Диссертация (1137342), страница 35
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Numerator of (6.88) contains Kcontributions from twisted bosons corresponding to plus-cycles, and K 0 contributionsfrom twisted Ramond bosons corresponding to minus-cycles.so(2N + 1) twist fieldsThe W-algebra W (so(2N +1)) contains fermionic operator V (z) = Ψ1 (z) . . . Ψ2N +1 (z),bwhich cannot be expressed in terms of generators of so(2N+1)1 since they are all evenin fermions. It means that to construct a module of the W -algebra one should useentire fermionic algebra. Taking into account the fermionic nature of this W-algebraone can consider Z/2Z graded modules and define two different charactersχ− (q) = tr (−1)F q L0χ+ (q) = tr q L0 ,(6.90)where F is the fermionic number:(−1)F Uk (z) = Uk (z)(−1)F ,(−1)F V (z) = −V (z)(−1)F(6.91)One of the characters vanishes χ− (q) = 0 if at least one fermionic zero mode exists,since each state gets partner with opposite fermionic parity. Such fermionic zeromodes are always present for the Ramond fermions and η-fermions, so the only casewith non-trivial χ− (q) corresponds to:KYg = [1] [li , e2πiri ]+(6.92)i=1In this case our computation works as follows: take bosonization for the [l]+ -cycles interms of K twisted bosons (6.266), (6.80), then the fermionic operators produce the(i)zero-mode shifts e±Q with the fermionic number F = F b + F f = F b = 1, and the(i)Heisenberg generators Jn/li with the fermionic number F = F b = 0.
Moreover, wealso have an extra “true” fermion Ψ(z) with F = F f = 1. Therefore the total tracecan be computed, separating bosons and fermions, asbfbχ− (q) = tr q L0 (−1)F = tr q L0 (−1)F · tr q L0 (−1)Ff(6.93)where the traces over bosonic and fermionic spaces are given byKPPbbtr q L0 (−1)F =fqi=11(ni +li ri )22liKP(−1)nii=1n1 ,...,nK ∈ZK Q∞Qf(1 − q n/li )i=1 n=1∞Ytr q L0 (−1)F =1(1 − q n+ 2 )n=0167(6.94)6. Twist-field representations of W-algebras, exact conformal blocks and character identitiesHence, the final answer for this character is given byKKPP1122∞(n+lr)(n+lr)PQPii iii i2l2l (1 − q k+ 12 )q i=1 i−q i=1 iχ−g (q)=q∆0g~n∈QDK~n∈QD0k=0(6.95)KK Q∞Q(1 − q k/li )i=1 k=1where D- and D0 -lattices are defined in (6.205).Let us now turn to the computation of χ+ (q). Choose an element from NO(2N +1) (h)0g = [(−1)a+1KKYY2πiri] [li , e]+ [li0 ]−i=1(6.96)i=1(i)(i)where a = 0, 1.
The bosonized fermions eiϕ (z) contain elements of eQ . generatingthe BK root lattice, which together with contribution of the fermionic and Heisenbergmodes finally give0[ K2 ]20∆gχ+g (q) = qKPPqi=11(ni +li ri )22li~n∈QBKK Q∞Q∞Qa(1 + q k+ 2 )k=0(1 − q k/li )K0 Q∞Q(1 − q(6.97)(k+ 12 )/li0)i=1 k=0i=1 k=1whereK0Kδa,0 X li2 − 1 X 2li02 + 1++∆0g =1624li48li0i=1i=1(6.98)Here the only new part, compare to the DN -case, is extra factor corresponding to (Ror N S)∞δa,0 Ya(1 + q 2 +k )χf (q) = q 16(6.99)k=0fermionic contribution.Character identitiesIn sect.
6.4 we have classified the twist fields by conjugacy classes in NG (h). However itis possible, that two different elements g1 , g2 ∈ NG (h) in the normalizer of Cartan arenevertheless conjugated in the group G. Such twisted representations are isomorphic,and it gives an obviousTheorem 6.3.
If g1 ∼ g2 in G for different g1 , g2 ∈ NG (h), then χg1 (q) = χg2 (q).This leads sometimes to non-trivial identities and product formulas for the latticetheta-functions, and below we examine such examples.1686.5. Characters for the twisted modulesgl(N ) case. Here any element is conjugated to a product of cycles of length one:l−1 2πir Y∼[1, e2πivj ] ,l, e(6.100)j=0where vj = r +1−l+2j.2lOne gets therefore an identityPq12NPKP(vi +ki )2Pi=1k1 +...+kN =0=η(q)Nq i=11(ni +li ri )22lin1 +...+nK =0KQ(6.101)η(q 1/li )i=1where all conformal dimensions for vanishingr-charges are conveniently absorbed byQthe Dedekind eta-functions η(q) = q 1/24 ∞(1− q n ).n=1This equality of characters can be checked by direct computation, see (6.221) inAppendix 6.9 for S = {0}.
For a single cycle K = 1 this gives a product formula forthe lattice AN −1 -theta function (6.220), which for N = 2P1(n+1/4)2q 16n∈Z qQ= Qk+1/2 )nk≥0 (1 − qn>0 (1 − q )(6.102)was known yet to Gauss and has been originally used by Al. Zamolodchikov in thecontext of twist-field representations of the Virasoro algebra.so(2N ) case. For the conjugacy classes of the first type we have again (6.100), orl−1Y 2πir l, e∼[1, e2πivj ]++(6.103)j=0which leads to very similar identities to the gl(N )-case. For example, one can easilyrederive the product formula [Mac] for the D-lattice theta functionX12q 2 (~n+~v) = ΘDN (~v |q) =~n∈QDNη(q)N +1 η(q 1/(N −1) )η(q 1/2 )η(q 1/2(N −1) )(6.104)for ~v = hρ~ , where the structure of product in the r.h.s.
again comes from the characteristic polynomial of the Coxeter element of the Weyl group W(DN ). Here h = 2(N − 1)is the Coxeter number, and ρ~ = (N − 1, N − 2, . . . , 1, 0) is the Weyl vector, corre(2N −1)sponding to the twist field with dimension ∆ = ∆0 = N48(N, and the easiest way to−1)derive (6.104) is to use (6.223) from Appendix 6.9.For another type of the conjugacy classes [l]− , the situation is more tricky.
Thecorresponding η-fermionX ηk1η(z) = z − 2k(6.105)2lk∈Z z1696. Twist-field representations of W-algebras, exact conformal blocks and character identitiescan be separated into the parts with fixed monodromies around zero:1η(a) = z − 2X ηa+2l·kaz 2l +kk∈Z,(6.106)so that the only non-trivial OPE is between η(a) and η(2l−a) . In particular, η(0) andη(l) are self-conjugated Ramond (R) and Neveu-Schwarz (NS) fermions, which canbe combined into new η̄ fermion, whereas all other components can be considered ascharged twisted fermions ψ̄, ψ̄ ∗ :ψ̄(a) (z) = η(a) (z),ψ̄ ∗ (z) = η(2l−a) (z), a = 1, . .
. , l − 1η̄(z) = η(0) (z) + η(l) (z)(6.107)Therefore one gets equivalencel−1Y[l]− ∼ [1]− · [1, e2πivj ] ,(6.108)j=1where vj = 2lj .Moreover, if we take the product of two cycles [1]− , then we can combine a pair ofR-fermions and a pair of N S-fermions into two complex fermions with charges 0 and1, therefore2[1]− [1]− ∼ [1, 1]+ [1, −1]+(6.109)This means literally that pair of η-fermions is equivalent to two charged bosons: onewith charge v = 0 and another one with charge v = 21 . Equivalence between thesetwo representations leads to the simple identity (6.247), (6.248):2q∞QP18(1 − q=n+ 12)2n=112 + 1 (k+ 1 )222q 2nk,n∈Z∞Q(6.110)(1 − q n )2n=1Using this identity we can remove a pair of [1]− cycles from (6.88) shifting K 0 7→ K 0 −2,and add two more directions to the lattice of charges BK 7→ BK+2 with correspondingr-charges 0 and 12 .so(2N ) case, K 0 = 0.
We have the consequence of identity (6.221) for the caseS = 2Z:NKK1liX 12 PYX P(vi +ki )2(ni +li ri )2η(q)2lii=1i=1q=·q(6.111)1li=1 η(q i ) ~~k∈QDn∈QDKNso(2N ) case, K 0 > 0; so(2N + 1), K 0 > 0. In these cases everything can beexpressed in factorized form using (6.223) and checked explicitly, so these cases arenot very interesting.1706.5. Characters for the twisted modulesso(2N + 1) case, NS fermion. Here in addition to all identities that we had inthe so(2N ) case, we have two more identities that appear because of the fact that wecan combine N S (or R) fermion with a pair of N S, R fermions to get one complexfermion with twist 0 (or twist 21 ) and one R-fermion (or NS-fermion).
Thus[1] · [1]− ∼ [−1] · [1, 1]+[−1] · [1]− ∼ [1] · [1, −1]+(6.112)Thanks to these identities in the cases K 0 6= 0 we can transform any character withN S fermion to a character with R fermion, and vice versa.Twist representations and modules of W-algebrasBy definition, all our twisted representations are representations of the W -algebra.As was explained in previous section it is sufficient to consider the case g ∈ H (otherelements of NG (h) are conjugated to H). In this case subspaces of Hg with all fixedfermion charges become representations of W-algebra6 .
The r-charges of the corregby root latticesponding representations are given by shifts of the vector ~r = log2πiof g.The explicit formulas are given below, but we want first to comment the irreducibility of representations. The Verma modules of W -algebras are irreducible if(α, r) 6∈ Z,(6.113)see [FKW], [Arakawa] (in particular Theorem 6.6.1) or [FL] (eq (4.4)). For genericr this condition is satisfied and all modules arising in the decomposition (subspacesof Hg with all fixed fermion charges) are Verma modules due to coincidence of thecharacters.If g comes from the element of NG (h) with nontrivial cyclic structure then r is notnecessarily generic. For gl(N ) case, as follows (6.100), the r-charges corresponding toa single cycle do satisfy (6.113), and for different cycles this condition also holds provided r are generic (no relations between r from different cycles).
The same argumentworks for so(N ) with “plus-cycles”, but if we have at least two “minus-cycles” thecorresponding r-charges can violate condition (6.113), and not only Verma modulesarise in the decomposition over irreducible representations.In any case we have an identity of charactersχg (q) = χ0 (q)χ̂g (q)(6.114)where χ0 (q) is the character of Verma module, and χ̂g (q) is the character of the spaceof highest vectors. Hence, there is a non-trivial statement, that all coefficients of thepower expansion of the ratios χg (q)/χ0 (q) are positive integers, which can be provenusing identities, derived in previous section.The list of characters of the Verma modules, appeared above is:6This is a common well-known procedure, see e.g.
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