense of [21], proving that the c = c = − HOST A LGEBRAS
The bosonic ghost system is generated by two (mutually bosonic) fields b (cid:0) z (cid:1) and g (cid:0) z (cid:1) , subject to the followingoperator product expansions: b ( z ) b ( w ) ∼ , b ( z ) g ( w ) ∼ − z − w , g ( z ) g ( w ) ∼ . (2.1)From these fields, one constructs a unique b gl ( ) current J (cid:0) z (cid:1) as follows: J ( z ) = : b ( z ) g ( z ) : . (2.2) This technology was originally developed for rational conformal field theories, but also applies, with almost no changes, to non-rational theoriesto which the standard module formalism applies. This formalism is refined, and the connection to simple currents outlined, in [13]. The importance of the parabolic modules appears to have been largely overlooked in previous studies [7, 15], leading to incorrect conclusionsconcerning modularity and the inapplicability of the Verlinde formula. Here, we are guided by [10, 16] where the analogous modules are found tobe crucial for a complete understanding of the b sl ( R ) − / model. OSONIC GHOSTS AT c = This current is lorentzian and it gives b (cid:0) z (cid:1) a charge of +
1, whereas g (cid:0) z (cid:1) is given charge −
1. In contrast, theconformal structure is not unique. The bosonic ghost system admits a one-parameter family of energy-momentumtensors T a ( z ) parametrised by a : T a ( z ) = (cid:0) a − (cid:1) : b ( z ) ¶g ( z ) : + (cid:0) a + (cid:1) : ¶b ( z ) g ( z ) : . (2.3)Although we may take a ∈ R (or C ), we will restrict a to be in Z for technical reasons to be discussed shortly. Weremark that J ( z ) is only a conformal primary when a =
0. The central charge and conformal weights assigned to b (cid:0) z (cid:1) and g (cid:0) z (cid:1) are then c a = a − ∈ Z ; h a b = − a ∈ Z , h a g = + a ∈ Z . (2.4)Note that h a b + h a g =
1, in accordance with (2.1).Expanding the ghost fields (in the untwisted sector) as b (cid:0) z (cid:1) = ∑ n ∈ Z − h a b b n z − n − h a b , g (cid:0) z (cid:1) = ∑ n ∈ Z − h a g g n z − n − h a g , (2.5)the commutation relations corresponding to (2.1) are (cid:2) b m , b n (cid:3) = , (cid:2) b m , g n (cid:3) = − d m + n = , (cid:2) g m , g n (cid:3) = . (2.6)We will denote by G the infinite-dimensional complex Lie algebra spanned by the b n , g n and the central element , equipped with the Lie brackets (2.6). We also identify with the unit of the universal enveloping algebra of G and assume that it acts as the identity operator on any G -module. The subspace C will be referred to as the Cartansubalgebra of G .The Lie algebra G admits several useful automorphisms that preserve this Cartan subalgebra. In particular, wemention the conjugation automorphism c and the spectral flow automorphisms s ℓ which act on the generators b n and g n as follows: c (cid:0) b n (cid:1) = g n , c (cid:0) g n (cid:1) = − b n ; s ℓ (cid:0) b n (cid:1) = b n − ℓ , s ℓ (cid:0) g n (cid:1) = g n + ℓ . (2.7)Note that c s ℓ = s − ℓ c . We remark that conjugation does not have order 2 as one might expect, but instead has order4. We also note that c preserves the mode index so that, for example, if b n has n ∈ Z − h a b , then g n = c (cid:0) b n (cid:1) has n ∈ Z − h a b = Z + h a g , rather than n ∈ Z − h a g . It follows that unless h a b and h a g belong to Z , conjugation will notpreserve the untwisted sector. This is why we are explicitly assuming that a ∈ Z . Similarly, the spectral flowautomorphism s ℓ will only preserve the untwisted sector if ℓ ∈ Z .These automorphisms may be used to construct families of G -modules by twisting the action on any given module.So, let M be a G -module and define new G -modules c (cid:0) M (cid:1) and s ℓ (cid:0) M (cid:1) as follows. First, we define c (cid:0) M (cid:1) and s ℓ (cid:0) M (cid:1) as vector spaces isomorphic to M : c (cid:0) M (cid:1) = n c (cid:0) v (cid:1) : v ∈ M o , s ℓ (cid:0) M (cid:1) = n s ℓ (cid:0) v (cid:1) : v ∈ M o . (2.8)Here, the symbols c (cid:0) v (cid:1) and s ℓ (cid:0) v (cid:1) are formal so that the isomorphisms are given by v c (cid:0) v (cid:1) and v s ℓ (cid:0) v (cid:1) ,respectively. These vector spaces are then equipped with the following G -action: a · c (cid:0) v (cid:1) = c (cid:0) c − (cid:0) a (cid:1) v (cid:1) , a · s ℓ (cid:0) v (cid:1) = s ℓ (cid:0) s − ℓ (cid:0) a (cid:1) v (cid:1) , for all a ∈ G . (2.9)We will refer to the module c (cid:0) M (cid:1) as the conjugate of M and to the s ℓ (cid:0) M (cid:1) as the spectral flow images of M .Of course, one can always identify the vector space underlying M with that of c (cid:0) M (cid:1) and the s ℓ (cid:0) M (cid:1) , instead ofmaking the isomorphism explicit. Then, one needs to distinguish the G -action, for example by making the repre-sentation explicit. In particular, if r denotes the representation of G on the vector space M , then the conjugate and As the action of c may be identified with that of − , conjugation still defines an order 2 permutation on modules, as one would expect. Equivalently, the conjugate of an untwisted module will be twisted in general. There is no inconsistency here, but it does simplify matters if weassume that h a b , h a g ∈ Z . More importantly, we know of no physical applications of ghost systems with conformal weights not in Z . D RIDOUT AND S WOOD spectrally flowed representations are defined by r c (cid:0) a (cid:1) = r (cid:0) c − (cid:0) a (cid:1)(cid:1) , r ℓ (cid:0) a (cid:1) = r (cid:0) s − ℓ (cid:0) a (cid:1)(cid:1) . (2.10)As we prefer the language of modules over representations, we will keep the vector space isomorphisms explicit. Itis not hard to translate between the two languages if desired.The induced action of the conjugation and spectral flow automorphisms on the current and Virasoro modes is mosteasily computed by lifting the automorphisms to the level of fields. The results are: c (cid:0) J n (cid:1) = − J n − a d n = , c (cid:0) L an (cid:1) = L an + anJ n , s ℓ (cid:0) J n (cid:1) = J n + ℓ d n = , s ℓ (cid:0) L an (cid:1) = L an − ℓ J n − ℓ (cid:0) a + ℓ (cid:1) d n = . (2.11)We note that the charge and conformal weight of a weight vector v ∈ M change as follows upon conjugating orapplying spectral flow: J v = jv , L a v = hv ⇒ J c (cid:0) v (cid:1) = − ( j + a ) c (cid:0) v (cid:1) , L a c (cid:0) v (cid:1) = h c (cid:0) v (cid:1) , J s ℓ (cid:0) v (cid:1) = ( j − ℓ ) s ℓ (cid:0) v (cid:1) , L a s ℓ (cid:0) v (cid:1) = (cid:2) h + ℓ j + ℓ (cid:0) a − ℓ (cid:1)(cid:3) s ℓ (cid:0) v (cid:1) . (2.12)The fact that the s ℓ do not preserve L a , hence the conformal weights, is the origin of the name “spectral flow”.We close this section by remarking that the elements of the one-parameter family T a ( z ) may be viewed as defor-mations of the element with a =
0. Indeed, T a ( z ) = T ( z ) + a ¶ J ( z ) . (2.13)It follows that each bosonic ghost system shares the same representation content, independent of a . We may thereforechoose a convenient representative to study in detail. As mentioned above, the theory with a = c = − b sl ( R ) − / , essentially because h b = h g = leads to twisted modules whose characters are not eigenvectorsfor the modular T-transformation. In what follows, we will instead specialise to a = − and set h b = h g = c =
2, dropping the label a from all quantities for simplicity. This choice facilitates a direct investigation of thespectrum, modular properties of the characters, and fusion rules. It also has the advantage of being of significantmathematical and physical interest through the Wakimoto free field realisations of affine Kac-Moody algebras [2].3. R EPRESENTATION T HEORY
As mentioned above, we will now choose the conformal structure, once and for all, so that a = − , h b = h g = c =
2. The ghost algebra G is not a (generalised) Kac-Moody algebra, but it does admit triangular decompositionswith Cartan subalgebra C . In particular, we introduce a family of triangular decompositions, parametrised by ℓ ∈ Z ,wherein the positive subalgebra is spanned by the b n − ℓ and the g n + ℓ + , with n >
0. These decompositions are clearlymapped into one another by the spectral flow automorphisms (and conjugation). We will refer to the triangulardecomposition with ℓ = normal decomposition.Given a triangular decomposition, we may construct Verma modules. As the Cartan subalgebra is spanned by thecentral element , which we assume always acts as the identity operator, there is a unique Verma module for eachdecomposition. We shall denote this Verma module by V in the case that the decomposition is the normal one. Thismodule is generated by a state W which is annihilated by the b n and g n + , for n >
0, hence it is annihilated by J , L and L − . We may therefore take W to be the (translation-invariant) vacuum of the bosonic ghost theory. Thevacuum Verma module V is simple because any vector annihilated by the positive modes is also annihilated by L , sohas conformal weight 0, and the vectors g n W ∈ V of conformal weight 0 are easily checked to be cyclic. As is wellknown, this vacuum module admits the structure of a vertex operator algebra.The Verma modules obtained from the other triangular decompositions are then precisely the spectral flow imagesof the vacuum Verma module V . We remark that the vector w = c (cid:0) W (cid:1) ∈ c (cid:0) V (cid:1) has charge 1 and conformal weight In fact, the evidence at hand not only suggests that the modules over the ghost vertex operator algebras form equivalent abelian categories, butthat the equivalence extends to tensor categories. In other words, the fusion rules of the ghost theories are also identical. We hope to make thismore precise in the future.
OSONIC GHOSTS AT c =
0, by (2.12), so the module it generates is not isomorphic to V . The ghost vacuum module is therefore not self-conjugate. Indeed, it is easy to check that the s ℓ (cid:0) V (cid:1) are all mutually non-isomorphic and that c (cid:0) V (cid:1) ∼ = s − (cid:0) V (cid:1) . As s is an automorphism of G , all the Verma modules s ℓ (cid:0) V (cid:1) are simple.In categorical terms, the vacuum module V is the only simple object in the analogue of category O . There are alsothe twisted versions O ℓ whose unique simple object is s ℓ (cid:0) V (cid:1) . As s is an automorphism of G , spectral flow definesexact functors between the O ℓ . Each of these categories is semisimple because V admits no non-split self-extensionon which the Cartan element acts as the identity operator. For in such an extension, 0 → V → W → V →
0, any W ′ ∈ W projecting onto the highest weight vector W of the quotient V would be cyclic, so there would exist U in theuniversal enveloping algebra of G for which U W ′ = W , the highest weight vector of the submodule V . As U W = V is a Verma module, U is a sum of terms of the form U ′ b n or U ′′ g n + , where n >
0. But, b n W ′ = g n + W ′ = n , because V has no non-zero vectors of ( J , L ) -eigenvalues ( , − n ) and ( − , − n − ) , respectively.However, the representation theory of the ghost vertex operator algebra is not limited to Verma modules and twistedversions of category O . One can also consider parabolic Verma modules; indeed, we shall see in Section 5 that wemust. Recall that a subalgebra of a Lie algebra with triangular decomposition g = g − ⊕ g ⊕ g + is said to be parabolicif it contains the Borel subalgebra g ⊕ g + . Given the normal triangular decomposition, say, there turn out to beinfinitely many parabolic subalgebras because we may extend the normal Borel subalgebra by any combination of thenegative b n modes and the non-positive g n modes. Parabolic subalgebras containing the other Borel subalgebras maythen be obtained through spectral flow (and conjugation).This plethora of parabolic subalgebras turns out to be surplus to our needs. For the analysis to follow, we will onlyrequire one of the parabolic subalgebras extending the normal Borel subalgebra, as well as its cousins obtained byapplying spectral flow. The reason for ignoring the remaining parabolic subalgebras, and their associated parabolicVerma modules, will not be detailed here. Suffice to say, the point is that we want these structures to be compatiblewith the entire mode algebra of the ghost vertex operator algebra, not just G . The normal parabolic subalgebra thatwe require corresponds to extending the normal Borel subalgebra by b . It is therefore spanned by the b n and g n , with n >
0, and ; we will denote it by p . For this choice, the parabolic Verma modules (also known as generalised Vermamodules) are obtained by inducing any module over the subalgebra G spanned by b , g and , this module beinglifted to a module over p by letting the modes with positive index act as zero.We therefore study the G -modules that are the direct sum of their eigenspaces under J = g b , these being theobvious candidates for weight modules over G : Proposition 1. x (1) The only highest weight G -module is V = C [ g ] W , generated by a highest weight vector W satisfying b W = ,hence J W = . This module is simple. (2) The only lowest weight G -module is c (cid:0) V (cid:1) = C [ b ] w , generated by a lowest weight vector w satisfying g w = ,hence J w = w . This module is simple. (3) There is, in addition, a continuous family of G -modules parametrised by [ l ] ∈ C / Z . They have a basis consistingof vectors u j , with j ∈ [ l ] = Z + l , satisfying J u j = ju j . (a) When [ l ] = [ ] , these modules are simple and are denoted by W l . We may realise W l on C [ b ] u l ⊕ C [ g ] g u l , noting that W l = W m when l − m ∈ Z . (b) When [ l ] = [ ] , there are two inequivalent indecomposable modules, W + and W − , whose isomorphismclasses are determined by the following short exact sequences ( Ext (cid:0) c (cid:0) V (cid:1) , V (cid:1) = Ext (cid:0) V , c (cid:0) V (cid:1)(cid:1) = C ): −→ V −→ W + −→ c (cid:0) V (cid:1) −→ , −→ c (cid:0) V (cid:1) −→ W − −→ V −→ . (3.1) Both may be realised on the space C [ g ] u ⊕ C [ b ] u , where b u = and g u = a + u , for W + , and b u = a − u and g u = , for W − . We may normalise the basis vectors so that a + = a − = . This classification is well known because G is the Weyl algebra A , also known as the canonical commutation relationsalgebra. Indeed, Block classified all simple modules over A in [22]. However, the proof for simple weight modulesis quite easy, see [23, Sec. 3.4] for a similar proof for sl (cid:0) (cid:1) , so we present a sketch for completeness. D RIDOUT AND S WOOD
Proof (sketch).
As the Cartan subalgebra is spanned by , which always acts as the identity, there is a unique Vermamodule and it is easy to verify that it is simple. This takes care of (1). (2) now follows by applying conjugation.For (3), we need to know that a simple weight G -module has one-dimensional weight spaces. This follows byconsidering each weight space as a module over C [ J ] and showing that these modules are simple. The argument isby contradiction and uses only the Poincar´e-Birkhoff-Witt theorem: If a weight space has a proper non-zero C [ J ] -submodule, then it generates a proper non-zero G -module. (3a) now follows because we may normalise the weightvectors u j ∈ W l so that g u j = u j − and then, J u j = ju j implies that b u j = ju j + . The existence of the W l followsfrom their explicit construction. (3b) likewise follows, with the extension groups being essentially parametrised bythe coefficients a ± .We remark that because we do not seem to need complex weights in physical theories, we will throughout restrict theparameter [ l ] appearing in item (3) above (and elsewhere) to lie in R / Z .Inducing V and c (cid:0) V (cid:1) recovers the usual Verma modules V and c (cid:0) V (cid:1) , respectively, over G . However, inducing the W l and W ± results in new parabolic Verma modules that we shall denote by W l and W ± , respectively. These mayalso be regarded as examples of relaxed highest weight modules in the spirit of [24]. It follows from Proposition 1that these new modules are simple for l / ∈ Z and are otherwise characterised by the exact sequences0 −→ V −→ W + −→ c (cid:0) V (cid:1) −→ , −→ c (cid:0) V (cid:1) −→ W − −→ V −→ . (3.2)We also have W l = W m whenever l − m ∈ Z . Twisting by spectral flow now realises the parabolic Verma mod-ules, s ℓ (cid:0) W l (cid:1) , s ℓ (cid:0) W + (cid:1) and s ℓ (cid:0) W − (cid:1) , that correspond to other parabolic subalgebras of G . These parabolic Vermamodules are all mutually non-isomorphic.The category O is therefore a full subcategory of the category P of parabolic highest weight modules corre-sponding to the parabolic subalgebra p . An analogous statement holds for the categories obtained by twisting by s ℓ .Note that P has an uncountable family W l , [ l ] ∈ C / Z , [ l ] = [ ] , of inequivalent simple objects, as well as V and c (cid:0) V (cid:1) . This category is not semisimple because of (3.2), but the only non-semisimple block corresponds to [ l ] = [ ] .However, we shall see in Section 5 that the physically relevant category must include not only P , but also each ofits spectrally-flowed versions, in order that the ghost characters span a representation of the modular group SL (cid:0) Z (cid:1) .We will also see in Section 7 that closure under fusion leads to extensions between parabolic modules with differentspectral flow indices.To summarise (without categories), and to make contact with the standard module formalism of [8, 13], we haveconstructed a continuous family of simple G -modules s ℓ (cid:0) W l (cid:1) , parametrised by [ l ] ∈ R / Z , [ l ] = [ ] , and ℓ ∈ Z .These parabolic Verma modules are the typical modules. The module conjugate to s ℓ (cid:0) W l (cid:1) is c (cid:0) s ℓ (cid:0) W l (cid:1)(cid:1) = s − ℓ (cid:0) W − l (cid:1) . There are, moreover, two discrete families of indecomposable, but reducible, G -modules, s ℓ (cid:0) W + (cid:1) and s ℓ (cid:0) W − (cid:1) , with simple composition factors s ℓ (cid:0) V (cid:1) and s ℓ (cid:0) c (cid:0) V (cid:1)(cid:1) . These modules are all atypical and are alsorelated by conjugation: c (cid:0) s ℓ (cid:0) W + (cid:1)(cid:1) = s − ℓ (cid:0) W − (cid:1) and c (cid:0) s ℓ (cid:0) V (cid:1)(cid:1) = s − ℓ − (cid:0) V (cid:1) . As the vacuum module V is atypical,we expect that ghost theories will all be logarithmic. The standard modules of the theory are the typicals s ℓ (cid:0) W l (cid:1) and the indecomposable atypicals s ℓ (cid:0) W + (cid:1) and s ℓ (cid:0) W − (cid:1) . As we shall see, there is a uniform character formula forthe standard modules and the corresponding modular S-transformations are straightforward to determine.4. C HARACTERS
Being a Verma module, the character of the vacuum module V is easily found:ch (cid:2) V (cid:3)(cid:0) z ; q (cid:1) = tr V z J q L − c /
24 ! = q − / ∏ ¥ i = ( − zq i ) ( − z − q i − ) = − i z / h ( q ) J (cid:0) z ; q (cid:1) . (4.1)Here, the “ ! = ” indicates that we are (temporarily) ignoring convergence regions by identifying the characters, whichare formal power series, with their meromorphic continuations to z ∈ C ∪ { ¥ } and | q | <
1. The character of theconjugate module c (cid:0) V (cid:1) is similarly determined to bech (cid:2) c (cid:0) V (cid:1)(cid:3)(cid:0) z ; q (cid:1) ! = zq − / ∏ ¥ i = ( − z − q i ) ( − zq i − ) = − i z / h ( q ) J (cid:0) z − ; q (cid:1) = + i z / h ( q ) J (cid:0) z ; q (cid:1) . (4.2) OSONIC GHOSTS AT c = It is not hard to check that these formulae are consistent with the identification c (cid:0) V (cid:1) ∼ = s − (cid:0) V (cid:1) using the propertiesof Jacobi theta functions and the relationsch (cid:2) c (cid:0) M (cid:1)(cid:3)(cid:0) z ; q (cid:1) = z ch (cid:2) M (cid:3)(cid:0) z − ; q (cid:1) , ch (cid:2) s ℓ (cid:0) M (cid:1)(cid:3)(cid:0) z ; q (cid:1) = z − ℓ q − ℓ ( ℓ + ) / ch (cid:2) M (cid:3)(cid:0) zq ℓ ; q (cid:1) , (4.3)valid for any G -module M . However, they do lead to the suspicious identity of meromorphically-continued charactersch (cid:2) V (cid:3) + ch (cid:2) c (cid:0) V (cid:1)(cid:3) ! = W + and W − must vanish identically.This erroneous conclusion is corrected [18] by considering the difference between regarding characters as formalpower series and regarding them as meromorphic functions. The Dedekind eta and Jacobi theta functions convergefor | q | <
1, but the character formula (4.1) has poles whenever z = q i , for some i ∈ Z . Thus, the character as a formalpower series will only converge, upon interpreting z and q as complex numbers, to the given meromorphic function onone of the annuli in which the magnitude of z is bounded between the magnitudes of two consecutive poles. Indeed,the region of convergence of the vacuum character (4.1) is | q | < , < | z | < | q | − . (4.5)In general, the character of s ℓ (cid:0) V (cid:1) is only convergent in the region | q | < , | q | − ℓ < | z | < | q | − ℓ − . (4.6)The regions of convergence of ch (cid:2) V (cid:3) and ch (cid:2) c (cid:0) V (cid:1)(cid:3) = ch (cid:2) s − (cid:0) V (cid:1)(cid:3) are therefore disjoint, so that while (4.4) mayhold at the level of meromorphic functions, it makes no sense at the level of the characters (which are formal powerseries) themselves. We therefore conclude that it is incorrect to treat characters as meromorphic functions in thiscase. Instead, we shall treat these formal power series as distributions over Laurent polynomials in q and z . This issuggested by the character formula for the typical modules W l which obviously diverges everywhere if one tries tointerpret it as a meromorphic function:ch (cid:2) W l (cid:3) = ∑ n ∈ Z z n + l q − / ∏ ¥ i = ( − zq i ) ( − z − q i ) = ∑ n ∈ Z z n + l q − / ∏ ¥ i = ( − q i ) = z l h ( q ) ∑ n ∈ Z z n . (4.7)Here, we remark that the denominators in expressions such as these should be regarded as shorthand notation forthe corresponding (geometric) power series. This formula follows from the fact that a basis for the parabolic Vermamodule W l may be chosen to consist of the parabolic highest weight vectors u j , j ∈ Z + l , being acted upon freelyby the negative modes b n and g n , n <
0. We have also noted that ∑ n ∈ Z z n − zq i = ∑ n ∈ Z ¥ ∑ k = z n + k q ik = ∑ m ∈ Z ¥ ∑ k = z m q ik = ∑ m ∈ Z z m − q i . (4.8)As an identity of formal power series (distributions), (4.7) also holds for the atypical standards W + and W − upon substituting l = Setting z = e p i z now results in the divergent sum in (4.7) being recognised as a singulardistribution supported at z ∈ Z , that is z = ∑ n ∈ Z z n = ∑ n ∈ Z e p i n z = ∑ m ∈ Z d ( z = m ) . (4.9)Equation (4.4) is therefore replaced, in this distributional setting, bych (cid:2) V (cid:3) + ch (cid:2) c (cid:0) V (cid:1)(cid:3) = ch (cid:2) W (cid:3) = ∑ m ∈ Z d ( z = m ) h ( q ) , (4.10) We also mention that it does not seem possible to instead consider characters as meromorphic functions with a given region of convergence. Oneconceptual objection to this is that the modular S-transformation does not respect these convergence regions in any way, so it is not clear thatcharacters with convergence regions may be subjected to modular analysis. We will often drop the label “ ± ” when considering the characters of the atypical standard modules. D RIDOUT AND S WOOD demonstrating that the right-hand side is not 0, but is rather a singular distribution supported at z =
1. We remark that z = V , and its spectral flow images, as distributionsrather than as meromorphic functions. This is achieved by splicing the exact sequences (3.2) with their spectrally-flowed counterparts to obtain resolutions · · · −→ s (cid:0) W + (cid:1) −→ s (cid:0) W + (cid:1) −→ s (cid:0) W + (cid:1) −→ V −→ , · · · −→ s − (cid:0) W − (cid:1) −→ s − (cid:0) W − (cid:1) −→ W − −→ V −→ −→ V −→ s (cid:0) W − (cid:1) −→ s (cid:0) W − (cid:1) −→ s (cid:0) W − (cid:1) −→ · · · , −→ V −→ W + −→ s − (cid:0) W + (cid:1) −→ s − (cid:0) W + (cid:1) −→ · · · . (4.11b)We thereby deduce two character formulae for the vacuum module as a formal power series (distributions):ch (cid:2) V (cid:3) = ¥ ∑ ℓ = ( − ) ℓ − ch (cid:2) s ℓ (cid:0) W (cid:1)(cid:3) , ch (cid:2) V (cid:3) = ¥ ∑ ℓ = ( − ) ℓ ch (cid:2) s − ℓ (cid:0) W (cid:1)(cid:3) . (4.12)The convergence of these expressions is meant in the following sense: For each weight ( j , h ) , only a finite number ofterms in either sum contribute to the multiplicity of z j q h . We shall not dwell on the implication that the difference ofthese two expressions, a bi-infinite alternating sum of the atypical standard characters, vanishes. Suffice to say thatwe regard either of these formulae as deciding on an appropriate topological completion of the span of the standardcharacters. It is straightforward to check that the results which follow will not depend on which formula, hence whichcompletion, we choose. 5. M ODULAR T RANSFORMATIONS
We prepare for computing S-transformations by calculating the character of a general standard module usingEquations (4.3) and (4.7):ch (cid:2) s ℓ (cid:0) W l (cid:1)(cid:3)(cid:0) z ; q (cid:1) = z − ℓ q − ℓ ( ℓ + ) / z l q ℓ l h ( q ) ∑ n ∈ Z z n q n ℓ = z l q ℓ l + ℓ ( ℓ − ) / h ( q ) ∑ n ∈ Z z n q n ℓ . (5.1)Writing q = e p i t and z = e p i z , this simplifies toch (cid:2) s ℓ (cid:0) W l (cid:1)(cid:3)(cid:0) z (cid:12)(cid:12) t (cid:1) = e i p ℓ ( ℓ − ) t h ( t ) ∑ n ∈ Z e p i n l d ( z + ℓ t = n ) . (5.2) Theorem 2.
The standard characters (5.2) have S-transformation ch (cid:2) s ℓ (cid:0) W l (cid:1)(cid:3)(cid:0) z / t (cid:12)(cid:12) − / t (cid:1) = A (cid:0) z (cid:12)(cid:12) t (cid:1) ∑ m ∈ Z Z R / Z S (cid:2) s ℓ (cid:0) W l (cid:1) → s m (cid:0) W m (cid:1)(cid:3) ch (cid:2) s m (cid:0) W m (cid:1)(cid:3)(cid:0) z (cid:12)(cid:12) t (cid:1) d m , (5.3a) where A (cid:0) z (cid:12)(cid:12) t (cid:1) = | t |− i t e − i pz / t e i pz / t e − i pz , S (cid:2) s ℓ (cid:0) W l (cid:1) → s m (cid:0) W m (cid:1)(cid:3) = ( − ) ℓ + m e − p i ( ℓ m + m l ) . (5.3b)This theorem may be verified by direct substitution. We omit the details.Recall from (4.12) that all characters may be expressed as (infinite) linear combinations of the standard characters(5.2). The latter therefore form a (topological) basis for the space of characters. In this basis, which we call thestandard basis, the S-transformation is manifestly symmetric and unitary: S (cid:2) s ℓ (cid:0) W l (cid:1) → s m (cid:0) W m (cid:1)(cid:3) = S (cid:2) s m (cid:0) W m (cid:1) → s ℓ (cid:0) W l (cid:1)(cid:3) , (5.4a) ∑ m ∈ Z Z R / Z S (cid:2) s ℓ (cid:0) W l (cid:1) → s m (cid:0) W m (cid:1)(cid:3) S (cid:2) s n (cid:0) W n (cid:1) → s m (cid:0) W m (cid:1)(cid:3) ∗ d m = d n = ℓ d ( n = l mod 1 ) . (5.4b)Its square may also be identified with conjugation at the level of the standard characters: ∑ m ∈ Z Z R / Z S (cid:2) s ℓ (cid:0) W l (cid:1) → s m (cid:0) W m (cid:1)(cid:3) S (cid:2) s m (cid:0) W m (cid:1) → s n (cid:0) W n (cid:1)(cid:3) d m = d n = − ℓ d ( n = − l mod 1 ) . (5.4c) OSONIC GHOSTS AT c = These three familiar properties lead us to expect that substituting this integration kernel into a Verlinde formula willresult in the Grothendieck fusion coefficients.Before doing this, we need to determine the S-transformation for the atypical characters. This follows readily fromthe character formulae (4.12) and Theorem 2.
Corollary 3.
The simple atypical characters have S-transformations ch (cid:2) s ℓ (cid:0) V (cid:1)(cid:3)(cid:0) z / t (cid:12)(cid:12) − / t (cid:1) = A (cid:0) z (cid:12)(cid:12) t (cid:1) ∑ m ∈ Z Z R / Z S (cid:2) s ℓ (cid:0) V (cid:1) → s m (cid:0) W m (cid:1)(cid:3) ch (cid:2) s m (cid:0) W m (cid:1)(cid:3)(cid:0) z (cid:12)(cid:12) t (cid:1) d m , (5.5a) where S (cid:2) s ℓ (cid:0) V (cid:1) → s m (cid:0) W m (cid:1)(cid:3) = ( − ) ℓ + m + e − p i ( ℓ + / ) m e i pm − e − i pm . (5.5b)Here, the denominator should also be regarded as shorthand for a formal power series in e p i m . In fact, it arises fromsumming a geometric series at its radius of convergence, a fact which may be useful to remember for the Verlindecomputations to come. We remark that both the character formulae of (4.12) conveniently yield the same atypicalS-transformation kernel when expressed using denominators (though the respective convergence regions are disjoint).Finally, we address the automorphy factor A (cid:0) z (cid:12)(cid:12) t (cid:1) appearing in the transformation rules (5.3) and (5.5). This factordoes not depend upon the labels characterising the modules in the S-transformation kernel and, as with a similar (butless complicated) factor appearing in the S-transformation of integrable Kac-Moody module characters [25], it maybe absorbed by augmenting the definition of characters by another variable y which tracks the eigenvalue of the Cartanelement . This eigenvalue is always 1, so we end up multiplying all G -module characters by y = e p i q . Proposition 4.
The transformations S : ( q | z | t ) (cid:18) q + z t − z t + z + p (cid:16) arg t − p (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) zt (cid:12)(cid:12)(cid:12)(cid:12) − t (cid:19) , T : ( q | z | t ) (cid:18) q + (cid:12)(cid:12)(cid:12)(cid:12) z (cid:12)(cid:12)(cid:12)(cid:12) t + (cid:19) (5.6) define an action of the modular group SL (cid:0) Z (cid:1) . That is, S = ( ST ) = C and C is the identity. The proof is a straightforward verification that S and ( ST ) map ( q | z | t ) to ( q + z |− z | t ) ; this obviously squares tothe identity. We remark that the term involving arg t in (5.6) accounts for the factor of | t | / − i t in A (cid:0) z (cid:12)(cid:12) t (cid:1) . It nowfollows that inserting y into characters and transforming as in (5.6) will cancel the factor A (cid:0) z (cid:12)(cid:12) t (cid:1) in (5.3) and (5.5).This justifies our separation of this automorphy factor from the S-transformation kernel.6. T HE V ERLINDE F ORMULA
We define a product ⊠ on the (appropriate topological completion of the) span of the standard characters bych (cid:2) M (cid:3) ⊠ ch (cid:2) N (cid:3) = ∑ n ∈ Z Z R / Z (cid:20) s n (cid:0) W n (cid:1) M N (cid:21) ch (cid:2) s n (cid:0) W n (cid:1)(cid:3) d n , (6.1a)where the coefficients appearing in the integrand are determined by the following variant of the Verlinde formula: (cid:20) s n (cid:0) W n (cid:1) M N (cid:21) = ∑ r ∈ Z Z R / Z S (cid:2) M → s r (cid:0) W r (cid:1)(cid:3) S (cid:2) N → s r (cid:0) W r (cid:1)(cid:3) S (cid:2) s n (cid:0) W n (cid:1) → s r (cid:0) W r (cid:1)(cid:3) ∗ S (cid:2) V → s r (cid:0) W r (cid:1)(cid:3) d r . (6.1b)We will demonstrate shortly that this product, which we call the Verlinde product , is indeed well-defined — (6.1a)always gives a finite linear combination of standard characters or infinite alternating sums, the latter being interpretedas atypical simple characters. For now, we note that the Verlinde product is commutative and associative. The unitarity(5.4b) of the S-transformation implies that the unit is the vacuum character ch (cid:2) V (cid:3) . Lemma 5.
The Verlinde product satisfies ch (cid:2) s ℓ (cid:0) M (cid:1)(cid:3) ⊠ ch (cid:2) s m (cid:0) N (cid:1)(cid:3) = s ℓ + m (cid:0) ch (cid:2) M (cid:3) ⊠ ch (cid:2) N (cid:3)(cid:1) , (6.2) This t -dependent factor was also present in the modular S-transformations of the standard characters of admissible level b sl ( ) [10, 26], but wasargued to be inconsequential as phases cancel when considering modular invariants and Verlinde computations. A more satisfactory explanation isto absorb it into the automorphy factor A (cid:0) z (cid:12)(cid:12) t (cid:1) as we have done here for the standard ghost characters. where the right-hand side is to be interpreted as evaluating the Verlinde product in the standard basis and applyingspectral flow to each basis element uniformly.Proof. This follows by noting that the S-transformation kernels (5.3b) and (5.5b) may be factored as S (cid:2) s ℓ (cid:0) M (cid:1) → s r (cid:0) W r (cid:1)(cid:3) = ( − ) ℓ e − p i ℓ r S (cid:2) M → s r (cid:0) W r (cid:1)(cid:3) , (6.3)where M is either W l or V . Applying this factorisation to the kernels for M and N appearing in (6.1b), and thenabsorbing both phases into the kernel for s n (cid:0) W n (cid:1) , we arrive at (cid:20) s n (cid:0) W n (cid:1) s ℓ (cid:0) M (cid:1) s m (cid:0) N (cid:1)(cid:21) = (cid:20) s − ℓ − m + n (cid:0) W n (cid:1) M N (cid:21) . (6.4)Replacing n by ℓ + m + n now gives the desired result. Theorem 6.
The Verlinde product rules take the form ch (cid:2) s ℓ (cid:0) V (cid:1)(cid:3) ⊠ ch (cid:2) s m (cid:0) V (cid:1)(cid:3) = ch (cid:2) s ℓ + m (cid:0) V (cid:1)(cid:3) , (6.5a)ch (cid:2) s ℓ (cid:0) V (cid:1)(cid:3) ⊠ ch (cid:2) s m (cid:0) W m (cid:1)(cid:3) = ch (cid:2) s ℓ + m (cid:0) W m (cid:1)(cid:3) , (6.5b)ch (cid:2) s ℓ (cid:0) W l (cid:1)(cid:3) ⊠ ch (cid:2) s m (cid:0) W m (cid:1)(cid:3) = ch (cid:2) s ℓ + m (cid:0) W l + m (cid:1)(cid:3) + ch (cid:2) s ℓ + m − (cid:0) W l + m (cid:1)(cid:3) . (6.5c) In particular, the Verlinde multiplicities (6.1b) are non-negative integer multiples of delta functions.Proof.
By Lemma 5, we may assume that ℓ = m =
0. Then, (6.5a) and (6.5b) follow from the vacuum characterch (cid:2) V (cid:3) being the unit of the Verlinde product. We therefore turn to the rule (6.5c) and compute the coefficient (cid:20) s n (cid:0) W n (cid:1) W l W m (cid:21) = ( − ) n + ∑ r ∈ Z e − p i ( l + m − n ) r Z R / Z (cid:16) e p i ( n + ) r − e p i n r (cid:17) d r = ( d n = + d n = − ) d ( n = l + m mod 1 ) . (6.6)The result now follows by substituting into (6.1a).Because the multiplicities appearing in the Verlinde product rules are non-negative integers, the product ⊠ endowsthe (completion of the) Z -span of the standard characters with a ring structure. We call this ring the Verlinde ring.The following assumption and conjecture are now very plausible: Conjecture 1.
Let × denote the fusion product on the Z -span of the indecomposable G -modules (where addition isdirect sum). We assume that fusing with any given G -module defines an exact functor from this fusion ring to itself,hence that the fusion product descends to a well-defined product ⊠ on the Grothendieck group: (cid:2) M (cid:3) ⊠ (cid:2) N (cid:3) = (cid:2) M × N (cid:3) . (6.7)We conjecture that the product on the resulting Grothendieck ring may be identified with the Verlinde product underthe group isomorphism (cid:2) M (cid:3) ch (cid:2) M (cid:3) . In other words, we conjecture that this constitutes an isomorphism betweenthe Verlinde and Grothendieck fusion rings.This conjecture holds for rational conformal field theories [27]. We will assume from now on that this conjecture holdsfor the c = (cid:2) M × N (cid:3) = ch (cid:2) M (cid:3) ⊠ ch (cid:2) N (cid:3) . (6.8)Of course, if the right-hand side of a Grothendieck fusion rule is the character of a simple module, it may be liftedto a genuine fusion rule. More generally, if a Grothendieck product is a sum of characters of modules among whichno non-trivial extensions are possible, then we may again lift the result to a genuine fusion rule. In the latter case,consideration of charges and conformal weights modulo 1 is often sufficient to rule out indecomposable extensions.Such considerations lead us to the following fusion rules: OSONIC GHOSTS AT c = Corollary 7.
Assuming Conjecture 1, the Verlinde product rules of Theorem 6 imply the following fusion rules: s ℓ (cid:0) V (cid:1) × s m (cid:0) V (cid:1) = s ℓ + m (cid:0) V (cid:1) , (6.9a) s ℓ (cid:0) V (cid:1) × s m (cid:0) W m (cid:1) = s ℓ + m (cid:0) W m (cid:1) ( m / ∈ Z ), (6.9b) s ℓ (cid:0) W l (cid:1) × s m (cid:0) W m (cid:1) = s ℓ + m (cid:0) W l + m (cid:1) ⊕ s ℓ + m − (cid:0) W l + m (cid:1) ( l + m / ∈ Z ). (6.9c)We remark that fusing the module s ℓ (cid:0) V (cid:1) with its conjugate c (cid:0) s ℓ (cid:0) V (cid:1)(cid:1) = s − ℓ (cid:0) c (cid:0) V (cid:1)(cid:1) = s − ℓ − (cid:0) V (cid:1) does not give backthe vacuum, but rather its conjugate c (cid:0) V (cid:1) = s − (cid:0) V (cid:1) . This is consistent with the one-point function of the identityfield vanishing and that of its conjugate w ( z ) being non-vanishing.7. F USION
In this section, we compute the remaining fusion product involving simple modules, that of the typicals W l and W − l (so [ l ] = [ ] ). Theorem 6 and Equation (6.8) give the character of this fusion product if we assume (and we do)that Conjecture 1 holds: ch (cid:2) W l × W − l (cid:3) = ch (cid:2) s − (cid:0) V (cid:1)(cid:3) + (cid:2) s − (cid:0) V (cid:1)(cid:3) + ch (cid:2) V (cid:3) . (7.1)We illustrate the (convex hull of the) weights of the composition factors of this fusion product in Figure 1 (left); here,the charge increases horizontally from right to left and the conformal weight increases from top to bottom. To deducethe module structure, we turn to the Nahm-Gaberdiel-Kausch fusion algorithm [19, 20]. This constructs (an algebraiccompletion of) the fusion product of two modules as a quotient of their tensor product (over C ) [28], the action on theproduct being characterised by the following master equations: D ( b n ) = n ∑ m = (cid:18) nm (cid:19) b m ⊗ + ⊗ b n ( n > D ( b − n ) = ¥ ∑ m = (cid:18) m + n − n − (cid:19) ( − ) m b m ⊗ + ⊗ b n ( n > b − n ⊗ = ¥ ∑ m = n (cid:18) m − n − (cid:19) D ( b − m ) + ( − ) n − ¥ ∑ m = (cid:18) m + n − n − (cid:19) ⊗ b m ( n > D ( g n ) = n ∑ m = (cid:18) n − m − (cid:19) g m ⊗ + ⊗ g n ( n > D ( g − n ) = ¥ ∑ m = (cid:18) m + n − n (cid:19) ( − ) m − g m ⊗ + ⊗ g n ( n > g − n ⊗ = ¥ ∑ m = n (cid:18) mn (cid:19) D ( g − m ) + ( − ) n ¥ ∑ m = (cid:18) m + n − n (cid:19) ⊗ g m ( n > b − m and g − n , with m > n >
0; quotienting by its action defines the specialsubspace [19]. Unfortunately, the typical modules W l have trivial special subspaces because g acts surjectively. The s ℓ (cid:0) W l (cid:1) likewise have trivial special subspaces.The standard methodology therefore needs refining. We introduce a (commutative) subalgebra U of the universalenveloping algebra of G by U = C [ b − , b − , . . . , g − , g − , . . . ] (7.3)and claim that W l × W − l U ( W l × W − l ) ⊆ W l U ( W l ) ⊗ W − l U ( W − l ) , (7.4) x − x + yw b g − g b F IGURE
1. The structure of the fusion product W l × W − l = s − (cid:0) P (cid:1) . At left, the four compositionfactors are visualised with one vector of each marked. The weights ( j , h ) of the vectors x − , w , y and x + are ( , ) , ( , ) , ( , ) and ( , − ) , respectively. At right, the composition factors are “glued”together into an indecomposable module through the indicated action of the algebra modes.as vector spaces. Because we impose (7.2c) and (7.2f) as identities on W l ⊗ W − l , we may identify the left-handside with the corresponding tensor product quotient: W l × W − l U ( W l × W − l ) ∼ = W l ⊗ W − l h (7.2c) , (7.2f) , D ( U ) i ( W l ⊗ W − l ) ⊆ W l U ( W l ) ⊗ W − l U ( W − l ) , (7.5)It is this inclusion of quotients of tensor products that we shall actually prove.The proof amounts to showing that any u ⊗ v , with u ∈ W l and v ∈ W − l , representing the left-hand side may bewritten as a linear combination of the u j ⊗ v k that represent the right-hand side. Here, the u j ∈ W l and v k ∈ W − l arethe parabolic highest weight vectors that restrict to the basis vectors of the G -modules W l and W − l , respectively(see Proposition 1). As these modules are simple, we may parametrise them so that b u j = ju j + , g u j = u j − , b v k = kv k + , g v k = v k − ⇒ J u j = ju j , L u j = , J v k = kv k , L v k = j ∈ Z + l ),( k ∈ Z − l ). (7.6)The proof proceeds in four steps, starting with some arbitrary u ⊗ v ∈ W l ⊗ W − l and iterating each step on each ofthe terms, which we shall typically also denote by u ⊗ v , obtained in the previous step:(1) If u = b − n u ′ , with n >
1, then use (7.2c) to write u ⊗ v = ( − ) n − ∑ ¥ m = (cid:0) m + n − n − (cid:1) u ′ ⊗ b m v . Iterate this repeatedlyuntil the result is a finite linear combination of vectors of the form u ⊗ v , where each u cannot be written as b − n u ′ ,with n >
1. Termination is guaranteed as the conformal weight of the first factor decreases strictly with eachiteration.(2) If, in any of these u ⊗ v , we have u = g − n u ′ , with n >
1, then use (7.2f) to write each as the linear combination ( − ) n ∑ ¥ m = (cid:0) m + n − n (cid:1) u ⊗ g m v . Simplifying, and repeating for all terms, we arrive at a finite linear combination ofvectors of the form u j ⊗ v .(3) If v = b − n v ′ , with n >
1, then use (7.2b) and D ( b − n ) ( u j ⊗ v ′ ) = u j ⊗ v = − ju j + ⊗ v ′ . Repeat.(4) Finally, if v = g − n v ′ , with n >
1, then use (7.2e) and D ( g − n ) ( u j ⊗ v ′ ) = u j ⊗ v =
0. The final result isnow a finite linear combination of vectors of the form u j ⊗ v k , completing the proof.In principle, we could also apply (2) when u = g u ′ . However, all vectors u ∈ W l have this form, so repeating thisstep would lead to an infinite regress. Instead, we apply Equations (7.2e) and (7.2f), both for n =
0, to reduce thebasis (cid:8) u j ⊗ v k : j ∈ Z + l , k ∈ Z − l (cid:9) of the right-hand side of (7.5), giving an analogue of a spurious state: u j − ⊗ v k + = g u j ⊗ v k + = D ( g ) ( u j ⊗ v k + ) = u j ⊗ g v k + = u j ⊗ v k . (7.7)We therefore propose that a basis for the left-hand side is { u l ⊗ v k : k ∈ Z − l } . Being vector spaces, one may also regard the left-hand side of (7.4) as a quotient of the right-hand side. We present (7.4) as an inclusion as thisis how we will prove it. The equivalent point of view, where we instead regard the left-hand side as a quotient, is used when actually computing afusion product. Then, one first characterises the left-hand side by determining elements, called spurious states , of the right-hand side which mustbe set to 0 for the master equations (7.2) to have a well-defined action.
OSONIC GHOSTS AT c = The action of b and g on these basis vectors is easily computed using (7.2a), (7.2e) and (7.7): D ( b ) ( u l ⊗ v k ) = l u l + ⊗ v k + ku l ⊗ v k + = ( l + k ) u l ⊗ v k + , D ( g ) ( u l ⊗ v k ) = u l ⊗ v k − . (7.8)This is the same action as that of b and g on the quotient W + / U (cid:0) W + (cid:1) (which coincides with that on the G -module W + appearing in Proposition 1): g acts surjectively while b annihilates the vector u − k ⊗ v k of weight ( , ) . Wetherefore conclude that the fusion product W l × W − l has a quotient isomorphic to W + . This accounts for thecomposition factor V and one of the s − (cid:0) V (cid:1) factors appearing in (7.1). It also verifies the arrow labelled by g inFigure 1 (right).Because g − ∈ U acts surjectively on the composition factor s − (cid:0) V (cid:1) , every vector associated to this factor is setto 0 in the fusion quotient that we have computed. It therefore remains to account for the other composition factor s − (cid:0) V (cid:1) in (7.1). As no vectors associated to this factor are observed in the fusion quotient, they must be in the imageof U . In particular, the vector of weight ( , ) that is labelled by w in Figure 1 (left) must be in im U . Referring to thisfigure (or considering multiplicities from the character (7.1) of the fusion product), we see that the only way this canhappen is if w is a non-zero multiple of g − x + . This conclusion therefore verifies the arrow labelled by g − drawn inFigure 1 (right).We remark that if the basis proposed after (7.7) were incorrect, meaning that there were further spurious statesto find, then we would have to set some of the elements of W + / U (cid:0) W + (cid:1) to 0. However, this is impossible becauseFigure 1 makes it clear that there cannot be any U -descendants beyond those we have accounted for. The basis istherefore correct.To obtain the remaining arrows in Figure 1 (right), we change the subalgebra by whose action we quotient. Let U ′ denote the (commutative) subalgebra U ′ = C [ b , b − , b − , . . . , g − , g − , g − , . . . ] . (7.9)The claim is now that W l × W − l U ′ ( W l × W − l ) ⊆ W l U ( W l ) ⊗ W − l U ′ ( W − l ) ; (7.10)that is, that any u ⊗ v ∈ W l ⊗ W − l may be reduced to a linear combination of vectors of the form u j ⊗ g m − v k . Theproof again proceeds as above, with the same proviso regarding Equations (7.2c) and (7.2f), though (2) and (4) arenow only performed when n >
2. Moreover, we need an additional step after (2):(2’) As u = g ℓ − u j , we use (7.2e) and (7.2f) to write g ℓ − u j ⊗ v = g ℓ − − u j ⊗ g − v − ∑ ¥ m = m g ℓ − − u j ⊗ g m v , when ℓ > u j ⊗ v .We may again reduce the basis for the right-hand side of (7.10) by computing analogues of spurious states:0 = D ( b ) (cid:0) v j ⊗ g m − w k (cid:1) = jv j + ⊗ g m − w k + kv j ⊗ g m − w k + , (7.11a) v j ⊗ g m − w k = g v j + ⊗ g m − w k = ( D ( g ) + D ( g − )) (cid:0) v j + ⊗ g m − w k (cid:1) = v j + ⊗ g m − w k − + v j + ⊗ g m + − w k . (7.11b)Applying (7.11b) repeatedly lets us reduce the power of g − to 0, then (7.11a) lets us fix j = l . Our proposed basis istherefore { u l ⊗ v k : k ∈ Z − l } . We now compute D ( b ) ( v l ⊗ w k ) = l v l + ⊗ w k = − kv l ⊗ w k + , (7.12a) D ( g − ) ( v l ⊗ w k ) = v l ⊗ g − w k = v l − ⊗ w k − v l ⊗ w k − = − l + k − k − v l ⊗ w k − (7.12b)and a little work shows that this action matches that on the quotient s − (cid:0) W − (cid:1) / U ′ (cid:0) s − (cid:0) W − (cid:1)(cid:1) . This verifies thearrow labelled by b in Figure 1 (right) and that labelled by b is obtained by noting that the missing vector of weight ( , ) can only be a U ′ -descendant of the vector of weight ( , ) .It remains only to determine if there are any ambiguities in the structure that we have uncovered for this fusionproduct W l × W − l . The analysis amounts to considering the four vectors labelled in Figure 1 (left): • First, choose x + = ( , − ) . • Then, define w = g − x + so that w has weight ( , ) . Let y and w be linearly independent in this weight space. • Fix x − , of weight ( , ) , by requiring that b x − = w . We will fix the normalisation of y shortly. For now, we note that J y = ( g b + g − b ) y = y + ( b g + g − b ) y , L y = − g − b y = , (7.13)so that ( J − ) y = ( b g + g − b ) y and L y are proportional to w (see Figure 1). The Virasoro zero mode thereforehas a Jordan block of rank 2 indicating that W l × W − l is a staggered module in the sense of [8, 21]. We may nownormalise y so that • L y = w ,noting that this fixes y up to adding multiples of w . The structure of the staggered module is then determined bycomputing b y = b + x + and g y = b − x − , as the constants b ± are independent of the remaining freedom in choosing y . We find that w = L y = − g − b y = − b + b − x + = − b + w ⇒ b + = − . (7.14)To compute b − , we note that the coproduct formula D ( J ) = J ⊗ + ⊗ J implies that J acts semisimply on thefusion product W l × W − l because it does on the typical modules. Thus, we deduce that0 = ( J − ) y = ( b g + g − b ) y = ( b − + b + ) w ⇒ b − = − b + = . (7.15)The analysis of the fusion product is complete and we summarise the result as follows: Theorem 8.
The ghost fusion rule W l × W − l = s − (cid:0) P (cid:1) (7.16) defines an indecomposable staggered module P with rank Jordan blocks that is determined up to isomorphism byeither of the following exact sequences or by its Loewy diagram: −→ s (cid:0) W − (cid:1) −→ P −→ W − −→ , −→ W + −→ P −→ s (cid:0) W + (cid:1) −→ , V s − (cid:0) V (cid:1) s (cid:0) V (cid:1) . VP (7.17)In other words, there are no logarithmic couplings [29] to determine in order to completely specify the isomorphismclass of P . We emphasise that this computation assumed Conjecture 1.It is extremely natural to generalise this result to the fusion rules of spectrally-flowed typical modules. Thisrequires the following standard conjecture, still unproven to the best of our knowledge, that lifts Lemma 5 to fusion: Conjecture 2.
The fusion product satisfies s ℓ (cid:0) M (cid:1) × s m (cid:0) N (cid:1) ∼ = s ℓ + m (cid:0) M × N (cid:1) . (7.18) Corollary 9.
Assuming Conjectures 1 and 2, Theorem 8 implies the following fusion rules: s ℓ (cid:0) W l (cid:1) × s m (cid:0) W − l (cid:1) = s ℓ + m − (cid:0) P (cid:1) . (7.19)Because the spectrum of the theory contains staggered modules, the bosonic ghost system at c = Corollary 10.
We have the following fusion rules, assuming Conjectures 1 and 2: s ℓ (cid:0) V (cid:1) × s m (cid:0) P (cid:1) = s ℓ + m (cid:0) P (cid:1) , (7.20a) s ℓ (cid:0) W l (cid:1) × s m (cid:0) P (cid:1) = s ℓ + m + (cid:0) W l (cid:1) ⊕ s ℓ + m (cid:0) W l (cid:1) ⊕ s ℓ + m − (cid:0) W l (cid:1) , (7.20b) s ℓ (cid:0) P (cid:1) × s m (cid:0) P (cid:1) = s ℓ + m + (cid:0) P (cid:1) ⊕ s ℓ + m (cid:0) P (cid:1) ⊕ s ℓ + m − (cid:0) P (cid:1) . (7.20c) OSONIC GHOSTS AT c = We remark that there are many other indecomposables whose fusion rules have not been determined, the atypicalstandards W ± and the length 3 subquotients of P , for example. We expect that computing these fusion productsiteratively will fill out a complete set of indecomposables for the c = b gl ( | ) [30]. As the results determined above seem to suggest that the typical modules s ℓ (cid:0) W l (cid:1) and staggeredmodules s ℓ (cid:0) P (cid:1) form an ideal in the fusion ring, we make the following conjecture: Conjecture 3.
Let C be the abelian category of ghost vertex operator algebra modules generated, by imposingclosure under extensions, from the typicals s ℓ (cid:0) W l (cid:1) and the simple atypicals s ℓ (cid:0) V (cid:1) (we still insist that ∈ G actas the identity on these extensions). Then, in C , the typical module s ℓ (cid:0) W l (cid:1) is simple and projective, whereas thestaggered module s ℓ (cid:0) P (cid:1) is the projective cover of the simple atypical module s ℓ (cid:0) V (cid:1) .The category C of ghost vertex operator algebra modules is then closed under fusion and conjugation. Moreover,we will see shortly that one can construct modular invariant partition functions from the characters of its modules.We therefore think of C as being the physically relevant module category for bosonic ghost (logarithmic) conformalfield theories. It seems very likely to us that this category is rigid, so that, for example, fusing with any given moduledefines an exact functor from C to itself. Fusion would then define a well-defined product of the Grothendieck group,proving half of Conjecture 1. We hope to return to this question of rigidity in the future.Finally, we remark that in order to explicitly observe the Jordan block for L using the Nahm-Gaberdiel-Kauschalgorithm, one would have to construct a quotient in which w =
0. This would require excluding all powers of b and g − from the subalgebra by whose action we quotient; the largest such subalgebra is that generated by the b n and g n − with n −
1. Unfortunately, the quotient of W l × W − l by the action of this subalgebra has infinite-dimensionalsubspaces of constant charge. Thus, linear algebra would not suffice to determine the existence of the Jordan block,leading one instead into the world of abstract analysis. We will also leave this technical endeavour for the future.8. M ODULAR I NVARIANTS
Since the S-transformation is symmetric and unitary in the standard basis (Section 5), the diagonal partition func-tion Z diag. ( y ; z ; q ) = ∑ ℓ ∈ Z Z R / Z (cid:12)(cid:12)(cid:12) ch (cid:2) s ℓ (cid:0) W l (cid:1)(cid:3)(cid:0) y ; z ; q (cid:1)(cid:12)(cid:12)(cid:12) d l (8.1)is (formally) modular invariant. Here, it is important to augment the characters by the additional variable y as inthe discussion surrounding Proposition 4. According to the proposals of [26, 31], the corresponding bulk state spaceshould have the form H = B ⊕ M ℓ ∈ Z ⊖ Z R / Z s ℓ (cid:0) W l (cid:1) ⊗ s ℓ (cid:0) W l (cid:1) d l , (8.2)where B is an indecomposable atypical bulk module whose structure is described by the following (partial) Loewydiagram in which the solid and dotted arrows represent the action of the two copies of G : ··· B ··· s − ⊗ s − s − ⊗ s − ⊗ s ⊗ s s ⊗ s s − ⊗ s − s − ⊗ s − s − ⊗ ⊗ s − ⊗ s s ⊗ s ⊗ s s ⊗ ss − ⊗ s − s − ⊗ s − ⊗ s ⊗ s s ⊗ s Here, we represent the bulk composition factor s ℓ (cid:0) V (cid:1) ⊗ s m (cid:0) V (cid:1) by the automorphism s ℓ ⊗ s m for brevity. Thecharacter ch (cid:2) B (cid:3) = ∑ ℓ ∈ Z ch (cid:2) s ℓ (cid:0) V (cid:1)(cid:3) ∗ ch (cid:2) s ℓ (cid:0) P (cid:1)(cid:3) = ∑ ℓ ∈ Z ch (cid:2) s ℓ (cid:0) P (cid:1)(cid:3) ∗ ch (cid:2) s ℓ (cid:0) V (cid:1)(cid:3) (8.3)underscores the similarity between this proposal and the standard decompositions of the regular representations offinite-dimensional associative algebras and compact Lie groups. We note that the nilpotent part of the actions of L and L both map each vector associated with the head of this module (the top composition factors) to the same vectorin its socle (the bottom composition factors). Locality, meaning the single-valuedness of bulk correlators, is thussatisfied for this proposed bulk module structure [32]. Note that the charge conjugate partition function is likewise formally modular invariant, but the correspondingatypical bulk module does not have a submodule isomorphic to V ⊗ V because V ≇ c (cid:0) V (cid:1) . In particular, the chargeconjugate bulk state space would possess no vacuum state, so its physical consistency is not clear to us. It wouldbe interesting to know whether these bulk state space proposals may be interpreted in terms of coends as advocatedin [33].There are nevertheless many other modular invariants of simple current type. Indeed, the fusion rules (6.9a) showthat each of the s p (cid:0) V (cid:1) is a simple current of infinite order. The vacuum module V of the corresponding simplecurrent extension E p (we take p > V ∼ = M r ∈ Z s rp (cid:0) V (cid:1) , (8.4)when restricted to a G -module. It is easy to check that the charges and conformal weights of the vectors in V areintegers and that this continues to hold for all atypical indecomposables. The same is not true for the typical extendedalgebra modules. The module W l ∼ = M r ∈ Z s rp (cid:0) W l (cid:1) (8.5)turns out to be untwisted, meaning that the extended algebra fields have trivial monodromy, if and only if p l ∈ Z .When p is even, the characters of the (untwisted) standard extended algebra modules s ℓ (cid:0) W j / p (cid:1) , for j , ℓ = , , . . . , p −
1, span a finite-dimensional representation of the modular group. In particular,ch (cid:2) s ℓ (cid:0) W j / p (cid:1)(cid:3) S p p − ∑ m , k = ( − ) ℓ + m e − p i ( ℓ k + m j ) / p ch (cid:2) s m (cid:0) W k / p (cid:1)(cid:3) (8.6)and the extended S-matrices are easily checked to be symmetric and unitary. The partition function Z p = p − ∑ ℓ, j = (cid:12)(cid:12)(cid:12) ch (cid:2) s ℓ (cid:0) W j / p (cid:1)(cid:3)(cid:12)(cid:12)(cid:12) = ∑ ℓ, r ∈ Z p − ∑ j = ch (cid:2) s ℓ (cid:0) W j / p (cid:1)(cid:3) ∗ ch (cid:2) s ℓ + rp (cid:0) W j / p (cid:1)(cid:3) (8.7)is therefore modular invariant for p even. We remark that the corresponding theories are always logarithmic.Finally, we mention that although these simple current extensions define formal modular invariants, their modularproperties are unsatisfactory in general. In particular, it is not clear how to evaluate the S-transforms of the simpleatypical characters — the obvious manipulations lead to a divergence due to the pole in Equation (5.5). It wouldbe interesting to understand this because the standard examples of C -cofinite logarithmic theories, whose modularproperties are similarly unsatisfactory, may likewise be realised as simple current extensions [34]. It would also beinteresting to classify all ghost modular invariants; we hope to return to these questions in the future.A CKNOWLEDGEMENTS
We thank J¨urgen Fuchs and Christoph Schweigert for illuminating discussions regarding parabolic Verma mod-ules and the organisers of the Erwin Schr¨odinger Institute programme “Modern trends in topological quantum fieldtheory” for their hospitality. DR’s research is supported by the Australian Research Council Discovery ProjectDP1093910. SW’s work is supported by the Australian Research Council Discovery Early Career Researcher AwardDE140101825. A
PPENDIX
A. F
USION FOR THE c = − OSONIC G HOST S YSTEM
In this appendix, we quickly recall the fusion rules of the bosonic ghosts with a =
0, hence central charge c = − b sl ( R ) − / fusion rules computed there, seealso [10, Sec.2.3]. We include them here for comparison with the c = When p is odd, these untwisted characters are transformed by S into a linear combination of twisted characters. We expect that in this case theextended algebra E p is fermionic in nature. OSONIC GHOSTS AT c = Corollary 10. The equivalence of the results then supports our assertion that the tensor structure on the category ofghost modules is independent of the central charge. As noted in Section 2, the different ghost systems only differ in the choice of conformal structure, so their modulecategories are equivalent (as abelian categories). However, there is one important difference: Because h b = h g = when a =
0, one should also consider spectral flow twists s ℓ where ℓ is a half-integer. This translates into integerspectral flow twists for the Z -orbifold b sl ( R ) − / . The results of [11] assumed Conjecture 2 and may be stated inthe following form: s ℓ (cid:0) L (cid:1) b × s m (cid:0) L (cid:1) = s ℓ + m (cid:0) L (cid:1) , s ℓ (cid:0) L (cid:1) b × s m (cid:0) E l (cid:1) = s ℓ + m (cid:0) E l (cid:1) , s ℓ (cid:0) L (cid:1) b × s m (cid:0) S (cid:1) = s ℓ + m (cid:0) S (cid:1) , (A.1a) s ℓ (cid:0) E l (cid:1) b × s m (cid:0) E m (cid:1) = s ℓ + m (cid:0) S (cid:1) if l + m ∈ Z , s ℓ + m + / (cid:0) E l + m + / (cid:1) ⊕ s ℓ + m − / (cid:0) E l + m − / (cid:1) otherwise, (A.1b) s ℓ (cid:0) E l (cid:1) b × s m (cid:0) S (cid:1) = s ℓ + m + (cid:0) E l (cid:1) ⊕ s ℓ + m (cid:0) E l (cid:1) ⊕ s ℓ + m − (cid:0) E l (cid:1) , (A.1c) s ℓ (cid:0) S (cid:1) b × s m (cid:0) S (cid:1) = s ℓ + m + (cid:0) S (cid:1) ⊕ s ℓ + m (cid:0) S (cid:1) ⊕ s ℓ + m − (cid:0) S (cid:1) . (A.1d)Here, b × denotes the fusion product of the c = − L denotes the vacuum module, the E l constitute a family ofparabolic highest weight modules parametrised by [ l ] ∈ R / Z whose elements are simple if [ l ] = , and S denotes astaggered module whose Loewy diagram L s − (cid:0) L (cid:1) s (cid:0) L (cid:1) LS (A.2)fixes its structure up to isomorphism.The equivalence between these results and those that we have derived for c = L ←→ V , s − / (cid:0) E l + / (cid:1) ←→ W l , S ←→ P . (A.3)The twist by s − / for the standard modules should not be surprising: Equation (2.13) implies that conformal weightsat c = − a =
0) and c = a = − ) are related by L = L − / − J . (A.4)Thus, the parabolic highest weight vectors of W l , which all have conformal weight 0, will no longer have constantconformal weight upon changing the conformal structure. The shift of l by likewise accounts for the fact that theatypical point is [ l ] = [ ] for c = −
1, rather than [ l ] = [ ] . We view this identification as providing strong evidencefor the equivalence of the c = − c = L − , which is independent of the choice of conformal structure because of (2.13) and ( ¶ J ) − =
0. R
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EPARTMENT OF T HEORETICAL P HYSICS , R
ESEARCH S CHOOL OF P HYSICS AND E NGINEERING AND M ATHEMATICAL S CIENCES I NSTITUTE , A
USTRALIAN N ATIONAL U NIVERSITY , A
CTON , ACT 2601, A
USTRALIA
E-mail address : [email protected] (Simon Wood) D EPARTMENT OF T HEORETICAL P HYSICS , R
ESEARCH S CHOOL OF P HYSICS AND E NGINEERING , A
USTRALIAN N ATIONAL U NIVERSITY , A
CTON , ACT 2601, A
USTRALIA
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