Диссертация (1137342), страница 27
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Weshould also point out that only local deformations of the positions of the branch points{qα } are allowed, since the global ones – due to nontrivial monodromies – can changethe global structure of the cover π : C → P1 . This leads in particular to the fact thatin the case of non-abelian monodromy group the positions of the branch points {qα }cannot play the role of the global coordinates on the corresponding Hurwitz space.2The picture of the 3-sheeted cover with the most simple branch cuts looks likeat fig.5.1, where we have shown explicitly three (dependent) cycles in H1 (C) corresponding to the cuts between the positions of the fields, labeled by mutually inversepermutations.
To understand our notations better we present also at fig.5.2 the picture of the vicinity of the branch-point (of the 6-sheeted cover) of the cyclic types ∼ [3, 1, 2] with several independent permutation cycles.2Although, sometimes the Hurwitz space of our interest occurs to be rational, and in this caseone can choose some global coordinates – but not the positions of the branch points. An explicitexample is considered below in sect.5.6.1245.2. Twist fields and branched coversFigure 5.1: Covering Riemann surface C with simplest cuts between the positions ofcolliding twist-fields.
Sum of the shown cycles of A-type vanishes in H1 (C).Figure 5.2: Vicinity of a ramification point of a general type.1255. Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformationsCorrelators with the currentConsider a permutation of the cyclic type s ∼ [l1 , ..., lk ], which corresponds to theramification at z = q (for simplicity we put q = 0) with k preimages q i , π(q i ) = qwith multiplicities li . The coordinates in the vicinity of these points can be chosen asξi = z 1/li . One can write down a general expression for the expansion of current J(z)on the coverli −1 X (i)k−1 X (j)k XXan−vi /li · hi,vi Xbn · Hj+J(z) =(5.10)1+n−v/liizz n+1j=1 n∈Zi=1 v =1 n∈Ziwhere hi,vi and Hj form the orthogonal basis in h out of the eigenvectors of thepermutation s, and in coordinates (related to the weights {ei }) they have the formh1,v1 = (1, e2πi·v1 /l1 , ..., e2πi(l1 −1)·v1 /l1 ; 0, ..., 0; ...; 0, ..., 0)h2,v2 = (0, ...0; 1, e2πi·v2 /l2 , ..., e2πi(l2 −1)·v2 /l2 ; 0, ..., 0; ...; 0, ..., 0)X (i)(1)(1) (2)(2)(k)(k)Hj = (yj , ..., yj ; yj , ..., yj ; ....; yj , ..., yj )li yj = 0(5.11)iwith hi,vi , corresponding to non-zero eigenvalues of the permutation cycles si , whileHj – to the trivial permutations.(i) (j)The expansion modes satisfy usual Heisenberg commutation relations [au , av ] =(i) (j)uδu+v δij , [bu , bv ] = uδu+v δij , up to possible inessential numerical factors which canbe extracted from the singularity of the OPE J(z)J(z 0 ).
The condition that fieldOs (q) is primary for the W-currents means in terms of the corresponding state that(j)a(i)ui |si = bn |si = 0,ui > 0, n > 0, ∀ i, j(5.12)(j)and this state is also an eigenvector of the zero modes b0 ∀ j. The correspondingeigenvalues are extra quantum numbers – the charges, which have to be included intothe definition of the state |si → |s, ri (and Os (q) → Os,r (q)) and fixed by expansionof the h-valued 1-form dzJ(z)|si at z → 0, i.e.Ndzdz X iJ(z)|s, ri =r ei |s, ri + reg.zz i=1(5.13)where r1 = .
. . = rl1 , rl1 +1 = . . . = rl1 +l2 , etc: the U (1) charges are obviously thesame for each point of the cover, they also satisfy the slN conditionNXrαi = 0,∀α(5.14)i=1for each branch point q ∈ {qα }. It means that G1 (z)dz on the cover C has only poleswith prescribed by (5.13) singularities, so one can writeg2LXG1 (ξ|q)dξ X=dΩrα +aI dωI = dSG0 (q)α=1I=1126(5.15)5.2.
Twist fields and branched coversand we shall call this 1-form as the Seiberg-Witten (SW) differential, since its periodsover the cycles in H1 (C) play important role in what follows. Here {dωI }, I = 1, . . . , gare the canonically normalized first kind Abelian holomorphic differentials˛1dωJ = δIJ2πi AI(in slightly unconventional normalization of [Dub] as compare to [Fay, Mumford]),whileNXdΩrα =rαi dΩqαi ,p0i=1is the third kind meromorphic Abelian differential with the simple poles at all preimiages of qα (with the expansion dΩrα =i lαi rαi dξξiα + reg. in corresponding local coorαpαdinates) and vanishing A-periods.
We denote by qαi = π −1 (qα ), i = 1, . . . , N thepreimages on C of the point qα , with such conventions the point of multiplicity lαi hasto be counted lαi times ( Res piα dΩrα = lαi rαi ).The A-periods of the differential (5.15)˛˛11dξG1 (ξ|q)aI =dS =, I = 1, . . . , g(5.16)2πi AI2πi AI G0 (q)are determined by fixed charges in the intermediate channels due to (5.7). The numberof these constraints is ensured by the Riemann-Hurwitz formula χ(C) = N · χ(P1 ) −#BP for the cover π : C → P1 , org=L XkαXljαLX(N − kα ) − N + 1−1 −N +1=α=1 j=1(5.17)α=1where kα stands for the number of cycles in the permutation sα . One can easily seethis in the “weak-coupling” regime, when we can apply (5.7) in the limit q2α−1 → q2α ,so that−1G0 (q1 , ..., q2L )|θ = hOs1 (q1 )Os−1(q)...O(q)O(q)2 sL 2L−12L i ∼sL1θ1∼q2α−1 →q2αhLYθL(5.18)Vθα (q2α )i + . .
.α=1PLand the charge conservation law0 gives exactly N − 1 constraints toα=1 θ α = Pthe parameters {θ α }, whose total number is Lα=1 (N − kα ), since for each pair ofcolliding ends of the cut (i.e. α = 1, . . . , L) there are kα linear relations for the Nintegrals over the contours, encircling two colliding ramification points, see fig.5.1(this procedure also gives a way to choose convenient basis in H1 (C) as shown on thispicture). For the dual B-periods of (5.15) one gets˛DaI = dS = TIJ aJ + UI , I = 1, . .
. , g(5.19)BI1275. Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformationswhere the last term can be transformed using the Riemann bilinear relations (RBR)asX˛XUJ =dΩrα =rαm AJ (qαm ), J = 1, . . . , g(5.20)´pα BJα,mwhere AJ (p) = p0 dωJ is the Abel map of a point p ∈ C, and UJ do not depend onthe reference point p0 ∈ C due to (5.14).Stress-tensor and projective connectionSimilarly the 2-differential from (5.8) is fixed by its analytic properties and one canwrite1G2 (p0 , p|q)dξp0 dξp = dS(p0 )dS(p) + K(p0 , p) − K0 (p0 , p)(5.21)G0 (q)Nwhere˛dξp0 dξp00K(p , p) = dξp0 dξp log E(p , p) =+ reg.,K(p0 , p) = 0(5.22)20(ξp − ξp )AIis the canonical meromorphic bidifferential on C (the double logarithmic derivative ofthe prime form, see [Fay]), normalized on vanishing A-periods in each of two variables,whiledπ(ξ)dπ(ξ 0 )K0 =(5.23)(π(ξ) − π(ξ 0 ))20dzdz1is just the pull-back π ∗ of the bidifferential (z−z0 )2 from P .
Formula (5.21) is fixedby the following properties: in each of two variables it has almost the same structureas G1 (ξ)dξ, but with extra singularitydiagonalPp0 = p, which comes from (5.3), itP onijalso satisfies an obvious conditionG2 (z, z 0 ) = G2ij (z, z 0 ) = 0ijNow one can define [Fay] the projective connection tx (p) by subtracting the singular part of (5.22)1dx(p0 )dx(p)02tx (p)dx =K(p , p) −(5.24)2(x(p0 ) − x(p))2 p0 =pIt depends on the choice of the local coordinate x(p), and it is easy to check thattx (p)dx2 − tξ (p)dξ 2 =ξxxxξx32ξxxξx1{ξ, x}dx212(5.25)2−is the Schwarzian derivative.where {ξ, x} = (Sξ)(x) =It is almost obvious that expression (5.24) is directly related with the averageof the Sugawara stress-tensor T (z) (5.4) of conformal field theory (with extendedW-symmetry), since normal ordering of free bosonic currents exactly results in subtraction of its singular part.
One gets in this way from (5.21) thath: 12 Ji (z)Ji (z) : Os1 (q1 )Os−1(q2 ) . . . OsL (q2L−1 )Os−1 (q2L )i1LhOs1 (q1 )Os−1(q2 )...OsL (q2L−1 )Os−1 (q2L )i1L2i1 dS(z )= tz (z i ) +2dz128=(5.26)5.3. W-charges for the twist fieldswhere z = z(p) is the global coordinate on P1 , and we have used that after subtraction(5.24) one can substitute K 7→ 2tz (p)dz 2 and K0 7→ 0, leading tohT (z)iO =(q2 ) . . . OsL (q2L−1 )Os−1 (q2L )ihT (z)Os1 (q1 )Os−11L(q2 ) .
. . OsL (q2L−1 )Os−1 (q2L )ihOs1 (q1 )Os−11! L2X1 dS(p)=tz (p) +2dz=(5.27)π(p)=zwhere sum in the r.h.s. computes the pushforward π∗ , appeared here as a result ofsummation in (5.4).W-charges for the twist fieldsConformal dimensions for quasi-permutation operatorsUsing the OPE with the stress-tensor T (z)T (z)Os,r (q) =∆(s, r)Os,r (q) ∂q Os,r (q)++ reg.(z − q)2z−q(5.28)one can extract from the singularities of (5.27) the dimensions of the twist fields.Following [BR] we first notice from (5.24) that near the branch point (e.g. at q = 0)the local coordinate is ξi = z 1/li , so that 2l2 − 1 1dξ1(5.29)tz (p) = tξ (p)+ {ξ, z} = tξ (p)z 2/li −2 +dz1224l2 z 2The first term in the r.h.s. cannot contain z12 -singularity, since tξ (p) is regular in localcoordinate on the cover C.
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