Диссертация (1137342), страница 36

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[MMMO] and references therein.1716. Twist-field representations of W-algebras, exact conformal blocks and character identities• gl(N ), so(2N ) (NS sector). Algebra is generated by N bosonic currents, eachof them producing Q 1(1−qn ) , so the character isn>01n Nn=1 (1 − q )χ0 (q) = Q∞(6.115)• so(2N ) (R sector). One of these currents, V (z), becomes Ramond, with halfinteger modes:χ0 (q) = Q∞1n=1 (1− q n )N −1Q∞n=0 (1(6.116)1− q 2 +n )• so(2N + 1) (NS sector).

One current, V (z) becomes Neveu-Schwarz fermion, sotaking into account its parity we getQ∞1(1 ± q 2 +n )±n=0(6.117)χ0 (q) = Q∞n Nn=1 (1 − q )• so(2N + 1) (R sector). In the case of Ramond fermion V (z) character χ−0 (q)vanishes because fermionic zero mode produces equal numbers of states withopposite fermionic parities:Q∞n+n=1 (1 + q )Qχ0 (q) = 2 ∞n N(6.118)n=1 (1 − q )−χ0 (q) = 0gl(N ) case. Any element is conjugated to a product of cycles of length 1, soX 120χ̂g (q) = q ∆gq 2 (v+~n)(6.119)~n∈AN −1Q2πvj]+ , soso(2N ) case, K 0 = 0.

Any element is conjugated to Nj=1 [1, eX 102χ̂g (q) = q ∆gq 2 (v+~n)(6.120)~n∈DNso(2N ) case, K 0 > 0, NS-sector. Again, any element is conjugated tosoX 1K002χ̂g (q) = 2 2 −1 q ∆gq 2 (v+~n)Q[1, e2πvj ]+ ,(6.121)~n∈BNso(2N ) case, R-sector. Here any element is conjugated to [1]−X 1K002χ̂g (q) = 2[ 2 ] q ∆gq 2 (v+~n)QN −1j=1[1, e2πvj ]+ , so(6.122)~n∈BN −1because contribution from the cycle [1]− to the denominator cancels contribution fromthe Ramond boson V (z).1726.6. Characters from lattice algebras constructionsso(2N + 1) case, K 0 = 0, NS fermion.

Here one has two non-trivial charactersX 12∆0gχ̂+(q)=qq 2 (v+~n)g~n∈BNχ̂−g (q)=q∆0gXq1(v+~n)22−Xq1(v+~n)22(6.123)0~n∈DN~n∈DNso(2N + 1) case, K 0 > 0 This case gives nothing interesting as compared to DNsituation.X 102[ K2 ] ∆0gqq 2 (v+~n)χ̂+(q)=2g(6.124)~n∈BN−χ̂g (q) = 0Characters from lattice algebras constructionsTwisted representation of bg1Now we reformulate the results of previous sections using the notion of twisted representations of vertex algebras. Recall the corresponding setting (following, for example,[BK]). Let V be a vertex algebra (equivalently vacuum representation of the vertexalgebra), and σ be a automorphism of V of finite order l.

Then V = ⊕Vk , whereVk = {v ∈ V |σv = exp(2πik/l)v}. The σ-twisted module is a vector-space M endowed with a linear map from V to the space of currentsXam (v)z −m−1 , v ∈ V, am (v) ∈ End(M ).v 7→ Av (z) =(6.125)1m∈ l ZSuch correspondence should be σ-equivariant, namelyAσv (z) = Av (e2πi z)(6.126)giving the boundary conditions for the currents, and agree with the vacuum vectorand relations in V . In particular, it follows from the σ-equivariancy (6.126), that ifv ∈ Vk then Av (z) ∈ z −k/l C[[z, z −1 ]].Consider now a Lie group G (either GL(N ) or SO(2N ), N ≥ 2), with g = Lie(G)being the corresponding Lie algebra.

Denote by V(g) the irreducible vacuum representation of bg of the level one. This space has a structure of the vertex algebra i.e.for any v ∈ V(g) one can assign the current Av (z), this space of currents is generatedby the currents Jαβ (z) from sect. 6.3.The vertex algebra V(g) is a lattice vertex algebra. Let Qg denote the root latticeof g, and introduce rank of g bosonic fields with thej (w) = −δij log(z −P OPE ϕi (z)ϕ12w) + reg, and the stress-energy tensor T (z) = − 2 j : ∂ϕj (z) :, then the currents ofV(g) can be presented in the bosonized formYX:∂ ai,m ϕi exp(iαi ϕi (z)) :,(6.127)i,m1736.

Twist-field representations of W-algebras, exact conformal blocks and character identitieswhere α = (α1 , . . . , αn ) ∈ Qg and ai,m are any positivePintegers, while the stress-energy12|0i = τ ∈ V(g) (here Jj,ntensor corresponding to standard conformal vector 2 Jj,−1are modes of the field i∂ϕj (z)). The group G acts on V (g), and in order to use latticealgebra description we consider only the subgroup NG (h) ⊂ G which preserves theCartan subalgebra.In [BK] the representations of the lattice vertex algebra, twisted by automorphisms, arise from isometries of the lattice Qg .

Here we restrict ourself to the isometries provided by action of the Weyl group W (this case was actually considered in [KP]without language of twisted representations). Let s ∈ W be an element of the Weylgroup, by g we denote its lifting to G, in other words g ∈ NG (h) such that adjointaction g on h coincides with s. We consider representation twisted by such g. Settingof [BK] and [KP] works for special g, for example such g should have finite order,but we will expand this to the generic g ∈ NG (h).

Clearly, the conformal vector τ isinvariant under the adjoint action of NG (h).The g-twisted representations of V (g) in [BK] are defined as a direct sum of twistedrepresentations of bh. By {e2πiθAdj,k } we denote eigenvalues of s, or of the adjoint actiongadj on h, we set −1 < θAdj,k ≤ 0, by {Jk ∈ h} - the corresponding eigenvectors, anddefine the currentsXJk (z) =Jk,θAdj,k +n z −θAdj,k −n−1(6.128)n∈ZA g-twisted representations of the Heisenberg algebra bh is a Fock module Fµ with thehighest weight vector vµJk,θAdj,k +n vµ = 0, n > 0,Jk,0 vµ = µ(Jk )vµ , if θAdj,k = 0.(6.129)generated by creation operators Jk,θAdj,k +n , n ≤ 0.

Here µ ∈ h∗0 , where h0 is gadj invariant subspace of h.It has been proven in [BK] that twisted representations of V (g) have the structureM (s, µ0 ) = ⊕µ∈µ0 +πs Qg Fµ ⊗ Cd(s)(6.130)for certain finite set of µ0 ∈ h∗0 . Here πs denotes projection from h∗ to h∗0 , corresponding to the element s ∈ W for the chosen adjoint action gadj .

For any root α thecorresponding current Jα (z) acts from Fµ to Fµ+πs α and equals to the linear combination of vertex operators. Number d(s) denotes the defect of the element s ∈ W , itssquare is defined by(6.131)d(s)2 = |(Qg ∩ h⊥0 )/(1 − s)Pg |.Here Pg denotes weight lattice of g, h⊥0 denotes the space of linear functions vanishingon h0 , | · | stands for number of elements in the group. It can be proven that for anys the numbers d(s) is integer. In our case (GL(N ) and SO(n) groups) this numberalways equals to some power of 2.Formula (6.130) allows to calculate the character of module M i.e.

the trace ofL0q . First, notice that the character of the Fock module Fµ equalsq ∆µθAdj,i +n )n=1 (1 − qχµ (q) = Q Q∞i174(6.132)6.6. Characters from lattice algebras constructionswhere ∆µ is an eigenvalue of L0 on the vector vµ . The value of ∆µ consists of twocontributions. The first comes from the terms with θAdj = 0, and, as follows from(6.129), is equal to 12 (µ, µ). The second contribution comes from the normal ordering.The vectors Jk ∈ h, corresponding to θAdj,k 6= 0 can be always arranged into orthogonalpairs (J1 , J10 ), (J2 , J20 ), . .

. with complementary eigenvalues θAdj,k + θAdj,k0 = −1 7 .After normal ordering of the corresponding currents one getsJk (z)Jk0 (w) =X Jk,n+θ Jk0 ,m−θX Jk,n+θ Jk0 ,m−θ=+z n+θ+1 wm−θ+1 n∈Z,m≥0 z n+θ+1 wm−θ+1n,m∈ZXwn+θ−1Jk0 ,m−θ Jk,n+θ+(n+θ)+wm−θ+1 z n+θ+1 n>0z n+θ+1n∈Z,m<0(6.133)Xwhere θ = θAdj,k . The last term in the r.h.s., which appears due to [Jk,n+θ , Jk0 ,m−θ ] =(n + θ)δn+m,0 also gives a nontrivial contribution to the action of L0 on highest vectorvµ , sinceX(1 + θ)wθ z −θ + (−θ)w1+θ z −1−θwn+θ−1=(n + θ) n+θ+1 =z(z − w)2n>0(6.134)1θ(1 + θ)=−+ regz→w (z − w)22w2Altogether one getsX θAdj,k (1 + θAdj,k )1∆µ = (µ, µ) −24k(6.135)and therefore, finally for the character of (6.130)L0 Trq =qM (s,µ0 )− 14PP1(µ,µ)2d(s)µ∈µ0 +πw Q qk θAdj,k (1+θAdj,k )QN Q∞θAdj,i +n )n=1 (1 − qi=1(6.136)Recall that in the the initial weight µ0 in the setting of [BK] should belong to thefinite set in h∗0 (or h∗0 /πW Q).

But we will generalize such representations and take anyµ0 ∈ h∗0 . This can be viewed as a twisting by more general elements g ∈ NG (h), whichcan have infinite order. Actually the corresponding elements are representatives ofthe conjugacy classes of NG (h) used in sect. 6.4.Calculation of charactersGL(N ) case The root lattice Qgl(N ) = QAN −1 is generated by vectors {ei − ej },where {e1 , . . .

, eN } denote the vectors of orthonormal basis in RN . Assume that s ∈ Wis product of disjoint cycles of lengths l1 , . . . , lK , then without loss of generality theaction of such elements can be defined as (e1 7→ e2 7→ . . . 7→ el1 7→ e1 ), (el1 +1 7→el1 +2 7→ . . . 7→ el1 +l2 7→ el1 +1 ), . . ..In this case h∗0 (the s-invariant part of h∗ ) is generated by the vectorsf1 = e1 + . . . + el1 , f2 = el1 +1 + .

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