Диссертация (Isomonodromic deformations and quantum field theory), страница 3

. . Jαk (z) :(1.18)α1 <...<αkClearly, there are only N such currents. It happens so that their commutators areactually non-linear functions of the initial generators. For example, in the N = 3 casethey look schematically likeT ·T ∼TT · W3 ∼ W3W3 · W3 ∼ T + (T T )(1.19)This algebra is very complicated in the general case, but nevertheless it can be studiedwith the help of various free-field techniques.The field V~a (z) introduced above is also an example of a W-primary field since itsOPE with Wk (z) is given by the following formula:Wk (z)V~a (w) =ek (~a)V~a (w)+ less singular,(z − w)k(1.20)where the {ek } are elementary symmetric polynomials.

The main difference withthe usual conformal symmetry (1.8) is that, in general, coefficients near the lowerorders of this expansion are not given in terms of V~a (z). This causes one of the mainproblems of W-algebras: their vertex operators are not defined uniquely in the generalsituation. We propose some solution of this problem in Chapter 3.41.1. Basic conceptsFree fermionic CFTAnother very important concept in two-dimensional physics is the boson-fermion correspondence, which relates free bosonic and free fermionic theories. The transformation between these two theories can be given approximately (precise expressions arewritten in the main text) by the following formulas:ψα∗ (z) ≈ : eiφα (z) :ψα (z) ≈ : eiφα (z) :Jα (z) = : ψα∗ (z)ψα (z) :(1.21)Here ψα (z) and ψα∗ (z) are N -component fermionic fields with the following OPEs andanticommutation relations:δαβ+ reg.z−w∗{ψα,p, ψβ,q } = δαβ δp+q,0ψα∗ (z)ψβ (w) =(1.22)The conformal dimensions of both ψ and ψ ∗ are the same and equal to ( 12 , 0).From many points of view the fermionic description is much better.

For example,instead of the complicated non-linear generators (1.20) the W-algebra has another setof nice fermionic operators which are just bilinear:W̃k (z) =NX: ∂ k−1 ψα∗ (z)ψα (z) :(1.23)α=1Such a representation also gives us a better understanding of what is W-symmetry.Namely, its action on fermions is given by formulaW̃k (z)ψα∗ (w) =∂ k−1 ψα∗ (w)+ reg.z−wIt may also be rewritten using (1.7) in terms of the modes W̃k,n =hi∗W̃k,n , ψ (w) = wn+k−1 ∂ k−1 ψα∗ (w)(1.24)12πi¸W̃k (z)z k+n−1 dz:(1.25)The above calculation demonstrates that the analogy between the vector fields −z n+1 ∂and the Virasoro generators Ln can be continued to an analogy between arbitrarydifferential operators z n+k−1 ∂ k and W-generators W̃k,n .Another important concept in the free-fermionic theory are the group-like ele∗ments: such operators act on the generators of the Clifford algebra ψα,n , ψα,nin alinear wayXO−1 ψα,p O =Cα,n;β,q ψβ,q(1.26)β,qSuch operators were widely used before in the literature to construct solutions of integrable hierarchies, like KP, Toda, and their multi-component generalizations.

Here weshow that they also appear in conformal theory: we find the general vertex operatorsfor the W-algebra in such a form.51. IntroductionA remarkable property of a group-like element is the fact that any of its matrixelements can be expressed as a determinant of just two-particle ones. As an immediateconsequence of this property any correlation function of the group-like elements canbe expressed as some Fredholm determinant.The AGT relationThere is an important object in conformal field theory, called the conformal block.For simplicity we consider the 4-point one:F(∆0 , ∆t , ∆1 , ∆∞ ; ∆0t ; c|t) = h∆∞ |φ∆1 (1)P∆0t φ∆t (t)|∆0 i(1.27)To explain the meaning of this definition one has to recall that the symmetry algebraof the theory is the Virasoro algebra, and since it acts on the Hilbert space of thetheory, this Hilbert space decomposes into the sum of its highest-weight irreduciblerepresentations.

In the general position they are Verma modules, i.e. modules withhighest weight |∆i such thatLk>0 |∆i = 0L0 |∆i = ∆|∆i(1.28)and the module itself is spanned by the vectors L−k1 . . . L−kn |∆i with k1 ≥ k2 ≥ . . . ≥kn .Now one can say that projector P∆0t is a projector onto the Verma module withhighest weight (dimension) ∆, and any 4-point correlation function in conformal fieldtheory can be expanded over conformal blocks since its Hilbert space can be expandedinto Verma modules.Conformal blocks itself are purely algebraic universal objects that can be computedjust from the commutation relations in the Virasoro algebra (1.12) and from thedefinition of the primary field (1.11).

However, for the more complicated W-algebracase they can be computed algebraically only for the cases when two charges 2 , ~at and~a1 have a very special form: a~1 = (a1 , b1 , . . . , b1 ), a~t = (at , bt , . . . , bt ). Fields with suchcharges are called semi-degenerate.Virasoro conformal block is in general a concrete, but very complicated specialfunction, and until 2009 there were only two ways to compute it: by doing order-byorder computations in the Virasoro algebra or by using the Zamolodchikov recursionformula.

The situation changed in 2009 when Alday, Gaiotto and Tachikawa proposedthe correspondence between 2D CFT and 4D N = 2 supersymmetric gauge theories.In this approach the conformal blocks become equal to the so-called Nekrasov instantonic partition functions. For our purposes the most important fact is that anycoefficient in the expansion of the instantonic partition function, and so of the conformal block, is given by an explicit combinatorial formula (we will write it for simplicity2As we have seen in (1.20) for the free bosonic field, the action of the W-generators was expressedin terms of elementary symmetric polynomials ek (~a). It turns out that it is useful to use thisparametrization not only for the free case.61.1. Basic conceptsonly for c = 1) of the following kind:2F(a20 , a2t , a21 , a2∞ ; σ0t; c = 1; |t) = (1 − t)2a0 at ×Y Zb (at + sa0 − s0 σ0t |Ys , Ys0 )Zb (a1 + sσ0t − s0 a∞ |Ys , Ys0 ) (1.29)X|Y1 |+|Y2 |×tZb ((s − s0 )σ0t |Ys , Ys0 )Y ,Ys,s0 =±+−In this formula Y+ , Y− are two Young diagrams, and Zb (ν|Y1 , Y2 ) is some explicit factorized combinatorial expression depending on two Young diagrams and one complexnumber.This formula for a conformal block was proved in 2010 by Alba, Fateev, Litvinovand Tarnopolsky.

In their proof they presented such a basis that any matrix elementof the Virasoro vertex operator can be expressed in terms of Zb . What matters for usis that for c = 1 their basis is exactly the free-fermionic one. This is one more hintthat a fermionic description of conformal field theory is better than a bosonic one.Isomonodromic deformationsThere is a story from the beginning of 20th century when mathematicians started tostudy N × N matrix linear systems with first-order singularities:nXAkdΦ(z)= Φ(z)dzz − zkk=1wherenP(1.30)Ak = 0.

One may ask, at first, when such a system can be solved explicitlyk=1in terms of some known special function. I present below an important list of someexamples which, however, does not cover everything:• N = 2, n = 3. Always solvable in terms of hypergeometric function 2 F1 .• n = 3, N – arbitrary, but the spectral type of A1 is special: A1 ∼ diag(a1 , b1 , . .

. , b1 ).Always solvable in terms of N FN −1 . Here the analogy with the semi-degeneratefields is absolutely not accidental.• n – arbitrary, but the monodromy group is a semidirect product of a permutationgroup and the diagonal matrices (quasi-permutation group). Always solvable interms of higher-genus theta-functions. This case corresponds at the CFT sideto the twist fields and is considered in Chapters 5, 6.For n > 4 and for general A’s the Fuchsian system cannot be solved explicitly (thoughin Chapter 4 we give the formula that can give its explicit expansion in some regionof parameters). Instead of this it is reasonable to ask about the monodromy of sucha system. Namely, if we take some solution and continue it analytically around theloop γ encircling some singular point, we get another solution.

Now any two solutionsof the system are connected by linear transformations, so we haveγ : Φ(z) 7→ Mγ Φ(z)7(1.31)1. IntroductionThe matrices Mγ ∈ GL(N ) generate the monodromy group of the system, and analyticcontinuation around closed loops generates a map from π1 (C \ {z1 , . . . , zn }) to GL(N )with the image coinciding with the monodromy group.The problem of finding the monodromy group for a given system is also complicated. Instead of this one may look for such transformations of the systems thatpreserve the monodromy, the so-called isomonodromic transformations.

It happensso that in general position we are able to move all singular points and to make somemodifications of the matrices Ak that preserve the monodromy: in this setting all matrices Ak become functions of {z1 , . . . , zn }. Such a functional dependence is describedby a non-linear system of matrix equations, the Schlesinger equations:[Aj , Ak ]∂Aj=∂zkzj − zkX [Aj , Ak ]∂Aj=−∂zjzj − zkk6=j(1.32)There is also a non-trivial statement that can be verified explicitly that any solutionof the Schlesinger system gives some function of the {zk }, the tau-function, definedby its derivatives:∂ log τ (z1 , . . .