Диссертация (Isomonodromic deformations and quantum field theory), страница 39

These cyclestype [l]+ produce l pairs of A-cycles Aiare permuted by σ (6.174), so they actually form l − 1 independent odd combinations,giving contribution to the r.h.s. of (6.177). For two mutuallyP inverse elements of00the type [l ]− one gets instead 2l A-cycles with constrainti Ai = 0, and with00the action of involution σ : Ai 7→ Ai+l , giving l independent odd combinations{Ai − Ai+l0 }, arising in the r.h.s. of (6.177), while the extra term −N corresponds tocharge conservation in the infinity.Hence, we got g− odd A-cycles, whose projections to P1 encircle pairs of thecolliding twist fields Oh·g (q2i−1 )Og−1 (q2i ) for i = 1, . .

. , M , so that the integrals of˛1dS = aI , I = 1, . . . , g−(6.179)2πi AIgive the W-charges in the intermediate channels of conformal block (6.163). ThereforedS can be expandedg−2MXX(6.180)dS =aI dωI +dSrii=1I=1over the odd holomorphic differentials, and meromorphic differentials of the 3-rd kindcorresponding to the nonvanishing r-charges.Now, for the bidifferential dΩ2 (ξ, ξ 0 ) one can writedΩ2 (ξ, ξ 0 ) = K(ξ, ξ 0 ) − K(σ(ξ), ξ 0 ) = 2K(ξ, ξ 0 ) − K̃(ξ, ξ 0 )(6.181)where K(ξ, ξ 0 ) is the canonical meromorphic bidifferential on Σ (the double logarithmic derivative of the prime form, see [Fay]), normalized on vanishing A-periods ineach of two variables, andK̃(ξ, ξ 0 ) = K(ξ, ξ 0 ) + K(σ(ξ), ξ 0 )(6.182)is actually a pullback of the canonical meromorphic bidifferential on Σ̃.

Indeed, considerδK(ξ, ξ 0 ) = K(ξ, ξ 0 ) − K(σ(ξ), σ(ξ 0 ))(6.183)1856. Twist-field representations of W-algebras, exact conformal blocks and character identities¸which is already holomorphic at ξ = ξ 0 , and Ai δK(ξ, ξ 0 ) = 0, since due to (6.174)normalization conditions do not change under involution. Thus δK(ξ, ξ 0 ) = 0 and thecanonical bidifferential is σ-invariantK(σ(ξ), σ(ξ 0 )) = K(ξ, ξ 0 )(6.184)K̃(ξ, ξ 0 ) = K̃(σ(ξ), ξ 0 ) = K̃(ξ, σ(ξ 0 ))(6.185)Moreover, sinceexpression (6.182) actually defines the canonical bidifferential on Σ̃.Computation of conformal blockNow we use the technique from [ZamAT87, ZamAT86, ApiZam, GMtw] to computethe conformal block (6.163). For the vacuum expectation value of the stress-energytensor (6.168) one gets from (6.172), (6.181)hT (z)Oh1 ·g1 (q1 )Og1−1 (q2 ) .

. . OhM ·gM (q2M −1 )Og−1 (q2M )iG0−1 =M 2XX1 dS=tz (ξ) −t̃z (ζ) +4 dzπ2N (ξ)=z(6.186)πN (ζ)=zwhere tz and t̃z are the regularized parts of the bidifferentials K and K̃ on diagonalin coordinate z:dπ2N (ξ)dπ2N (ξ 0 )10lim K(ξ , ξ) −tz (ξ)dξ =2 ξ→ξ0(π2N (ξ 0 ) − π2N (ξ))21dπN (ζ)dπN (ζ 0 )02t̃z (ζ)dζ =lim K̃(ζ , ζ) −2 ζ→ζ 0(πN (ζ 0 ) − πN (ζ))22(6.187)Expanding (6.186) at z → qi one gets11l2 − 1{ζ; z} + reg.

=+ reg.z→qi 12(z − qi )2 24l2114l02 − 1tz (ξ) ={ξ; z} + reg. =+ reg.z→qi 12(z − qi )2 96l02t̃z (ζ) =186(6.188)6.7. Exact conformal blocks of W (so(2N )) twist fields0in local co-ordinates ξ 2l = ζ l = z − qi , which gives for the conformal dimensionsthe fields Og (with generic o(2N ) twist field of the type (6.86))∆g =KXlj2 − 124ljj=10+KX2lj02 + 148lj0j=1+KX12i=1li ri2=∆0g+KX1i=12li ri2 ,11of(6.191)dzwhere the last term in the r.h.s. comes from the expansion dS ≈ ri z−q+.

. .. Withouticontributions of r-charges this formula is equivalent to (6.142), (6.191).From the first order poles we obtainX∂qi log G0 (q1 , . . . , q2M ) =Res tz (ξ)dξ −π2N (ξ)=qi1+4π2N (ξ)=qiRes t̃z (ζ)dζ+πN (ζ)=qi(dS)2Res,dzXX(6.192)i = 1, . . . , 2MThis system of equations for conformal block is obviously solved, so that we canformulate:Theorem 6.4. Conformal blocks (6.163) for generic W (o(2N )) twist fields are givenby(6.193)G0 (a, r, q) = τB (Σ|q)τB−1 (Σ̃|q)τSW (a, r, q)whereX∂qi log τB (Σ|q) =Res tz (ξ)dξπ2N (ξ)=qi∂qi log τB (Σ̃|q) =XRes t̃z (ζ)dζ(6.194)πN (ζ)=qii = 1, . . . , 2Mand14∂qi log τSW (a, r, q) =∂log τSW =∂aI11˛XResπ2N (ξ)=qi(dS)2,dz(6.195)AI ◦ BJ = δIJ ,dS,i = 1, . . .

, 2MI, J = 1, . . . , g−BIThe counting here works astz − t̃z → 2KXj=1ljKK 2lj2 − 1 Xlj2 − 1 Xlj − 1−l=j2224lj24lj24ljj=1j=1(6.189)for the [l]+ -cycles, and0tz − t̃z →KXj=102lj00KK4lj02 − 1 Xl02 − 1 X2lj02 + 10 j−l=j96lj0224lj0248lj0j=1j=1for the [l0 ]− -cycles.187(6.190)6. Twist-field representations of W-algebras, exact conformal blocks and character identitiesEquations (6.194) define so-called Bergmann tau-functions [KK04] for the curvesΣ and Σ̃ respectively, while the so-called Seiberg-Witten tau-function (6.195) can beread literally from [GMtw]g−g−1X11 XaI TIJ aJ +aI UI (r) + Q(r)log τSW (a, r, q) =4 I,J=12 I=14(6.196)where TIJ is the g− × g− “odd block” of the period matrix of Σ, or the period matrixof corresponding Prym variety [Fay], the “odd” vector˛XUJ (r) = dΩr =riα AJ (qiα ), J = 1, .

. . , g−(6.197)i,αBJwhere qiα are preimages of qi and riα – corresponding r-charges, andQ(r) =Xriα rjβlog θ∗ (A(qiα )−A(qjβ ))−Xliα (riα )2qiαqiα 6=qjβαd(z(q) − qi )1/li log αh2∗ (q)q=qi(6.198)where θ∗ is some odd Riemann theta-function for the curve Σ, A is the Abel map,andgX∂θ∗ (0)2h∗ (z) =dωI (z)(6.199)∂ZII=1Relation between W (so(2N )) and W (gl(N )) blocksIt is interesting to compare the formulas from previous section with the formulasfrom [GMtw] for the exact W (gl(N )) conformal blocks. Since, as we already discussed W (so(2N )) ⊂ W (gl(N )), any vertex operator of the W (gl(N )) algebra is avertex operator of its subalgebra W (so(2N )),Q and it is clear from our construction,that twist fields Og for the elements g ∼ [l, e2πir ]+ , are also the twist fields forW (gl(N )).

Moreover, the corresponding Verma modules, generated by W (so(2N ))and by W (gl(N )), actually coincide 12 , and it means that corresponding conformalblocks of such fields in these two theories should coincide as well.Indeed, in such a case Σ = Σ̃ t Σ̃, and therefore K(ξ, ξ 0 ) = 0 if ξ 0 , ξ are on differentcomponents, and K(ξ, ξ 0 ) = K̃(ξ, ξ 0 ) if they are on the same component, hencetz (z) = 2t̃z (z)(6.200)For holomorphic and meromorphic differentials, one has in this case in natural basis˛˛aI =dS =dS,I = 1, . . . , g̃(6.201)(1)(2)AAIriαI= Res qiα dS = Res σ(qiα ) dS12These two modules coincide due to dimensional argument: they are both irreducible and havethesame characters.

Irreducibility follows from the fact that null-vector condition can be written aslog gα, 2πi ∈ Z for a simple root α and generic r’s, see also comments in sect. 6.5.6.1886.8. Conclusionfor the preimages {qiα } on Σ̃, and the period matrix of Σ consists of two nonzero g̃ × g̃blocks:(6.202)T (11) = T (22) = T̃Under such conditions formula (6.4) turns intoG0 (a, r, q) = τB (Σ̃|q)τ̃SW (a, r, q)wherelog τ̃SW (a, r, q) =12g̃XaI T̃IJ aJ +g̃XaI ŨI (r) + 12 Q̃(r)(6.203)(6.204)I=1I,J=1with corresponding obvious modifications of formulas (6.197) and (6.198), which givesexactly the W (gl(N )) conformal block in terms of the data on smaller curve Σ̃.ConclusionWe have considered in this chapter the twist fields for the W-algebras with integer Virasoro central charges, which are labeled by conjugacy classes in the Cartan normalizers NG (h) of corresponding Lie groups.

In addition to the most common WN -algebras,corresponding to A-series (or W (gl(N )) = WN ⊕ H, coming from G = GL(N )), wehave extended this construction for the G = O(n) case, which includes in addition toD-series the non simply-laced B-case with the half-integer Virasoro central charge.In terms of two-dimensional conformal field theory our construction is based onthe free-field representation, where generalization to the D-series and B-series exploitsthe theory of real fermions, which in the odd B-case cannot be fully bosonized, sothat in addition to modules of the twisted Heisenberg algebra one has to take intoaccount those of infinite-dimensional Clifford algebra.

This construction producesrepresentations of the W-algebras (that are at the same time twisted representationsof corresponding Kac-Moody algebras), which can be decomposed further into Vermamodules. To find this decomposition we have computed the characters of twistedrepresentations, using two alternative methods.The first one comes from bosonization of the W-algebra or corresponding KacMoody algebra at level one, dependently on particular element from NG (h) it identifiesthe representation space with a collection of the Fock modules for untwisted or twistedbosons. The essential new phenomenon, which appears in the case of orthogonalgroups is presence of different [l]− cycles in g ∈ NG (h) and necessity to use in suchcases “exotic” bosonization for the Ramond-type fermions with non-local OPE on thecover.Alternative method for computation of the characters uses pure algebraic construction of the twisted Kac-Moody algebras and the Weyl-Kac formula in principalgradation.There are examples of elements g1 , g2 that are not conjugated in NG (h), but conjugated in G.

Since two different constructions with elements g1 and g2 give differentformulations of the same representation, computation of corresponding charactersχg1 (q) and χg2 (q) leads to some simple but nontrivial identities for the corresponding lattice theta-functions, χg1 (q) = χg2 (q), which have been also proven by directmethods.1896. Twist-field representations of W-algebras, exact conformal blocks and character identitiesWe have also derived an exact formula for the general conformal block of the twistfields in D-case, which directly generalizes corresponding construction for commonWN -algebra. The result, as is usual for Zamolodchikov’s exact conformal block, isexpressed in terms of geometry of covering curve (here with extra involution), andcan be factorized into the classical “Seiberg-Witten” part, totally determined by theperiod matrix of the corresponding Prym variety, and the quasiclassical correction,expressed now in terms of two different canonical bi-differentials.