Диссертация (1137342), страница 28

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The second source of the second-order pole in (5.27) comesfrom the poles of the Seiberg-Witten differential (5.15), which look asdS ≈ ri lidzdξi+ reg. = ri + reg.ξiz(5.30)Taking them into account together with (5.29) one comes finally to the formula∆(s, r) =kXl2 − 1ii=124li+kX1i=12li ri2(5.31)which gives the full conformal dimension for the twist fields with r-charges.Since we are going to use this formula intensively below, let us illustrate first, howit works in the first two nontrivial cases:• N = 2: there are only two possible cyclic types:– s ∼ [1, 1], then l1 = l2 = 1, r1 = −r2 = r, so ∆(s, r) = r2 is only given bythe r-charges;1295.

Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations– s ∼ [2], then the only l1 = 2, the single r-charge must vanish, so one just1gets here the original Zamolodchikov’s twist field with ∆(s, r) = 16.• N = 3: here one has three possible cyclic types:– s ∼ [1, 1, 1], then l1 = l2 = l3 = 1, r1 +r2 +r3 = 0, ∆(s, r) = 21 (r12 + r22 + r32 )– s ∼ [2, 1], then l1 = 2, l3 = 1, r1 = r2 = r, r3 = −2r, ∆(s, r) =116+ 3r2– s ∼ [3], then l1 = 3, the single r-charge again should vanish, so that thedimension is ∆(s, r) = 19 .Quasipermutation matricesThe hypothesis of the isomonodromy-CFT correspondence [Gav] relates the constructed above twist fields to the quasipermutation monodromies (we return to thisissue in more details later).

This correspondence relates the WN charges of the twistfields to the symmetric functions of eigenvalues of the logarithms of the quasipermutation monodromy matricesMα ∼ e2πiθα ,α = 1, . . . , 2L ,(5.32)Nbeing the elements of the semidirect product SN n (C× ) (here we consider only thematrices with det Mα = 1).

An example of the quasipermutation matrix of cyclictype s ∼ [3, 2] is0a1 e2πir1000 00a2 e2πir100  2πir10000 M =  a3 e(5.33)2πir2  0000b1 e000b2 e2πir20where a1 a2 a3 = 1, b1 b2 = −1, 3r1 + 2r2 = 0 to get det M = 1. A generic quasipermutation is decomposed into several blocks of the sizes {li }, each of these blocks is givenbyiπe2πiri × e li(li )sli ,i = 1, . .

. , kwhere sli is the cyclic permutation of length li , (l) = 0 for l-odd and (l) = 1 forl-even. It is easy to check that eigenvalues of such matrices are2πiθi,viv2πi ri + l ii ,λi,vi = e=ei = 1, . . . , kli − 1li − 11 − li 1 − livi =,+ 1...,− 1,2222(5.34)According to relation (5.32) the conformal dimension of the corresponding field is2 Xkk1X 21Xvili2 − 1 X 1 2∆(M ) =θi,vi =ri +=+li ri22li24l2ii=1i=1130(5.35)5.3. W-charges for the twist fieldswhere we have used thatPvi = 0 for any fixed i = 1, . .

. , k, andl(l2 − 1)=12(l−1)/2P−(l−1)/2(l−1)/2Pv2l = 2m + 1 (v ∈ Z)21)2(5.36)vl = 2m (v ∈ Z +−(l−1)/2for both even or odd l ∈ {li }. The calculation (5.35) for the quasipermutation matricesreproduces exactly the CFT formula (5.31), confirming the correspondence.W3 currentOne can also perform a similar relatively simple check for the first higher W3 -current.An obvious generalization of (5.35) givesw3 (M ) =X(ra +a<b<cvavbvc1Xva(ra + )3 =)(rb + )(rc + ) =lalblc3 alakk1 X 3 X va2 X 1 3 X li2 − 1ra 2 =r +li r +ri=3 a ala3 i12liai=1i=1(5.37)To extract such formulas from conformal field theory one has to analyze the multicurrent correlation functions in presence of twist operators and action of the corresponding modes of the Wk (z) currents.

For W = W3 (z), following (5.8) one can firstdefineG3ijk (z, z 0 , z 00 |q)dzdz 0 dz 00 = G3ijk (z, z 0 , z 00 |q1 , ..., q2L )dzdz 0 dz 00 == hJi (z)Jj (z 0 )Jk (z 00 )Os1 (q1 )Os−1(q2 )...OsL (q2L−1 )Os−1 (q2L )idzdz 0 dz 001(5.38)Land write, similarly to (5.21)G3 (p00 , p0 , p|q)dξp00 dξp0 dξp = dS(p00 )dS(p0 )dS(p)+G0 (q)11000000000+dS(p ) K(p , p) − K0 (p , p) + dS(p ) K(p , p) − K0 (p , p) +NN1+dS(p) K(p00 , p0 ) − K0 (p00 , p0 )N(5.39)where the r.h.s. has appropriate singularities at all diagonals and correct A-periods ineach of three variables. Extracting singularities and using (5.4), (5.24) one can writehW (z)iO =Xπ(p)=z13dS(p)dz(p)3dS(p)+ 2tz (p)dz(p)!(5.40)It is easy to see that due to (5.29), (5.30) this formula gives the same result as (5.37).1315.

Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformationsFormula (5.34) also shows, how the charges of the twist fields can be seen withinthe context of W-algebras. It is important, for example, that for the complete cyclepermutation one would get its WN charges w2 (θ), w3 (θ), . . . , wN (θ), where1 N −1 N −11−N1−Nρ=,− 1, . . . ,+ 1,(5.41)θ=NN2222i.e. the vector of charges is proportional to the Weyl vector of g = slN . Such fieldsare non-degenerate from the point of view of the WN algebra, since for degeneratefields the charge vector always satisfy the condition (θ, α) ∈ Z for some root α.It means that here we are beyond the algebraically defined conformal blocks, andfurther investigation of descendants W−1 O etc can shed light on the structure ofgeneric conformal blocks for the W-algebras.

We are going to return to this issueelsewhere.Higher W-currentsFor the higher W-currents (Wk (z) with k > 3) the situation becomes far more complicated. We discuss here briefly only the case of W4 (z), which already gives a hint onwhat happens in generic situation.

An analog of (5.35), (5.37) gives for the quasipermutation matricesX1vbvcvd1va(ra + )(rb + )(rc + )(rd + ) = ∆(M )2 − Aw4 (M ) =(5.42)llll24aa<b<c<dbcdwith ∆(M ) given by (5.35) andA=N Xa=1vara +la4=kXi=1li ri4+6kXi=1l2ri2 ik− 1 X (li2 − 1)(3li2 − 7)+12li240li3i=1(5.43)To get this from CFT one needs just the most singular part of the correlation function4dz4(5.44)hW4 (z)iO (dz) = w4+ ...z→qz−qwhich is a particular case of the current correlatorsRi1 ,...in (z1 , .

. . zn ) = h: Ji1 (z1 ), . . . Jin (zn ) :iO dz1 . . . dzn(5.45)and the technique of calculation of such expressions is developed in Appendix 5.8.From the definition of the W4 (z) current (5.4) it is clear, that one should take onlythe most singular parts of the correlation functions of four currentsi t tii t tii t tiRiiii (z, z, z, z) =+ 6·+ 3·=i t tii t tii t ti= dS(z i )4 + 6dS(z i )2 K̂ii (z, z) + 3K̂ii (z, z)2and132(5.46)5.3.

W-charges for the twist fieldsitRiijj (z, z, z, z) =+ 4·tjittjittjitit++tjtjittjittjittj++ 2·ittjittjittjittj+== dS(z i )2 dS(z j )2 + K̂ii (z, z)dS(z j )2 + K̂jj (z, z)dS(z i )2 +(5.47)+4K̂ij (z, z)dS(z i )dS(z j ) + K̂ii (z, z)K̂jj (z, z) + 2K̂ij (z, z)2taken at the coinciding values of all arguments. It means, that one has to substitutedS(z i ) = ridz+ ...z(5.48)(we again put here q = 0 for simplicity) and do the same for the propagator K̂ij (z1 , z2 ) =K(z1i , z2j ) − δij K0 (z1 , z2 ) (see Appendix 5.8 for details), i.e.

to substitute into (5.46),(5.47)l2 − 1 (dz)2dzdz̃ dz 1/l dz̃ 1/l+...=−+ ...K̂ii (z, z) = 1/l(z − z̃ 1/l )2 (z − z̃)2 z→z̃12l2 z 2(5.49)1ζ i dz 1/l ζ j dz 1/lζ i−j(dz)2+ ... = 2+ ...K̂ij (z, z) = i 1/l(ζ z − ζ j z 1/l )2l (1 − ζ i−j )2 z 2PPwhere ζ = exp 2πi. In order to compute − 41 Riiii (z, z, z, z) + 81 Riijj (z, z, z, z)lii,jit is useful to move the term Kij (z, z)2 from the second expression to the first one,which givesXXXdS(z i )4 + 6dS(z i )2 K̂ii (z, z) + 3K̂ii (z, z)2 −ii−Xi2K̂ij (z, z)ij→A(5.48),(5.49)dzzwhile the rest from (5.47) gives rise to!2XXXi 2K̂ij (z, z)dS(z i )dS(z j )dS(z ) +K̂ii (z, z) + 4iiij(5.50)4→(5.48),(5.49)4∆2Ndzz4(5.51)after using (5.48), (5.49) and several nice formulas like(l−1)/2l−1l−1X v21Xζj1Xe2πij/l==−l j=1 (1 − ζ j )2l j=1 (1 − e2πij/l )2l2v=(1−l)/22(l−1)/2(l−1)/2l−1l−12j4πij/l2XXXX v4112v ζe= 3=−l3 j=1 (1 − ζ j )4l j=1 (1 − e2πij/l )4ll2l4v=(1−l)/2133v=(1−l)/2(5.52)5.

Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformationsHere the sum over the roots of unity can be performed using the contour integrall−1Xj=1ζ jm1=j2m(1 − ζ )2πi˛d logzmzl − 1·=z − 1 (1 − z)2mz6=1(5.53)zmz−1·= Res z=1 d log lz − 1 (1 − z)2mand the result indeed allows to identify the coefficients at maximal singularities in(5.50), (5.51) with the expressions (5.43). It means that the conformal charge (5.44)of the twist field indeed coincide with the corresponding symmetric function (5.42) ofthe eigenvalues of the permutation matrix, but it comes here already from a nontrivialcomputation.It is known from long ago that already a definition of the higher W-currents is anontrivial issue (see e.g.

[FZ, FL, Bil, MarMor, FLitv12]). Here it was important toconsider the particular (normally ordered) symmetricfunction of the currents (5.4),P4since, for example, another natural choice: Ji (z) : is even not contained in theialgebra generated by T (z), W3 (z) and W4 (z). However, the so defined W4 (z)-currentis not a primary field of conformal algebra, we discuss this issue in Appendix 5.9.Conformal blocks and τ -functionsConsider now the next singular term from the OPE (5.28), which immediately allowsto extract from (5.27) the accessory parameters∂log G0 (q1 , ..., q2L ) =∂qαXi )=qπ(qαα1Res tz dz +2XiqαResi )=qπ(qααiqα(dS)2dz(5.54)Computing residues in the r.h.s.

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