Диссертация (Isomonodromic deformations and quantum field theory), страница 8

These terms belong toVk,2k , thereforek=0log τ (t) ∈∞Xk=025Vk,2k(2.45)2. Isomonodromic τ -functions and WN conformal blocksFigure 2.2: Filtration Q•sl2Note that these estimates are too rough, since we have not taken into account thata number of the commutators actually vanish. The actual result turns out to be thesame for all three functions∞Xlog τ, Ã0 , A1 ∈Vk,k ,k=0and it can be checked numerically. Moreover, it turns out that the expansion of theτ -function itself is even more restricted:Xt−χ τ (t) ∈V 1 (w,w),w ,(2.46)2w∈Qgwhich in fact provides an evidence for the 2d CFT description: different fractionalpowers come from t∆ for the different ∆’s, but the conformal dimension ∆ = 12 (σ +w, σ + w) is a quadratic function of w leading to the structure (2.46).sl2 caseIn this case we illustrate all procedures, definitions and statements using the exactsolution of [GIL12].The Lie algebra sl2 is given by 3 generators Eα , E−α , Hα , such that[Eα , E−α ] = Hα ,[Hα , E±α ] = ±2E±α .(2.47)The root lattice Qsl2 is shown in Fig.2.2.

It is spanned by one root α. Q0sl2 is theempty square, Q1sl2 is the red rectangle, Q2sl2 is green and Q3sl2 is blue.All monomials have the form tn+(σ,w) = tn+m(σ,α) , and therefore can be depictedby the points of a two-dimensional lattice. Note that in our normalization (α, α) = 2.Several examples of the elements of this filtration are presented in Fig.2.3.Here the blue region represents V0,0 , red corresponds to V1,1 and green is V3,4 .We can also show the “true” and “naive” lattice supports of the quantities Ã0 (t),A1 (t), log τ (t) and t−χ τ (t). See Fig.2.4: green region is the “naive” support of A1 (t),the blue region is the true support of Ã0 (t), A1 (t), log τ (t), which can be derivedexperimentally. Now one can use an exact formula for the tau function expansion[GIL12] (cf (2.50) below) to see thatX2222τ (t) = tσ −θ0 −θtt2σn tn fn (t) ,(2.48)k∈Zwhich in turn implies222tθ0 +θt −σ τ (t) ∈∞Xk=026Vk2 ,k .(2.49)2.3.

Iterative solution of the Schlesinger systemFigure 2.3: Filtration V•,•Figure 2.4: Support of the solutionsIt looks like a miracle and means that a huge number of terms cancel out whenwe exponentiate, but this answer confirms the conjecture (2.46). This phenomenonis illustrated in Fig.2.5 in two ways. Upper bold numbers account for the degree inτ (t) (blue region), lower numbers correspond to the degree in log τ (t) (green region).Horizontal coordinate corresponds to the position in the sl2 root lattice.94101493210123Figure 2.5: Supports of τ (t) and log τ (t): circles correspond to the integral points ofthe x-axis, numbers inside show the y-coordinates of the cone and parabola.272.

Isomonodromic τ -functions and WN conformal blocksLet us take the main formula from [GIL12]:X222τ (t) =sn Cn(0t) (θ0 , θt , σ0t )Cn(1∞) (θ1 , θ∞ , σ0t )t(σ0t +n) −θ0 −θt B({θi }, σ0t + n; t) ,n∈Z(2.50)where B(. . . ; t) is the c = 1 Virasoro conformal block andQ=Cn(0t) (θ0 , θt , σ0t )Cn(1∞) (θ1 , θ∞ , σ0t ) =G(1 + θt + θ0 + 0 (σ0t + n))G(1 + θ1 + θ∞ + 0 (σ0t + n))=±,0 =±G(1 − 2σ0t )G(1 + 2σ0t )(2.51).Here (θν , −θν ) are the eigenvalues of the matrices Aν in the linear system (2.5),(e2πiσµν , e−2πiσµν ) are the eigenvalues of Mµ Mν , s is the only variable depending on σ1t(in a complicated way). The main properties of (2.50) and (2.51) can be summarizedas follows:1.

The support of τ (t) is as indicated in (2.49).2. Relative twist parameter enters only via the factor sn in the structure constants.3. The 3-point coefficients Cn factorize with respect to the pants decompositionparametrization.We are now going to check these important properties in the sl3 case.sl3 caseFigure 2.6: Filtration Q•sl3Fig.2.6 illustrates the filtration on the sl3 root lattice. The red hexagon corresponds to Q1sl3 , Q2sl3 is shown in green and Q3sl3 is blue.

It is difficult to visualize Vm,n ,282.3. Iterative solution of the Schlesinger systemsince one would then need a 3d picture. One can however think of∞PVk,k as being ak=0cone with hexagonal section.Let us perform the numerical study of the 3 × 3 Schlesinger system. We firstdetermine which degrees (k, w) are present in log τ (t) and in τ (t) (Fig.2.7).?????44444??77??433334?74347? ?4322234?731137? ?43211234??4114?????043211234 5 6 70?731137? ?43211234?74347? ?4322234??77??433334?????4 4 4 4 4Figure 2.7: Degrees present in t−χ τ (t) and in log τ (t).

Number χ is given by (2.44).As above, the upper bold numbers correspond to degrees in t−χ τ (t) and the lowerones to log τ (t). We mark with “?” sign those values which are obtained at the limitof machine precision or which are greater then 7 (so that they are not seen in thesolution up to the 7th order). Carefully analyzing this picture, one deduces thatlog τ (t) ∈∞XVk,k ,k=0−χtτ (t) ∈X(2.52)V 1 (w,w),w .2w∈Qsl3It means that nonzero monomials of τ (t) fill a paraboloid, and not the naively expectedcone. In other words, a lot of nontrivial cancellations take place, which providesfurther evidence for the conjecture (2.46). We now list other nontrivial properties ofτ (t) revealed by our experimental study.1. The expansion has the formX(0t)(1∞)τ (t) =e(β,w) Cw(θ 0 , θ t , σ 0t , µ0t , ν0t )Cw(θ 1 , θ ∞ , σ 0t , µ1t , ν1t )×(2.53)w∈Q×t1(σ 0t +w,σ 0t +w)− 21 (θ 0 ,θ 0 )− 21 (θ t ,θ t )2Bw ({θ i }, σ 0t , µ0t , ν0t , µ1∞ , ν1∞ ; t) .292.

Isomonodromic τ -functions and WN conformal blocks12. The non-zero coefficients of the expansion start from t 2 (w,w) .3. All the dependence on the relative twist parameters is hidden in β ∈ h, whichenters in a trivial way.4. The dependence of structure constants on the 3-point monodromy parametersis factorized.5. The first term in the expansion of conformal block has the formB0 = 1 + [α + βC1 (µ0t , ν0t ) + γ C̃1 (µ1∞ , ν1∞ ) + δC1 (µ0t , ν0t )C̃1 (µ1∞ , ν1∞ )]t + . . .This property is new, as compared to the N = 2 case, and we will see later thatit is very important.All these facts tell us that almost all properties of sl2 case hold in the sl3 case.This leads us toMain conjecture:B0 ({θ i }, σ 0t , µ0t , ν0t , µ1∞ , ν1∞ ; t) is a conformal block of W3 algebraThe corresponding dimensions and W -charges are given by1∆ν = (θ ν , θ ν )2r3Y(θ ν , ei ) .wν =2 i(2.54)The main advantage of the above definition of conformal block is that it dependsonly on 4 extra variables instead of a doubly-infinite set.It is easy to check this definition for the case when W3 -block can be definedalgebraically.

This becomes possible when θ t = at e1 and θ 1 = a1 e1 , where e1 isthe weight of the first fundamental representation. The best way to present thisconformal block is to use Nekrasov formulas [Nek] which can be applied to conformalfield theory in view of the extended AGT [AGT] correspondence, first established in[Wyll], [MirMor]. The most convenient (for c = 2) expression for the conformal blockcan be found in [FLitv12]:Bw (θ ∞ , a1 , σ, at , θ 0 ; t) = B(θ ∞ , a1 , σ + w, at , θ 0 ; t)X ~1t|Y | Zbif (−θ ∞ , a1 , σ|~0, Y~ )×B(θ ∞ , a1 , σ, at , θ 0 ; t) = (1 − t) 3 at a1(2.55)~Y×1Zbif (σ, 0, σ|Y~ , Y~ )Zbif (σ, at , θ 0 |Y~ , ~0) ,where3 YYaZbif (θ, a, θ |~ν , ~ν ) =−Eνi0 ,νj (i(θ, ej ) − i(θ , ei )|s) − i ×3i,j=1 s∈νi0Ya×Eνj ,νi0 (i(θ 0 , ei ) − i(θ, ej )|t) − i,3t∈ν000j30(2.56)2.4.

Remarks on W3 conformal blocksand the quantities E are defined byEλ,µ (x|s) = x − ilµ (s) − iaλ (s) − i .(2.57)It yields exactly the same result as our computations using iterative solution of theSchlesinger system.We have also conjectured in this case and checked experimentally a formula forthe structure constants, which is a straightforward generalization of (2.51):Q=ijG[1 −atN(1∞)(0t)(σ, a1 , θ ∞ ) =(θ 0 , at , σ)CwCw+ (ei , θ 0 ) − (ej , σ + w)]G[1 − aN1 + (ei , σ + w) + (ej , θ ∞ )] (2.58)Q.G[1 + (αi , σ + w)]iHere ei denote the weights of the first fundamental representation and αi are all rootsof slN (in our case N = 3). This formula was recently proved [GavIL] for generalN .