Диссертация (1137342), страница 31
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Notice that thisstatement is nevertheless nontrivial because we express five variables δi in terms ofonly four variables r̃i .1445.7. ConclusionsConclusionsWe have presented above an explicit construction of the conformal blocks of the twistfields in the conformal theory with integer central charges and extended W-symmetry.We have computed the W-charges of these twist fields and show that their Verma modules are non-degenerate from the point of view of W-algebra representation theory.The obtained exact formulas for the corresponding conformal blocks were derivedintensively using the correspondence between two-dimensional conformal and fourdimensional supersymmetric gauge theory.
We also checked that so constructed exactconformal blocks, when considered in the context of isomonodromy/CFT correspondence, give rise to the isomonodromic τ -functions of the quasipermutation type.We believe that it is only the beginning of the story and, finally, would like topresent a list (certainly not complete) of unresolved yet problems. For the conformalfield theory side these obviously include:• What is the algebraic structure of the W-algebra representations correspondingto the twist-field vertex operators, and in particular – what are the form-factorsor matrix elements of these operators?• Already for the twist fields representations the analysis of this chapter shouldbe supplemented by study of the W-analogs of the higher-twist representations[ApiZam] and of the W-representations at “dual values” of the central charges(an example of such block for the Virasoro case can be found in [ZamAT86]).• Finally, perhaps the most intriguing question is – what is the constructive generalization of these vertex operators to non-exactly-solvable case?However, the main intriguing part still corresponds to the side of supersymmetricgauge theory, where the resolution of these problems can help to understand theirproperties in the “unavoidable” regime of strong coupling, where even the Lagrangianformulation is not known.
We are going to return to these questions elsewhere.AppendixDiagram techniqueIn order to compute the correlators of the currents (5.45) the first useful observationis that one can embed slN ⊂ glN and introduce an extra current h(z), commutingwith Ji (z), such thath(z)h(z 0 ) =Introduce the glN currents1/N+ reg.,(z − z 0 )2h(z)O(q) = reg.J˜i (z) = Ji (z) + h(z)(5.107)(5.108)which satisfy the OPEJ˜i (z)J˜k (z 0 ) =δjk+ reg.(z − z 0 )2145(5.109)5. Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformationsand their normally-ordered averages are the same as for Ji (z) sinceh: Ji1 (z1 ) . . .
Jim (zm )h(zm+1 ) . . . h(zn ) :iO == h: Ji1 (z1 ) . . . Jim (zm ) :iO · h: h(zm+1 ) . . . h(zn ) :i = 0(5.110)Hence, one can simply to replace Ji (z) → J˜i (z) in (5.45), so below we just computethe averages for the glN currents.The normal ordering for two currents at colliding points is given byδij dz dz 0=(z − z 0 )2= Ji (z)Jj (z 0 )dz dz 0 − δij K0 (z, z 0 ): Ji (z)Jj (z 0 ) : dz dz 0 = Ji (z)Jj (z 0 )dz dz 0 −(5.111)i.e. it is defined by subtracting the canonical meromorphic bidifferential on the basecurve, since it corresponds to the vacuum expectation value of the Gaussian fields.Normal ordering for the correlators of many currents is defined, as usual, by the Wicktheorem.Similarly to (5.8) consider nowhJi1 (z1 ) : Ji2 (z2 ) . .
. Jin (zn ) :iO dz1 . . . dzn == dS(z1i1 )h: Ji2 (z2 ) . . . Jin (zn ) :iO dz2 . . . dzn ++nXiK(z1i1 , zjj )h:(5.112)cJi2 (z2 ) . . . J\ij (zj ) . . . Jin (zn ) :iO dz2 . . . dzj . . . dznj=2where by zki = π −1 (zk )i we have denoted the preimages on the cover. This formulais again obtained just from the analytic structure of this expression as 1-form in thefirst variable. The next formula comes from the application of the Wick theorem and(5.111)hJi1 (z1 ) : Ji2 (z2 ) .
. . Jin (zn ) :iO dz1 . . . dzn = h: Ji1 (z1 ) . . . Jin (zn ) :iO dz1 . . . dzn +nXc+δij K0 (z1 , zj )h: Ji2 (z2 ) . . . J\ij (zj ) . . . Jin (zn ) :iO dz2 . . . dzj . . . dznj=2(5.113)Subtracting them, one gets the recurrence relationh: Ji1 (z1 ) .
. . Jin (zn ) :iO dz1 . . . dzn = dS(z1i1 )h: Ji2 (z2 ) . . . Jin (zn ) :iO dz2 . . . dzn +nXc+K̂i1 ij (z1 , zj )h: Ji2 (z2 ) . . . J\ij (zj ) . . . Jin (zn ) :iO dz2 . . . dzj . . . dznj=2(5.114)where we have introduced the “propagator”K̂ij (z1 , z2 ) = K(z1i , z2j ) − δij K0 (z1 , z2 )Graphically for the result this recurrence produces one can write146(5.115)5.9. W4 (z) and the primary fieldtiRi (z1 ) =titiRij (z1 , z2 ) =Rijk (z1 , z2 , z3 ) =itjt+tjtk= dS(z1i )tjit+jt= dS(z1i )dS(z2j ) + K̂ij (z1i , z2j )tkit+t"j"tk+iHtjtH tk== dS(z1i )dS(z2j )dS(z3k ) + K̂ij (z1 , z2 )dS(z3k ) + K̂jk (z2 , z3 )dS(z1i ) + K̂ik (z1 , z3 )dS(z2j )These expressions have very simple meaning: the full correlation function is expressed through the only possible connected parts Rc , which are Rci (z1 ) = dS(z1i ),Rcij (z1 , z2 ) = K̂ij (z1 , z2 ), while Rcijk (z1 , z2 , z3 ) = 0 and all higher connected partsvanish.
The so constructed four point functions Rijkl (z1 , z2 , z3 , z4 ) at coinciding arguments (and at least pairwise coinciding labels of the sheets of the cover) were usedin sect. 5.3.4 for computation of the higher W-charges.W4(z) and the primary fieldHere we study the OPE of W4 (z) with T (z) and show an explicit correction whichmakes this field primary.XC ijkl : Ji (z)Jj (z)Jk (z)Jl (z) :W4 (z) =(5.116)ijkl1when i 6= j 6= k 6= l andwhere C ijkl is completely symmetric tensor, C ijkl = 24ijklC= 0 otherwise.X610δij −C ijkl : Jk (z)Jl (z) : +T (z)W4 (z ) =(z − z 0 )4 ijklN(5.117)∂W4 (z 0 )4W4 (z 0 )++ reg.+(z − z 0 )2z − z0The first sum equals6XijUsing now the fact that1δij −NPCijkl = −(N − 2)(N − 3)(1 − δij )4N(5.118)Ji (z) = 0 we getiT (z)W4 (z 0 ) =(N − 2)(N − 3) T (z 0 )4W4 (z 0 )∂W4 (z 0 )+++ reg.8N(z − z 0 )4 (z − z 0 )2z − z0147(5.119)5.
Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformationsFigure 5.4: Degenerate hyperelliptic curve with chosen basis in H1 .There¸ dw is also another well-known field Λ(z) = (T T )(z) −T (w)T (z), with the OPEz w−z3 2∂ T (z),10where (T T )(z) =22T (z 0 )4Λ(z 0 )∂Λ(z 0 )T (z)Λ(z ) = c ++++ reg.5 (z − z 0 )4 (z − z 0 )2z − z00(5.120)One can therefore cancel an anomalous term in (5.119) just introducingW̃4 (z) = W4 (z) −(N − 2)(N − 3)Λ(z)8(N + 17)5(5.121)which is already a primary conformal field. Its charge therefore is given by the formulaw̃4 = w4 −(N − 2)(N − 3)∆(5∆ + 1)8(5N + 17)(5.122)Degenerate period matrixHere we compute the period matrix of the genus g hyperelliptic curve (see fig. 5.4)ggYYy = (z − R) (z − q2I )(z − q2I + I ) = (z − R) (z − q2I )(z − q2I−1 )2I=1(5.123)I=1in the degenerate limit I → 0, R → ∞ up to the terms of order O(I ) and O R1 (thisequivalence will be denoted by “≈”).
The normalized first kind Abelian differentialswith such accuracy areYpdz(z − q2K ) ,dωI = q2I − RyK6=Isince √z−qI(z−qI )(z−qI +I )12πi˛dωI ≈ δIJ(5.124)AJ≈ 1 when z goes far from qI . First we compute the off-diagonalmatrix element TIJ for J > I˛TIJ =ˆR rdωI ≈ −2BJq2I − R dzq2J − q2I≈ −2 log 4 + 2 logz − R z − q2IRqJ148(5.125)5.10. Degenerate period matrixand then a little bit more complicated diagonal element˛ˆR rTII =dωI ≈ −2q2IBIˆR≈2r1−q2I − Rz−R!q2I − Rdzp≈z−R(z − q2I )(z − q2I + I )dz−2z − q2Iq2IˆRq2Idzp=(z − q2I )(z − q2I + I )= −2 log 4 − 2 log 4 + 2 log(5.126)IRwhere we have used the fact, that for our purposes in the expressions √f (z)(z−q2I )(z−q2I +I )one can drop I if f (q2I ) = 0.Now using (5.87) we can compute in this limitτSW = exp≈ ·4−Pa2I −(PaI )21XaI TIJ aJ2 I<J!≈ggYYP22(q2I − q2I−1 )aI(q2I − q2J )2aI aJ R−( aI )I=1(5.127)I>JThe result for τB (q) in this simple hyperelliptic example can be taken from [ZamAT87]2g+1τB (q) =Y1(qi − qj )− 8 × detIJi<j12πi˛zJ−1y− 21dz (5.128)AIwhere the determinant can be easily computed using (5.123)1detIJ 2πi˛AIg Yz J−1 dzq J−1(qI − qJ )−1≈ det √ Q 2I= R− 2IJyR(q2I − q2J )I>J(5.129)J6=IAltogether this gives the formula (5.92) for the degenerate form of the hyperellipticZamolodchikov exact conformal block.1496Twist-field representations of W-algebras,exact conformal blocks and characteridentitiesAbstractWe study twist-field representations of the W-algebras and generalize the constructionof the corresponding vertex operators to D- and B-series.
We demonstrate how thecomputation of characters of such representations leads to the nontrivial identitiesinvolving lattice theta-functions. We propose a construction of their exact conformalblocks, which for D-series express them in terms of geometric data of correspondingPrym variety.IntroductionRepresentation theory for the W-algebras [ZamW, FZ, FL] is still the subject withmany open questions.
These questions often arise in the context of a two-dimensionalconformal theory (CFT) with extended symmetry, and due to a nontrivial recentlyfound correspondence [LMN, NO, AGT, Nek15] may be important for multidimensional supersymmetric gauge theories.The main object of this study is a conformal block, which for generic W-algebrabeyond the Virasoro case is not fixed by its algebraic properties. Actually there areat least two different meanings of the term “conformal block” in the literature:• Space of all functionals on the product of Verma modules at given points of theRiemann surface that solve Ward identities (see e.g. [FBZv] where the languageof coinvariants was used).
Such spaces form a bundle over moduli space ofcurves with fixed points. We prefer to call this a ”space of conformal blocks”,and reserve the word ”conformal block” for another meanings following [BPZ].• Concrete section of this bundle. Usually the corresponding functionals are computed on the highest weights in each of the Verma modules.
For the Virasoro algebra the conformal blocks are specified by definite intermediate dimensions, or,equivalently, by the asymptotic behavior of the conformal block on the boundaryof the moduli space;1516. Twist-field representations of W-algebras, exact conformal blocks and character identitiesThe first object (which we call the “space of conformal blocks”) is defined for anyvertex algebra, but the problem is to specify a concrete section of this bundle, it isno longer enough to do this by fixing quantum numbers in the intermediate channels.Even for three points on sphere, the vector space of conformal blocks becomes infinitedimensional for WN algebras with N > 2.However, for certain particular cases this conformal block can be constructed explicitly applying some extra machinery. In what follows we first restrict ourselves tothe case of integer and sometimes half-integer Virasoro central charges, when representations W-algebras are more directly related to the representations of the corresponding Kac-Moody (KM) algebras (of level k = 1), and the corresponding fieldtheories can be directly described by free fields [FK].Even in such situation the most general case cannot be formulated explicitly,one of the recent methods [GMfer] reduces the problem here to a Riemann-Hilbertproblem, arising in the context of the isomonodromy/CFT correspondence [GIL12,ILTe, Gav].
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