Families of Singular and Subsingular Vectors of the Topological N=2 Superconformal Algebra

Transcription

1 IMAFF-9/40, NIKHEF hep-th/ Famiies of Singuar and Subsinguar Vectors of the Topoogica N=2 Superconforma Agebra Beatriz ato-rivera a,b and Jose Ignacio Rosado a a Instituto de Matemáticas y Física Fundamenta, CSIC, Serrano 12, Madrid 2800, Spain b NIKHEF-H, Kruisaan 409, NL-1098 SJ Amsterdam, The Netherands ABSTRACT We anayze severa issues concerning the singuar vectors of the Topoogica N=2 Superconforma agebra. First we propose an agebraic mechanism to decide which types of singuar vectors exist, regarding the reative U(1) charge and the BRST-invariance properties, finding four different types in chira (incompete) Verma modues and thirty-three different types in compete Verma modues. Then we investigate the famiy structure of the singuar vectors, every member of a famiy being mapped to any other member by a chain of simpe transformations invoving the spectra fows. The famiies of singuar vectors in chira Verma modues foow a unique pattern (four vectors) and contain subsinguar vectors. We write down these famiies unti eve, identifying the subsinguar vectors. The famiies of singuar vectors in compete Verma modues foow infinitey many different patterns, grouped roughy in six main kinds. We present a particuary interesting tweve-member famiy at eves and 4, as we as the compete set of singuar vectors at eve 1 (twenty-eight different types). Finay we anayze the Dörrzapf conditions eading to two ineary independent singuar vectors of the same type, at the same eve in the same Verma modue, and we write down four exampes of those pairs of singuar vectors, which beong to the same tweve-member famiy. December 199 e-mai addresses: bgato,

2 1 Introduction In the ast few years, singuar vectors of infinite dimensiona agebras corresponding to conforma fied theories are drawing much attention. Far from being empty objects that one simpy woud ike to get rid of, they rather contain an amazing amount of usefu information. For exampe, as a genera feature, their decouping from a other states in the corresponding Verma modue gives rise to differentia equations which can be soved for correators of conforma fieds. Aso, their possibe vanishing in the Fock space of the theories is directy connected with the existence of extra states in the Hibert space that are not primary and not secondary (not incuded in any Verma modues) [1]. In some specific theories the corresponding singuar vectors are, in addition, directy reated to Lian-Zuckermann states [2], [], etc... Regarding the construction of singuar vectors, using either the fusion method or the anaytic continuation method, expicit genera expressions have been obtained for the singuar vectors of the Virasoro agebra [4], the S(2) Kac-Moody agebra [5], the Affine agebra A (1) 1 [], the N=1 Superconforma agebra [7], the Antiperiodic N=2 Superconforma agebra [8], [9], and some W agebras [10]. There is aso the method of construction of singuar vertex operators, which produce singuar vectors when acting on the vacuum [11]. In some cases it is possibe to transform singuar vectors of an agebra into singuar vectors of the same or a different agebra, simpifying notaby the computation of the atter ones. For exampe, Kac-Moody singuar vectors have been transformed into Virasoro ones, by using the Knizhnik-Zamoodchikov equation [12], and singuar vectors of W agebras have been obtained out of A (1) 2 singuar vectors via a quantum version of the highest weight Drinfed-Sokoov gauge transformations [1]. The singuar vectors of the Twisted Topoogica N=2 Superconforma agebra have been considered ony in chira (incompete) Verma modues so far. Some interesting features of such topoogica singuar vectors have appeared in a series of papers, starting in the eary nineties. For exampe, in [14] and [15] it was shown that the uncharged BRST-invariant singuar vectors, in the mirror bosonic string reaization (KM) of the Topoogica agebra, are reated to Virasoro constraints on the KP τ -function. In [1] an isomorphism was uncovered between the uncharged BRST-invariant singuar vectors and the singuar vectors of the S(2) Kac- Moody agebra (this was proved unti eve four). In [] topoogica singuar vectors were reated to Lian-Zuckermann states. Some properties of the topoogica singuar vectors in the DDK and KM reaizations were anayzed in [17]. In [18] the compete set of topoogica singuar vectors at eve 2 (four types) was written down, together with the spectra fow automorphism which transforms topoogica singuar vectors into each other. In this paper we investigate severa issues reated to the singuar vectors of the Topo- 1

3 ogica N=2 Superconforma agebra, considering chira as we as compete Verma modues. The resuts are presented as foows. In section 2 we first derive the possibe types of topoogica singuar vectors which may exist, taking into account the reative U(1) charge and the BRST-invariance properties of the vector and of the primary state on which it is buit. For this we make the conjecture that the cascade effect, an agebraic mechanism expained in detai in Appendix A, is the key fact underying the existence or non-existence of the singuar vectors. We find four different types of singuar vectors in chira Verma modues and thirty-three different types in compete Verma modues. The conjecture turns out to be rigorousy correct for the singuar vectors of generic types (tweve types), as is proved in section 4. Secondy, we anayze a set of mappings which transform topoogica singuar vectors into each other (of a different or of the same Verma modue). These mappings give rise to famiy structures which depend on the types of singuar vectors and Verma modues invoved. In section we derive the famiy structure corresponding to singuar vectors in chira Verma modues. We find a unique structure consisting of four singuar vectors, one of each type at the same eve, invoving genericay two different chira Verma modues. We write down the compete famiies unti eve. These famiies contain subsinguar vectors; we identify them in the given famiies and we conjecture an infinite tower of them for higher eves. In section 4 we derive the famiy structure corresponding to singuar vectors in compete Verma modues. We find an infinite number of different patterns which can be roughy grouped in six main kinds (two generic and four specia). Then we derive the spectra of conforma weights and U(1) charges h corresponding to the Verma modues which contain generic and chira singuar vectors (chira singuar vectors ony exist in compete Verma modues). From the ast spectra we deduce the corresponding one for the chira and antichira singuar vectors of the Antiperiodic NS agebra. In addition, we anayze some conditions under which the (chains of) mappings act inside a Verma modue (in most cases they interpoate between different Verma modues), transforming some types of singuar vectors into singuar vectors of exacty the same types at the same eve. With this motivation in mind we then anayze the Dörrzapf equations, originay written for the untwisted Antiperiodic NS agebra, eading to two ineary independent singuar vectors of the same type, at the same eve in the same Verma modue. We present exampes which prove that some (at east) of those singuar vectors are mapped into each other by the transformations described here, i.e. they beong to the same famiies, and we conjecture that the same is true for a of them; that is, that the two partners in each Dörrzapf pair beong to the same famiy. Section 5 is devoted to concusions and fina remarks. Finay, in Appendix A we give a detaied account of the cascade effect, in Appendix B we write down the whoe set of singuar vectors at eve 1 in compete Verma modues, finding twenty-eight different types, and in Appendix C we present a particuary interesting tweve-member famiy of singuar vectors at eves and 4, connected to a branch of ten secondary singuar vectors, which contains four Dörrzapf pairs. 2

4 2 Singuar Vectors and Mappings 2.1 Basic Concepts and Notation The Topoogica N=2 Superconforma agebra reads [19] [L m, L n ] = (m n)l m+n, [H m,h n ] = cmδ m+n,0, [L m, n ] = (m n) m+n, [H m, n ] = m+n, [L m, n ] = n m+n, [H m, n ] = m+n, [L m, H n ] = nh m+n + c (m2 +m)δ m+n,0, { m, n } = 2L m+n 2nH m+n + c (m2 +m)δ m+n,0, m, n Z. (2.1) where L m and H m are the bosonic generators corresponding to the energy momentum tensor (Virasoro generators) and the topoogica U(1) current respectivey, whie m and m are the fermionic generators corresponding to the BRST current and the spin-2 fermionic current respectivey. This agebra can be viewed as a rewriting of the untwisted Antiperiodic NS agebra, using one of the two possibe twists of the superconforma generators: L (1) m = L m + 1(m +1)H 2 m, H m (1) = H m, m (1) = +, m+ 1 2 (1) m =, m 1 2 (2.2) or L (2) m = L m 1(m +1)H 2 m, H m (2) = H m, (2) m = m+ 1 2, (2) m = + m 1 2. (2.) whichwedenoteast W1 and T W 2, these twists being mirrored under the interchange H m H m, + r r. In what foows and h wi denote the conforma weight and U(1) charge of nonsinguar primary states, h, to which we wi refer simpy as primary states, or primaries. The singuar primary states, i.e. both primary and secondary, wi be denoted as singuar vectors.

5 Of specia interest are the topoogica chira primaries, annihiated by both 0 and 0. From the anticommutator { 0, 0 } =2L 0 one deduces that the conforma weight of the topoogica chira primaries is zero. The ony quantum number carried thus by such primaries is the U(1) charge h: H 0 0, h, = h 0, h,. From that anticommutator it is aso easy to deduce that any topoogica state (primary or secondary) with conforma weight different from zero can be decomposed as a 0 -cosed state pus a 0 -cosed state, and aso that 0 -cosed ( 0 -cosed) topoogica states with conforma weight different from zero are 0 -exact ( 0 -exact) as we. The topoogica states with zero conforma weight, however, can be 0 -cosed (satisfying 0 0 χ =0), or 0 -cosed (satisfying 0 0 χ = 0), or both (chira), or neither (satisfying 0 0 χ = 0 0 χ ). We see therefore that ony the topoogica states which are chira, or 0 - cosed with zero conforma weight, may be physica ( 0 -cosed but not 0 -exact). For simpicity, from now on we wi denote as 0 -cosed and 0 -cosed the states which are annihiated by 0 or by 0, by not by both, using ony the term chira for the states wich are both 0 -cosed and 0 -cosed. The states which are neither 0 -cosed nor 0 -cosed wi be denoted as no-abe. Regarding the twists (2.2) and (2.), a key observation is that ( + 1/2, 1/2 )resutsin ( (1) 0, (1) 0 )and( 1/2,+ 1/2 )gives((2) 0, (2) 0 ). This brings about two important consequences. First, the topoogica chira primaries Φ (1) and Φ (2) correspond to the antichira primaries (i.e. 1/2 Φ(1) = 0) and to the chira primaries (i.e. + 1/2 Φ(2) =0)of the Antiperiodic NS agebra, respectivey. Second, the highest weight (h.w.) conditions ± 1/2 χ NS = 0, of the Antiperiodic NS agebra, read 0 χ = 0 after the corresponding twistings. Therefore, any h.w. state of the Antiperiodic NS agebra resuts in a 0 -cosed or chira state of the Topoogica agebra, which is aso h.w. as the reader can easiy verify by inspecting the twists (2.2) and (2.). Conversey, any 0 -cosed or chira h.w. topoogica state (and ony these) transforms into a h.w. state of the Antiperiodic NS agebra. So far ony the topoogica primaries which are chira, with their corresponding chira (incompete) Verma modues, have been considered in the iterature. In our anaysis we wi consider aso the topoogica primaries without additiona constraints, with their corresponding compete Verma modues, i.e. either 0 -cosed or 0 -cosed primaries, denoted as, h and, h, or no-abe primaries, denoted as 0, h. As a first cassification of the topoogica secondary states one considers their eve, their reative U(1) charge q and their transformation properties under 0 and 0 (BRSTinvariance or anti-brst-invariance). The reative charge q is defined as the difference between the U(1) charge of the secondary state and the U(1) charge h of the primary state on which it is buit. It is given by the number of modes minus the number of modes in each term. Hence the topoogica secondary states wi be denoted as χ (q), or χ (q),or χ (q), (chira), or χ (q) (no-abe), where the abes and/or indicate 4

6 that the state is annihiated by 0 and/or 0. For convenience we wi aso indicate the conforma weight, the U(1) charge h, and the BRST-invariance properties of the primary state on which the secondary is buit. Observe that the conforma weight and the tota U(1) charge of the secondary state are given by + and h + q, respectivey. For a given eve it is easy to see that the maximum possibe vaue q max of the reative charge, for chira Verma modues, is given by an integer n such that n or, equivaenty q max = n =[ ] 0,where[] 0 indicates integer part from beow. As a resut, on chira Verma modues the secondary states at eve 2 can ony have reative charges q =0orq=±1. At eves, 4 and 5 there are aso secondaries with q = ±2; at eves, 7, 8 and 9 there are secondaries with reative charges q = ±, etc. For compete Verma modues the zero modes 0 and 0 aso contribute to buid secondary states, without adding to the eve, so that q max = n +1. When the secondary states are singuar vectors, however, the possibe vaues of the reative charge q are very restricted: q < 2 for singuar vectors in chira Verma modues and q < for singuar vectors in compete Verma modues. We wi anayze this issue in detai in the next subsection. 2.2 Types of singuar vectors Let us now investigate which types of topoogica secondary states can be singuar, a given type being defined by the reative charge q, the BRST-invariance properties of the state itsef, and the BRST-invariance properties of the primary on which it is buit. First of a one has to take into account that the topoogica states (primary or secondary) with conforma weight different from zero are either 0 -cosed or 0 -cosed (or a inear combination of those), whereas the topoogica states with zero conforma weight can be either 0 -cosed, or 0 -cosed, or chira, or no-abe. As a consequence there are four different types of topoogica primaries to be considered:, h,, h, 0, h, and 0, h. The first two types are interchangeabe when 0 since they are mapped into each other by 0 and 0 :, h = 0, h 1,, h = 0, h +1. (2.4) In other words, for 0, the h.w. vector is degenerate in the Verma modue. For = 0, however, the action of 0 or 0 produces a eve-zero chira singuar vector instead of a non-singuar primary, which cannot come back to the h.w. vector by acting with the agebra. As a consequence, for = 0 the h.w. vector is not degenerate in the Verma modue. Namey, 0 0, h is a eve-zero chira singuar vector, denoted as χ (1), 0, 0,h, 5

7 since it is annihiated by 0 and 0 : 0 0 0, h = 2L 0 0,h = 0 (in addition of satisfying the h.w. conditions). Simiary, 0 0, h is a eve-zero chira singuar vector, denoted as χ ( 1), 0, 0,h. As to the secondary states χ (q), χ (q), χ (q), and χ (q) (the atter two satisfying + = 0), taking into account that there are four kinds of topoogica primaries to be considered, a naive estimate woud give sixteen different types of them for every aowed vaue of the reative charge q, which in turn is determined by the eve as we pointed out before. This is incorrect, however, because the chira primaries 0, h, have no secondaries of types χ (q), and χ (q) and the no-abe primaries 0, h have no secondaries of type χ (q), and ony one of type χ (q), (corresponding to = 0,q = 0). Observe that there are no chira secondary states in chira Verma modues. Thus there are roughy tweve different types of secondary states for every aowed vaue of q. However, ony a few of these types admit singuar vectors. The reason is that, when one imposes the h.w. conditions, the aowed vaues of q reduce drasticay and, in addition, not a the tweve types do exist for a given aowed vaue of q. We have not found a rigorous method so far to deduce which types of singuar vectors do exist. We have identified an agebraic mechanism, however, which we beieve is the key fact underying whether or not a given type of topoogica secondary state admits singuar vectors. This mechanism, which we denote the cascade effect, consists of the vanishing in cascade of the coefficients of the woud-be singuar vector when the h.w. conditions are imposed, aone or in combination with the BRST and/or anti-brstinvariance conditions 0 χ = 0 and/or 0 χ = 0. As is expained in Appendix A, the starting of the cascade effect, which occurs in most of the possibe types, is very easy to determine. To prove that it goes on unti the very end, getting rid of a the coefficients of the woud-be singuar vectors, is a non-trivia probem which we have not soved yet (athough we have checked in many exampes that this is indeed the case). The cascade effect takes pace in a cases for q > 2, and aso for q = 2 in chira Verma modues. The types of singuar vectors for which the cascade effect does not take pace, i.e. with no starting points for the vanishing of the coefficients, are the foowing. - Four types in chira (incompete) Verma modues buit on chira primaries 0, h, : χ (1), χ ( 1), χ (0) and χ (0). - Tweve types in generic Verma modues buit on 0 -cosed primaries, h : χ ( 2) χ ( 1),, χ ( 1), χ (0),, χ ( 1), χ ( 2), χ (0), χ (0), χ ( 1), χ (0) and χ (1)., χ (1), and for = aso the types - Tweve types in generic Verma modues buit on 0 -cosed primaries, h : χ ( 1) χ (0),, χ (0), χ (1),, χ (0), χ (1), χ (1), χ ( 1), χ (0), χ (1),and χ (2)., χ (2), and for = aso the types

8 - Nine types in specia (compete) Verma modues buit on no-abe primaries 0, h : χ ( 2), χ ( 1), χ ( 1), χ (0), χ (0), χ (1) (Observe that the ast type ony exists for eve zero)., χ (1), χ (2) and χ (0), 0. However not a these types of singuar vectors are inequivaent since, for 0,the primaries of types, h and, h are interchangeabe, as we expained, producing a modification of ±1 in the U(1) charges h and q, so that the tota U(1) charge remains the same. For exampe, the singuar vectors of type χ (q) are equivaent (ony for,,h 0) to singuar vectors of type χ (q+1) if we express, h =,,h 1 0, h 1, etc... Observe that the BRST-invariance properties of the singuar vectors remain unchanged under this procedure. It turns out that some of these types of singuar vectors ony exist for 0. Namey a eight types of no-abe singuar vectors χ (2), χ ( 2), χ (1), χ (1),,,h,,h,,h,,h χ ( 1), χ ( 1), χ (0), χ (0), and the uncharged chira vectors χ (0),,,h,,h,,h,,h,,h and χ (0),. Therefore, the singuar vectors of type χ (2) can be expressed as,,h,,h χ (1), the singuar vectors of type χ ( 2) can be expressed as χ ( 1),the,,h+1,,h,,h 1 singuar vectors of type χ (1) can be expressed as χ (0), the singuar vectors,,h,,h+1 of type χ ( 1) can be expressed as χ (0), and the other way around. That,,h,,h 1 is, the eight types of no-abe singuar vectors reduce to four inequivaent types. The uncharged chira singuar vectors χ (0),,,h and χ (0),,,h can be expressed as charged chira singuar vectors χ (1),,,h 1 and χ ( 1),,,h+1 respectivey, which do exist for = 0, however. An important observation is that certain spectra fow mappings distinguish between some types of singuar vectors and the equivaent types. Therefore, for practica purposes the pairs of equivaent singuar vectors cannot be regarded as competey equivaent and must be taken into account separatey (this issue wi be discussed in next subsection). The types of topoogica singuar vectors aowed by the cascade effect (i.e. for which this effect does not take pace) seem to be the ony types which actuay exist. On the one hand, the existence of these types is consistent with the mappings that we wi anayze in next subsection and in sections and 4. That is, there are no mappings connecting (necessariy) aowed singuar vectors with forbidden singuar vectors, for which the cascade effect does take pace. On the other hand, this is rigorousy true for the case of generic singuar vectors, which are either 0 -cosed or 0 -cosed in generic Verma modues of types V (, h )andv(,h ). The reason is that the existence of any other type of generic singuar vector is in contradiction, via the mappings and the topoogica untwistings, with the we known resuts for the untwisted Antiperiodic NS agebra. Namey, as we wi discuss in section 4, any type of generic singuar vector besides the tweve types isted above woud impy the existence of singuar vectors of the 7

9 NS agebra with reative charges q > 1, which do not exist [25], [2], [27]. Based on these facts, and on many exampes, we conjecture that the cascade effect goes on unti the very end, once it starts, and is therefore a key agebraic mechanism underying whether or not a given type of singuar vector exists. Consequenty we conjecture that the topoogica singuar vectors come ony in the types isted above for which the cascade effect does not take pace. That is, four different types in chira Verma modues V ( 0, h, ), and thirty-three different types in compete Verma modues: twenty-four types in generic Verma modues V (, h )andv(,h ), from which six are 0 -cosed, six are 0 -cosed, four are chira and eight are no-abe, and nine types in no-abe Verma modues V ( 0, h ). In subsection. we present the compete set of singuar vectors in chira Verma modues at eves 1, 2 and, and in Appendix B we write down the compete set of singuar vectors at eve 1 in compete Verma modues, finding twenty-eight different types with a tota of thirty-two different soutions (whie in chira Verma modues at eve 1 there are ony four types with a tota of four soutions). To finish, an usefu observation is that the chira singuar vectors χ (q), regarded as particuar cases of 0 -cosed singuar vectors χ (q) cases of 0 -cosed singuar vectors χ (q) can be and aso as particuar. That is, the 0 -cosed and 0 -cosed singuar vectors may become chira (athough not necessariy) when the conforma weight of the singuar vector turns out to be zero, i.e. + = 0. Notice, however, that the singuar vectors of types χ (1),,h correct). and χ ( 1),,h cannot become chira (if our conjecture is 2. Mappings between Singuar Vectors Inside a given Verma modue V (, h) and for a given eve the topoogica singuar vectors are connected by the action of 0 and 0 in the foowing way: 0 χ (q) 0 χ (q) χ (q 1), 0 χ (q) χ (q+1) (2.5) χ (q 1), 0 χ (q) χ (q+1) (2.) These arrows can be reversed (up to a constant), using 0 and 0 respectivey, ony if the conforma weight of the singuar vectors is different from zero, i.e Otherwise, on the right-hand side of the arrows one obtains secondary singuar vectors which cannot come back to the primitive singuar vectors on the eft-hand side and are at eve zero with respect to these. In particuar this aways happens to the no-abe singuar vectors of type χ (q), in the second row, since they aways satisfy + = 0, whie this never happens to singuar vectors in chira Verma modues, for which 0 + >0. 8

10 The spectra fow automorphism A of the topoogica agebra, on the other hand, which is the twisted version of the other spectra fow [20], given by [18] AL m A 1 = L m mh m, AH m A 1 = H m cδ m,0, A m A 1 = m, A m A 1 = m. (2.7) with A 1 = A, maps singuar vectors into singuar vectors changing the Verma modue. More precisey, it transforms the (L 0, H 0 ) eigenvaues (, h) of the states as (, h c ), reversing the reative charge of the secondary states and eaving the eve invariant, as a consequence. In addition, A aso reverses the BRST-invariance properties of the states (primary as we as secondary) mapping 0 -cosed ( 0 -cosed) states into 0 -cosed ( 0 - cosed) states, and chira states into chira states. For chira Verma modues the action of A resuts therefore in the mappings [18] A χ (q), h χ ( q), h c, A χ (q), h χ ( q), h c, (2.8) where h denotes the chira primary 0, h,. Observe that these two mappings are each other s inverse, the reason being that A 1 = A. Hence at any given eve the singuar vectors of type χ (q) are directy reated to singuar vectors of types χ (q+1) h, χ ( q) h c vectors of type χ (q) h χ ( q+1) h c, as the diagrams show. and χ ( q 1) h c h ; equivaenty, the singuar are reated to singuar vectors of types χ (q 1) h, χ ( q) h c and χ (q), h 0 χ (q 1), h χ (q), h 0 χ (q+1), h A A or A A (2.9) χ ( q), h c 0 χ ( q+1), h c χ ( q), h c 0 χ ( q 1), h c Observe that the two diagrams are equivaent in two ways: the upper parts as we as the ower parts are equivaent because the action of 0 and the action of 0 are inverse to each other (up to a constant), and aso the upper part of each diagram is equivaent to the ower part of the other diagram under redefinitions of h and q. Now comes a subte point. As was expained in [18], the action of the spectra fow operators U ±1 on the Topoogica agebra is identica to the action of the spectra fow 9

11 automorphism A (2.7), except for the fact that U ±1 connect to each other the generators corresponding to the two different topoogica twistings of the Antiperiodic NS agebra, given by (2.2) and (2.), i.e. U 1 L (2) m U 1 1 = L (1) m mh(1) m, U 1 H (2) m U 1 1 = H (1) m c δ m,0, U 1 (2) m U 1 1 = (1) U 1 (2) m m, U 1 1 = (1) m. (2.10) with U 1 1 = U 1. Therefore, if we substitute A by U 1 or U 1 in diagrams (2.9), then the upper part corresponds to topoogica singuar vectors and generators abeed by (1), for exampe, whie the ower part corresponds to topoogica singuar vectors and generators abeed by (2) (or the other way around). For compete Verma modues the action of A resuts in the mappings: A χ (q),, h χ ( q),, h c, A χ (q),, h χ ( q),, h c, A χ (q),,, h χ ( q),,, h c, A χ (q),,h χ ( q),, h c, (2.11) and their inverses. In the case of singuar vectors of types χ (q) and χ (q) we get diagrams simiar to (2.9), which are aso equivaent (but ony in one way). When + =0, however, the arrows given by 0 and 0 cannot be reversed and one gets secondary singuar vectors on the right-hand side of the diagrams (eve-zero chira secondary singuar vectors instead). In the case of chira singuar vectors χ (q), the four-member box reduces to a vertica ine, whie in the case of no-abe singuar vectors χ (q) the righthand side of the box consists of eve-zero secondary singuar vectors (one 0 -cosed and one 0 -cosed), since + =0. There exists an additiona spectra fow transforming the singuar vectors of types χ (q), and χ (q),, into singuar vectors of types χ (q), and χ (q),, (both), and the other way around, whie transforming a other types of singuar vectors, incuding those in chira Verma modues, into various kinds of states which are not singuar vectors. The argument goes as foows [21]. The spectra fow (2.10) acting on the twisted topoogica generators is obtained [18] from the spectra fow U θ acting on the untwisted generators [22][2] U θ L m U 1 θ = L m + θh m + c θ2 δ m,0, U θ H m Uθ 1 = H m + cθδ m,0, U θ + r Uθ 1 = + r+θ, U θ r Uθ 1 = r θ, (2.12) 10

12 (satisfying U 1 θ = U θ ) by using different twists on the right and on the eft-hand sides. By appying the same twist on both sides one finds a different spectra fow W θ which does not map chira primary states into chira primary states for any vaue of θ. However, for θ = 1 it maps primary states annihiated by 0 into primary states annihiated by 0, and the other way around for θ = 1. This spectra fow ooks amost identica to (2.12): W θ L m Wθ 1 = L m + θh m + c(θ + θ2 )δ m,0, W θ H m Wθ 1 = H m + cθδ m,0, W θ m Wθ 1 = m+θ, W θ m W 1 θ = m θ, (2.1) and satisfies Wθ 1 = W ( θ). It transforms the (L 0, H 0 ) eigenvaues (, h) of the primary states as ( θh + c (θ2 θ), h c θ), and the eve of the secondary states as θq, etting invariant the reative charge q. Under W 1 a 0 -cosed, or chira, singuar vector at eve with reative charge q, buit on a primary of type, h, is transformed into a 0 -cosed singuar vector at eve q with reative charge q, buit on a primary of type h, h c. This 0 -cosed singuar vector may become chira if h = q, and is certainy chira if the 0 -cosed singuar vector was annihiated by 1. Therefore there are four possibiities: χ (q),,h W 1 χ (q) q, h,h c, χ (q),,, h W 1 χ (q) q, h, h c, (2.14) χ (q),, h W 1 χ (q), q, q, h c, χ (0),,, 0 W 1 χ (0),,, c, (2.15) An interesting observation is that chira singuar vectors are transformed, in genera, into non-chira singuar vectors (athough annihiated by 1, as the reader can easiy verify). The mapping into another chira singuar vector ony occurs under very restricted conditions: q = 0, h= 0, and the requirement that the singuar vector is annihiated by 1. The resuting chira singuar vector, in turn, is annihiated by 1. These are the ony types of singuar vectors transformed under W 1 into singuar vectors, and the other way around using the inverse W 1. The reason is that ony h.w. vectors (either primary or singuar) annihiated by 0 are transformed into h.w. vectors under W 1, and ony h.w. vectors annihiated by 0 are transformed into h.w. vectors under W 1. The chira primaries 0, h, are transformed under W 1 (or W 1 ) into (nonchira) primaries annihiated by 1 (or 1 ). Primaries with such constraints, which generate incompete Verma modues, are beyond the scope of this paper (uness they are singuar, obviousy). 11

13 We see that W ±1 distinguish between the different types of singuar vectors in a rather drastic way. Regarding the pairs of equivaent singuar vectors for 0, χ (q),, h and χ (q+1) on the one side, and χ (q) and χ (q 1) on the other side, W,, h 1,, h,,h+1 1 and W 1, respectivey, transform ony one member of the pair into a singuar vector. The situation is more invoved for the pairs of equivaent singuar vectors which are chira. Namey, W 1 transforms the chira singuar vector χ (0), into a singuar vector of,,h type χ (0), h,h c (annihiated by 1 ) whie W 1 transforms the equivaent singuar vector χ (1), into a singuar vector of type χ (1),, h 1 +1, +h 1+ c, h 1+ c, and simiary with the pair of equivaent chira singuar vectors χ (0), and χ ( 1),.,, h,,h+1 Famiies of Topoogica Singuar Vectors in Chira Verma Modues.1 Famiy Structure Now et us appy the resuts of ast section to anayze the famiy structure of the topoogica singuar vectors in chira Verma modues, i.e. buit on chira primaries 0, h,. First of a we found that in chira Verma modues there are ony four types of topoogica singuar vectors: χ (0), χ (0), χ (1) and χ ( 1). At a given eve, these four types are connected to each other by the action of 0, 0 and A in the way shown by the diagram: χ (0), h A 0 χ ( 1), h A (.1) χ (0), h c 0 χ (1), h c Hence the topoogica singuar vectors buit on chira primaries 0, h, come in famiies of four: one of each kind at the same eve. Two of them, one charged and one uncharged, beong to the chira Verma modue V (h), whereas the other pair beong to a different chira Verma modue V ( h c ). This impies that there are no oose singuar vectors, i.e. once one of them exists the other three are generated just by the action of 0, 0 and A. As a consequence it is sufficient to compute ony one of these four singuar vectors from scratch, the other three being obtained straightforwardy. For h = c the 12

14 two chira Verma modues reated by A coincide. Therefore, if there are singuar vectors for this vaue of h (see next subsection), they must come four by four in the same chira Verma modue: one of each kind at the same eve. As we expained, the action of U ±1 (2.10) is identica to the action of A (2.7) except that it distinguishes (and interchanges) the topoogica generators of the two twisted theories, (2.2) and (2.), i.e. it exchanges the abes (1) and (2). Thus one can substitute A by U 1 or U 1 in diagram (.1). This amounts to differentiate the upper part and the ower part of the diagram as corresponding to topoogica singuar vectors and generators abeed by (1) or by (2), respectivey. For exampe χ (1) (0), h (1) 0 χ (1) ( 1), h U 1 U 1 (.2) χ (2) (0), h c (2) 0 χ (2) (1), h c (one can reverse the arrows U 1 using U 1 ). Therefore, the topoogica singuar vectors in chira Verma modues come actuay in famiies of four pus four states, four of them abeed by (1) and the other four identica states abeed by (2), the two sets being connected through the action of U ±1. From the point of view of the Topoogica agebra (2.1) the two sets of singuar vectors and generators ook identica, so that one can erase the abes and consider a unique set of four singuar vectors, as we did before. However, the topoogica generators (1) and (2) are different with respect to the untwisted Antiperiodic NS agebra. As a consequence diagrams (.1) and (.2) give different resuts under the two (un)twistings given by (2.2) and (2.). Now comes a word of caution. Since the singuar vectors of types χ (0) and χ (1) are together in the same Verma modue at the same eve, mapped into each other by 0 and 0, and simiary, the singuar vectors of types χ (0) and χ ( 1) are together in the same Verma modue, one might regard one of the singuar vectors in each pair as primitive and the other as secondary at the same eve as the primitive singuar vector. This point of view is incorrect because a secondary singuar vector is not supposed to come back to the primitive singuar vector by acting with the agebra. Finay et us make some comments about the untwisting of these famiies of topoogica singuar vectors. The 0 -cosed singuar vectors in chira Verma modues are transformed into singuar vectors of the Antiperiodic NS agebra, with the same U(1) charge and buit on antichira primaries under T W 1 (2.2), and with the reverse U(1) charge and buit on chira primaries under T W 2 (2.) (whereas the 0 -cosed singuar vectors are transformed into states which are not singuar vectors). As a consequence, the conjec- 1

15 ture that the four types of singuar vectors in diagram (.1) are the ony existing ones in chira Verma modues, predicts [24] that the singuar vectors of the Antiperiodic NS agebra buit on chira primaries have ony reative charges q = 0 or q = 1, whie those buit on antichira primaries have q =0orq= 1 (we have checked this prediction at ow eves). In addition, if a singe singuar vector of a different type, i.e. forbidden by the cascade effect, does exist, then a whoe four-member famiy of forbidden singuar vectors must exist which contain 0 -cosed singuar vectors with reative charges q > 1. These, in turn, woud resut in singuar vectors of the Antiperiodic NS agebra, in chira and antichira Verma modues, with q > 1. In compete Verma modues of the NS agebra this resut cannot hod since it has been proved [27] that q 1 (this was conjectured previousy in [25] and [2]). However there are no rigorous resuts for chira and antichira Verma modues of the NS agebra, from which to derive rigorous resuts for the chira Verma modues of the Topoogica agebra. Observe that if one proves that q 1 for the singuar vectors in chira and antichira Verma modues of the Antiperiodic NS agebra, then one automaticay proves our conjecture that the four types of topoogica singuar vectors in diagram (.1) are the ony existing ones in chira Verma modues of the Topoogica agebra. Finay, et us point out a resut for the Antiperiodic NS agebra which is easier to see from the Topoogica agebra. It is the fact that there are no chira singuar vectors in chira Verma modues neither antichira singuar vectors in antichira Verma modues. To see this one ony needs to untwist the fact that there are no chira topoogica singuar vectors in chira Verma modues. The reason is that a chira state can ony exist if + = 0, but = 0 in chira Verma modues, so that the eve must be zero. It is impossibe, however, to construct secondary states (singuar or not) at eve zero on a chira primary state, since it is annihiated by both 0 and 0..2 Spectrum of h and Subsinguar Vectors Let us discuss now the spectrum of U(1) charges h corresponding to the topoogica chira primaries 0, h, which contain singuar vectors in their Verma modues. This spectrum foows directy from the spectrum corresponding to the singuar vectors of the Antiperiodic NS agebra on antichira Verma modues. Namey, the 0 -cosed topoogica singuar vectors on chira primaries become singuar vectors of the NS agebra under the untwistings: on antichira primaries with the same U(1) charge, using T W 1, (2.2) or on chira primaries with the opposite U(1) charge using T W 2 (2.), and the other way around under the twistings. Hence the spectrum of U(1) charges corresponding to the topoogica chira Verma 14

16 modues which contain 0 -cosed singuar vectors is identica to the corresponding spectrum of U(1) charges for the antichira Verma modues of the NS agebra. This spectrum has been conjectured in [24] by anayzing the roots of the determinant formua for the Antiperiodic NS agebra in a rather non-trivia way (because the determinant formua does not appy directy to chira or antichira Verma modues, but ony to compete Verma modues). The resut is as foows. Let us introduce two positive integers r and s (s even). Then a topoogica chira primary with U(1) charge given by h (0) r,s = c (r +1) s 2 (.) has a singuar vector of type χ (0), and consequenty aso a singuar vector of type χ ( 1), at eve rs in its Verma modue. Simiary, a topoogica chira primary with 2 U(1) charge given by h (1) r,s = c (r 1) + s 2 1 (.4) has a singuar vector of type χ (1), and consequenty aso a singuar vector of type χ (0), at eve rs in its Verma 2 modue. These expressions have been checked unti eve 4. Observe that between both expressions there exists the spectra fow reation h (1) h (0) rs c. Therefore the two expressions coincide for the specia case h(1) rs The soutions to this give discrete vaues of c <. Namey rs = = h(0) rs = c. c = (r s +1) r, r Z +, s 2Z + (.5) For exampe, for s = 2 the soutions corresponding to r =1,2, are c =0,/2,2 respectivey. Therefore for the discrete vaues (.5) the topoogica chira Verma modues with h = c rs have four singuar vectors, one of each type, at the same eve =. 2 Now comes an important issue. It happens that most of the singuar vectors given by the spectrum (.4), are subsinguar vectors in the compete Verma modue V ( 0, h ) and most of the singuar vectors given by the spectrum (.), are subsinguar vectors in the compete Verma modue V ( 0, h ). The argument goes as foows (a the statements about spectra in compete Verma modues can be checked in subsection 4.2). The spectrum h (1) r,s (.4), corresponding to singuar vectors of types χ (1) and χ (0) in chira Verma modues, is the fusion of two different spectra, denoted by h k and ĥr,s. The spectrum h k gives h (1) r,s for s =2inthewayh =h (1),2, being the eve of the singuar vectors. It coincides with the spectrum of singuar vectors of types χ (1) and χ (0), 0,h, 0,h The spectrum (.4) for the uncharged 0 -cosed topoogica singuar vectors χ (0) was aso written down in [1] just by fitting the known data (unti eve 4), without any derivation or further anaysis. 15

17 in the compete Verma modues V ( 0, h ). The spectrum ĥr,s, on the other hand, gives h (1) r,s for s>2intheway: ĥr,s 2 = h (1) r,s>2, where the eve of the singuar vector is given by = rs/2. It turns out that ĥr,s, givenby ĥ r,s = c (r 1) + s 2, (.) corresponds to haf the spectrum of singuar vectors of types χ (0) and χ ( 1) in, 0,h, 0,h the compete Verma modues V ( 0, h ) (the other haf being given by (.)), with the eve given by = rs/2. Therefore ĥr,s 2 = h (1) r,s>2 of types χ (1) and χ (0) singuar vectors of types χ (0), 0,h corresponds either to singuar vectors in chira Verma modues V ( 0, h, ), at eve = rs/2, or to and χ ( 1) in compete Verma modues V ( 0, h ), 0,h at eve = r(s 2)/2. It happens, at east at eves 2 and, that the atter singuar vectors vanish, whie the former ones appear from secondaries which become singuar, once one imposes the chiraity condition 0 0, h = 0 on the primary 0, h, turning it into the chira primary 0, h, as a resut (observe that 0 0, h is a singuar vector). Conversey, switching off the chiraity condition on the primary 0, h,, turning it into 0, h, the singuar vectors of types χ (0), 0,h whereas the singuar vectors of types χ (1) satisfying some of the h.w. conditions. and χ ( 1) appear from nowhere, 0,h and χ (0) become non-singuar secondaries These secondaries are nu and, however, are ocated outside the Verma modues buit on top of the singuar vectors. Some of them, but not a, descend to the singuar vectors under the action of the positive modes which fai to annihiate them (observe that the eve of the non-singuar secondaries is higher that the eve of the singuar vectors). These secondaries are therefore subsinguar vectors [28], i.e. singuar vectors in the quotient of the Verma modue V ( 0, h ) by the submodue generated by the singuar vector 0 0, h. Simiary, the spectrum h (0) r,s (.), corresponding to singuar vectors of types χ (0) r,2 which and χ ( 1) in chira Verma modues, is the fusion of two different spectra: h (0) corresponds to singuar vectors of types χ (0) and χ ( 1),andh (0), 0,h, 0,h r,s>2 which, together with h (1) r,s, corresponds to singuar vectors of types χ (0) and χ (1) in the compete, 0,h, 0,h Verma modues V ( 0, h ). Therefore, the singuar vectors of types χ (0) and χ ( 1) in chira Verma modues are subsinguar vectors in the compete Verma modues V ( 0, h ) for the vaues h (0) r,s>2; that is, they are not singuar vectors in V ( 0, h ) but they are singuar vectors in the quotient of V ( 0, h ) by the submodue generated by the singuar vector 0 0, h. These resuts we have checked at eves 2 and and we conjecture that they hod at any eve. In the next subsection we show the famiies of singuar vectors in chira Verma modues unti eve and we identify the subsinguar vectors contained in them. 1

18 . Famiies at Leves 1, 2 and Let us write down the compete famiies of topoogica vectors in chira Verma modues unti eve. Some of these vectors have been pubished before: χ (0) 2 and χ (0) were written in [14] and [15] respectivey, whie the compete famiy at eve 2 was given in [18]. The fact that most of these singuar vectors are subsinguar vectors in the compete Verma modues V ( 0, h )orv( 0,h ) was not reaized, however. The famiies of topoogica singuar vectors in chira Verma modues are the foowing. At eve 1: χ (0) 1 =(L 1 +H 1 ) 0, c,, χ (0) 1 = L 1 0, 0,, (.7) At eve 2: χ ( 1) 1 = 1 0, c,, χ (1) 1 = 1 0, 0,. (.8) χ (0) 2 =(θl 2 +αl 2 1 +ΓH 1L 1 +βh 2 1 +γh 2+δ 1 1 ) 0,h, (.9) 1 c 2 h = c+, α = β = c c 12 c c+, θ = 0 9 c c 2, γ = c+, Γ=, δ = 18 c c c 1 2 (.10) χ (0) 2 =(L 2 +αl 2 1+ΓH 1 L Γ 1 1 ) 0,h, (.11) c h = 1, α = c c, Γ= c 1 (.12) χ (1) 2 =( 2 +αl 1 1 +ΓH 1 1 ) 0,h, (.1) c h = 1, α = c c, Γ= c 1 (.14) 17

19 χ ( 1) 2 =( 2 +αl 1 1 +βh 1 1 ) 0,h, (.15) 1 c h = 2 c+, α = c c, β = 12 c c+ (.1) At eve : χ (0) =(αl 1 +θl 2L 1 +βh +γh 2 L 1 +δl + ɛh 1 L 2 +µh 2 1L 1 +νh 1 L 2 1+κH 1 H 2 +ρh 1+ a 2 1 +e 1 2 +fl gh ) 0,h, h = δ = µ = 2c c+ (c ) 2 ( c) 1 2 c c 2 +c 4( c), α =, γ = c c 12 1, a = ρ = 5c 12 c c 2 +c 4( c) c 18 c c 2 +12c+27 12( c), θ = 15 2c c 1 2, ɛ =, e = c c, f =, β = 4c c 1 2, ν =, κ = 9 2( c) 4, g = c c 2 +12c+27 ( c) 18 c c+ 4 9 c 2 +12c+27 4( c) 21 2( c) c+15 4( c) (.17) χ (0) =(αl 1 2L 2L 1 +γh 2 L 1 +δl +2eH 1 L 2 +gh 2 1 L 1 +2fH 1 L a e fl gh ) 0, h, c h = c a = 2 c 1 2, α = c c 12 1, e = c, γ =, f = c c 1 c+9 2 c 1 9 2c 4, δ =, g = c 9+c c c c (.18) χ (1) =(αl βL 1 2 +ɛl 2 1 +γh 2 1 +θ + eh 1 2 +fh 1 L 1 1 +gh ) 0,h, 18

20 c h = 2 c 2( c) ɛ = e =, α =, γ = c 2( c) c 24, f = c 2(c ) c+9 4(c ) 9 2(c ) 4, β =, θ = c 2( c) 4, g = c c c 27 c 4( c) c c (.19) χ ( 1) =(αl βL 1 2 +ɛl 2 1 +γh 2 1 +θ +eh 1 2 +fh 1 L 1 1 +gh ) 0,h, ɛ = e = h = 18 c c 2 9c 18 c 9c+45 2( c) 2c c+, γ =, α = 9 c c 4 9 c 2 +12c+27 4( c) 45, f = c c+ 2, β =, θ =, g = c c 9 2 c 2 c c 81 c 2( c) 54 c c 2 +12c+27 4( c) (.20) At eve 1 there are no subsinguar vectors because (r, s) =(1,2), i.e. the ony possibe vaue of s is 2. At eve 2 the singuar vectors χ (1) 2 and χ (0) 2, for h (1) 1,4 =1,are subsinguar in the compete Verma modue V ( 0, 1 ). That is, ˆχ (1) (0) ˆ 2, 0,1 and ˆχ 2, 0,1 given by and ˆχ (1) 2, 0,1 =( 2 + c L 1 1 H 1 1 ) 0,1 (.21) ˆχ (0) ˆ 2, 0,1 =(L 2 + c L2 1 H 1 L ) 0,1 (.22) are subsinguar vectors (the hat denotes that they are not h.w. vectors whie ˆ denotes that the vector is not 0 -cosed anymore). The positive mode 1 brings ˆχ (1) 2, 0,1 down to the singuar vector χ (0) 1, 0,1 = 1 0 0, 1 for c 9since 1 ˆχ (1) 2, 0,1 = (9 c) 1 0 0, 1. Simiary, the positive modes L 1 and H 1 bring ˆχ 2, 0,1 down to the 19 (0) ˆ

21 singuar vector χ (0) 1, 0,1. From this one it is not possibe to reach the subsinguar vectors acting with the negative modes, however. That is, the subsinguar vectors sit outside the (incompete) Verma modue buit on top of the singuar vector. Simiary, the singuar vectors χ (0) 2 and χ ( 1) 2, for h (0) 1,4 = c+, are subsinguar in the compete Verma modue V ( 0, c+ ). Both of them descend to the singuar vector χ (0) = 1, 0, c , c+ at eve 1, for c 9. At eve the singuar vectors χ (1) and χ (0), for h (1) 1, = 2, are subsinguar in the compete Verma modue V ( 0, 2 ) whereas the singuar vectors χ (0) and χ ( 1), for h (0) 1, = c+ cases, however, the subsinguar vectors do not descend to any singuar vectors (in spite of the fact that in V ( 0, 2 )andv( 0, c+ ) there are singuar vectors at eve 2)., are subsinguar in the compete Verma modue V ( 0, c+ ). In these 4 Famiies of Topoogica Singuar Vectors in Compete Verma Modues 4.1 Famiy Structure Now et us appy the resuts of section 2 to derive the famiy structure of the singuar vectors in compete Verma modues. One can expect a much richer structure than in the case of chira Verma modues since, on the one hand, there are many more types of singuar vectors, thirty-three versus four, and, on the other hand, there is an additiona spectra fow mapping at work, W ±1, which can extend the four-member subfamiies of singuar vectors given by the actions of 0, 0 and A (two-member subfamiies rather if the singuar vectors are chira). Let us start with the generic Verma modues of types V (, h )andv(,h ). The types of singuar vectors aowed by the cascade effect are the foowing twenty-four: a eight possibe types of uncharged singuar vectors χ (0), χ (0), χ (0), χ (0), χ (0),, χ (0),, χ (0) and χ (0) vectors χ (1), χ (1), χ (1) q = 1 singuar vectors χ ( 1), pus six types of charge q = 1 singuar, χ (1),, χ (1) and χ (1), χ ( 1), χ ( 1), χ ( 1),, pus six types of charge, χ ( 1) and χ ( 1), We remind the reader that chira singuar vectors χ (q),, i.e. annihiated by both 0 and 0,do not exist in chira Verma modues, they exist in compete Verma modues of generic types V (, h )and V(,h )with=, and at eve zero aso in the Verma modues V ( 0, h ): χ (0), = 0, 0,h 0 0 0, h. 20

22 pus the charge q = 2 singuar vectors singuar vectors χ ( 2) and χ ( 2) χ (2) and χ (2), and the charge q = 2. Therefore in generic Verma modues there are tweve types of generic singuar vectors (six 0 -cosed and six 0 -cosed), pus four types of chira singuar vectors, pus eight types of no-abe singuar vectors. First et us anayze how the generic and chira types of singuar vectors are organized into famiies. The key fact is that using W ±1 one can extend a topoogica subfamiy of four (or two) singuar vectors given by the actions of 0, 0 and A, resuting in, at east, two subfamiies at eves and q, respectivey, ocated in four different Verma modues, two each. These two subfamiies wi be denoted as the skeeton-famiy. As we wi expain, it is possibe to extend this skeeton-famiy further, in the most genera case, by attaching to it a finite, or infinite, number of four-member subfamiies, this number depending on the vaues of, h and c. In some cases, one or more of these four-member subfamiies reduces to a coupe of chira singuar vectors. We shoud keep in mind that chira singuar vectors are particuar cases of 0 -cosed singuar vectors, as we as particuar cases of 0 -cosed singuar vectors, which may appear (athough not necessariy) when the conforma weight of the singuar vector is zero ( + =0). For exampe, any singuar vector of type χ (q) is connected, at east, to seven singuar vectors of the types: χ (q 1), χ ( q), χ ( q+1), χ (q),,, h,, h,, h c/,, h c/ q, h,h c/ χ (q+1), χ ( q),and χ ( q 1), athough these can reduce q, h,h c/ q, h, h q, h, h to five if one coupe of vectors connected by A turns out to be chira. Let us start with an uncharged singuar vector of type χ (0), in the Verma modue V (, h ). As diagram (4.1) shows, the members of its skeeton-famiy are, in the most genera case, singuar vectors of the types χ (0), and χ ( 1),, in both the Verma modues V (, h )andv( h, h ), and singuar vectors of the types χ (0), and, in both the Verma modues V ( h, h c/ )andv(, h c/ ). This χ (1), skeeton-famiy contains therefore the singuar vectors of the four types χ (0), χ (0) and χ ( 1), a of them at the same eve. In the specia case h =, χ (1), however, the conforma weight of the corresponding singuar vectors is zero so that one of the foowing two possibiities must happen: a) The singuar vectors denoted by χ (0) and χ (0) in diagram, h, h c/, h, h (4.1) turn out to be chira, i.e. of types χ (0), and χ (0), instead, so, h,h c/, h, h that the singuar vectors χ ( 1) and χ (1) are absent., h, h, h,h c/ b) The singuar vectors χ (0) and χ (0) are not chira, therefore the, h,h c/, h, h singuar vectors denoted by χ (1) and χ ( 1) in diagram (4.1) must be, h,h c/, h, h substituted by secondary chira singuar vectors at eve zero with respect to the primitive ones, i.e. of the types χ (1), and χ ( 1), instead. 0, 0, h c/ 0, 0, h 21

23 W 1 χ (0), h, h 0 A χ ( 1), h, h A χ (0), h,h c 0 χ (1), h,h c W 1 (4.1) χ (0),,h 0 A χ ( 1),,h A χ (0),, h c 0 χ (1),, h c Simiary, if = the corresponding arrows 0, 0 cannot be reversed, and on the right-hand side one has secondary chira singuar vectors χ (1), and χ ( 1), 0, 0, h c/ 0, 0, h instead. The upper row and the ower row of diagram (4.1) are connected by W 1, as indicated by the arrow on top, since W 1 AW 1 A = I. It has therefore the topoogy of a circe. There are no arrows W ±1 coming in or out of the singuar vectors on the right-hand side because these types do not transform into singuar vectors under W ±1, as is expained in section 2. In the most genera case, it wi be possibe to cure this shortcoming and attach more subfamiies to the skeeton-famiy, as is expained a few paragraphs beow. If we start with an uncharged chira singuar vector in the Verma modue V (, h ), with =, then the skeeton-famiy is as in diagram (4.1) but with the ower subdiagram reduced to the coupe of chira singuar vectors χ (0), and χ (0),.,, h,, h c/ In addition, if h = 0 the singuar vectors in the upper subdiagram have zero conforma weight aso, so that they contain chira singuar vectors (primitive or secondary), as we. Therefore, in the case =, h = 0, the skeeton-famiy of diagram (4.1) may reduce to four (primitive) uncharged chira singuar vectors attached to each other by A and W ±1 in a circuar way. Finay, one can consider one coupe of charged vectors in diagram (4.1) as primitive, from the very beginning, drawing the arrows 0, 0 from right to eft (interchanging 0 0 ). For = it may happen that the uncharged singuar vectors on the eft must be substituted by secondary charged chira singuar vectors, or that the charged singuar vectors on the right are chira, with no arrows 0, 0 to the eft. In this ast case the coupe of charged chira singuar vectors is decouped from any other vectors in diagram (4.1) and beong rather to diagram (4.2), which we wi study next. 22