Диссертация (1137342), страница 32
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Below, following [GMtw], we are going to restrict ourselves to the case ofso-called twist fields [ZamAT87, ZamAT86, ApiZam], when the representations of theW-algebras become related to the twisted representations of the corresponding KMalgebras 1 [KacBook, BK].The chapter is organized as follows. We start from the formulation of the representations of KM and W-algebras in terms of free bosons and fermions, remind firstthe GL(N ) case and extend it to the D- and B- series, using real fermions. We definethen the twist representations, and show that they are parameterized by the conjugacyclasses in the correspondent Cartan’s normalizer NG (h). We classify the conjugacyclasses g ∈ NG (h) for G = GL(N ) and G = O(n) (for n = 2N and n = 2N + 1) anddefine the twist fields Og in terms of the boundary conditions in corresponding freetheory.Bosonization rules allow to compute easily the characters χg (q) of the corresponding representations.
For the twist fields of “GL(N ) type” this goes back to the oldresults of Al. Zamolodchikov and V. Knizhnik, and we develop here similar techniquein the case of real fermions and another class of twist fields, arising in D- and Bseries. The character formulas include summations over the root lattices, reflectingthe fact that we deal here with the class of lattice vertex algebras. Dependently of theconjugacy class g ∈ NG (h) of a twist field the lattice can be reduced to its projectionto the Weyl-invariant part, in this case the “smaller” lattice theta functions show up,or we find even a kind of “exchange” between those for D- and B- series.If two different classes g1,2 ∈ NG (h) are nevertheless conjugated g1 ∼ g2 in G (butnot in NG (h)) this gives a nontrivial identity χg1 (q) = χg2 (q) between two characters,involving lattice theta-functions. Such character identities go back to 1970’s (see[Mac], [Kac78]) and even to Gauss, but our derivation gives probably the new ones,involving in particular the theta functions for D- and B-root lattices.We propose construction of the exact conformal blocks of the twist fields for Walgebras of D-series, generalizing approach of [ZamAT87, ZamAT86, ApiZam, GMtw],and obtain an explicit formula, expressing multipoint blocks in terms of the algebro1To prevent the reader’s confusion we should notice that “twist field representation” is differentfrom “twisted representation”: the latter one implies that the algebra itself is changed (twisted),wheres the first one only reflects the way – how this representation was constructed.1526.3.
W-algebras and KM algebras at level onegeometric objects on the branched cover with extra involution.W-algebras and KM algebras at level oneBoson-fermion construction for GL(N)We start from standard complex fermionsψα∗ (z)=∗X ψα,pp∈ 21 +Zzp+ 12,X ψα,pψα (z) =(6.1)1p∈ 21 +Zz p+ 2with the operator product expansions (OPE’s)δαβ+ reg.z−wψα (z)ψβ (w) = ψα∗ (z)ψβ∗ (w) = reg.ψα∗ (z)ψβ (w) = −ψβ (w)ψα∗ (z) =(6.2)equivalent to the following anticommutation relations∗{ψα,p, ψβ,q } = δαβ δp+q,0 ,∗∗{ψα,p , ψβ,q } = {ψα,p, ψβ,q} = 0,p, q ∈12+Z(6.3)\) algebra by the currentsOne can introduce the Kac-Moody gl(N1Jαβ (z) =: ψα∗ (z)ψβ (z) :(6.4)where the free fermion normal ordering moves all {ψr } and {ψr∗ } with r > 0 to theright.
These currents have standard OPE’s:Jαβ (z)Jγδ (w) =δβγ δαδδβγ Jαδ (w) − δαδ Jβγ (w)++ reg.2(z − w)z−w(6.5)and when expanded into the (integer!) powers of zJαβ (z) =X Jαβ,nn∈Z(6.6)z n+1we get the standard Lie-algebra commutation relations[Jαβ,n , Jγδ,m ] = nδn+m,0 δβγ δαδ + δβγ Jαδ,m+n − δαδ Jβγ,m+n ,n, m ∈ Z(6.7)\) . TheThis set contains zero modes Jαβ,0 , generating the subalgebra gl(N ) ⊂ gl(N1W (gl(N )) = WN ⊕ H algebra can be defined in a standard way - as a commutant of\) ).gl(N ) in the (completion of the) universal enveloping U (gl(N1This basis of the generators of W (gl(N )) = WN ⊕ H algebra can be chosen inseveral different ways. In what follows the most convenient for our purposes is to usefermionic bilinears∞ N Xtk−1ψα∗ z + 21 t ψα z − 12 t =+Uk (z)t(k−1)!α=1k=1NX153(6.8)6.
Twist-field representations of W-algebras, exact conformal blocks and character identitiesor, using the Hirota derivative Dzn f (z) · g(z) = (∂z1 − ∂z2 )n f (z1 )g(z2 )|z1 =z2 =z ,Uk (z) =Dzk−1NX: ψα∗ (z) · ψα (z) :(6.9)α=1while another useful basis is the bosonic representationXXWk (z) =: Jα1 α1 (z) . . . Jαk αk (z) :≡: Jα1 (z) . . .
Jαk (z) :α1 <...<αk(6.10)α1 <...<αkThe formula (6.10) is equivalent to quantum Miura transform from [ZamW, FZ, FL].To explain that the formula (6.9) is actually equivalent to (6.10) one can use description of W (gl(N )) as centralizer of screening operators which coincide with gl(N ) inthis case. It is already proven, that (6.10) is centralizer of screening operators [FF],so it remains to show that (6.9) is centralizer as well, what can be done in severalsteps:QQ1. Consider all normally-ordered fermionic monomials : i ∂ ki ψαi (z) i ∂ li ψβ∗i (z) :,which transform as tensors under the action of GL(N ). By First fundamentaltheorem of invariant theory [Weyl] the only invariants in suchareQrepresentationPgiven by all possible contractions, so they can be written as : i ( ∂ ki ψα (z)∂ li ψα∗ (z)) :.α2.
Any such expressionbe obtained by taking regular products of the “elemenP cantary elements” :∂ k ψα (z)∂ l ψα∗ (z) :, sinceα!:Y X∂ ki ψα (z)∂ li ψα∗ (z)αi::X∂ k ψβ (z)∂ l ψβ∗ (z) :=β!=:Y Xi∂ ki ψα (z)∂ li ψα∗ (z)αX(6.11)∂ k ψβ (z)∂ l ψβ∗ (z) : + lower termsβTherefore one can perform this procedure iteratively and express everything asregular products of bilinears.P k3. Any element :∂ ψα (z)∂ l ψα∗ (z) : can be expressed as a linear combination of0α∂ l +1 Uk0 (z) for different l0 and k 0 with l0 + k 0 = l + k.Hence, the generators {Uk (z)} are expressible in terms of {Wk (z)} (and vice versa)by some non-linear triangular transformations, but we do not need here these explicitformulas 2 .Formally there is an infinite number of generators in (6.8) and (6.9), since all ofthem are expressed in terms of N generators (6.10)they are not independent: we haveUN +n (z) = Pn ({∂ k Ul≤N })2(6.12)The fact, that nonlinear W-algebra generators can be expressed through just bilinear fermionicexpressions is well-known, and was already exploited in [LMN, NO] (see also [GMfer] and referencestherein).1546.3.
W-algebras and KM algebras at level onefor some polynomials {Pn }, and this is the origin of the non-linearity of the Walgebra [FKRW]. The relation between fermions and bosons are given by well-known[FK, KVdL] bosonization formulas!!X Jα,nX Jα,nψα∗ (z) = exp −exp −eQα z Jα,0 α (J 0 ) =nnnznzn<0n>0= eiϕα,− (z) eiϕα,+ (z) eQα z Jα,0 α (J 0 )!!X Jα,nX Jα,nψα (z) = expexpe−Qα z −Jα,0 α (J 0 ) =nnnznzn<0n>0(6.13)= e−iϕα,− (z) e−iϕα,+ (z) e−Qα z −Jα,0 α (J 0 )where α (J 0 ) =Qα−1J0,ββ=1 (−1)and diagonal Jαα,n ≡ Jα,n form the Heisenberg algebra[Jα,n , Jβ,m ] = nδαβ δm+n,0 ,[J0,α , Qβ ] = δαβ(6.14)One can also express all other generators in terms of (positive and negative parts of)the bosonsX Jα,nX Jα,niϕ+,α (z) = −,iϕ(z)=−−,α(6.15)nz nnz nn>0n<0namelyβ−1PJγ,0 +θ(β−α)Jαβ (z) = eiϕ−,α −iϕ−,β eiϕ+,α −iϕ+,β eQα −Qβ z Jα,0 −Jβ,0 (−1)γ=α−1Jαα (z) = Jα (z) = i∂ϕ+,α (z) + i∂ϕ−,α (z),α 6= β (6.16)Real fermions for D- and B- seriesNow we can almost repeat the same construction for the orthogonal series, BN andDN , which correspond to the W-algebras W (so(2N +1)) and W (so(2N )), respectively.The corresponding Kac-Moody algebras at level one can be realized in terms of thereal fermions (see e.g.
[AWM]) with the OPE’sΨi (z)Ψj (w) =δij+ reg.,z−wi, j = 1, . . . , n(6.17)(here dependently on the case we put either n = 2N or n = 2N +1), which correspondsto anti-commutation relations{Ψi,p , Ψj,q } = δij δp+q,0 ,p, q ∈12+Z(6.18)One can say that these OPE’s and commutation relations are define by the metrics onnPdΨ2i . The Kac-Moodyn-dimensional space given by δij , or symbolically by ds2 =i=1currents are again expressed by bilinear combinations(1)Jik (z) =: Ψi (z)Ψj (z) :155(6.19)6. Twist-field representations of W-algebras, exact conformal blocks and character identities(1)(1)and satisfy usual commutation relations together with Jij (z) = −Jji (z).
It is alsoconvenient to pass to the complexified fermions (α = 1, . . . , N )1ψα∗ (z) = √ (Ψ2α−1 (z) + iΨ2α (z)) ,21ψα (z) = √ (Ψ2α−1 (z) − iΨ2α (z))2(6.20)which due to (6.17) have the standard OPE’s given by (6.2). Let us point out thatBN -series (i, j = 1, .
. . , 2N + 1) differs from DN -series (i, j = 1, . . . , 2N ) by remainingsingle real fermion Ψ2N +1 (z) = Ψ(z).Using the complexified fermions the generators (6.19) can be re-written asi (1)1 (1)(1)(1)Jαβ =: ψα∗ (z)ψβ (z) := (J2α−1,2β−1 + J2α,2β ) + (J2α,2β−1 + J2β,2α−1 )22(6.21)together withJαβ̄ = ψα∗ (z)ψβ∗ (z),Jᾱβ = ψα (z)ψβ (z)ψα∗ (z)Ψ(z),Jᾱ,Ψ = ψα (z)Ψ(z)Jα,Ψ =(6.22)\) ⊂ so(n)[ . Note also, that elements Jαα (z) = Jα (z)so that we see explicitly gl(N11again form the Heisenberg algebra, and its zero modes Jα,0 correspond to the Cartansubalgebra of so(n).[ .As before, we define the W-algebra W (so(n)) as commutant of so(n) ⊂ so(n)1In contrast to the simple-laced cases we find this commutant for B-series not in com\pletion of the U (so(2N+ 1)1 ), but in the entire fermionic algebra.
An inclusion ofalgebras gl(N ) ⊂ so(2N ), acting on the same space, leads to inverse inclusionW (so(2N )) ⊂ W (gl(N ))(6.23)Similarly to (6.9) one can present the generators of the W (so(n))-algebra explicitly,using the real fermionsnX1Uk (z) = Dzk−1: Ψj (z) · Ψj (z) :,2j=1V (z) =nYΨj (z)(6.24)j=1The last current is bosonic in DN case and fermionic for BN . These expressions areobtained analogously to (6.9) with the help of invariant theory, the only importantdifference is that for SO(n) case there is also completely antisymmetric invarianttensor. We can rewrite these expressions using complex fermions (for the DN caseone should just put here Ψ(z) = 0 in the expressions for U -currents and Ψ(z) = 1 inthe expressions for the V -current)Uk (z) = 21 Dzk−1NX(ψα∗ (z) · ψα (z) + ψα (z) · ψα∗ (z)) + 21 Dzk−1 Ψ(z) · Ψ(z)α=1V (z) =NY(6.25): ψα∗ (z)ψα (z) : Ψ(z)α=11566.4.
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