Диссертация (1137342), страница 34

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It is enough just to drop the λdependence and to consider only the even number of minus-cycles (the latter conditioncorresponds to the fact that (extension of) the Weyl group lies inside the connectedcomponent of identity in O(n), whereas another component corresponds to exteriorautomorphism of the Dynkin diagram). In [BK] the Weyl group has been extended toW̃ = W n (Z/2Z)N −1 ⊂ NG (h), which corresponds to breaking the Cartan torus H ⊂ Gdown to (Z/2Z)N −1 .Twist fields and bosonization for gl(N )Take an element (6.45), whose action on fermions (in the fundamental and antifundamental representations), say for a single cycle, is∗g : (ψα∗ (z), ψα (z)) 7→ (e2πir ψα+1(z), e−2πir ψα+1 (z)),mod l(6.57)while the corresponding (adjoint) action on the Cartan is justgAdj : Jα (z) 7→ Jα+1 (z),mod l(6.58)Such formulas have simple geometric interpretation [Knizhnik]: there is the branchedcover in the vicinity of the point z = 0 given by ξ l = z, so that all these (fermionicand bosonic) fields are actually defined on different sheets ξ (α) = z 1/l e2πiα/l of thecover:pp√√ψα∗ (z) dz = ψ̃ ∗ (ξ (α) ) dξ (α) , ψα (z) dz = ψ̃(ξ (α) ) dξ (α)(6.59)˜ (α) )dξ (α)Jα (z)dz = J(ξ (α) )dz = J(ξ1616.

Twist-field representations of W-algebras, exact conformal blocks and character identitiesUsing these formulas one can write down expansions for the fields on the cover, whoseOPE’s would be locally given byψ̃ ∗ (ξ)ψ̃(ξ 0 ) =1+ reg.,ξ − ξ0˜ J(ξ˜ 0) =J(ξ)1+ reg.(ξ − ξ 0 )2Now one write for the mode expansionr11ψpψpdξz 2l − 2 X1 X√ψ̃(ξ) = √=ψ(z) =111p++σ+dzl p∈Z+ 1 ξ 2l p∈Z+ 1 z 2 l (p+σ)22r1∗− 12 XX2lψpψp∗dξ ∗1z∗ψ (z) ==√ψ̃ (ξ) = √111dzl p∈Z+ 1 ξ p+ 2 −σl p∈Z+ 1 z 2 + l (p−σ)2(6.60)(6.61)2Due to (6.57) one should have ψ ∗ (e2πil z) = e2πilr ψ ∗ (z) and ψ(e2πil z) = e−2πilr ψ(z),, so that:therefore one can take, for example, σ = lr + 1−l21ψ(z) = √Xψp1ψ (z) = √∗1 1l p∈Z+ 1 z l ( 2 +p)+r2ψp , ψp∗0 = δp+p0 ,0ψp∗X1 1l p∈Z+ 1 z l ( 2 +p)−r2(6.62)or the mode expansion is shifted by the r-charges, corresponding to given cycles.The same procedure gives for the twisted bosons1 1 −1 X Jn/l1 1 ˜1 1 X Jn/l1 X Jn/ll=zJ(z) = z l −1 J(ξ)= z l −1=1(n+1)llξ n+1ll n∈Z z 1l n+1ln∈Zn∈Z zwith the commutation relations between the modes of these bosons beingJn/l , Jm/l = nδn+m,0n, m ∈ Z(6.63)(6.64)These twisted bosons provide one of the convenient languages for the twist field representations.

The other one is provided by bosonization of the constituent fermionswith the fixed fractional parts of the power expansions in (6.62)1ψ(z) = √Xψ(a) (z),al a∈Z/lZ1 X ∗ψ ∗ (z) = √ψ(a) (z),l a∈Z/lZψ(a) (e2πiz ) = e−2πir−2πi l ψ(a) (z)(6.65)a∗∗ψ(a)(e2πiz ) = e2πir+2πi l ψ(a)(z)The corresponding bosons (see (6.266) in Appendix) X In(a)1a∗(z)ψ(a) (z) =+r+I(a) (z) = ψ(a)z n+1 zln∈Zalways have integer mode expansion.162(6.66)6.4. Twist-field representations from twisted fermionsTwist fields and bosonization for so(n)Let us mention first, that there is a difference between the groups NO(n) (h) andNSO(n) (h), since the action of the first one can also map V (z) 7→ −V (z), so thatone of the generators of the W-algebra V (e2πi z) = −V (z) becomes a Ramond field,and we allow this extra minus sign below 5 .In addition to the conjugacy classes [l, λ]+ , similar to those of gl(N ), we now alsohave to study [l]− ’s.

First one has to identify the action of NO(n) (h) on the fermions,where just by definition:(1, ~σ , 1) : ψα 7→ ψα∗σα = −1 :(6.67)This means that the element of our interest is the complete cycle[l]− : ψ1 7→ ψ2 7→ . . . 7→ ψl 7→ ψ1∗ 7→ . . . 7→ ψl∗ 7→ ψ1(6.68)Therefore 2N complex fermions can be realized as a pushforward of a single realfermion η(ξ), living on a 2l-sheeted branched coverp√ψα (z) dz = η(ξ (α) ) dξp√ψα∗ (z) dz = η(ξ (l+α) ) dξ(6.69)Here the branched cover z = ξ 2l can be realized as a sequence of two covers π2 : ξ 7→ζ = ξ 2 and πl : ζ 7→ ζ l = z, and it leads to more tricky global construction of theexact conformal blocks, see sect.

6.7 below.An important fact is that there is an element σ ∈ (NO(n) (h)/H) in the center ofthis groupσ = (1, (−1, −1, . . . , −1))(6.70)which generates the global automorphism of the cover of order two, which is continued to the global automorphism of algebraic curve during the consideration of exactconformal blocks in sect. 6.7. It acts locally by ξ 7→ −ξ.

Using this element one canwrite the OPE of η(ξ) in the form:η(ξ)η(σ(ξ 0 )) =1+ reg.ξ − ξ0(6.71)Now the analytic structure of this field can be obtainedsψ(z) =11ηp+ 1ηp+ 1dzz 4l − 2 X1 X22√=η(ξ) = √1111dξ2l p∈Z+ 1 z 2l (p+ 2 +σ)2l p∈Z+ 1 z 2l (p+σ)+ 22(6.72)2ψ ∗ (z) = ψ(e2πil z)5Note that in case of n = 2N the action of NSO(2N ) (h) on h is given by Weyl group action, butadditional element from NO(2n) (h) gives external (diagram) automorphism. Corresponding twisted(2)representations could be viewed as a representation of twisted affine Lie algebra DN .1636.

Twist-field representations of W-algebras, exact conformal blocks and character identitiesIn order to ensure right monodromies (6.68) for ψ, ψ ∗ one should get powers 2l1 Z inthe r.h.s., which means, that σ ∼ l − 12 ∼ 21 , and η(ξ) turns to be a Ramond fermionwith the extra ramificationη(ξ) =X ηnn∈Zξ,n+ 12(−)l X (−)n ηnψ ∗ (z) = √n12l n∈Z z 2l + 21 X ηnψ(z) = √n1 ,2l n∈Z z 2l + 2Let us now construct (a twisted!) boson from this fermion byJ(z) = (ψ ∗ (z)ψ(z)) = ψ(e2iπl z)ψ(z)(6.73)(6.74)This boson behaves like follows under the action of twist field:J1 7→ J2 7→ . . .

7→ Jl 7→ −J1 7→ . . . 7→ −Jl(6.75)To realize this situation we may take the Ramond boson on the cover in variable ζ:1z l −1 X Jr/l1 X Jr/ldζ X Jr/l==J(z) =r1dzζ r+1llz l +1z l (r+1)r∈Z+ 1r∈Z+ 1r∈Z+ 122where commutation relations of modes are given byJr/l , Jr0 /l = rδr+r0 ,0 r, r0 ∈ Z +212Inverse bosonization formula for this real fermion looks likeX ηn√σ1 iφ− (z) iφ+ (z)zψ(z) =en = √ e2lz2n∈Zwith the Pauli matrix σ1 =0 11 0(6.76)(6.77)(6.78), and it is discussed in detail in Appendix (6.10.1).Characters for the twisted modulesNow we turn directly to the computation of characters, using bosonization rules. Inorder to do this one has to apply the following heuristic “master formula” for thetraceχZM (q)χg (q) = tr Hg q L0 “ = ” Q Q∞(6.79)(1 − q θAdj,k (g)+n )k n=1over the space Hg which is the minimal space closed under the action of both Walgebra and twisted Kac-Moody algebra.

For simply-laced cases, gl(N ) and so(2N ),Hg is the module of corresponding Kac-Moody algebra, whereas in the so(2N + 1)case it should be entire fermionic Fock module due to presence of the fermionic Wcurrent. Explicit descriptions of Hg are the following: for gl(N ) it is the subspacewith fixed total fermionic charge, for so(2N ) it is the subspace with fixed parity oftotal fermionic charge, and for so(2N + 1) it is entire space.1646.5. Characters for the twisted modulesDenominator of (6.79) collects the contributions from the Fock descendants oftwisted bosons (parameters θAdj,k (g) are the eigenvalues of adjoint action of g on theCartan subalgebra), and the numerator – contribution of the zero modes. This formulais heuristic, moreover in some important cases we also get contribution from extrafermion, sometimes it is more informative to consider super-characters etc.

Below weprove the followingTheorem 6.2. The characters of twisted representations are given by the formulas(6.85), (6.88), (6.95), (6.97).gl(N ) twist fieldsKQTo be definite, let us fix an element g =[lj , e2πirj ] from (6.45) which, accordingj=1to (6.57) performs the permutation of fermions with simultaneous multiplication bye±2πirj . In this setting N fermions can be bosonized in terms of K twisted bosons(see detail in Appendix 6.10.3), and here we just present the final formulas1−lP(k)J0(j) 2πiα(j) 2πiα1 (j)z 2l(j)= √ eiφ− (e z) eiφ+ (e z) eQ (e2πiα z) l J0 (−1)k<jlP (k)1−lJ01 (j)z 2l −iφ(j)2πiα z) −iφ(j) (e2πiα z) −Q(j)(eJ2πiα−k<je +e(ez) l 0 (−1)ψα (z) = √ e −lψα∗ (z)(6.80)for α ∈ Z/lj Z, labeling the fields within [lj ]-cycle.

For the conformal dimensionone gets therefore (see (6.262), and computation by alternative methods in (6.139),(6.191))L0 =KXlj2 − 1j=124lj+KX1 (j) 2(J0 ) + . . .lij=1(6.81)\) ,and since we are computing character of the space, obtained by the action of gl(N1we have to take into account all vacua arising after the action of the shift operators(i)(j)eQ −Q , i.e. labeled by AK−1 root lattice. Hence, the character (6.79) for this caseis given byKPKPχg (q) = q j=1l2j −124ljPq i=11(r l +ni )22li i in1 +...+nK =0K Q∞Q(6.82)(1 − q k/lj )j=1 k=1In this formula the numerator collects contribution from the highest vectors χZM (they(i)(i)differ by the value of zero modes J0 ) of the Heisenberg algebras with generators Jn/li ,whereas the denominator contains the contributions from the descendants.1656. Twist-field representations of W-algebras, exact conformal blocks and character identitiesso(2N ) twist fields, K 0 = 0Consider now the twist fields (6.56) for g ∈ NO(2N ) (h), and take first K 0 = 0, so ourtwist has no minus-cyclesKYg=[lj , e2πirj ]+(6.83)j=1The only difference from the previous situation with the gl(N ) case is that now onealso has extra currents Jαβ̄ = ψα∗ (z)ψβ∗ (z) and Jᾱβ = ψα (z)ψβ (z).

It means that due(i)(j)to bosonization (6.266), (6.80) possible charge’s shifts now include e±(Q +Q ) , so thefull lattice of the zero-mode charges (one zero mode for each cycle [li , e2πiri ]+ ) containsall points withKX(6.84)ni ∈ 2Z, {ni } ∈ ZKi=1or is just the root lattice QDK . After corresponding modification of numerator andthe same contribution of the twisted Heisenberg algebra to denominator, the formulafor the character in this case acquires the formKPl2j −124ljj=1KPχg (q) = qPq j=112lj(nj +lj rj )2~n∈QDK(6.85)K Q∞Q(1 − q n/lj )j=1 n=1so(2N ) twist fields, K 0 > 0Take0KKYY2πirjg=[lj , e]+ [lj0 ]−j=1(6.86)j=1Now we have extra cycles of type [li0 ]− , so we have extra η-fermions that have to bebosonized in a different way (6.241):1(k)11J0z− 22l2l(6.87)γiηi (z) = √ eiφ− (z i ) eiφ+ (z i ) (−1) k2 lwhere {γi , γj } = 2δij are gamma-matrices (or generators of the Clifford algebraClK 0 (C)) in the smallest possible representation, which make different fermions anticommuting.

Due to presence of K 0 cycles of type [li0 ]− , the zero-mode χZM (q) gener(i)ating operators include now γj eQ , which perform integer shifts of i-th bosonic zeromode together with inessential action on fermionic vacua – now we do not have to(i)imply that the number of shifts by eQ should be even. Hence, instead of DK -latticefrom (6.85) the numerator includes now summation over the root lattice QBK , i.e.P0[ K 2+1 ]−121(ni +li ri )22liq i=1~n∈QBK0χg (q) = q ∆gKPPK Q∞Q(1 − q k/li )i=1 k=1∞K0 QQ(6.88)(1 − qi=1 k=0166(k+ 21 )/li0)6.5. Characters for the twisted modulesK 0 +1where factor 2[ 2 ]−1 corresponds to the dimension of the smallest representation of0so(K 0 ), generated by γi γj . Another simple factor q ∆g contains the minimal conformaldimension (without contribution of the “r-charges”)∆0g=KXl2 − 1i24lii=10+KX2l02 + 1ii=1(6.89)48li0which will be computed below in (6.142), (6.191).

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