Диссертация (1137342), страница 30
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Hence, the same methods, which giveπ(P )=qβPrise to explicit formula for the main part τSW (a, q) of the exact conformal block,ensure also the consistency of definition of the quasiclassical correction τB (q).Isomonodromic τ -functionThe full exact conformal block equals therefore!G0 (q|a) = τB (q) exp12XaI TIJ (q)aJ +IJXaI UI (q, r) + 21 Q(r)(5.87)IAccording to [GIL12, Gav] the τ -functions of the isomonodromy problem [SMJ] onsphere with four marked points 0, q, 1, ∞ can be decomposed into a linear combinationof the corresponding conformal blocks 3 .
This expansion looks asτIM (q) =X(0q)(1∞)e(b,w) Cw(θ 0 , θ q , a, µ0q , ν0q )Cw(θ 1 , θ ∞ , a, µ1∞ , ν1∞ )×(5.88)w∈Q(slN )×q1(σ 0t +w,σ 0t +w)− 21 (θ 0 ,θ 0 )− 21 (θ t ,θ t )2Bw ({θ i }, a, µ0q , ν0q , µ1∞ , ν1∞ ; q)and can be tested, both numerically and exactly for some degenerate values of theW-charges θ of the fields [Gav, GavIL].
In (5.88) the normalization of conformal(•)block Bw (•; q) is chosen to be Bw (•; q) = 1 + O(q) and Cw (•) as usually denote thecorresponding 3-point structure constants (all these quantities in the case of W (slN ) =WN blocks with N > 2 depend on extra parameters {µ, ν}, being the coordinates onthe moduli space of flat connections on 3-punctured sphere, and for their genericvalues the conformal blocks Bw (•; q) are not defined algebraically, see [Gav] for moredetails).3This relation has been predicted in [Knizhnik], see also [Nov] for a slightly different observationof the same kind.1405.6.
ExamplesWe now conjecture that such decomposition exists also for conformal blocks considered above. Moreover, then a natural guess is, that the structure constants havesuch a form that111(0q)(1∞)Cw(θ 0 , θ q , a, µ0q , ν0q )Cw(θ 1 , θ ∞ , a, µ1∞ , ν1∞ )q 2 (a+w,a+w)− 2 (θ0 ,θ0 )− 2 (θq ,θq ) ··Bw ({θ i }, a, µ0q , ν0q , µ1∞ , ν1∞ ; q) = G0 ({θ i }, a + w; q)(5.89)i.e.
they are absorbed into our definition of the W-block of the twist fields, and thiscan be extended from four to arbitrary number of even 2L points on sphere. Thisconjecture can be easily checked in the N = 2 case, where the structure constantsfor the values, corresponding to the Picard solution [GIL12, ILTe], coincide exactlywith given by degenerate period matrices in (5.87), when applied to the case of theZamolodchikov conformal blocks [GMqui] (see sect. 5.6 and Appendix 5.10).It means that in order to get isomonodromic τ -functions from the exact conformalblocks (5.87) one has just to sum up the series (for the arbitrary number of points onehas to replace the root lattice of Q(slN ) = ZN −1 by the lattice Zg , where g = g(C) isthe genus of the cover)XτIM (q|a, b) =G0 (q|a + n)e(n,b) = τB (q) exp 21 Q(r) ×n∈ZgX×n∈Zgexp1(a + n, T (a + n)) + (U , a + n) + (b, n)2 a1= τB (q) exp 2 Q(r) Θ(U )b=(5.90)which is easily expressed through the theta-function.
One gets in this way exactly theKorotkin isomonodromic τ -function, where the only difference of this expression withproposed in [K04, formula 6.10] is in the term Q(r), which is not expressed globallythrough the coordinates of the branch points in the case of non-abelian monodromygroup. This fact supports both our conjectures: about the form of the structureconstants, and about the general correspondence between the isomonodromic deformations and conformal field theory.Formula (5.90) has also clear meaning in the context of gauge theory/topologicalstring correspondence.
It has been noticed yet in [NO], that the CFT free fermionrepresentation exists only for the dual partition function, which is obtained from thegauge-theory matrix element (conformal block) by a Fourier transform 4 . We plan toreturn to this issue separately in the context of the free fermion representation for theexact W-conformal blocks.ExamplesThere are several well-known examples of the conformal blocks corresponding toAbelian monodromy groups. All of them basically come from the Zamolodchikov4The fact, that only the Fourier-Legendre transformed quantity can be identified with partitionfunction in string theory has been established recently in quite general context from their transformation properties in [CWM].1415. Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformationsexact conformal block [ZamAT87, formula 3.29] for the Ashkin-Teller model, definedon the families of hyperelliptic curves2y =2LY(z − qα )(5.91)α=1with projection π : (y, z) 7→ z.
Parametersr are absent here, so the result is justPG0 (q) = τB (q) exp 12 aI TIJ (q)aJ , where for the hyperelliptic period matricesIJone gets from (5.59) the well-known Rauch formulas (see e.g. [GMqui] and referencestherein).When the hyperelliptic curve degenerates (see Appendix 5.10), this formula gives−G0 (q) ≈ 4PPa2I −( aI )2gYa2I − 18(q2I − q2I−1 )I=1≈ 4−Pa2I −(P2aI )·gY(q2I − q2J )2aI aJ R−(I>JgYP2a2 − 1I I 8 R−( aI )I=1·gYPaI )2≈(5.92)(q2I − q2J )2aI aJI>JHere in the r.h.s. the second factor comes from the OPE (5.7), i.e. O(q2I −I )O(q2I ) ∼Qa2 − 1I I 8 VaI (q2I ) + .
. ., while the third one is just the correlator h VaI (q2I )i. Hence, thefirst most important factor corresponds to the non-trivial product of the structureconstants in (5.89), which acquires here a very simple form. The main point of thisobservation is that normalization of (5.87) automatically contains not only q # factors,but also the structure constants, and we have already exploited such conjecture forgeneral situation in sect. 5.5, since the argument with degenerate tau-function can beeasily extended.These observations have an obvious generalization for the ZN -curvesNy =2LY(z − qα )kα(5.93)α=1with the same projection π : (y, z) 7→ z.
The main contribution to the answerPτSW = exp 21 aI TIJ (q)aJ comes just from a general reasoning as in sect. 6.4IJand to make it more explicit one can use the Rauch formulas for ZN -curves, whichexpress everything in terms of the coordinates {q} on the projection, since there is nosumming over preimages in formulas like (5.59).Let us now turn to an elementary new example with non-abelian monodromygroup. Notice, first, that a simple genus g(C) = 0 curvey 3 = x2 (1 − x)(5.94)gives rise to the curve with non-abelian monodromy group if one takes a different(from Z3 -option πx : (y, x) 7→ x) projection πy : (y, x) 7→ y. For the curve C, whichis just a sphere or P1 itself, one gets here two essentially different (and unrelated!)setups, corresponding to differently chosen functions x or y.1425.6.
ExamplesIn the first case our construction leads, for example, to the formulashT (x)iO =where x =11+ξ 3hT (x)Os (0)Os−1 (1)i11= {ξ; x} = 2hOs (0)Os−1 (1)i49x (x − 1)2(5.95)in terms of the global coordinate ξ on C, and this formula fixes theinsertions at x = 0, 1 to be the twist operators for s = (123), with ∆(s) =l2 −1= 1.24l l=3 9However, for a similar correlator on y-sphereQhT (y) A=0,1,2,3 Õ(yA )i1 + 54y 3=hT (y)iÕ ==Q(27y 3 − 4)2 y 2h A=0,1,2,3 Õ(yA )iX uA1+=216(y−yy − yAA)A=0,1,2,3y0 = u0 = 0,(5.96)3yk = −8uk = 22/3 e2πi(k−1)/3 , k = 1, 2, 3one has to insert the twist operators for s̃ = (12)(3) of dimension ∆(s̃) =l̃2 −124l̃=l̃=21.16The r.h.s.
here follows from summation of1ξ(ξ 3 + 4)(1 + ξ 3 )4ξ 5 (3y + ξ){ξ; y} ==122(2ξ 3 − 1)42y(2ξ − 3y)4(5.97)whereξ, ξ ∈ C = P1(5.98)1 + ξ3To get (5.96) one has to sum (5.97) over π(ξ) = y, or three solutions of the equationR(ξ) = ξ 3 − y1 ξ + 1 = 0, i.e. 5Xξ (3y + ξ)1 X (β)resξ=ξ(β){ξ ; y} =d log R(ξ) =hT (y)iC =412 β2y(2ξ−3y)β(5.99) 5 ξ (3y + ξ)R0 (ξ)11 + 54y 3=−resξ=3y/2 + resξ=∞dξ =2y(2ξ − 3y)4 R(ξ)(27y 3 − 4)2 y 2y=in contrast to the sum over three sheets of the cover π(ξ) = x, which gives only afactor hT (x)iO = 3 · {ξ; x}/12.To analyze the simplest nontrivial τ -function for non-abelian monodromy group,let us consider the deformation of the formula from (5.98) for z = 1/y = ξ 2 + 1/ξ, i.e.the cover π : C = P1ξ → P1z given by 1-parametric familyz=(2ξ − t2 + 1)2 (ξ − 4)(t − 3)2 (t2 − 2t − 3)ξ(5.100)The parametrization is adjusted in the way that the branching points dz = 0 are atξ = 12 (t2 − 1), z = 0ξ = 1 + t, z = 1(t + 3)3 (t − 1)ξ = 1 − t, z = q(t) =(t − 3)3 (t + 1)143(5.101)5.
Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformationstogether with ξ = ∞, z = ∞.One also has non-branching points above the branched ones ξ = 4, z = 0; ξ =(t − 1)2 , z = 1; ξ = (t + 1)2 , z = q(t). Now we rewrite these points in our notationξ01 = 4,ξq1 = (t + 1)2 ,ξ11 = (t − 1)2 ,1ξ∞= 0,ξ02 = 12 (t2 − 1),ξq2 = 1 − t,ξ12 = 1 + t,2ξ∞= ∞,ξ03 = 21 (t2 − 1)ξq3 = 1 − tξ13 = 1 + t3ξ∞=∞(5.102)Using an explicit formula (5.79) and the definition (5.56) of τB one can write downthe result for the τ -function1τ (t) = τB (t) expQr (t) =2(5.103)11δ−3 + 24δ3 − 31δ1 − 81 δ0 + 24δ−1= (t − 3)(t − 1)t(t + 1) (t + 3)where δν = δν (r) are given by some particular quadratic forms2δ3 = 9rq2 − 9r∞2δ1 = r02 − 4r0 r1 + 4r12 + 8r0 rq − 4r1 rq + rq2 − 4r0 r∞ + 8r1 r∞ − 4rq r∞ + 4r∞δ0 = −9r12 − 9rq2(5.104)2δ−1 = 4r02 + 8r0 r1 + 4r12 − 4r0 rq + 7rq2 − 4r0 r∞ − 4r1 r∞ + 8rq r∞ + r∞δ−3 = −9r02 − 9rq2while their “semiclassical” shifts come from the Bergman τ -function.
Notice thatisomonodromic function (5.103) looks very similar to the tau-functions of algebraicsolutions of the Painlevé VI equation [GIL12, examples 5-7], but depends on essentially more parameters.An interesting, but yet unclear observation is that in this example τB (t) itself canbe represented as1Q(r̃)(5.105)τB (t) = exp2for several particular choices of parameters r̃, e.g.√√√77i7(r̃0 , r̃q , r̃1 , r̃∞ ) = ( √ , − √ , √ , √ )12 3 12 3 4 3 12 3√iii5(r̃0 , r̃q , r̃1 , r̃∞ ) = ( √ , √ , √ ,)12 3 12 3 12 3 12(5.106)whereas all other (altogetherare√obtained after the action of the√ eight) √solutions√Galois group generated by 3 7→ − 3, 5 7→ − 5 and i 7→ −i.
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