Диссертация (Isomonodromic deformations and quantum field theory), страница 12

It is also important to noticethat commutation relations between these generators are linear, the only place whenthe non-linearity appears are the relations between these generators.Using the bosonization rules (3.53) one can rewrite these generators in the conventional form. To perform explicit splitting of this algebra into WN ⊕ H it is convenienttoP redefine Jα (z) 7→ Jα (z) + j(z), where the new currents already satisfy the conditionJα = 0 and the operator product expansions (OPE)j(z)j(w) =1N(z − w)2+ reg.Jα (z)Jβ (w) =δαβ − N1+ reg.(z − w)2(3.70)Now we take the bilinear expressionXαXttttttψ̃α (z + )ψα (z − ) =: eiϕ(z+ 2 )+iφα (z+ 2 ) :: e−iϕ(z− 2 )−iφα (z− 2 ) : =22α1 iϕ(z+ t )−iϕ(z− t ) X iφα (z+ t )−iφα (z− t )22 :22 := :e:etα(3.71)with j(z) = i∂ϕ, Jα (z) = i∂φα (z) and expand it into the powers of t.

Comparing1One can also notice at the level of the generating functions (3.68) that11vk,0 (p) = uk (p + ) − uk (p − ) .22First polynomials are given explicitly byu1 (p) = p,p3+3p53pu5 (p) =+ p3 + ,510u2 (p) =p4p2u4 (p) =+ ,42p2,249u3 (p) =p,6...(3.67)3. Free fermions, W-algebras and isomonodromic deformationswith (3.65) we get the following formulas:N 2U2 (z) = T (z) +: j (z) :,2N1 23U3 (z) = W3 (z) + 2N T (z)j(z) +: j (z) : + ∂ j(z) ,341U4 (z) = −W4 (z) + (T T )(z) + 3W3 (z)j(z) + 3 : j 2 (z) : T (z)+2N: j 4 (z) : + : j(z)∂ 2 j(z) : ,U5 (z) = . . .+4U1 (z) = N j(z),(3.72)where T (z) = −W2 (z) is the stress-energy tensor, and (AB)(z) is the “interacting”normal ordering˛dw(AB)(z) =A(w)B(z)z w−zOne find therefore, that one basis is related with the other by some complicated,though explicit and triangular transformation.

Here we can see that generators Uk (z)are actually dependent, namely, if N = 3, then W4 (z) = 0 and U4 (z) becomes somenon-linear expression of the lower generators.It is also easy to see that for the states (3.58)σJα,0|n, σi = (σα + nα )|n, σi ,σUk,0|n, σi = uk (σ + n)|n, σi ,(3.73)σUk,m>0|n, σi = 0.ItLis sometimes useful to decompose the whole Hilbert space into the sectorsHσ =LσσHnwith fixed h ∈ gl(N ) charges and also into the sectors Hlσ = PHnwithn∈ZNnα =lfixed overall u(1) = gl(1) charge. Summarizing all these facts we can formulate thefollowing\) , and for general σ spacesTheorem 3.1. Spaces Hlσ are representations of gl(N1σare the Verma modules of WN ⊕ H algebra with the highest weight vectors |σ, niHnand with basis vectors |Y , n, σi, ∀Y .\) generators have zero fermionic gl -charge, WN ⊕Proof is extremely simple: gl(N11H generators have zero charges with respect to the whole Cartan subalgebra h, so theσspaces Hlσ and Hnare closed under the action of these algebras.

We also know from(3.73) that |σ, ni are the highest weight vectors of WN ⊕ H, so we have a non-zeroσmap from the Verma module to Hn, but this Verma module is generally irreducibleL0and has the same character tr q , so we actually have an isomorphism.Free fermions and representations of W-algebrasLet us now illustrate how can free fermions appear in the theory with WN -symmetryat integer central charges after inclusion of extra Heisenberg algebra. Constructionbelow is a straightforward generalization of the bosonization procedure from [ILTe].It is well-known [FZ, FL] that conformal theory with WN -symmetry contains twodegenerate fields Vµ1 (z) and VµN −1 (z), such that their W -charges are determined by503.3. Non-Abelian U (N ) theorythe highest weights of the fundamental (N) and antifundamental (N̄) representations,respectively.

Their dimensions are∆(µi ) = 21 µ2i =N −1,2Ni = 1, N − 1(3.74)and they have the following fusion rules with arbitrary primary field[µ1 ] ⊗ [σ] = ⊕Nα=1 [σ + eα ](3.75)[µN −1 ] ⊗ [σ] = ⊕Nα=1 [σ − eα ]where {±eβ } is the set of all weights of N and N̄. One can define now the vertexoperatorsXXΨα (z) =Pσ+eα Vµ1 (z)Pσ , Ψ̃α (z) =Pσ−eα VµN −1 (z)Pσ(3.76)σσwhich, due to extra projector operators, act only from one Verma module to another,just extracting the corresponding term from the fusion rules (3.75). Using the generalstructure of the OPE of two initial degenerate fields1+N1−NVµ1 (z)VµN −1 (w) = 1 · (z − w) N + # · (z − w) N T (w) +X1(3.77)cα Vα (w) + . . .

,+(z − w) Nα∈roots(glN )1−None finds, that Ψ̃α (z)Ψβ (w) = δαβ 1 · (z − w) N + reg., i.e. these fields look almostlike fermions, except for the wrong power in the OPE. To fix this let us add an extrascalar field φ(z), such thatφ(z)φ(w) = −1log(z − w) + .

. .N(3.78)and define the new, the true fermionic, vertex operatorsψα (z) = e−iφ(z) Ψα (z),ψ̃α (z) = eiφ(z) Ψ̃α (z),α = 1, . . . , N(3.79)which have the canonical OPE (cf. with (3.52))ψα (z)ψ̃β (w) =ψα (z)ψβ (w) = reg.δαβ+ reg.z−wψ̃α (z)ψ̃β (w) = reg.(3.80)The rest is to understand, how to express the W-algebra generators in terms of thesefree fermions. One can easily write for the structure of the sum(z − w)−1/NXα2Ψ̃α (z)Ψα (w) =1+ # · (z − w)(L−2 1)(w)+z−w+#(z − w) · (L−1 L−2 1) + #(z − w)2 · (W−3 1)(w) + .

. . ==1+ # · (z − w)T (w) + # · (z − w)2 ∂T (w) + # · (z − w)2 W (w) + ...z−w51(3.81)3. Free fermions, W-algebras and isomonodromic deformationswith some coefficients (and where we have used obvious notations for the descendants).We do not need their exact numeric values at the moment, just the very fact thatonly the unit operator 1 enters the r.h.s. of this OPE together with its descendants.Using additionally the OPE of the U (1) factors(z − w)1/N eiφ(z) e−iφ(w) =112 23 3=: exp i(z − w)∂φ(w) + i(z − w) ∂ φ(w) + (z − w) ∂ φ(w) :=2611= 1 + (z − w)j(w) + (z − w)2 ∂j(w) + (z − w)3 ∂ 2 j(w)+26111+ (z − w)2 : j(w)2 : + (z − w)3 : j(w)∂j(w) : + (z − w)3 j(w)3 + .

. .226(3.82)one can get1112ψ̃α (z)ψα (w) =+ j(w) + (z − w) # · T (w) + j(w) + : j(w) : +z−w22α111 232+(z − w) # · W (w) + # · j(w)T (w) + ∂ j(w) + : j(w)∂j(w) : + : j(w) : + ...626(3.83)This formula states, how the standard W-generators can be expressed via the fermionicbilinears by some triangular transformation, and its symmetric form is equivalent to(3.71), (3.72).XVertex operators and Riemann-Hilbert problemVertex operators and monodromiesLet us now turn to general construction of the monodromy vertex operator2Vν (t) : Hσ → Hθ(3.84)Actually one can define only the operator Vν (1) due to conformal Ward identityVν (t) = t−∆ν tL0 Vν (1)t−L0(3.85)and the operator Vν (1) is defined by the following three properties:• Vν (1) is a (quasi)-group element, i.e.Vν (1)Hσ (Vν (1))−1 ⊆ Hθ ,(Vν (1))−1 Hθ Vν (1) ⊆ HσAs we discussed already in sect.

3.2 this fact actually implies that all correlatorsof fermions in the presence of such an operator can be computed using the Wicktheorem.2Notice, that we have here only the conservation of the “total charge”and apart of that their values are arbitrary.52Pασα +Pανα =Pα θα ,3.4. Vertex operators and Riemann-Hilbert problem• hθ|Vν (1)|σi = 1, which is a kind of convenient normalization. Notice, however,that vertex operator is defined by the adjoint action only up to some diagonalfactor S = exp(β), β ∈ h ⊂ gl(N ).

In what follows we shall restore thesediagonal factors when necessary.• All two-fermionic correlators give the solution for the 3-point Riemann-Hilbertproblem in the different regionshθ|Vν (1)ψ̃ασ (z)ψβσ (w)|σi = Kαβ (z, w),|z| ≤ 1, |w| ≤ 1hθ|ψ̃α̇θ (z)ψβ̇θ (w)Vν (1)|σi= Kα̇β̇ (z, w),|z| ≥ 1, |w| ≥ 1hθ|ψ̃α̇θ (z)Vν (1)ψβσ (w)|σi = Kα̇β (z, w),|z| ≥ 1, |w| ≤ 1−hθ|ψβ̇θ (w)Vν (1)ψ̃ασ (z)|σi = Kαβ̇ (z, w),(3.86)|z| ≤ 1, |w| ≥ 1In terms of some matrix kernels K(z, w) = Kν (z, w), where we have used {α, β}and {α̇, β̇} to denote matrix indices, corresponding to different bases, associatedwith the points z = 0 and z = ∞ respectively.By this moment the only claim is that this operator is uniquely defined by thproperties listed above, and this follows from the fact, that all matrix elements ofthe quasi-group Vν (1) element are given by certain determinants of the matrices withthe entries, constructed from K(z, w).

Existence of this operator is therefore obvious,since one can compute all its matrix elements using the Wick theorem.Now, we would like to specify the kernels K(z, w) first by their monodromy properties. We associate the basis at z = 0 with the eigenvectors of M0 ∼ e2πiσ , while thebasis at z = ∞ with the eigenvectors of M∞ ∼ e2πiθ (only the conjugacy classes ofthese two matrices are fixed, and certainly in general [M0 , M∞ ] 6= 0). We propose anexplicit form of the kernelKαβ (z, w) =[φ(z)φ(w)−1 ]αβz−wgiven in terms of the solution to the linear systemdA0A1φ(z) = φ(z)+= φ(z)A(z)dzzz−1(3.87)(3.88)with A0 ∼ σ, A1 ∼ ν, A∞ ∼ θ and prescribed monodromiesXγ0 : φαi (z) 7→(M0 )αβ φ(z)βiβγ∞ : φαi (z) 7→X(3.89)(M∞ )αβ φ(z)βiβPalso implying monodromy around z = 1, i.e.

γ1 : φαi (z) 7→ β (M1 )αβ φ(z)βi , withM1 ∼ e2πiν and M0 M1 M∞ = 1. Solutions for a linear system (3.88) can be expressedthemselves in terms of a fermionic correlators, namelyφαγ (z) =z · hθ|Vν (1)ψ̃α (z)| − 1γ , σiφ−1γβ (z) =z · hθ|Vν (1)ψβ (z)|1γ , σi53(3.90)3. Free fermions, W-algebras and isomonodromic deformationsfor some fixed normalization at z → 0.