Диссертация (1137342), страница 29
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one gets the set of differential equations (α =1, . . . , 2L), which define the correlation function of the twist fields G0 (q1 , ..., q2L ) itself.A non-trivial statement [KriW, GMqui, KK04, KK06] is that these equations arecompatible, moreover (5.54) defines actually two different functions τSW (q) and τB (q),where1 X(dS)2∂log τSW (q1 , ..., q2L ) =Res qαi(5.55)∂qα2 idzπ(qα )=qαand∂log τB (q1 , ..., q2L ) =∂qαXRes qαi tz dz(5.56)i )=qπ(qααso that G0 (q) = τSW (q) · τB (q), and the claim of [Mtau, KK04, KK06] is that boththem are well-defined separately.Seiberg-Witten integrable systemLet us concentrate attention on τSW = τSW (a, q) or the Seiberg-Witten prepotentialF = log τSW , which is the main contribution to conformal block, and the only one,1345.4. Conformal blocks and τ -functionswhich depends on the charges in the intermediate channel.
According to [Mtau,GMqui] F(a, q), up to some possible only a-dependent term, satisfies also another setof equations∂(5.57)log τSW = aDI = 1, . . . , gI ,∂aIwhere the dual periods aDI are defined in (5.19). The total system of equations (5.55),(5.57) is also integrable [KriW, Mtau, GMqui] due to the Riemann bilinear relations.Moreover, in our case this system of equations can be easily solved due toTheorem 5.1. Functionlog τSW =12XaI TIJ aJ +XI,JaI UI + 12 Q(r)(5.58)IP(dΩ)2i=Ressolves the system (5.55), iff Q(r) solves the system ∂Q(r)forqα dz∂qαi )=qπ(qααPα = 1, .
. . , 2L, dΩ =dΩrα and other ingredients in the r.h.s. are given by (5.16),α(5.20) and the period matrix of C.One can check this statement explicitly, using the definitions (5.15) and (5.20)˛XXdωI dωJ∂ωI∂ωIRes qαiRes qαi=−dωJ = −dωJ =dz∂qα∂C ∂qαiiπ(qα )=qαπ(qα )=qα(5.59)˛∂TIJ∂dωJ =,=∂qα BI∂qαIIwhere we have first applied the formula ∂ω+hol. and then the RBR. Similarly,= − dω∂qαdzfor the second term:˛XX∂ΩrαdωI dΩrα∂ΩrαRes qαi=−Res qαidωI = −dωI =dz∂qα∂C ∂qαi )=qi )=qπ(qαπ(qααα(5.60)˛∂∂UIdΩrα ==,∂qα BI∂qαwhile the last term Q(r), vanishing after taking the a-derivatives, should be computedseparately, and the proof will be completed in next section.Quadratic form of r-chargesIn the limit aI = 0 equation (5.55) gives us the formula∂Q(r) =∂qαwheredΩ =XαXi ∈π −1 (q )qααdΩrα =dΩ2dz(5.61)rαi dΩqαi ,p0(5.62)Res qαiXα,i1355.
Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformationsFigure 5.3: Integration path for Q(r)~Theorem 5.2. Regularized expression for Q(r)X ˆ (qαi )αQ(r)~ =dΩrαiα,i(5.63)p̃0satisfies (5.61) in the limit → 0Proof: It is useful to introduce the differential with shifted polesXdΩ~ =rαi dΩ(qαi )α ,p̃0(5.64)α,iNote that due to conditions (5.14) nothing depends on the reference points p0 , p̃0 .The regularized points (qiα )α are defined in such a way thatz ((qiα )α ) = z (qiα ) − α = qα − α(5.65)and this is the only place where the coordinate z on P1 enters the definition of Q(r). Allother parts of τSW do not depend explicitly on the choice of the coordinate z becausethey are given by the periods of some meromorphic differentials on the covering curve.Expression (5.63) can now be rewritten equivalently˛1Ω~ dΩQ(r)~ = −(5.66)2πi Cwhere contour C (see fig.5.3) encircles the branch-cuts of Ω~, while the poles of dΩare left outside.
Taking the derivatives one gets˛ 1∂Ω∂Ω~∂Q(r)~ =dΩ~ −dΩ(5.67)∂qα2πi C ∂qα∂qαwhere each of the terms in r.h.s. contains only the poles at the points qαi and (qαi )αcorrespondingly. One can therefore shrink the contour of integration in the firstterm onto the points qαi (up to the integration over the boundary of cut Riemannsurface, which vanishes due to the Riemann bilinear relations for the differentialswith vanishing A-periods), and in the second – to the points (qαi )α , henceXX∂∂Ω∂Ω~Q(r)~ = −Res qαidΩ~ −Res (qαi )αdΩ∂qα∂q∂qααii136(5.68)5.4.
Conformal blocks and τ -functionsNear the point piα one can choose the local coordinate ξ such that z = qα + ξ l , so thatexpansion of Abelian integrals can be written asΩ = ri log(z − qα ) + c0 (q) + c1 (q)(z − qα )1/l + c2 (q)(z − qα )2/l + . . .Ω~ = c̃0 (q) + c̃1 (q)(z − qα )1/l + c̃2 (q)(z − qα )2/l + . . .(5.69)giving rise to∂ΩdΩ ∂c0 (q)=−++ O (z − qα )1/l∂qαdz∂qα(5.70)∂Ω~dΩ ∂c̃0 (q)1/l=−++ O (z − qα − α )∂qαdz∂qαSince the differential dΩ~ is regular near z = qα , one can ignore the regular part whencomputing the residues:XX∂dΩdΩ~Q(r)~ =Res qαidΩ~ +Res (qαi )αdΩ =∂qαdzdzii(5.71)X 1 ˛dΩdΩ~=2πi qαi ,(qαi )α dziThe r.h.s. of this formula has a limit when α → 0, so extracting the singular partfrom Q(r)~ (easily found from the explicit formula below)XQ(r) = Q(r)~ − 2∆α log α(5.72)one gets from (5.71) exactly the formula (5.61).
This also completes (together with(5.59), (5.60)) the proof of (5.58).Using the integration formula for the third kind Abelian differentials [Fay]ˆ bE(c, b)E(d, a)dΩc,d = logE(c, a)E(d, b)aone gets from (5.63) an explicit expressionQ(r)~ =Xrαi rβjα,i,β,j=XlogE((qαi )α , qβj )E(p̃0 , p0 )E((qαi )α , p0 )E(p̃0 , qβj )rαi rβj log E(qαi , qβj ) +X=Xrαi rβj log E((qαi )α , qβj ) =α,i,β,j(5.73)(rαi )2 lαi log E((qαi )α , qαi )α,ii 6=q jqαβThe first term in the r.h.s. is regular, while for the second one can useiii1/lα1/lα1/lα(z−q+)−(z−q)ααααE((qαi )α , qαi ) = p≈iii1/l1/l1/lαd (z − qα )d(z − qα + α ) α d(z − qα ) α(5.74)z→qαThereforeQ(r) =Xi 6=q jqαβrαi rβjlog E(qαi , qβj )−Xα,ii 2 i1/l (rα ) lα log d[(z − qα ) ]137(5.75)z→qα5. Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformationsSubstituting expression of the prime formE(p, p0 ) =Θ∗ (A(p) − A(p0 ))h∗ (p)h∗ (p0 )(5.76)in terms of some odd theta-function Θ∗ , the already defined above Abel map A(p),and holomorphic differentialh2∗ (p) =X ∂Θ∗ (0)I∂AIdωI (p)(5.77)one can write more explicitlyQ(r) =Xrαi rβj log Θ∗ (A(qαi ) − A(qβj )) −Xiqαi 6=q jqαβ(rαi )2 lαi logd(z(q) − qα )h2∗ (q)i1/lαiq=qα(5.78)0If cover C has zero genus g(C) = 0 itself, the prime form is just E(ξ, ξ ) =interms of the globally defined coordinate ξ, and formula (5.78) acquires the formi 1/lαXXd(z(ξ) − qα )Q(r) =rαi rβj log(ξαi − ξβj ) −(rαi )2 lαi log(5.79)dξ iiξαpi 6=pj0α√√ ξ−ξdξ dξ 0ξ=ξαβBelow we are going to apply this formula to explicit calculation of a particular examplefor a genus zero cover, but with a non-abelian monodromy group.
The result of thecomputation clearly shows that τ -function (5.79) cannot be expressed already in suchcase as a function of positions of the ramification points z = qα on P1 , which meansthat the corresponding formula for Q(r) from [K04] can be applied only in the caseof Abelian monodromy group.Bergman τ -functionThe Bergman τ -function, was studied extensively for the different cases [Knizhnik,BR, ZamAT87] from early days of string theory, mostly using the technique of freeconformal theory.
Modern results and formalism for this object can be found in [KK04,KK06]. Already from its definition (5.56) τB can be identified with the variation w.r.t.moduli of the complex structure of the one-loop effective action in the free field theoryon the cover.We are not going to present here an explicit formula for the general Bergmanτ -function, it can be found in [KK06, formula 1.7]. We would like only to point out,that for our purposes of studying the conformal blocks this is the less interestingpart, since it does not depends on quantum numbers of the intermediate channels(it means in particular, that it can be computed just in free field theory).
Below insect. 5.6 we present the result of its direct computation in the simplest case with nonabelian monodromy group. The result shows that it arises just as some quasiclassicalrenormalization of the term (5.79) in the classical part.1385.4. Conformal blocks and τ -functionsHowever, as for the SW tau-function, the definition (5.56) is easily seen to beconsistent. Taking one more derivative one gets from this formula∂ 2 log τB (q)∂ X1dz(p0 )dz(p)0=Reslim K(p , p) −=p dz(p) p0 →p∂qα ∂qβ∂qβ(z(p0 ) − z(p))2π(p)=qα=XRespπ(p)=qα× lim0p →pX∂K(p0 , p)11lim=Res×p dz(p)dz(p) p0 →p ∂qβπ(p)=qαXResπ(p00 )=qβPK(p0 , p00 )K(p, p00 )=dz(p00 )XRes00p,pπ(p)=qαπ(p00 )=qβ(5.80)K(p, p00 )2,dz(p)d(p00 )where we have used the Rauch variational formula [Fay92, formula 3.21] for the canonical meromorphic bidifferential, computed in the points p and p0 with fixed projectionsX∂K(p0 , p)=∂qβResPπ(P )=qβK(p0 , P )K(p, P )dz(P )(5.81)so that the expression in r.h.s.
of (5.80) is symmetric w.r.t. α ↔ β.This is certainly a well-known fact, but we would like just to point out here,that the Rauch formula (5.81), which ensures integrability of (5.56) can be easilyderived itself from the Wick theorem, using the technique, developed in sect. 6.4 andAppendix 5.8. Indeed,=∂ G2ij (z 0 , z|q) 0i j∂K(z 0i , z j )=dz dz =∂qβ∂qβ G0 (q)!∂G ij (z 0 , z|q) G ij (z 0 , z|q) ∂q∂β G0 (q)∂qβ 22−dz 0i dz jG0 (q)G0 (q)G0 (q)(5.82)as follows from (5.21) for the conformal block with two currents inserted G2ij (z 0 , z|q) =G2ij (z 0 , z|q)0 = hJi (z 0 )Jj (z)O(q)i0 when projected to the vanishing a-periods (5.16) orthe charges in the intermediate channels (note, that the Bergman tau-function doesnot depend on these charges).
Proceeding with (5.82) and using ∂q∂β = Lβ−1 one getstherefore!hJi (z 0 )Jj (z)Lβ−1 O(q)i0 hJi (z 0 )Jj (z)O(q)i0 hLβ−1 O(q)i0∂K(z 0i , z j )dz 0i dz j=−∂qβhO(q)i0hO(q)i0hO(q)i0(5.83)where we have used the obvious notationshO(q)i0 = h2LYOα (qα )i0 = hOs1 (q1 )Os−1(q2 )...OsL (q2L−1 )Os−1 (q2L )i01Lα=1hLβ−1 O(q)i0 = h(L−1 Oβ (qβ ))YOα (qα )i0 =α6=βhJi (z 0 )Jj (z)Lβ−1 O(q)i0 =12˛ Xqβ12˛ Xqβdζh: Jk2 (ζ) : O(q)i0kdζhJi (z 0 )Jj (z) : Jk2 (ζ) : O(q)i0k139(5.84)5.
Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations¸where the integration qβ dζ is performed on the base P1 . Applying now in the r.h.s.the Wick theorem (see Appendix 5.8 for details), one gets1hJi (z 0 )Jj (z)2: Jk2 (ζ) : O(q)i0 hO(q)i0 = 12 hJi (z 0 )Jj (z)O(q)i0 h: Jk2 (ζ) : O(q)i0 ++hJi (z 0 )Jk (ζ)O(q)i0 hJj (z)Jk (ζ)O(q)i0(5.85)which means for (5.83), that˛ X∂K(z 0i , z j )hJi (z 0 )Jk (ζ)O(q)i0 hJj (z)Jk (ζ)O(q)i0 0i j=dζdz dz =∂qβhO(q)i0hO(q)i0qβ k˛ XXK(z 0i , ζ k )K(z j , ζ k )K(z 0i , P )K(z j , P )==ResPdζdz(P )qβ k(5.86)π(P )=qβwhere we have used that¸ Pqβ=kPRes .
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