Диссертация (Isomonodromic deformations and quantum field theory), страница 9

One can also observe a similarity between this formula and Toda 3-point functioncomputed in [FLitv07].Remarks on W3 conformal blocksGeneral conformal blockHere we consider for simplicity the c = 2 case, but the generalization to arbitrary c isstraightforward. First we explain how the WN conformal block is defined algebraically.For that let us compute the following expression:B(θ ∞ , θ 1 , σ, θ t , θ 0 ; t) = h−θ ∞ |φθ1 (1)Pσ φθt (t)|θ 0 i ,(2.59)where |θ 0 i and h−θ ∞ | are the highest-weight vectors with the charges given by (2.54),Pσ is the projector onto the whole Verma module (2.2) with given highest weight. Thisconformal block can be computed by inserting the resolution of the identity in theVerma module.

One can take, for instance, the naive basis (2.2), or (if we do notnecessarily want to preserve the L0 grading) the basis from [BW], or (if we wish toadd the Heisenberg algebra) the AGT basis from [FLitv12], [BBFLT]. Let us call thevectors of this basis |σ, Y~ i and suppose thatL0 |σi = (∆(σ) + |Y~ |)|σ, Y~ i .Their scalar products will be Kσ (Y~ , Y~ 0 ) = hσ, Y~ |σ, Y~ 0 i. This allows to express conformal block asX ~B(θ ∞ , θ 1 , σ, θ t , θ 0 ; t) = tχt|Y | h−θ ∞ |φθ1 (1)|σ, Y~ iK −1 (Y~ , Y~ 0 )hσ, Y~ 0 |φθt (1)|θ 0 i =~ ,Y~0Y= tχX~t|Y | Q(Y~ )Q̃(Y~ ) ,~Y(2.60)312. Isomonodromic τ -functions and WN conformal blockswhere Q̃(Y~ ) =PK −1 (Y~ , Y~ 0 )hσ, Y~ 0 |φθt (1)|θ 0 i and Q(Y~ ) = hθ ∞ |φθ1 (1)|σ, Y~ i and χ~0Yis given by (2.44).

The claim of [BW] is that Q(Y~ ) and Q̃(Y~ )~Q(Y~ ) = Q(Y~ |C1 , . . . , C|Y~ | ) = γ0 (Y~ ) +|Y |Xγk (Y~ )Ck ,k=1(2.61)~Q̃(Y~ ) = Q(Y~ |C̃1 , . . . , C̃|Y~ | ) = γ̃0 (Y~ ) +|Y |Xγ̃k (Y~ )C̃k ,k=1are “triangular” “affine” linear functions of infinitely many arbitrary parametersCk , C̃k defined byC̃k = hσ|W1k φθt (1)|θ 0 i .k|σi,Ck = h−θ ∞ |φθ1 (1)W−1(2.62)Degenerate fieldLet us consider the case θ t = e1 (the weight of the first fundamental representation).The fusion rules for such fields are known to be given by[e1 ]OM[θ + ek ] .[θ] =(2.63)k2Let us also shift θ 0 7→ θ 0 − en , multiply the conformal block by t 3 = t(e1 ,e1 ) , anddefine the quantityΦnk (t) = t(e1 ,e1 ) B(θ ∞ , θ 1 , θ 0 + ek − en , e1 , θ 0 − en ; t) =X ~t|Y | Q(Y~ , C1 , .

. . , C|Y~ | )q̃(Y~ ) ,= t(θ0 ,ek )+(1−δkn )(2.64)~Ywhere q̃(Y~ ) do not contain any free parameters [BW].Now denote the degenerate field φe1 (t) by ψ(t) and consider the correlation functiont(e1 ,e1 ) h−θ ∞ |φθ1 (1)ψ(t)|θ 0 − ek i .In the region t → 0 (s-channel) it can be expanded in the basis of conformal blockswritten above. But if we set t → 1 or t → ∞ (t- and u-channel), then we will havethe following OPEsψ(t)φθ1 (1) =Xh−θ ∞ |ψ(t) =XCeθ11,θ+e1 k · (t − 1)(θ1 ,ek ) (φθ1 +ek (1) + descendants) ,k(e1 ,e1 )t−(θ ∞ ,ek )∞ +ekCθθ∞(h−θ ∞ − ek | + descendants) .,e1 · t(2.65)kThese formulas suggest that the space of conformal blocks involving ψ(t) is 3-dimensionaland near each point we have a basis with asymptotics prescribed by θ ν .

It is clear that322.4. Remarks on W3 conformal blocksupon analytic continuation of Ψ1k (t) around 0, 1, ∞ one gets some linear combinationsof the basis elementsXγ0 : Φ1k (t) 7→Φ1k0 (t)(M0 )k0 k ,k0γ1 : Φ1k (t) 7→XΦ1k0 (t)(M1 )k0 k ,(2.66)k0γ∞ : Φ1k (t) 7→XΦ1k0 (t)(M∞ )k0 k .k0In our case M0 = diag(e2πi(θ0 ,e1 ) , e2πi(θ0 ,e2 ) , e2πi(θ0 ,e3 ) ).

That these formulas must holdcan be expected on general grounds (crossing symmetry) and from the fact that thespace is 3-dimensional. However, looking at the formula (2.64), the freedom in choice∞Pfl tl with arbitrary fn ’s. It means that W of Ck can give us Φ1k = t(θ0 ,e1 )+(1−δk1 )k=0algebra itself does not account for the global structure of conformal blocks and thisinformation should be introduced as an extra input.Now suppose that we have some globally-defined multivalued functions Φ1k .

Thenwe have three monodromies M0 , M1 , M∞ and one can solve the refined 3-point RiemannHilbert problem. Suppose that its solution is given by the matrix F (t) such thatA0A1dF (t) =+F (t) ,(2.67)dttt−1A0 = diag ((θ 0 , e1 ), (θ 0 , e2 ), (θ 0 , e3 )) and F (t)Pis normalized in such a way that F (t) =A0t (1 + O(t)). Next let us compute Ri (t) =Φ1k (t)(F (t)−1 )ki . This vector has thektrivial monodromies around all singular points, it is regular there and R(0) = (1, 0, 0),so that R(t) = (1, 0, 0). It means thatΦ1k (t) = F1k (t) .(2.68)This formula allows us to fix all constants Ck . This is done in the following way:we solve the 3-point Riemann-Hilbert problem, take F11 (t) and read the coefficients ofconformal block from its series.

These coefficients are triangular linear combinations(2.61) of Ck (i.e., kth term of the conformal block expansion involves only Cj≤k ).This construction thus expresses Ck via the moduli (µ, ν) of flat connections on the3-punctured sphere.(2.69)Ck = Ck (µ, ν) .All constants are expressed in terms of only two parameters. If we now recall the 5thexperimental property of the τ -function, its origin can be easily understood: the firstterm of the conformal block (with the structure constants fixed above) depends onlyon C1 (µ, ν) and C̃1 (µ, ν) and this dependence is at most bilinear.Verlinde loop operatorsHere we can slightly modify our point of view: now all possible vertex operatorsdefined by (2.3) and (2.4) have to be considered simultaneously. They form some ∞dimensional vector space, which can be identified with the space of 3-point conformal332.

Isomonodromic τ -functions and WN conformal blocksblocks (and which was one-dimensional in the Virasoro case). One can define theaction of the Verlinde loop operators on this space in the same way as it was done in[CGTe]. This action is given by some operators V̂ (γ) depending on the loop γ.If we now look at the results of [ILTe] then we realize that (2.50) can be definedalternatively as the common eigenvector of all possible Verlinde loops.

One can actin the same way for the case of 3-point conformal blocksV̂ (γ) · hY |φθ1 ,µ,ν (1)|Y 0 i = Mγ (µ, ν) · hY |φθ1 ,µ,ν (1)|Y 0 i(2.70)This procedure defines the basis of the “right” vertex operators φθ1 ,µ,ν (1) characterizedby some Ck (µ, ν). It looks more natural in this approach that τ -function constructedfrom such operators should solve the Riemann-Hilbert problem.The question about interpretation in c 6= N − 1 case is still open: the problem iscaused by non-commutativity of the algebra of V̂ (γ). Moreover, even in the minimal2for integer k, when the algebra ismodel-like cases c = N − 1 − N (N 2 − 1) (k−1)kcommutative again, the relation to the isomonodromic deformations becomes unclear(see “concluding remarks” in [BShch] for discussion of the Virasoro case).ConclusionsWe have discovered several important properties of the isomonodromic τ -functionsin higher rank, which can be interpreted as signatures of the isomonodromy-CFTcorrespondence for the WN case.

This allows to give a definition of the general WNconformal block, depending only on a finite number of parameters. It is also possibleto prove [GavIL] that the algebraic way to define well-known conformal blocks forsemi-degenerate fields agrees with the above definition.We have also considered a particular conformal block with degenerate field andshown that its global structure is not fixed algebraically.

The requirement of thecorrect global behavior of this object yields an expression for the whole infinite seriesof constants in the W3 conformal block in terms of the solution of the 3-point RiemannHilbert problem.These expressions can be written in terms of coordinates on the moduli space offlat connections on sphere with 3 punctures. This is expected to be universal andwork for any conformal block (not only for those with degenerate fields). We havechecked experimentally some properties, which support this conjecture.Finally, let us list some remaining open problems:• One needs to check that the procedure of fixing Ck is self-consistent.• If the constants Ck can be fixed in such a way, we may try to prove that theτ -function can be given as a sum of the general WN conformal blocks.• A constructive solution of the 3-point Riemann-Hilbert problem is still missing.• It would be interesting to understand the meaning of Zbif (θ t , σ, θ 0 ; µ, ν|Y~ , Y~ 0 )in the context of isomonodromy-CFT correspondence.

It can be done for thecase Y~ 0 = ~0 and it is interesting what happens for the arbitrary Young diagrams.342.5. Conclusions• There as an approach to the definition of conformal blocks of the light fields inthe limit c → ∞ [FR]. In that case explicit integral expression for the conformalblock was derived. All the information about the 3-point functions enters thisdefinition via several functions of one variable.

The open problem is to obtainthe monodromy properties of such conformal blocks and to identify the choiceof 3-point functions that gives the conformal blocks arising in our approach.• It is also important to understand the meaning of the results [BMPTY] aboutpartition functions of TN theories without lagrangian description (which arebelieved to be the counterparts of the general WN 3-point functions) from theisomonodromic point of view.353Free fermions, W-algebras andisomonodromic deformationsAbstractWe consider the theory of multicomponent free massless fermions in two dimensionsand use it for construction of representations of W-algebras at integer Virasoro centralcharges.