Диссертация (Isomonodromic deformations and quantum field theory), страница 4

, zn ) X tr Ak Aj=(1.33)∂zkzk − zjj6=kThis function is simpler than the fundamental solution itself. For example, for n = 33Qthe singular points it can be given explicitly by τ (z1 , z2 , z3 ) = (zi − zj ) tr Ai Aj , whilei<jthe fundamental solution is still unknown in general.

One of the first interesting casesis n = 4, N = 2: this tau-function solves Painlevé VI equation and gives actuallyits general solution. This fact is one of the motivations to study isomonodromicdeformations: they give a convenient framework to study the equations from thePainlevé family.One of the achievements in the study of this tau-function for the Painlevé VI casewas the work of Jimbo in 1982 where he obtained the first 3 terms of the tau-functionin terms of the monodromy. The next breakthrough in this direction was done byGamayun, Iorgov, Lisovyy and Teschner when they gave the general formula for theN = 2 tau-function, including arbitrary number of points, in terms of conformalblocks, which easily recovers the Jimbo formula. In the present thesis, see Chapters2-4, I present the generalization of their result for arbitrary N .

In particular, I givein Chapter 4 a rigorous proof of this result without using any field theory.Isomonodromy-CFT correspondenceVarious parts of the correspondences between isomonodromic deformations, Painlevéequations and quantum field theory (QFT) have been found in the late 70’s by Sato,Miwa and Jimbo. Archetypal formulas of such correspondence look like follows:τ (x1 , . . . , xn ) = hO(x1 ) .

. . O(xn )i,Φ(y, y0 ) =8y − y0 ∗hψ (y)ψ(y0 )O(x1 ) . . . O(xn )iτ(1.34)1.1. Basic conceptsHere the O(x) are some disorder fields in some free field theory (like spin variable inthe Ising model), ψ(y), ψ ∗ (y0 ) are initial free fields, and Φ(y, y0 ) is a solution of somelinear problem. Such a correspondence was found for various massive and masslessbosonic and fermionic models. The only problem was that this correspondence wasfound 5 years before the creation of conformal field theory, otherwise this researchcould be related to CFT at that time.There were several guesses that belong to Knizhnik and Moore that CFT is actuallyrelated to isomonodromic deformations, but they were not developed to get a finalexplicit answer.

Such a development was done by Gamayun, Iorgov and Lisovyy in2012, when they gave the general solution of the Painlevé VI equation as a linearcombination of c = 1 conformal blocks:τ (t) =Xsn0t t(σ0t +n)2 −θ 2 −θ 2t02Cn (σ0t , {θν })F(θ02 , θt2 , θ12 , θ∞; (σ0t + n)2 |t)(1.35)n∈ZTogether with the AGT formula (1.29) this gave the general tau-function as an explicitseries. To explain this formula I give below the short dictionary of the correspondence:Painlevé VI1tr A2ν = θν22tr M0 Mt = 2 cos πσ0tsome function of tr Mµ Mν , tr M ντ (t)[Φ(z)Φ(z0 )−1 ]αβz−z0nPAk 2tr ()z−zkk=1CFT∆ν = θν2∆ = (σ0t + n)2s0th∆∞ |φ∆1 (1)φ∆t (t)|∆0 i1h∆∞ |φ∆1 (1)φ∆t (t)ψα∗ (z)ψβ (w)|∆0 iτ (t)1h∆∞ |φ∆1 (1)φ∆t (t)T (z)|∆0 iτ (t)So the main rule is the following: dimensions (or higher W-charges) are symmetricfunctions of the eigenvalues of logarithms of the monodromy matrices.Formula (1.35) was proved in several different ways, it was also generalized toarbitrary number of points with 2 × 2 matrices.In this thesis I present the same construction for the N × N case which relatesisomonodromic tau-function to a linear combination of conformal blocks of the Walgebra.

In Chapter 2 we solve Schlesinger system numerically and conjecture thegeneral form of the tau-function, in Chapters 3, 4 we prove it using two differentapproaches. In Chapter 3 we construct explicitly W-primary fields as some fermionicgroup-like elements with given monodromy and then find the Fredholm-determinantformula for their correlator; in Chapter 4 we give the generalization of this formulato an arbitrary number of points and prove it.Twist fieldsThe archetypal example of a twist field is Zamolodchikov’s construction of conformal1field with dimension ∆ = 16in c = 1 CFT. The first ingredient is the expression of91. Introductionthe Virasoro algebra in terms of the half-integer Heisenberg algebra:hi1Jn+ 1 , J−m− 1 = (n + )δm,n222Xδn,0Ln =+: Jk Jn−k :161(1.36)k∈Z+ 2The usual bosonic representation (Fock module) is reducible, and it is expanded overthe infinite series of Verma modules with dimensions ( 14 + n)2 .

This statement can beobtained from the computation of characters with the help of the well-known Gaussformula:P ( 1 +n)2q 41q 16n∈Z(1.37)= Q∞Q∞kk+ 12)k=1 (1 − q )k=0 (1 − qThe picture corresponding to this situation looks as follows: there is a bosonic fieldJ(z) which has monodromy around the origin J(e2πi z) = −J(z). This monodromyis actually related to the twist field O(0) sitting in the point z = 0.

Its dimension1.equals to 16Another ingredient of the construction concerns the corresponding vertex operator:the field O(x) sitting in the arbitrary point and changing the sign of J(z) when it goesaround. The great discovery of Zamolodchikov was an exact formula for the conformalblock of such fields. For example, the 4-point block is given by simple formula:F((16t−1 )∆ eiπ∆τ (t)1 1 1 1, , , ; ∆; c = 1|t) =116 16 16 16(1 − t) 8 θ3 (0|τ (t))(1.38)where τ (t) is a period of the elliptic curve y 2 = z(z − t)(z − 1).As far as we have the isomonodromy-CFT correspondence, we can use this conformal block in (1.35): this leads us to so-called Picard solution of Painlevé VI. Fromthe point of view of the monodromy it corresponds to the quasi-permutations thatwere mentioned above: in this case one can find explicitly the general solution of then-point system.In this thesis I present the generalization of Zamolodchikov’s construction to thecase of W-algebra.

In contrast to the previous situation, here there is a richer collectionof twist fields that are labelled by the elements of the permutation group. Theypermute the bosonic currents leaving the W-generators untouched:Jk (e2πi z)Os (0) = Js(k) (z)Os (0)(1.39)In Chapter 5 we construct such fields and find the generalization of Zamolodchikov’sformula for their conformal blocks.

We also show that using extended isomonodromyCFT correspondence we can construct the tau-function from these conformal blocksand then identify it with the known tau-function found by Korotkin.In Chapter 6 we find many generalizations of the character formula (1.37), wealso find a very close relation between the construction of the W-algebra twist fields\) .and the Lepowski-Wilson construction of the integrable representations of sl(N1We also relate this construction to the free-fermionic approach from Chapter 2. Incontrast to the previous considerations, here we also touch upon the W-algebras forthe orthogonal series and generalize all results related to twist-fields to this case.101.2. OutlineOutlineHere I list the most important results of the thesis and then explain how the differentparts of the text are related to each other.List of the key results• Formula 2.53:P (β,w) (0t)(1∞)τ (t) =eCw (θ 0 , θ t , σ 0t , µ0t , ν0t )Cw (θ 1 , θ ∞ , σ 0t , µ1t , ν1t ) ×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)This formula describes the conjectural form of the general N = 3, n = 4 taufunction.• Formula 2.58:(0t)(1∞)Cw (θ 0 , at , σ)CwQ(σ, a1 , θ ∞ ) =aaijG[1− Nt +(ei ,θ 0 )−(ej ,σ+w)]G[1− N1 +(ei ,σ+w)+(ej ,θ ∞ )]QG[1+(αi ,σ+w)]iThis formula gives the conjectural form of the structure constants for two semidegenerate fields.• Theorem 3.2: Vν (t) is a primary field of the conformal WN ⊕ H algebra withthe highest weights uk (ν).• Theorem 3.5:Solution of the linear problem with n marked points is given by(z − w)Kαβ (z, w) withhθ ∞ |Vθn−2 (tn−2 ) .

. . Vθ1 (t1 )ψ̃αθ0 (z)ψβθ0 (w)|θ 0 iKαβ (z, w) =hθ ∞ |Vθn−2 (tn−2 ) . . . Vθ1 (t1 )|θ 0 i(1.40)whereas its isomonodromic tau-function is defined byτ (t1 , . . . , tn−2 ) = hθ ∞ |Vθn−2 (tn−2 ) . . . Vθ1 (t1 )|θ 0 i(1.41)• Formula 3.136: τ (t) = det (1 + Rt )This formula expresses the 4-point isomonodromic tau-function as a Fredholmdeterminant with explicitly given kernel.• Theorem 4.22: Fredholm determinant τ (a) giving the isomonodromic taufunction τJMU (a) can be written as a combinatorial seriesτ (a) =XXn−2Y~Y~ k−1,QZ Y~k−1~,QkT [k] ,k~ 1 ,...Q~ n−3 ∈QN Y~1 ,...Y~n−3 ∈YN k=1Q~~ k−1Y,Q[k]where Z Y~k−1Tare expressed by (4.66), (4.63) in terms of matrix ele~k ,Q kments of 3-point Plemelj operators in the Fourier basis.111.

Introduction• Theorem 4.32: This theorem describes the relation of our general Fredholmdeterminant and the particular hypergeometric one found before by Borodinand Deift.• Theorem 5.1:Functionlog τSW =12XaI TIJ aJ +XI,JaI UI + 12 Q(r)IP2Res qαi (dΩ)solves the system (5.55), iff Q(r) solves the system ∂Q(r)=∂qαdzi )=qπ(qααPfor α = 1, . .