Диссертация (1137342), страница 37
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. . + el1 +l2 , . . .7There is also “degenerate” case Jk = Jk0 for θAdj,k = θAdj,k0 = − 21 .175(6.137)6. Twist-field representations of W-algebras, exact conformal blocks and character identitieswhile πs Qgl(N ) is generated by vectors l1i fi − l1j fj , so one can present any elementP1Pof πs Qgl(N ) asnj fj withnj = 0 and identify with that from Qgl(K) .
LetljPµ0 = j rj fj . Then the formula (6.136) takes here the formTr(q L0 )M (s,µ0 )=q∆0s1j 2ljPPQgl(K) qQK Q∞j=1(nj +lj rj )2n/lj )n=1 (1 − q,(6.138)where, since for any length l cycle θAdj,k = −k/l,∆0sljK XKXi(lj − i) X lj2 − 1==24l24ljjj=1 i=1j=1(6.139)This formula coincides with (6.82), andQthe reason is that the corresponding element2πirjfrom NGL(N ) (h) is exactly (6.45), g = K]. Indeed, let α = ea − eb , wherej=1 [lj , ea belongs to the cycle j and b belongs to the cycle j 0 then the current Jα (z) shifts L0grading by rj − rj 0 + [rational number with denominator lj , lj 0 ].SO(2N ) case The root lattice Qso(2N ) = QDN is generated by the vectors {ei −ej , ei + ej }, where again e1 , .
. . , eN denote the basis in RN . As we already discussed insect. 6.4, there are two types of the Weyl group elements, the first type just permutesei , while the second type permutes ei together with the sign changes.The first case almost repeats the previous paragraph, without loss of generalitywe assume that the Weyl group element acts as (e1 7→ e2 7→ . . . 7→ el1 7→ e1 ), (el1 +1 7→el1 +2 7→ . . .
7→ el1 +l2 7→ el1 +1 ), . . ., where l1 , . . . , lK are again the lengths of the cycles.The s-invariant part of h∗0 is generated by the same “averaged” vectors (6.137), whileπs QDN is generated by the vectors l1i fi − l1j fj , l1i fi + l1j fj . In other words πs QDN consistPP njf , where (n1 , . . . , nk ) ∈ Qso(2K) . Let µ0 = j rj fj , then the characterof vectorslj jformula (6.136) for this case acquires the formPPTr(q L0 )M (s,µ0 )=q∆0sQso(2K) qQK Q∞j=11j 2lj(nj +lj rj )2(6.140)n/lj )n=1 (1 − q0and coincides with (6.85).
Herein (6.139). The corresponding elementQK∆s is definedfrom NSO(2N ) (h) has the form j=1 [lj , e2πirj ]+ in the notations of sect. 6.4 (see (6.56)).For the second type (the corresponding element from NSO(2N ) (h) has the formQKQ 02πirj]+ · Kj=1 [lj , ej 0 =1 [lj 0 ]− ) one can present the Weyl group element as product ofK disjoint cycles of lengths l1 , . . . , lK which just permutes ei and K 0 cycles of lengthsl10 , . . .
, lK 0 which permutes ei with signs, see (6.56). Now, without loss of generality,we assume that s acts as (e1 7→ e2 7→ . . . 7→ el1 7→ e1 ), (el1 +1 7→ el1 +2 7→ . . . 7→el1 +l2 7→ el1 +1 ), . . ., (eL+1 7→ e2 7→ . . . 7→ eL+l10 7→ −e1 ), (eL+l10 +1 7→ eL+l10 +2 7→.
. . 7→ eL+l10 +l20 7→ −eL+l10 +1 ), . . ., where L = l1 + . . . + lK . The s-invariant part ofh∗0 is generated by the same vectors (6.137), while πs QDPis generated by the vectorsNnj1f . One can say that πs QDN consists of the vectorsf , where (n1 , . . . , nk ) ∈li ilj j1766.6. Characters from lattice algebras constructionsQso(2K+1) = QBN , so that for the character formula one getsTr(q L0 )M (s,µ0 )P 12Pj 2lj (nj +lj rj )K 0 /2−12qQ0so(2K+1),= q ∆s Q K Q ∞QK 0 Q∞(2n−1)/2lj0n/lj))(1−q(1−qj=1n=1j=1n=1(6.141)where, since in addition to [l]+ -cycles with θAdj,k = −k/l one now has [l0 ]− -cycles with0= −(k − 21 )/l0 ,θAdj,kl00ljjKK XXi(lj − i) X X (2i − 1)(2lj0 − 2i + 1)0+=∆s =4lj216lj02j=1 i=1j=1 i=1=KXlj2 − 1j=124lj0+KX2lj02 + 1j=1(6.142)48lj00This formula coincides with (6.88).
The number 2K /2−1 equals to d(σ), this is thefirst case where this number is nontrivial. Note, that we consider here only internalautomorphisms, i.e. K 0 is even.Recall also (see sect. 6.5.5)that if g, g 0 ∈ N G (h) are conjugate in G then corresponding characters Tr(q L0 )M (s,µ0 ) and Tr(q L0 )M (s0 ,µ0 ) are equal.0Characters from principal specialization of the Weyl-Kac formulaFix element g ∈ G of finite order l.
The g-twisted representations of V (g) are representations if the affine Lie algebra twisted by g. Recall that these twisted affine Lieb g) are defined in [KacBook, Sec 8] as g invariant part of g[t, t−1 ] ⊕ Ckalgebras L(g,where g acts asg(tj ⊗ J) = −j tj ⊗ gJg −1 , where = exp(2πi/l), g(k) = k.(6.143)b g) is isomorphicBy definition g is an internal automorphism, therefore the algebra L(g,b g)to bg (see Theorem [KacBook, 8.5]), though natural homogeneous grading on L(g,differs from the homogeneous grading on bg.Therefore the g-twisted representations of V (g) as a vector spaces are integrablerepresentations of bg 8 .
Their characters can be computed using the Weyl-Kac character formula. This formula has simplest form in the principal specialization, i.e.∨b Here ρ∨ ∈ h ⊕ Ck such that αi (ρ∨ ) = 1, for allcomputed on the element q ρ ∈ G.affine simple roots αi (including α0 ) . Then the character of integrable highest weightmodule with the highest weight Λ equals (see [KacBook, eq.
(10.9.4)])Tr(qρ∨ /h∨Y 1 − q (Λ+ρ,α∨ )/h mult(α )Λ(ρ∨ )/h,)L =qΛ1 − q (ρ,α∨ )/h∨∨(6.144)α ∈∆+8Note that we get only level 1 integrable representation of bg since V (g) was defined above as alattice vertex algebra i.e. vacuum representation of the level k = 11776. Twist-field representations of W-algebras, exact conformal blocks and character identitieswhere ∆∨+ is the set of all positive (affine) coroots. Here h is the Coxeter number, it∨∨will be convenient to use q ρ /h instead of q ρ . The weight ρ is defined by (ρ, αi∨ ) = 1for all simple coroots αi (including affine root α0 ).The grading above in this section was the L0 grading and it was obtained usingthe twist by the element g ∈ NG h.
Now we take certain g such that g-twisted L0grading coincides with principal grading in (6.144). We take g in Cartan subgroup Hand as was explained above choice g corresponds to the choice of µ0 in (6.136).In the principal grading used in (6.144) deg Eαi = h1 for all simple roots Eαi(including affine root α0 ). Therefore µ0 ∈ Pg + h1 ρ, where Pg is the weight lattice forg, ρ is defined by the formula (ρ, αi ) = 1 for all simple roots9 .Below we write explicit formulas for characters of twisted representation corresponding to such g (and such µ). In the simply laced case, computing the charactersusing two formulas (6.136) and (6.144) one gets an identity, which is actually theMacdonald identity [Mac].In notation for root system we follow [Mac] and [KacBook].
Below we considerroots as vectors in the linear space, generated by e1 , . . . , en , δ, Λ0 , and coroots – in thespace generated by e∨1 , . . . , e∨n , K, d. The pairing between these dual spaces given by(ei , e∨j ) = δij , (Λ0 , K) = (δ, d) = 1 while all other vanish.(1)(1)GL(N ) case. Root system is AN −1 (affine AN −1 ) dual root system is also AN −1 .Simple roots:α0 = δ − e1 + eN , αi = ei − ei+1 , 1 ≤ i ≤ N − 1∨Simple coroots: α0 = K + e∨N − e∨1 , αi∨ = e∨i − e∨i+1 , 1 ≤ i ≤ N − 1Real coroots:mK + e∨i − e∨j , m ∈ Z, i 6= jmK of multiplicity N, m ∈ Z.Imaginary coroots:Level k = 1 weights:Λ0 , Λj = Λ0 +jX(6.145)ei , 1 ≤ j ≤ N − 1i=1h = N, ρ =12NX(N − 2i + 1)ei + N Λ0 , ρ =i=112NX(N − 2i + 1)ei .i=1Note the multiplicity of imaginary roots in N instead on N − 1 since we considerG = GL(N ) instead of SL(N ), and the corresponding affine algebra differs by oneadditional Heisenberg algebra.The computation of the denominator in (6.144), using (6.145) givesY(1 − q(ρ,α∨ )/h mult(α∨ ))=α∨ ∈∆∨+∞Y(1 − q k/N )N(6.146)k=1while for the numerator (the same for all level k = 1 weights) one getsY(1 − q (Λ+ρ,α∨ )/h)mult(αα∨ ∈∆∨+∨)=∞Y(1 − q k/N )N −1(6.147)k=19Note the difference between ρ and ρ): first was defined by pairing with simple coroots (includingaffine one) and the second is defined by scalar products with (non affine) roots.
In the simply lacedcase conditions in terms of roots and coroots are equivalent and we have ρ = ρ + hΛ01786.6. Characters from lattice algebras constructionsso that the character (6.144) in principal specialization1∨∨q −Λ(ρ )/h Tr(q ρ /h )L = Q∞k/N )Λk=1 (1 − q(6.148)One can compare the last expression with the formula (6.136) using the choice of µ0 ,as explained above.
We get an identityP111q 2 (α+ N ρ,α+ N ρ)1α∈Qsl(N )q 2N 2 (ρ,ρ)(6.149)Q∞.= Q∞kNk/N)k=1 (1 − q )k=1 (1 − qwhich is a particular case of formula (6.101) from sect. 6.5.5, and again reproducesthe product formula for the lattice AN −1 -theta function (6.220).Recall that the r.h.s. of (6.149) also has an interpretation of a character of thetwisted Heisenberg algebra. This twist of the Heisenberg algebra emerges in therepresentation twisted by g with gAdj acting as the Coxeter element of the Weyl group,hence r.h.s. of (6.149) equals to the r.h.s.
of (6.138) for a single cycle K = 1, l = N .This g is conjugate to used above in computing of l.h.s., therefore the characters of thetwisted modules should be the same. The construction of level one representations interms of principal Heisenberg subalgebra is well-known, see [LW, KKLW]. Anotherinterpretation of the l.h.s in (6.149) is the sum of characters of the W -algebra namelyW algebra of gl(N ), (see sect. 6.5.6).(1)(1)SO(2N ) case. Root system DN (affine DN ), the dual root system is also DN .Simple roots: α0 = δ−e1 −e2 , αi = ei −ei+1 , 1 ≤ i < N, αN = eN −1 +eNSimple coroots: α0∨ = K − e∨1 −e∨2 , αi∨ = e∨i −e∨i+1 , 1 ≤ i < N, αN = e∨N −1 +e∨NReal coroots: mK+e∨i −e∨j , mK + e∨i + e∨j , mK−e∨i −e∨j , m ∈ Z, i 6= jImaginary coroots: mK of multiplicity N, m ∈ Zk = 1 weights: Λ0 , Λ1 =e1 +Λ0 ,ΛN −1 = 21NXei +Λ0 ,ΛN = 12i=1NXei −eN +Λ0i=1NNXXh = 2N − 2, ρ =(N − i)ei + (2N − 2)Λ0 , ρ =(N − i)ei .i=1i=1(6.150)Now we again just compute the denominatorY(1 − q(Λ+ρ,α∨ )/h mult(α∨ ))α∨ ∈∆∨+=∞Y(1 − q k/(2N −2) )N(6.151)k=1and the numerator (the same for all k = 1 weights)Y(ρ,α∨ )∨(1 − q 2N −2 )mult(α ) =α∨ ∈∆∨+=N−1 Y∞Y(1 − q2j−1k− 2N−2)N +1 (1 − qk− 2N2j−2j=1 k=1)N ·∞Y(6.152)(1 − qk=1179k− 12)6.
Twist-field representations of W-algebras, exact conformal blocks and character identitiesin (6.144), giving for the character∨ )/hq −Λ(ρ∨ /hTr(q ρ1)L = QN −1 Q∞Λj=1k=1 (1−q2j−1k− 2N−2)·Q∞k=1 (11− q k− 2 ).(6.153)As in previous case, comparing this with the formula (6.130), one gets an identityP1111(α+ hρ,α+ hρ)2q 2h2 (ρ,ρ)α∈QDN qQ∞(6.154).= QN −1 Q∞Q∞2j−1k Nk− 2Nk− 21−2 ) ·(1−q)(1−qk=1 (1 − q )k=1j=1k=1where the r.h.s. side can be interpreted as a character of the representation Heisenbergalgebra twisted by g such that gAdj is Coxeter element.
Again, this is the same asconstruction of level k = 1 representation in terms of principal Heisenberg subalgebrafrom [LW, KKLW]. The l.h.s formula (6.154) can be also interpreted as the sum ofcharacters of the W (so(2N ))-algebra, (see sect. 6.5.6).By now in this section we have considered only the simply laced case – the onlyone, when the algebra V (g) is lattice algebra or, in other words, when the level k = 1representations can be constructed as a sum of representations of the Heisenbergalgebra.
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