Staggered modules of N=2 superconformal minimal models
aa r X i v : . [ h e p - t h ] F e b STAGGERED MODULES OF N = 2 SUPERCONFORMAL MINIMALMODELS
CHRISTOPHER RAYMOND, DAVID RIDOUT AND JØRGEN RASMUSSEN
Abstract.
We investigate a class of reducible yet indecomposable modules over the N = 2superconformal algebras. These so-called staggered modules exhibit a non-diagonalisable actionof the Virasoro mode L . Using recent results on the coset construction of N = 2 minimalmodels, we explicitly construct such modules for central charges c = − c = −
6. We alsodescribe spectral-flow orbits and symmetries of the families of staggered modules which arisevia the coset. Introduction
Logarithmic conformal field theories (CFTs) are intimately related to the representation theoryof vertex operator algebras (VOAs), see [1] for a review. In the traditional setting of rationalCFT, the representations of the chiral symmetry algebra are completely reducible and are madeup of finitely many irreducible summands. In logarithmic CFT, we no longer have completereducibility; the state space includes reducible yet indecomposable modules. A subclass of thesemodules, known as staggered modules, has the property that the Virasoro zero-mode operator L exhibits rank-two Jordan blocks [2, 3]. The presence of these modules in the theory leads tothe logarithmic divergences in correlation functions after which logarithmic CFT is named [4].While logarithmic CFTs are fundamentally non-unitary, this does not preclude them from beingphysically important. Many examples of logarithmic CFTs have appeared in the literature. Anon-exhaustive list of examples includes theories based on the Virasoro algebra [3,5–8], fractionallevel WZW models based on b sl (2) [9–13] and b sl (3) [14], WZW models based on Lie supergroups[15–21], bosonic ghost systems [22–24], triplet algebras [25–31], the Heisenberg algebra [32], andthe N = 1 superconformal algebras [33–35].Many examples of logarithmic CFTs admit a unifying understanding within the standard-moduleformalism of [1, 36]. This formalism posits that certain VOAs admit a distinguished set ofrepresentations, called standard modules, whose characters form a (topological) basis for thespace of all (physically relevant) characters and moreover carry a representation of the modulargroup SL (2 , Z ). The parameter space of the standard modules is a measurable space and themodules are divided into two classes: typical ones which are irreducible and atypical ones whichare not. The latter are responsible for the logarithmic properties of the correlation functions.Determining a Verlinde formula [37] for Grothendieck (character) fusion in logarithmic CFTs isa key motivation of the standard-module formalism. Early attempts at applying the Verlindeformula to CFTs such as fractional-level WZW models resulted in negative integer fusion coef-ficients [38, 39]. The logarithmic nature of these theories was not realised until much later [9]and it took many more years before the subtle issue that led to the observed negative integerfusion coefficients was determined [40] and cured [41,42]. The standard-module formalism positsa Verlinde-like formula for the (Grothendieck) fusion coefficients of the modules of the CFT.Other Verlinde-like formulae for logarithmic CFTs have been studied in [43–49].A categorical understanding of the representation theory of logarithmic CFTs is an active area ofinvestigation, see [50] for example. The category will be non-finite and non-semisimple in general,yet admit a tensor structure with some generalised notion of modularity. The indecomposableprojective objects in such a category should then be the typical standard modules and theprojective covers of the atypical standard modules. It has been conjectured in many cases, see [51] for example, that these atypical projective covers are staggered modules. However, thishas so far only been established rigorously in a few cases [21, 24].In this paper, we use results on fractional-level b sl (2) models [9, 12, 40, 52–58], especially thoseestablished within the standard-module formalism [41, 42]. This includes the classification ofirreducible relaxed highest-weight modules [52, 56], the identification of the standard mod-ules [41, 42], the computation of their characters [58] and the determination of their modulartransformations and Grothendieck fusion rules [42]. We note that the latter are consistent withthe genuine fusion rules that have been computed for the models with levels k = − [9, 41] and k = − [12]. Moreover, the atypical staggered modules that arise in these models have beenstudied in detail.The b sl (2) minimal models and the N = 2 superconformal minimal models are related by a well-known coset (commutant) construction [59–61]. This coset realises the N = 2 superconformalminimal-model VOA M ( u, v ) as(1.1) M ( u, v ) = A ( u, v ) ⊗ bc H , u, v ∈ Z , u ≥ , v ≥ , gcd( u, v ) = 1 , where A ( u, v ) is the admissible-level b sl (2) minimal-model VOA of level k = − uv , bc is thefermionic ghost VOA and H is the Heisenberg VOA. In the case where k is non-integer, so v > M ( u, v ) minimal model is non-unitary with reducible but indecomposable modules [51].Recent work [62] on general Heisenberg cosets provides a concrete dictionary between the repre-sentation theories of the b sl (2) and N = 2 minimal models, in both the unitary and non-unitarycases. In particular, this dictionary is structure-preserving — the known (or conjectured) struc-tures of reducible yet indecomposable modules over the b sl (2) minimal-model algebras A ( u, v )lead to known (or conjectured) structures for the reducible yet indecomposable modules over M ( u, v ).This dictionary was explored in [51, 63], where the irreducible highest-weight modules of the N = 2 minimal-model VOAs M ( u, v ) were classified, their characters determined and their(Grothendieck) fusion rules computed. While this provides a lot of new knowledge about thesemodels, especially the non-unitary ones, it leaves many questions unanswered simply becausethe answers are not yet known for the b sl (2) minimal models. In particular, a proper studyof staggered M ( u, v )-modules is needed to improve our knowledge of the non-unitary N = 2minimal models and to provide new insights into their b sl (2) cousins.In this paper, building on the results of [51,62,63], we present the first investigation of staggeredmodules over the N = 2 superconformal algebras. We begin in Section 2 with an introductionto the N = 2 superconformal algebras, their highest-weight representation theory, and theirconjugation and spectral-flow morphisms. In Section 3, we introduce the rank-2 staggeredmodules discussed conjecturally in [51]. We recall their Loewy diagrams and introduce thestructure parameters known as logarithmic couplings [8, 64] in the N = 2 setting. Section 4presents explicit examples of staggered modules for the minimal model M (3 , b sl (2) minimal model A (3 ,
2) (level k = − ) is one of thebest understood logarithmic CFTs, see [1, 12, 40, 41, 54, 55].In Section 5, we discuss the effects of the conjugation and spectral-flow morphisms on general N = 2 staggered modules. We verify that the N = 2 staggered-module families producedby the coset branching rules are closed under the action of spectral flow and describe howspectral flow affects the logarithmic couplings. Moreover, we demonstrate a symmetry betweenparticular parameter values that allows identification of Loewy diagrams between staggeredmodule families.Section 6 gives two further explicit examples of staggered modules in the minimal model M (2 , b sl (2) minimal model A (2 ,
3) for which the staggeredmodules have been constructed [9, 10, 41]. Finally, Section 7 addresses a family of staggered
TAGGERED MODULES OF N = 2 SUPERCONFORMAL MINIMAL MODELS 3 modules in which the vacuum module of M ( u, v ) appears as a submodule (and as a quotient).Such a family exists in all non-unitary N = 2 minimal models M ( u, v ) and we refer to itsmembers as the vacuum staggered modules. These modules are closely tied to the Neveu-Schwarz sector modules investigated in Sections 4 and 6. We determine the structures of thesemodules in general and note that their spectral flows provide illustrative examples of some ofthe general results described in Section 5.2. N = 2 superconformal algebras The N = 2 superconformal algebras are a family of vertex operator superalgebras (VOSAs)parametrised by the central charge c ∈ C . The algebras are strongly generated by four fields: T ( z ) , J ( z ) are bosonic, and G ± ( z ) are fermionic. The field T ( z ) is the Virasoro field and gener-ates conformal symmetry. The other generating fields are Virasoro primary fields with conformalweights 1 , , , respectively. The defining operator product expansions (OPEs) for the universal N = 2 VOSA at central charge c are given by(2.1) T ( z ) T ( w ) ∼ c ( z − w ) + 2 T ( w )( z − w ) + ∂T ( w ) z − w , J ( z ) J ( w ) ∼ c ( z − w ) ,T ( z ) J ( w ) ∼ J ( w )( z − w ) + ∂J ( w ) z − w , T ( z ) G ± ( w ) ∼ G ± ( w )( z − w ) + ∂G ± ( w ) z − w ,J ( z ) G ± ( w ) ∼ ± G ± ( w ) z − w , G ± ( z ) G ∓ ( w ) ∼ c ( z − w ) ± J ( w )( z − w ) + 2 T ( w ) ± ∂J ( w ) z − w ,G ± ( z ) G ± ( w ) ∼ , where is the identity field corresponding to the vacuum vector.The universal N = 2 VOSA is non-simple if and only if [66](2.2) c = 3 (cid:18) − t (cid:19) , t = uv , gcd( u, v ) = 1 , u ∈ Z ≥ , v ∈ Z ≥ . We are primarily interested in the case in which the universal N = 2 VOSA is not simple. Werefer to the unique simple quotient of the universal VOSA as the N = 2 minimal-model algebraand denote it by M ( u, v ).The modes of the generating fields form an infinite-dimensional Lie superalgebra. Several choicesof mode labelling are possible, depending on the boundary conditions chosen for the fields inthe theory. In this paper, we focus on two choices: the Neveu-Schwarz algebra(2.3) NS = span C { L n , J m , G + r , G − s , | n, m ∈ Z , r, s ∈ Z + } and the Ramond algebra(2.4) R = span C { L n , J m , G + r , G − s , | n, m ∈ Z , r, s ∈ Z } . In this setting, the element is a central element of the Lie superalgebra. The non-zero(anti)commutation relations defining the algebra are(2.5) [ L m , L n ] = ( m − n ) L m + n + c m ( m − δ m + n, , { G ± r , G ∓ s } = 2 L r + s ± ( r − s ) J r + s + c
12 (4 r − δ r + s, , [ L m , J n ] = − nJ m + n , [ J m , J n ] = c mδ m + n, , (cid:2) L m , G ± r (cid:3) = (cid:16) m − r (cid:17) G ± m + r , (cid:2) J m , G ± r (cid:3) = ± G ± m + r . C RAYMOND, D RIDOUT, AND J RASMUSSEN
Representation theory.
Here we introduce highest-weight modules over the N = 2 su-perconformal algebras. Both the Neveu-Schwarz and Ramond algebras admit a triangular de-composition(2.6) g = g − ⊕ g ⊕ g + . For the Neveu-Schwarz algebra, we have(2.7) g − = span C { G − n , G + n , L n , J n | n < } , g + = span C { G − n , G + n , L n , J n | n > } , g = span C { L , J , } ;and for the Ramond algebra,(2.8) g − = span C { G − m , G + n , L n , J n | m ≤ , n < } , g + = span C { G + m , G − n , L n , J n | m ≥ , n > } , g = span C { L , J , } . Note that for the Ramond algebra, we consider G − as a creation operator, and G +0 an annihilator.Consider the 1-dimensional module C • , ± j, ∆ over the subalgebra g ⊕ g + , labelled by j, ∆ ∈ C , • ∈{ NS , R } . The label ± denotes the parity of the module, which is discussed in the followingparagraph. The modes L , J , act on C • , ± j, ∆ as multiplication by ∆ , j,
1, respectively, and allother modes act trivially. One can induce from C • , ± j, ∆ to a Verma module of weight ( j, ∆) overthe Neveu-Schwarz or Ramond algebra, which we denote by V • , ± j, ∆ . We denote the highest-weightvector of V • , ± j, ∆ by | j, ∆ i ± . The Verma module has a unique maximal proper submodule. We let L • , ± j, ∆ denote the irreducible quotient of V • , ± j, ∆ by its maximal proper submodule.As modules over a superalgebra, the N = 2 Verma modules are Z -graded, meaning that V • , ± j, ∆ = V ⊕ V , where J n and L n preserve the grading while G ± n maps between V and V . Thesubspaces V and V are referred to as the even subspace and odd subspace, respectively. Theparity of the module, denoted by ± , is determined by whether the highest-weight vector is inthe even ( | j, ∆ i + ) or odd ( | j, ∆ i − ) subspace. There is a functor Π : V • , ± j, ∆ → V • , ∓ j, ∆ which changesthe parity of the module.A singular vector | χ i is a non-zero vector which is annihilated by the action of g + and is asimultaneous eigenvector of g . If a singular vector generates a proper submodule, then we referto it as a proper singular vector. Representations of the N = 2 superconformal algebras mayalso contain subsingular vectors. A subsingular vector is a vector | w i in a representation V which is not singular in V , but which becomes singular in the quotient of V by a submodule.The Verma modules of the Neveu-Schwarz and Ramond algebras can be equipped with a bilin-ear form given by the Shapovalov form [67]. The highest-weight vector | j, ∆ i ± is normalisedrelative to the highest-weight vector ± h j, ∆ | of the contragredient dual module (cid:16) V ± , • j, ∆ (cid:17) ∗ suchthat ± h j, ∆ | j, ∆ i ± = 1. Moreover, the adjoint is given by(2.9) L † n = L − n , J † n = J − n , (cid:0) G ± r (cid:1) † = G ∓− r , † = . Proper submodules are null with respect to this form.A well-known result of [68] is the determinant formula for the Gram matrix of inner products forVerma modules over both the Neveu-Schwarz and Ramond algebras. For Neveu-Schwarz Vermamodules, the determinant for the Gram matrix at grade n is given by(2.10) det n ( V NS , ± j, ∆ ) = Y r ∈ Z > Y s ∈ Z > ≤ rs ≤ n (cid:0) f NS r,s (cid:1) P NS ( n − rs/ ,m ) · Y q ∈ Z + (cid:0) g NS q (cid:1) ˜ P NS ( n −| q | ,m − sgn( q ); q ) , TAGGERED MODULES OF N = 2 SUPERCONFORMAL MINIMAL MODELS 5 where the functions f r,s and g q are given by(2.11) f NS r,s ( j, ∆ , c ( t )) = ( st − r ) t − j − t − t , and(2.12) g NS q ( j, ∆ , c ( t )) = 2∆ − qj − t (cid:18) q − (cid:19) , with c ( t ) given as in (2.2). The functions P NS and ˜ P NS are partition functions whose preciseform, which we shall not need, are given in [68]. The vanishing of f NS r,s implies the existence of asingular vector with ( J , L )-eigenvalue ( j, ∆ + rs ), referred to as an uncharged singular vector.Likewise, a vanishing g NS q implies the existence of a singular vector with weight ( j +sgn( q ) , ∆+ q ),referred to as a charged singular vector.In the Ramond sector, the determinant formula is given by(2.13) det n ( V R , ± j, ∆ ) = Y r ∈ Z > Y s ∈ Z > ≤ rs ≤ n (cid:0) f R r,s (cid:1) P R ( n − rs/ ,m ) · Y q ∈ Z (cid:0) g R q (cid:1) ˜ P R ( n −| q | ,m − sgn( q ); q ) , where(2.14) f R r,s ( j, ∆ , c ( t )) = s − (cid:18) j − (cid:19) + 1 t (cid:18) − − rs (cid:19) + 1 t (cid:0) r − (cid:1) , and(2.15) g R q = 2∆ − q t − q (cid:18) j − (cid:19) + 12 t − . Again, the forms of the partition functions P R and ˜ P R may be found in [68]. The solutions to thevanishing equations imply the existence of singular vectors in the same way as for Neveu-Schwarzmodules.We remark that submodules of N = 2 Verma modules are not necessarily Verma modules. Anexample of this, which is relevant to our later discussions, is the Verma module V NS , +0 , withhighest-weight vector | , i + . Setting ( j, ∆) = (0 , g NS q vanishes for q = ± ,implying that the vectors G ±− / | , i + are singular. In the submodule generated by | w i = G + − / | , i + , there is no corresponding G + − / | w i , as (cid:0) G + − / (cid:1) = 0 in the universal envelopingalgebra. Thus, this submodule is not isomorphic to the Verma module V NS , − , / . Similar argumentshold for G −− / | , i + .2.2. Automorphisms.
The N = 2 superconformal algebras admit a number of well-studiedautomorphisms [69]. Here, we are interested in the conjugation automorphism denoted by γ ,and the spectral-flow family of automorphisms denoted by σ ℓ , for ℓ ∈ Z .The conjugation automorphism acts on the modes of the generating fields as γ ( L n ) = L n , γ ( J n ) = − J n , γ ( G ± r ) = G ∓ r , γ ( ) = , (2.16)while the action of spectral flow is given by σ ℓ ( L n ) = L n − ℓJ n + 16 ℓ δ n, c , σ ℓ ( J n ) = J n + ℓ δ n, c , σ ℓ ( G ± r ) = G ± r ∓ ℓ , σ ℓ ( ) = . (2.17)Spectral flow can be extended to ℓ ∈ Z . Non-integral parameters then define a family ofisomorphisms between the Neveu-Schwarz and Ramond algebras.We remark that spectral flow is not an automorphism of the corresponding VOSA (rather, onlythe vertex superalgebra), as it does not preserve the conformal vector L − | , i + , where | , i + is the vacuum vector of the vertex algebra. C RAYMOND, D RIDOUT, AND J RASMUSSEN
Modules over an algebra may be twisted by the action of an automorphism and are then referredto as twisted modules. Suppose we have a vector-space isomorphism ξ from a given g -module M to a new vector space ξ ( M ). This becomes a twisted module upon endowing it with the actionof g given by(2.18) xξ (cid:0) | v i (cid:1) = ξ (cid:0) ω − ( x ) | v i (cid:1) , for x ∈ g , | v i ∈ M, where ω is an automorphism of g . As ξ merely serves to distinguish the module M from its twist,which will not be isomorphic to M in general, we may replace ξ by the algebra automorphism ω . Moreover, we will refer to γ ( M ) and σ ℓ ( M ) as the conjugation of M and the spectral flow of M , respectively. Thus, the twisting action of the spectral-flow automorphisms will be writtenas(2.19) xσ ℓ (cid:0) | v i (cid:1) = σ ℓ (cid:0) σ − ℓ ( x ) | v i (cid:1) , x ∈ g . Combining (2.18) with (2.16) and (2.17), we can identify the action of these automorphisms onthe irreducible highest-weight modules, as in [51]. For conjugate modules, we have(2.20) γ (cid:16) L NS , ± j, ∆ (cid:17) ∼ = L NS , ±− j ;∆ , γ (cid:16) L R , ± j ;∆ (cid:17) ∼ = L R , ±− j ;∆ , if ∆ = c ,L R , ∓− j +1 , ∆ , otherwise , while for spectral flows of modules, we have σ (cid:16) L NS , ± j, ∆ (cid:17) ∼ = L R , ± j + c , ∆+ j + c , σ (cid:16) L R , ± j, ∆ (cid:17) ∼ = L NS , ± j + c , j + c , if ∆ = c ,L NS , ∓ j − c , ∆+ j − + c , otherwise . (2.21)For Ramond sector modules, the case ∆ = c is distinguished, as the vector G − | j, c i ∈ V R , ± j, c is singular. This is easily verified by computing(2.22) G +0 G − | j, c i = (cid:16) L − c (cid:17) | j, c i . Coset construction and staggered modules
We are interested in staggered modules over the N = 2 minimal-model algebras M ( u, v ). Fol-lowing [1, 64], we define a staggered module P over M ( u, v ) to be an indecomposable moduledescribed by a non-split short exact sequence(3.1) 0 → M L → P → M R → , where M L and M R are highest-weight M ( u, v )-modules, and the action of L on P exhibitsJordan blocks of rank 2 but not higher. The staggered modules we investigate arise throughthe coset construction (1.1) of the N = 2 minimal models as counterparts to staggered modulesover the b sl (2) minimal models A ( u, v ).3.1. Coset ingredients.
To better understand the background and notation, we begin thissection by giving a brief introduction to the coset construction of the non-unitary N = 2 minimalmodels, as described in [51]. Our review will not be comprehensive, focusing only on those resultswhich are integral to the presentation of our results. We give a brief introduction to the VOSAswhich appear in the particular coset, as well as the relevant representation theories. We mentionthat the authors of [51] analysed both the unitary and non-unitary minimal models of the N = 2superconformal algebras constructed via the coset (1.1).The Heisenberg VOA H is generated by a single field a ( z ) with defining OPE(3.2) a ( z ) a ( w ) ∼ t ( z − w ) , TAGGERED MODULES OF N = 2 SUPERCONFORMAL MINIMAL MODELS 7 where t ∈ C × is identified with the constant t introduced in (2.2). The commutation relationsbetween modes of the generating field are(3.3) [ a m , a n ] = 2 tmδ m + n, , m, n ∈ Z . Using this particular normalisation, the energy-momentum tensor of the theory is given by(3.4) T H ( z ) = 14 t : aa : ( z ) . This field generates a Virasoro algebra with central charge 1, and a ( z ) is a primary field ofconformal weight 1.The highest-weight modules of H are the Fock spaces of charge p ∈ C , denoted by F p , withhighest-weight vector | p i . The action of the zero-modes on | p i is given by(3.5) a | p i = p | p i , | p i = | p i , a n | p i = 0 , ∀ n ≥ . The Fock spaces are irreducible for all p . The module F is the vacuum module.The bc -ghost system is a VOSA generated by the two fermionic fields b ( z ) and c ( z ) with definingOPEs(3.6) b ( z ) c ( w ) ∼ z − w , b ( z ) b ( w ) ∼ c ( z ) c ( w ) ∼ . The anti-commutation relations between modes of the generating fields are(3.7) { b m , c n } = δ m + n, , { b m , b n } = { c m , c n } = 0 , where m, n ∈ Z + in the Neveu-Schwarz sector, and m, n ∈ Z in the Ramond sector. Thecorresponding energy-momentum tensor is(3.8) T bc ( z ) = (cid:0) : ∂bc : ( z ) − : b∂c : ( z ) (cid:1) . It generates a Virasoro algebra with central charge 1, and the fields b ( z ) and c ( z ) are primaryfields of conformal weight . The Heisenberg field Q ( z ) = : bc : ( z ) acts as a charge operator withOPEs(3.9) Q ( z ) b ( w ) ∼ b ( w ) z − w , Q ( z ) c ( w ) ∼ − c ( w ) z − w . Up to isomorphism, there are four highest-weight modules of bc , denoted by N i , i ∈ { , , , } ,all of which are irreducible. The vacuum module is N , a Neveu-Schwarz sector module whosehighest-weight vector has Q -eigenvalue 0 and conformal dimension 0. The highest-weight vec-tor of the Ramond Verma module N has Q -eigenvalue and conformal dimension . Theremaining modules are obtained by applying parity reversal: Π( N ) = N and Π( N ) = N .The VOAs associated to the affine Lie algebra b sl (2) are generated by the bosonic fields e ( z ), h ( z ), and f ( z ), and have defining OPEs h ( z ) e ( w ) ∼ e ( w ) z − w , h ( z ) h ( w ) ∼ k ( z − w ) , h ( z ) f ( w ) ∼ − f ( w ) z − w ,e ( z ) f ( w ) ∼ k ( z − w ) + h ( w ) z − w , e ( z ) e ( w ) ∼ f ( z ) f ( w ) ∼ , (3.10)where k ∈ C \ {− } is the level of the VOA. The resulting commutation relations between modesof the generating fields are given by[ h m , e n ] = 2 e m + n , [ h m , h n ] = 2 kmδ m + n, , [ h m , f n ] = − f m + n , [ e m , f n ] = h m + n + kmδ m + n, , [ e m , e n ] = [ f m , f n ] = 0 . (3.11)The energy-momentum tensor is given by(3.12) T b sl (2) ( z ) = 12 t (cid:18)
12 : hh : ( z ) + : ef : ( z ) + : f e : ( z ) (cid:19) , C RAYMOND, D RIDOUT, AND J RASMUSSEN where t = k + 2 is the same t as introduced for the Heisenberg algebra. We denote the modesof the field T b sl (2) ( z ) by L b sl (2) m with m ∈ Z . The field T b sl (2) generates a Virasoro algebra withcentral charge(3.13) c = 3 (cid:18) − t (cid:19) . The universal VOA associated to b sl (2) is non-simple if and only if t = uv for gcd( u, v ) = 1 , u ∈ Z ≥ , and v ∈ Z ≥ [65, 66]. The corresponding simple quotient for these values of t is known asan b sl (2) minimal model; we denote it by A ( u, v ).If v = 1, the level k is a non-negative integer and the corresponding minimal model is unitary.For v >
1, the resulting models are non-unitary. Here, we focus on the irreducible highest-weight modules over the non-unitary minimal-model algebras A ( u, v ), as it is these that areglued together to form staggered modules. A complete classification of positive-energy A ( u, v )-modules includes the so-called relaxed modules [56, 61, 70] which were classified in [52]; however,we will not need these modules here.The irreducible positive-energy representations are labelled by the h - and L b sl (2)0 -eigenvalues,denoted by λ r,s and ∆ aff r,s , respectively, of their highest-weight vectors. The classification of theserepresentations was given in [52]. We use the following standard parametrisations for λ r,s and∆ aff r,s :(3.14) λ r,s = r − − st, ∆ aff r,s = ( r − st ) − t , r, s ∈ Z . In this paper, we focus on the following classes of modules: • The irreducible highest-weight modules, denoted by L r, , where 1 ≤ r ≤ u −
1, withhighest weights ( λ r, , ∆ aff r, ). As λ r, ∈ Z ≥ for all r ≥
1, the space of lowest-energy(minimal L b sl (2)0 − eigenvalue) states forms a finite-dimensional representation over the sl (2) subalgebra spanned by the modes { h , e , f } . The vacuum module of the VOA is L , . • The irreducible highest-weight modules D + r,s , where 1 ≤ r ≤ u − ≤ s ≤ v − λ r,s , ∆ aff r,s ). In this case, λ r,s / ∈ Z and the space of lowest-energystates forms an infinite-dimensional irreducible Verma module over the sl (2) subalgebra. • The irreducible modules D − r,s , where 1 ≤ r ≤ u − ≤ s ≤ v −
1. These modulesare not highest-weight modules; rather, they are conjugate to the modules D + r,s . Corre-spondingly, the space of lowest-energy states forms an infinite-dimensional lowest-weightrepresentation over the sl (2) subalgebra, with lowest weight − λ r,s .3.2. Coset decompositions.
For L r, ⊗ N i and D ± r,s ⊗ N i , representations over A ( u, v ) ⊗ bc ,we have the following branching rules for the restriction to the subalgebra H ⊗ M ( u, v ) [51]:(3.15) ( L r, ⊗ N i ) ↓ ∼ = M p ∈ λ r, + i +2 Z F p ⊗ [ i ] C p ; r, , (cid:0) D ± r,s ⊗ N i (cid:1) ↓ ∼ = M p ∈ λ r,s + i +2 Z F p ⊗ [ i ] C p ; r,s , where we recall that i ∈ { , , , } , F p is the Fock space of weight p , and [ i ] C p ; r,s is an irreduciblerepresentation of M ( u, v ). The irreducibility of the representation [ i ] C p ; r,s follows from the resultsof [62].Through the coset, each irreducible A ( u, v )-module gives rise to an infinite family of irreducible M ( u, v )-modules, labelled by the weight p of its partner Heisenberg Fock space.The authors of [51] give a dictionary for translating between irreducible M ( u, v )-modules inthe notation [ i ] C p ; r,s and those introduced in Section 2, denoted by L • , ± j, ∆ . For the irreducible TAGGERED MODULES OF N = 2 SUPERCONFORMAL MINIMAL MODELS 9 L -type A ( u, v )-modules (those for which s = 0), the corresponding modules which appear inthe branching rule are given by [0] C p ; r, ∼ = L NS , • j, ∆ , p ∈ λ r, + 2 Z , • = − , j = pt + 1 , ∆ = ∆ N =2 p ; r, − p + r , p ≤ − r − , • = + , j = pt , ∆ = ∆ N =2 p ; r, , − r ≤ p ≤ r − , • = − , j = pt − , ∆ = ∆ N =2 p ; r, + p − r , p ≥ r + 1 , [1] C p ; r, ∼ = L R , • j, ∆ , p ∈ λ r, + 1 + 2 Z , • = + , j = pt + , ∆ = ∆ N =2 p ; r, − p + r + , p ≤ − r − , • = − , j = pt + , ∆ = ∆ N =2 p ; r, + , − r ≤ p ≤ r − , • = + , j = pt − , ∆ = ∆ N =2 p ; r, + p − r + , p ≥ r, where we have made use of(3.16) ∆ N =2 p ; r,s = ( r − st ) − t − p t . For s = 0 ( D + -type modules), the dictionary is given by [0] C p ; r,s ∼ = L NS , • j, ∆ , p ∈ λ r,s + 2 Z , ( • = + , j = pt , ∆ = ∆ N =2 p ; r,s , p ≤ λ r,s , • = − , j = pt − , ∆ = ∆ N =2 p ; r,s + p − λ r,s − , p ≥ λ r,s + 2 , [1] C p ; r,s ∼ = L R , • j, ∆ , p ∈ λ r,s + 1 + 2 Z , ( • = − , j = pt + , ∆ = ∆ N =2 p ; r,s + , p ≤ λ r,s − , • = + , j = pt − , ∆ = ∆ N =2 p ; r,s + p − λ r,s − + , p ≥ λ r,s + 1 . As the modules with i = 2 , i = 0 ,
1, respectively, theirdictionaries follow from those given above.For the b sl (2) minimal models A ( u, v ), staggered modules have been explicitly constructed asfusion products for ( u, v ) = (3 ,
2) and (2 , b sl (2), see [13]. Under the coset,each A ( u, v ) staggered module gives rise to a family of M ( u, v ) staggered modules, denoted by [ i ] P p ; r,s . The M ( u, v ) staggered modules are labelled by p , the weight of the accompanying Fockspace, i the label of the bc -ghost system irreducible module, and r, s where 1 ≤ r ≤ u − ≤ s ≤ v − [ i ] P p ; r,s displays the irreducible component modules of [ i ] P p ; r,s , witharrows describing the action of the algebra which “glues” component modules together. Fromthe conjectured Loewy diagrams of the staggered A ( u, v )-modules [20], we easily obtain thoseof the [ i ] P p ; r,s , using the general theory of [62]. These diagrams are presented in Figure 1.In order for the module dictionary to be compatible with these Loewy diagrams for all possible r, s, i, p , we require the additional isomorphisms (see [51])(3.17) [ i ] C p ; r, − ∼ = [ i +2] C p + t ; u − r,v − , [ i ] C p ; r,v ∼ = [ i − C p − t ; u − r, . The results of [62] imply that the spectral flow and conjugation of an M ( u, v ) staggered module isdetermined by twisting the irreducible component modules by the automorphism, maintainingthe gluing structure of the Loewy diagram. Using this fact, the authors of [51] deduce thefollowing isomorphisms between irreducible M ( u, v )-modules:(3.18) γ (cid:16) [ i ] C p ; r,s (cid:17) ∼ = [ − i ] C − p ; r,s , σ ℓ (cid:16) [ i ] C p ; r,s (cid:17) ∼ = [ i − ℓ ] C p − ℓ ; r,s . Here, the argument of [ · ] is taken modulo 4 and the parity reversal functor acts as Π (cid:0) [ i ] C p ; r,s (cid:1) = [ i +2] C p ; r,s .Following Figure 1, we denote the images of the highest-weight vectors of component modulesin the staggered module by | v h i , | v ℓ i , | v r i , | v s i for the head, left, right, and socle modules,respectively. We remark that these vectors are not necessarily highest-weight vectors in thestaggered module. Our diagrammatic convention is that J eigenvalues increase to the right,and L eigenvalues increase down the page. i ] P p ; r,s [ i − C p − t ; r,s +1[ i ] C p ; r,s [ i +2] C p + t ; r,s − i ] C p ; r,s Figure 1.
The Loewy diagram for a general staggered module [ i ] P p ; r,s . The com-ponent modules related by arrows are irreducible M ( u, v ) -modules. All arrowsare one way and indicate that the action of the algebra can map from states inone component module to states in another, but not back. We will refer to thealgebra action which maps between component modules as gluing. We refer to theirreducible submodule [ i ] C p ; r,s as the socle. The component module [ i ] C p ; r,s fromwhich all arrows point outward is referred to as the head. The remaining compo-nent modules, namely [ i − C p − t ; r,s +1 and [ i +2] C p + t ; r,s − , are referred to as the leftand right modules according to the J -weight of their component highest-weightvectors relative to that of the socle. Logarithmic couplings.
The structure of a staggered module may not be completely de-scribed by its Loewy diagram. Accordingly, we require information about the gluing of the headto the left and right modules. This is described by parameters known as logarithmic couplings .The value of these parameters for a given staggered module, along with its Loewy diagram, isbelieved to be sufficient to characterise the module up to isomorphism. (This “complete in-variant” property has only been proven rigorously for Virasoro staggered modules [64].) Wemotivate and define the logarithmic couplings in the remainder of this section.The non-diagonalisability of the Virasoro zero mode L on staggered M ( u, v )-modules is cap-tured by the relation(3.19) L | v h i = ∆ h | v h i + | v s i , exhibiting the vector | v h i as the logarithmic partner of | v s i . This relation does not define | v h i in the staggered module uniquely. There is a “gauge freedom” whereby we are free to add anyweight vector in the weight space of | v s i to | v h i , without changing the above relation.We are able to introduce a Shapovalov-like bilinear form on the left and right modules, nor-malised such that h v ℓ | v ℓ i = h v r | v r i = 1, and with adjoint given as in (2.9). This form can beextended to the indecomposable module formed by including the socle as a submodule of the leftand right modules. The form is therefore zero on the socle, as a proper submodule. However,as | v h i is only defined up to multiples of | v s i and other vectors in that weight space, we cannotnormalise the form on the head. However, as we shall see below, it is possible to partially extendthis form so that one of the vectors is associated to the head.Let U ( g ) denote the universal enveloping algebra of the Neveu-Schwarz or Ramond algebra. Thevector | v s i satisfies(3.20) U ℓ | v ℓ i = | v s i , U r | v r i = | v s i , for some U ℓ , U r ∈ U ( g ), such that U ℓ | j ℓ , ∆ ℓ i = U r | j r , ∆ r i = 0 in the corresponding irreduciblecomponent modules.The logarithmic couplings are then defined by the relations(3.21) U † ℓ | v h i = β ℓ | v ℓ i , U † r | v h i = β r | v r i , β ℓ , β r ∈ C , TAGGERED MODULES OF N = 2 SUPERCONFORMAL MINIMAL MODELS 11 with adjoint defined as in (2.9). Hence, the logarithmic couplings can be understood as gluingparameters between the head module and the left/right modules. Alternatively, in terms of theintroduced form,(3.22) h v s | v h i = h v ℓ | U † ℓ | v h i = β ℓ , h v s | v h i = h v r | U † r | v h i = β r . The motivation for defining the logarithmic couplings in this way is that they are invariant underthe “gauge” transformations discussed earlier. As an example to demonstrate this fact, considerthe simple gauge transformation | v h i 7→ | v h i + α | v s i for some α ∈ C . Then we have that(3.23) h v r | U † r ( | v h i + α | v s i ) = h v r | U † r | v h i + α h v r | U † r | v s i = β r + 0 , and a similar result follows for β ℓ .To determine the numerical values of the logarithmic couplings (see also [71]), one uses con-straints coming from vanishing singular vectors in the head module, which lift to relationsbetween vectors in the staggered module [72]. These constraints allow one to determine the ac-tion of the algebra on the staggered module. The logarithmic couplings, along with the Loewydiagram, are then expected to completely determine the structure of the module. The aboveprescription has been understood in the most detail for staggered modules over the Virasoroalgebra, where it was shown in [64] that the logarithmic couplings in fact parametrise the spaceof isomorphism classes of staggered modules. There is no similar result for the N = 2 algebrasas yet, but the use of logarithmic couplings is standard in logarithmic CFT.We remark that the choice of U ℓ , U r is only fixed up to normalisation, so the logarithmic couplingsdepend (in a trivial way) on this choice of normalisation. As such, when introducing a modulewe specify our choice of normalisation for U ℓ | v ℓ i = | v s i = U r | v r i , and hence U † ℓ , U † r .4. Staggered modules over M (3 , N = 2 superconformal algebras. We start with examples over the minimal-model VOA M (3 , c = −
1. The branching rules produce families of staggered modules [ i ] P p ; r,s ,labelled by p = λ r,s + i + 2 Z , for ( r, s ) = (1 , , (1 , , (2 , , (2 ,
1) and i ∈ { , , , } . We willlook at two examples in detail: [0] P , from the Neveu-Schwarz sector, and [1] P ;1 , from theRamond sector. These staggered modules are conjectured to be the projective covers of theirreducible highest-weight modules L NS , +0 , and L R , + , , respectively.4.1. The module [0] P , . We begin by defining the gluing action of the algebra. We startwith the fermionic vector | v r i of weight (cid:0) , − (cid:1) and normalise it so that h v r | v r i = 1. The vector G −− | , − i − is singular in the Verma module V NS , − , − and G −− | v r i generates the socle in [0] P , .We may thus choose the vector | v s i ∈ [0] P , to be | v s i = G −− | v r i .Similarly, we introduce a fermionic vector | v ℓ i , with weight (cid:0) − , − (cid:1) , such that G + − | v ℓ i = | v s i .We are free to normalise the product h v ℓ | v ℓ i = 1, as there is no operator x ∈ U ( g ) such that x | v r i = | v ℓ i . The vector | v s i also generates a proper submodule of the module generated by | v ℓ i , hence the form agrees (and is 0) on the intersection of the modules generated by | v ℓ i and | v r i .It remains to define the logarithmic partner of | v s i (up to gauge transformations). We choosethe action of L on | v h i to be(4.1) L | v h i = | v s i . The Loewy diagram of [0] P , and a diagram of the gluing action of the algebra are given inFigure 2. With these choices, the logarithmic couplings which are defined by the relations(4.2) G − | v h i = β ℓ | v ℓ i , G + | v h i = β r | v r i . P , L NS , − , − L NS , +0 , L NS , + − , − L NS , +0 , | v h i| v s i | v r i| v ℓ i (cid:0) − , − (cid:1) (cid:0) , − (cid:1) (0 , L G + G − G + − G −− Figure 2.
Loewy diagram and weight-space diagram of the module [0] P , . Both | v h i and | v s i have L eigenvalue , and J eigenvalue . Included in the weight-space diagram is the action of the algebra, with the chosen normalisation. The values of β ℓ and β r follow by determining the action of the algebra on the staggered module [0] P , . The quotiented singular vector relations of the head module lift to relations betweenvectors in the staggered module, which constrain the action of the algebra.The vectors G ±− |
0; 0 i + are singular in the Verma module V NS , +0 , , and have been set to 0 in theirreducible component modules L NS , +0 , . In the staggered module, this implies that(4.3) G ±− | v s i = 0 , as the socle component module is the submodule isomorphic to L NS , +0 , . Furthermore, the singularvectors of the irreducible component module corresponding to the head give rise to the followingrelations in the staggered module:(4.4) G + − | v h i = − ( α L − + α J − ) | v r i , G −− | v h i = − ( γ L − + γ J − ) | v ℓ i , for some α , α , γ , γ ∈ C , to be determined. Correspondingly, we introduce vectors of thestaggered module as follows:(4.5) | χ − i = G −− | v h i + ( γ L − + γ J − ) | v ℓ i , | χ + i = G + − | v h i + ( α L − + α J − ) | v r i . Relations resulting from the vanishing of these vectors then constrain the action of the algebraon [0] P , .To illustrate, acting with L on | χ ± i , we have(4.6) L | χ − i = L G −− | v h i + L ( γ L − + γ J − ) | v ℓ i = ( β ℓ − γ − γ ) | v ℓ i and(4.7) L | χ + i = L G + − | v h i + L ( α L − + α J − ) | v r i = ( β r − α + α ) | v r i , which leads to the relations(4.8) β ℓ − γ − γ = 0 , β r − α + α = 0 . Similarly, acting with J leads to(4.9) − β ℓ − γ − γ = 0 , β r + α − α = 0 . The relation which defines the logarithmic partner vector L | v h i = | v s i yields additional con-straints by observing that(4.10) L | v h i = (cid:0) { G + − , G − } + J (cid:1) | v h i = ⇒ β ℓ − α − α = 2 , TAGGERED MODULES OF N = 2 SUPERCONFORMAL MINIMAL MODELS 13[1] P ;1 , L R , − , − L R , + , L R , −− , − L R , + , | v h i| v s i | v r i| v ℓ i (cid:0) − , − (cid:1) (cid:0) , − (cid:1)(cid:0) , (cid:1) L U † r G − G + − U r Figure 3.
Loewy diagram and weight-space diagram for the module [1] P ;1 , .Here, U r = G −− − J − G − . and(4.11) L | v h i = (cid:0) { G −− , G + } − J (cid:1) | v h i = ⇒ β r − γ + γ = 2 . Hence, we have an inhomogeneous linear system of equations, the unique solution of which isgiven by(4.12) α = − , α = − , γ = − , γ = 1and(4.13) β ℓ = β r = . The logarithmic couplings β ℓ and β r then uniquely determine the action of the minimal-modelalgebra on the staggered module [0] P , , thereby fixing the structure of the module.4.2. The module [1] P ;1 , . Our second example is the staggered module [1] P ;1 , , which is amodule over the Ramond algebra. The Loewy diagram and weight-space diagram of the gluingaction are given in Figure 3.We begin with | v r i , a fermionic vector of weight (cid:0) , − (cid:1) , which we normalise such that h v r | v r i =1. The vector(4.14) (cid:0) G −− − J − G − (cid:1) | , − i − is subsingular in the Verma module V R , − , − , becoming singular after quotienting by the submodulegenerated by the singular vector(4.15) (cid:0) G + − G − + J − + L − (cid:1) | , − i − . We therefore set(4.16) | v s i = U r | v r i , U r = G −− − J − G − . We moreover define | v ℓ i by setting G + − | v ℓ i = | v s i and normalising it so that h v ℓ | v ℓ i = 1.Finally, the logarithmic partner of | v s i must satisfy(4.17) L | v h i = | v h i + | v s i . The logarithmic couplings are then defined by the relations(4.18) (cid:0) G +1 − G +0 J (cid:1) | v h i = β r | v r i , G − | v h i = β ℓ | v ℓ i . Solving for the logarithmic couplings proceeds as in the previous example. The Verma module V R , + , has charged singular vectors at level 1 with relative charges ±
1. These vectors are givenby(4.19) G + − | , i , (cid:0) L − G − + J − G − − G −− (cid:1) | , i . In the staggered module, we compute that the following vectors are zero:(4.20) | χ − i = (cid:0) L − G − + J − G − − G −− (cid:1) | v h i − (cid:0) J − − L − + J − (cid:1) | v ℓ i , | χ + i = G + − | v h i − (cid:0) G −− G + − + 1476 J − − L − + 1599 G + − G − + 246 J − (cid:1) | v r i . Using these relations, we determine that the generators { G − , G +0 , J } , which generate thepositive-mode subalgebra, act on | v h i as G +0 | v h i = − (cid:0) L − + J − (cid:1) | v r i , J | v h i = G − | v r i , G − | v h i = | v ℓ i . (4.21)We also have(4.22) U † r | v h i = (cid:0) G +1 − G +0 J (cid:1) | v h i = − J (cid:0) J − + 2 L − (cid:1) | v r i − G +0 G − | v r i = | v r i . Thus, the values of the logarithmic couplings for [1] P ;1 , are(4.23) β ℓ = 32303 , β r = 656909 . Symmetries of N = 2 staggered module families The examples above present the first concrete analyses of staggered modules over N = 2 su-perconformal algebras. However, the coset provides an infinite number of staggered modules [ i ] P p ; r,s , labelled by r, s, i, p , for each non-unitary minimal model. To better understand the fullspectrum of staggered modules, this section investigates their symmetries.First, we discuss spectral flows of staggered modules and describe the values of r, s, i, p for whichtwo staggered modules are related by spectral flow. Along with this, we analyse the action ofspectral flow on the value of the logarithmic couplings describing the module structure. Finally,we identify relations between Loewy diagrams of staggered modules for particular values of the r, s labels.5.1. Spectral flow.
We begin by considering the spectral flow of staggered modules. The resultsof [62] state that the spectral flow of a staggered module maintains the Loewy diagram; however,the component modules are spectral flows of the initial component modules. By applying (3.18)to [ i ] P p ; r,s , we determine that the modules σ ℓ (cid:0) [ i ] P p ; r,s (cid:1) and [ i − ℓ ] P p − ℓ ; r,s have the same Loewydiagrams. Recall that for [ i ] P p ; r,s , we have p ∈ λ r,s + i + 2 Z and that [ i ] is taken modulo 4. TheLoewy diagram for [ i ] P p ; r,s is given in Figure 1. We remark that equivalent Loewy diagrams arenot enough to determine an isomorphism of staggered modules.To each fixed choice of r, s , there are infinitely many staggered modules, labelled by i, p , obtainedby applying spectral flow (and parity reversal). We will refer to the set of modules related byspectral flow, for a choice of r, s , as a family .We would like to understand the effect of spectral flow on the logarithmic couplings. Considerthe relations (3.21) under the action of the spectral flow. Without loss of generality, we considerthe right module only, obtaining σ ℓ ( U r ) σ ℓ ( | v h i ) = σ ℓ ( U r | v h i ) = σ ℓ ( β r | v r i ) = β r σ ℓ ( | v r i ) . (5.1)Unfortunately, it is no longer guaranteed (in fact it can only occur rarely) that σ ℓ ( | v h i ) and σ ℓ ( | v r i ) are the highest-weight vectors of the spectral flow of the head and right component mod-ules, respectively. So the spectrally flowed relation does not describe the gluing of highest-weight TAGGERED MODULES OF N = 2 SUPERCONFORMAL MINIMAL MODELS 15 ◦• •• ◦• ••◦•• • ◦• ••◦•• • σ σ σ σ Figure 4.
The action of twisting by spectral flow on the relative weights of thecomponent modules. The highest-weight vectors of the component modules arerepresented by • , with the exception of that of the socle, which is represented by ◦ . The arrows demonstrate the action of the algebra gluing the component mod-ules. ◦ ◦ ◦ ◦•• β ˜ β βP σ ℓ ( P ) Figure 5.
The logarithmic coupling β between component highest-weight vectorsin the module P is mapped to a numerically identical coupling between extremalvectors in the module σ ℓ ( P ) . The actual logarithmic coupling ˜ β of σ ℓ ( P ) is de-termined by raising the relation described by β to a relation between componenthighest-weight vectors in σ ℓ ( P ) . vectors of the head- and right-component modules; rather, it describes the coupling between ex-tremal vectors of the component modules, so named [73] as they appear as the outermost vectorsof a module in a weight-space diagram.The extremal vectors of an N = 2 superconformal Verma module are the highest-weight vector | v i and the states(5.2) | x − n i = G −− n − · · · G −− G −− | v i , | x + n i = G + − n − · · · G + − G + − | v i , n ≥ , for Neveu-Schwarz modules, and(5.3) | x − n i = G −− n · · · G −− G − | v i , | x + n i = G + − n − · · · G + − G + − | v i , n ≥ , for Ramond modules. More formally, these are the states of minimal conformal dimension in each J -eigenspace. It is clear from the equations above that weight spaces containing an extremalvector are one-dimensional. It may occur that an extremal vector is (sub)singular in the Vermamodule, in which case, it is set to 0 in the irreducible module. The corresponding expressions forthe extremal vectors of the irreducible module are then a straightforward change of the indicesgiven in the above expressions. Clearly, twisting with spectral flow maps extremal vectors toextremal vectors [73].To determine the coupling between highest-weight vectors of component modules after spectralflow, we apply the adjoint raising generator to the coupled extremal vectors. As the action ofraising operators on the irreducible component modules is completely determined, the logarith-mic couplings of a module ( β ) and its spectral flow ( ˜ β ) are related by a calculable factor. Thisis presented diagrammatically in Figure 5.Consider the spectral flow σ ℓ ( | v i ) of the highest-weight vector | v i of a highest-weight module.Acting with a string of raising generators (without loss of generality, we consider a string of G + r modes), we have(5.4) G + G + · · · G + n − σ ℓ ( | v i ) = σ ℓ (cid:18) G + − ℓ G + − ℓ · · · G + n − ℓ − | v i (cid:19) . If ℓ ≥ n , then the action of the raising operators is generically non-zero and given as above. If n > ℓ , then the action must give zero. When ℓ = n , we act on an extremal vector with thelargest possible string of raising operators, such that the action is non-zero and the resultingvector is extremal. The resulting vector must be proportional to the highest-weight vector ofthe spectrally-flowed module.This determines exactly the element of U ( g ) one needs to act with to calculate the constant ofproportionality for the logarithmic couplings. For spectral flows with ℓ <
0, the same argumentholds with G − r modes, and this carries over straightforwardly to the Ramond sector.The implication is that it is sufficient to determine the logarithmic couplings for a single modulein any given staggered module family (that is, for one choice of r, s ). As the members of eachfamily are related by spectral flow, the logarithmic couplings of the member modules are allproportional to that of a representative and the constant of proportionality is calculated bycommuting the string of raising generators. We shall illustrate this with examples in Section 7.5.2. Label symmetry.
In the previous section we established that for each pair r, s , the stag-gered modules form a family related by spectral flow. Here, we attempt to determine symmetriesof the r, s labels which yield staggered modules with equivalent Loewy diagrams. We refer tothese symmetries as label symmetries .It is clear that two component modules [ i ] C p ; r ,s and [ i ] C p ; r ,s can label the same module L • , ± j, ∆ under the dictionary. We seek to understand for what values of the parameters p, r, s, i thecorresponding Loewy diagrams are equivalent up to parity of the component modules.For representations over the minimal-model algebra M (3 , r, s labels, namely ( r, s ) = (1 , , (1 , , (2 , , (2 , [2] P − , and [0] P − ;2 , ,as well as [0] P − , and [0] P − ;1 , , have equivalent Loewy diagrams.Moreover, we note that the modules [2] P − , and σ (cid:0) [0] P , (cid:1) , as well as [0] P − , and σ (cid:0) [0] P , (cid:1) ,have equivalent Loewy diagrams. This is suggestive of a symmetry between Loewy diagrams ofstaggered modules labelled by ( r,
0) and those labelled by ( u − r, v − σ r (cid:0) [0] P λ r, ; r, (cid:1) and [0] P λ u − r,v − ; u − r,v − are equivalent, where λ r, = r − λ u − r,v − = − r − t .We begin by identifying the Loewy diagrams of the modules σ r (cid:0) [0] P r − r, (cid:1) and [ − r ] P − r − r, ,using (3.18). We can see directly from Figure 6 that the left- and right-component modules of [ − r ] P − r − r, and [0] P − r − t ; u − r,v − are isomorphic. It remains to be shown that the head andsocle modules are also isomorphic, that is(5.5) [ − r ] C − r − r, ∼ = [0] C − r − t ; u − r,v − . Applying the module dictionary to [ − r ] C − r − r, , the weights of the corresponding module aregiven by(5.6) j = − r − t + 1 , ∆ = ∆ N =2 − r − r, + 12 = − r − t + 12 . Similarly, applying the module dictionary to [0] C − r − t ; u − r,v − , we have that the correspondingweights are given by(5.7) j = − r − tt , ∆ = ∆ N =2 − r − t ; u − r,v − = − r − t + 12 . TAGGERED MODULES OF N = 2 SUPERCONFORMAL MINIMAL MODELS 17[ − r ] P − r − r, − r +2] C − r − − t ; r, − r ] C − r − r, − r ] C − r − t ; u − r,v − − r ] C − r − r, P − r − t ; u − r,v − C − r − − t ; r, C − r − t ; u − r,v − C − r − t ; u − r,v − C − r − t ; u − r,v − Figure 6.
A comparison of Loewy diagrams for the staggered modules [0] P λ r, ; r, and [0] P − r − t ; u − r,v − . We have used the identifications [ − r +2] C − r − t ; r, − = [ − r ] C − r − t ; u − r,v − and [2] C − r − u − r,v = [0] C − r − − t ; r, to simplify the compo-nent modules so that the module dictionary can be applied. The weights of the module are indeed equal, establishing the isomorphism (5.5) and hence, anequivalence of the Loewy diagrams of the corresponding staggered module families.We remark that label symmetry can also be understood as arising via the coset. The equivalenceof Loewy diagrams arises as a consequence of the fact that the spectral flow of an L -type A ( u, v )-module gives rise to a D ± -type A ( u, v )-module.Although we have established that the Loewy diagrams are equivalent between module familieslabelled by ( r,
0) and ( u − r, v − Staggered modules over M (2 , N = 2 staggered modules which are coset counterparts to the A (2 ,
3) staggeredmodules analysed in [9, 41]. The minimal-model algebra M (2 ,
3) has c = −
6. There are threepossible labels: ( r, s ) = (1 , , (1 , , (1 , r, s ) = (1 , , The module [0] P , . We begin with the module [0] P , over the Neveu-Schwarz algebra.We choose vectors according to the same procedure as in the examples in Section 4, and presentthe definitions collectively to shorten the exposition: L | v h i = | v s i , G − | v h i = β ℓ | v ℓ i , G + | v h i = β r | v r i , G + − | v ℓ i = | v s i , G −− | v r i = | v s i . (6.1)The Loewy diagram and weight-space diagram are given in Figure 7.The quotiented singular vectors in the head component module are G ±− | v h i . Solving as inSection 4, the relations in the staggered module are(6.2) G −− | v ℓ i = (cid:0) L − − J − (cid:1) | v ℓ i , G + − | v h i = (cid:0) L − + J − (cid:1) | v r i . Using these relations, in conjunction with those coming from(6.3) { G + , G −− } | v h i = (2 L + J ) | v h i , { G + − , G − } | v h i = (2 L − J ) | v h i , we determine that(6.4) G − | v h i = − | v ℓ i , G + | v h i = − | v r i , and hence(6.5) β ℓ = β r = − . P , L NS , − , − L NS , +0 , L NS , −− , − L NS , +0 , | v h i| v s i | v r i| v ℓ i (cid:0) − , − (cid:1) (cid:0) , − (cid:1) (0 , L G + G − G + − G −− Figure 7.
The Loewy diagram and weight-space diagram for the module [0] P , . [1] P ;1 , L R, − , − L R, +0 , − L R, −− , − L R, +0 , − | v h i| v s i | v r i| v ℓ i (cid:0) − , − (cid:1) (cid:0) , − (cid:1)(cid:0) , − (cid:1) L G +0 G − G +0 G − Figure 8.
The Loewy diagram and weight-space diagram for the module [1] P ;1 , . The module [1] P ;1 , . For the family [ i ] P p ;1 , , we choose to explore the Ramond sectormodule [1] P ;1 , . The Loewy diagram and weight-space diagram for this module are presentedin Figure 8. In this staggered module, all the component modules have ∆ = c . The vector G − | v i , where | v i is the highest-weight vector, is singular in all Ramond sector Verma moduleswith ∆ = c .For the module [1] P ;1 , , we choose the gluing relations to be(6.6) L | v h i = − | v h i + | v s i , G − | v h i = β ℓ | v ℓ i , G +0 | v h i = β r | v r i ,G +0 | v ℓ i = | v s i , G − | v r i = | v s i . It is clear from weight-space considerations that(6.7) G − | v h i = J | v h i = 0 . As ∆ = c for the head module, the charged singular vector G − | v h i lifts to a relation on thestaggered module, namely G − | v h i = β ℓ | v ℓ i . The uncharged quotiented singular vector of thehead module appears relatively deep at grade 6 in the module. As such, determining this vectoris computationally challenging.We find a relation between the logarithmic couplings by considering(6.8) { G +0 , G − } | v h i = (cid:0) L + (cid:1) | v h i . Calculating, we have(6.9) (cid:0) L + (cid:1) | v h i = 2 | v s i TAGGERED MODULES OF N = 2 SUPERCONFORMAL MINIMAL MODELS 19 and(6.10) (cid:0) G +0 G − + G − G +0 (cid:1) | v h i = G +0 β ℓ | v ℓ i + G − β r | v r i = ( β ℓ + β r ) | v s i , arriving at(6.11) β ℓ + β r = 2 . The remaining constraint on the logarithmic couplings comes from the observation that themodule is conjugation invariant, that is, the Loewy diagrams of γ (cid:16) [1] P ;1 , (cid:17) and [1] P ;1 , areequivalent. Using (2.18) along with the equivalence of Loewy diagrams implies that we mayidentify β ℓ = β r . The action of conjugation simply interchanges their definitions, and maintainsthe relations which determine their values. Hence, we solve the system of coupled equations andfind(6.12) β ℓ = β r = 1 . The module [0] P , over M ( u, v )In Sections 4 and 6, we looked at the module [0] P , in the minimal-model algebra with c = − c = −
6, respectively. The weights of the component modules were equal in the two examples.In fact, the component modules have the same weights for all non-unitary c ( t ) (2.2). In thissection, we consider the module [0] P , for general u, v . As the head component module willbe the irreducible highest-weight module L NS , +0 , , we will refer to the staggered module [0] P , as the vacuum staggered module .The head component module of a vacuum staggered module has a quotiented singular vectorgiven by G ±− | v h i . Moreover, the vectors G ±− |∓ , − i − are singular in V NS , −∓ , − for all admissible c ( t ). Hence, for all vacuum staggered modules, we can choose the gluing action of the algebrato be(7.1) L | v h i = | v s i , G − | v h i = β r | v ℓ i , G + | v h i = β r | v r i , G + − | v ℓ i = | v s i , G −− | v r i = | v s i . As before, the singular vectors of the head module lift to relations in the staggered module:(7.2) G −− | v h i = ( γ L − + γ J − ) | v ℓ i , G + − | v h i = ( α L − + α J − ) | v r i . We compute the action of the same generators as in Section 4. For this example, the resultingrelations are dependent on c : β r − α + α = 0 , β r + α + c α = 0 ,β r − γ − γ = 0 , − β ℓ − γ + c γ = 0 ,β ℓ − α − α = 2 , β r − γ + γ = 2 . (7.3)Solving this system of equations yields(7.4) α = γ = 1 t , α = 1 , γ = − β ℓ = β r = t − t . As expected, the logarithmic couplings are indeed functions of t . Combining this result with ourprevious results, we have in principle determined the structure of the staggered module family [ i ] P p, , (and equivalently [ i ] P p + t,u − ,v − ) over the minimal-model algebras M ( u, v ).We can use these modules to better understand the action of spectral flow on the logarithmiccouplings. In Section 5, we noted that the logarithmic couplings change under spectral flow, byfactors which can be in principle calculated for any value of the flow parameter ℓ . The next twoexamples investigate this directly using the spectral flows of the vacuum staggered modules. σ (cid:0) [0] P , (cid:1) L R , − c +1 , c L R , + c , c L R , − c − , c − L R , + c , c | v h i| v s i | v r i| v ℓ i (cid:0) c − , c − (cid:1) (cid:0) c + 1 , c (cid:1)(cid:0) c , c (cid:1) L G +0 G − G + − G − Figure 9.
The Loewy diagram and weight-space diagram for σ (cid:0) [0] P , (cid:1) . The modules σ ± (cid:0) [0] P , (cid:1) . Applying σ to the vacuum staggered module [0] P , yieldsa staggered module over the Ramond algebra, with Loewy and weight-space diagram given inFigure 9.For this Ramond-algebra module, we choose the gluing relations and definitions of the logarith-mic couplings to be L | v h i = c | v h i + | v s i , G − | v h i = β ℓ | v ℓ i , G +0 | v h i = β r | v r i ,G + − | v ℓ i = | v s i , G − | v r i = | v s i . (7.6)Note that the right, head, and socle modules all have ∆ = c . We remark that this examplerealises the case where the spectral flow of highest-weight vectors of component modules (those inthe vacuum staggered module examples) are mapped to highest-weight vectors of spectral flowsof component modules. As noted in Section 5, spectral flow σ ℓ needs only preserve the propertyof being extremal, but it may happen, for sufficiently small values of ℓ , that highest-weightvectors are mapped to highest-weight vectors.The staggered module singular vectors are in this case of the form G − | v h i = ( γ L − + γ J − ) | v ℓ i , G + − | v h i = ( α L − + α J − ) | v r i . (7.7)The annihilation conditions of these vectors, along with relations given by(7.8) { G +0 , G − } | v h i = (cid:16) L − c (cid:17) | v h i , { G + − , G − } | v h i = (cid:16) L − J + c (cid:17) | v h i , lead to a linear system of equations for the unknown parameters. The resulting set of equationsagain has a solution for all admissible c ( t ), given by(7.9) α = γ = 1 t , α = 1 − t , γ = − − t and(7.10) β ℓ = β r = t − t . We see that indeed β ℓ , β r are unchanged under the action of σ , as expected from Section 5.1.We also consider the module σ − (cid:0) [0] P , (cid:1) for general admissible t . The Loewy diagram andweight-space diagram are given in Figure 10. We choose the following gluing: L | v h i = c | v h i + | v s i , G − | v h i = β ℓ | v ℓ i , G +1 | v h i = β r G − | v r i ,G +0 | v ℓ i = | v s i , G −− G − | v r i = | v s i . (7.11) TAGGERED MODULES OF N = 2 SUPERCONFORMAL MINIMAL MODELS 21 σ − (cid:0) [0] P , (cid:1) L R , −− − c , c L R , + − c , c L R , +2 − c , − c L R , + − c , c | v h i| v s i G − | v r i | v r i| v ℓ i (cid:0) − − c , c (cid:1) (cid:0) − c , c − (cid:1)(cid:0) − c , c (cid:1) G +0 L G +1 G − G +0 G −− G − Figure 10.
The Loewy diagram and weight-space diagram for σ − (cid:0) [0] P , (cid:1) . Note, we have set G +1 | v h i = β r G − | v r i . Applying spectral flow to the relation G + | v h i V = β r | v r i V in the vacuum staggered modules (subscript V denoting vacuum staggered modulevectors) yields(7.12) σ − (cid:0) G + | v h i V (cid:1) = σ − ( β r | v r i V ) = β r σ − ( | v r i V ) = G +1 σ − ( | v h i V ) , hence we expect to find β r = t − t . The gluing relation between right and socle componenthighest-weight vectors is given by G −− G − | v r i = | v s i . Recalling the notation ˜ β r of Section 5.1for the logarithmic coupling of the spectrally flowed module (see Figure 5), we have G +0 G +1 | v h i =˜ β r | v r i . We identify G +0 with the string of raising generators required to map from the imageof the highest-weight vector under spectral flow, to the highest-weight vector of the componentmodule.The relations coming from quotiented singular vectors are(7.13) G −− | v h i = ( γ L − + γ J − ) | v ℓ i , G +0 | v h i = (cid:0) α L − G − + α J − G − (cid:1) | v r i . We also require the relations coming from(7.14) { G +0 , G − } | v h i = (cid:16) L − c (cid:17) | v h i , { G +1 , G −− } | v h i = (cid:16) L + 2 J + c (cid:17) | v h i . Note that the second anti-commutation relation differs from that in the previous example.The solutions for the parameter functions are(7.15) α = γ = 1 t , γ = − t , α = 1 + 12 t , and(7.16) β ℓ = β r = t − t , confirming the expected value for β r . To calculate ˜ β r , we use that(7.17) G +0 G +1 | v h i = β r G +0 G − | v r i = t − t (cid:16) L − c (cid:17) | v r i = ( − t − t | v r i , from which it follows that(7.18) ˜ β r = − t − t . This confirms the general result of Section 5.1 that ˜ β r is related to β r , with the proportionalityconstant determined by the action of the raising operators: in this case, − Discussion
In this paper, we have presented a first investigation of staggered modules over the non-unitary N = 2 superconformal minimal-model algebras M ( u, v ). We have provided several concrete ex-amples, along with general results regarding symmetries of the spectral-flow families of staggeredmodules produced via the coset (1.1).A novel feature of the modules investigated here is the gluing of component modules into spacesgenerated by subsingular vectors, rather than simply those of singular vectors. By applying(2.21) to the weights of the component modules, one sees that this feature occurs generically instaggered M ( u, v )-modules. The difference between j -labels of the head/socle and either left orright irreducible component modules becomes 2. A difference of 2 is only possible if the gluingis into subsingular spaces, as all singular vectors for Neveu-Schwarz Verma modules appear with J -weight of {± , } relative to the Verma module highest-weight vector [68].We have assumed that the Loewy diagram in conjunction with the logarithmic couplings issufficient to describe the structure of a given N = 2 staggered module. This is well-motivatedby previous examples of logarithmic CFTs. In particular, the authors of [64] proved that thelogarithmic couplings of staggered modules over the Virasoro algebra in fact parametrise thespace of isomorphism classes of staggered modules. Pursuing a similar result for N = 2 staggeredmodules is a natural next step.The explicit examples of Sections 4, 6, and 7, provide strong evidence that the structure ofstaggered modules over the N = 2 minimal-model algebras M ( u, v ) is uniquely determined bytheir corresponding Loewy diagrams. If the Loewy diagrams were sufficient to determine thestructure, label symmetry would determine an isomorphism of staggered modules (and hence,staggered module families).This isomorphism of staggered modules would also follow if [ i ] P p ; r,s were proven to be the pro-jective cover of the irreducible module [ i ] C p ; r,s , as is conjectured [51]. As projective covers areunique, up to isomorphism, equivalence of Loewy diagrams (or simply isomorphisms betweenhead modules) implies the desired isomorphism. Should such an isomorphism exist, then theexplicit examples of modules provided in Sections 4 and 6 would be sufficient to understand thestructure of all staggered modules over M (3 ,
2) and M (2 ,
3) arising via the coset.In the paper [51], the authors give conjectural fusion rules for the atypical modules over M ( u, v )minimal-model algebras, coming from similar results for atypical A ( u, v )-modules. With theconcrete realisations of staggered modules over the N = 2 algebras presented here, anothernatural step would be to check that these staggered modules do indeed arise in these fusionrules for M (3 ,
2) and M (2 , A ( u, v ) minimal models also involve spectral flows ofrelaxed highest-weight modules [42]. Moreover, staggered modules over A (3 ,
2) and A (2 , A ( u, v ) minimal models using the standard module formalism requires modules that arenot positive-energy.The coset counterpart N = 2 modules are necessarily positive-energy as the fermionic generatorsof the N = 2 algebra square to zero. As such, it may be easier to determine properties ofinterest in the N = 2 setting, for instance genuine fusion rules and whether staggered modulesare projective. These results would then lift to similar results for the A ( u, v ) minimal models,by the results of [51, 62, 74]. TAGGERED MODULES OF N = 2 SUPERCONFORMAL MINIMAL MODELS 23 Acknowledgements
CR was supported by a University of Queensland Research Award and by the Australian Re-search Council under the Discovery Project scheme, numbers DP170103265 and DP200100067.DR’s research is supported by the Australian Research Council Discovery Project DP160101520.JR was supported by the Australian Research Council under the Discovery Project scheme,project number DP160101376.
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Mathematical Sciences Institute, Australian National University, Acton,Australia, 2601.
Email address : [email protected] (David Ridout) School of Mathematics and Statistics, University of Melbourne, Parkville,Australia, 3010.
Email address : [email protected] (Jørgen Rasmussen) School of Mathematics and Physics, The University of Queensland, St Lucia,Australia, 4072.
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