Диссертация (Isomonodromic deformations and quantum field theory), страница 11

One finds then, that(z − w)hθ|Vν (1)ψ̃ σ (z)ψ σ (w)|σi = φ(z)φ(w)−1is expressed actually through the solutions of a simple linear systemdφ(z)σν= φ(z)+dzz z−1(3.35)(3.36)It means that this linear system can be used to define all two-fermion matrix elements,e.g. in the region 1 > |z| > |w|X1hθ|Vν (1)ψ̃ σ (z)ψ σ (w)|σi =hθ|Vν (1)ψ̃pσ ψqσ |σi1(3.37)p+ 2 −σ q+ 12 +σwp,q zand together with the Wick theorem it defines all matrix elements, or just the vertexoperator Vν , uniquely – up to a numeric factor. In its turn the linear system itselfis determined by the monodromy properties (here very simple) of φ(z) at z = 0and z = 1 (and related to them monodromy at z = ∞).

Hence, the problem ofcomputation of the two-fermion correlation functions can be reformulated in terms ofa Riemann-Hilbert problem.433. Free fermions, W-algebras and isomonodromic deformationsRemarks• Formulas (3.28), (3.31) give a very explicit representation for the matrix elementand bi-fundamental Nekrasov function in terms of the Frobenius coordinatesof the corresponding Young diagrams (this representation, for example, is farmore adapted for practical computation, than the formulas (3.32)). However,it is sometimes not easy to see directly, that these formulas possess some niceproperties: for example satisfy the “sum rules” likeXZb (α1 |∅, Y )Zb (α2 |Y, ∅) X |Y |=t Z(α1 |∅, Y )Z(α2 |Y, ∅) =t|Y |Z0 (Y )YY(3.38)X|Y |iα1 φ(1)iα2 φ(1)α1 α2=t h0|e|Y, 0ihY, 0|e|0i = (1 − t)Ywhere the r.h.s. immediately follows from resolution of unity and the correlatorof two exponentialsh0|eiα1 φ(1) eiα2 φ(t) |0i = (1 − t)α1 α2(3.39)which is instructive to compare with the computation from [KKMST, Koz].• One can also easily extract some useful information from particular cases of(3.34), which include a nice identity (cf.

with [NO, BAW])∞X(−ν)a+1 (ν)b+1 a b1(1 − z)ν (1 − w)−ν=+z wz−wz − w a,b=0 (a + b + 1)νa!b!(3.40)containing some part of the matrix elements from (3.25).• According to (3.7)1ψ̃(z + t/2)ψ(z − t/2) = : exptˆ!z+t/2J(ξ)dξ:(3.41)z−t/2Expansion into the powers of t gives the infinite series of the currents of W1+∞algebra!!ˆ z+t/21: ψ̃(z + t/2)ψ(z − t/2) := : expJ(ξ)dξ − 1 :=tz−t/2(3.42)X tk−1=Uk (z)(k − 1)!k>0where explicitlyU1 (z) = J(z),U2 =122: J(z) :,1U3 =31 2: J(z) : + ∂ J(z) ,43...(3.43)and one implies bosonic normal ordering for the bosons and fermionic for thefermions. These formulas have been used many times (see e.g. [Pogr, OP, LMN,NO, Mint]) to relate the generators of the W1+∞ algebra with the fermionicbilinear operators, and we just recall them in order to generalize below to muchless trivial non Abelian case.443.3.

Non-Abelian U (N ) theoryNon-Abelian U (N ) theoryNekrasov functionsConsider now more general case of Nekrasov functions, corresponding to the U (N )non-Abelian theory. They can be expressed in terms of U (1) functions (3.31), (3.32)by the following product formula0NY0Ẑb (θ , ν, θ|Y , Y ) =Zb (ν − θα0 + θβ |Yα0 , Yβ )(3.44)α,β=1For the diagonal elements Ẑ0 (θ|Y ) = Ẑb (θ, 0, θ|Y , Y ) one getsẐ0 (θ|Y ) =NYZb (−θα + θβ |Yα , Yβ ) = ±YZb2 (−θα + θβ |Yα , Yβ ) ·Yαi<jα,β=1Z0 (Yα )(3.45)or, after taking the square root, just1Ẑ02 (θ|Y ) =YZb (−θα + θβ |Yα , Yβ ) ·Y1Z02 (Yα )(3.46)αα<βNow for simplicity it is better to replace θα0 − ν 7→ θα0 or θα + ν 7→ θα , then ν simplydisappears from (3.44).

Consider now the normalized matrix elementẐ(θ 0 , θ|Y 0 , Y ) =NQ=QẐb (θ 0 , θ|Y 0 , Y )11Ẑ02 (θ 0 |Y 0 )Ẑ02 (θ|Y )=Zb (−θα0 + θβ |Yα0 , Yβ )α,β=1α<βZb (−θα + θβ |Yα , Yβ )Q1Z02 (Yα ) ·Qαα<βZb (−θα0 + θβ0 |Yα0 , Yβ0 ) ·Q1Z02 (Yα0 )α(3.47)Using representation (3.28), (3.31) for the U (1) functions in terms of the Frobenius coordinates, one finds that the ratio of products of the elementary Cauchy determinantsfrom there is actually combined into more sophisticated unique Cauchy determinantẐ(θ 0 , θ|Y 0 , Y ) = detIJ×Yf1,α (θ01×x I − yJ0, θ, p0α,i )f2,α (θ 0 , θ, qα,i)f1,α (θ, θ 0 , pα,i )f2,α (θ, θ 0 , qα,i )(3.48)i,αwith two multi-sets of variables entering the determinant of the form0{xI } = {−qα,i− θα0 } ∪ {pα,i − θα }{yI } = {p0α,i − θα0 } ∪ {−qα,i − θα }45(3.49)3.

Free fermions, W-algebras and isomonodromic deformationsup to quite nontrivial diagonal part, which can be still read from (3.28) and (3.47),giving the following factors for (3.48)pY (θβ0 − θα )pα,i + 1 Yθβ − θα102qf1,α (θ, θ , pα,i ) =1(pα,i − 2 )! βθβ0 − θα β6=α (θβ − θα )pα,i + 12(3.50)pY (θα − θβ0 )qα,i + 1 Yθα − θβ102qf2,α (θ, θ , qα,i ) =(θα − θβ )qα,i + 1(qα,i − 12 )! βθα − θ0β6=αβ2Existence of the determinant formula (3.48) is very important, since it actually impliesthat Nekrasov functions Ẑ(θ 0 , θ|Y 0 , Y ) can be identified with the matrix elements ofsome vertex operator, characterized as in the Abelian U (1) case by its adjoint action,which is still a linear transformation but now of the N -component fermions. Weare going indeed to introduce this vertex operator below using the theory of (N component) free fermions, generalizing the Abelian case considered above.

In generalsituation this operator is characterized by solution to auxiliary linear problem onsphere with three marked points, while explicit formulas of this section just correspondto particular case of the hypergeometric-type solutions.N -component free fermionsHence, consider the generalization of the free-fermionic construction from U (1) to thenon-Abelian U (N ) case. First, introduce the algebra{ψα,r , ψβ,s } = 0 ,{ψ̃α,r , ψ̃β,s } = 0 ,{ψ̃α,r , ψβ,s } = δα,β δr+s,0r, s ∈ Z + 12 , α, β = 1, .

. . , N(3.51)of the canonical anticommutation Prelationsthe components of the fermionic fields´ for2dzψ̃with free first-order action S = π1 Nα ∂ψα , so that (3.51) are equivalent toα=1 Σthe operator product expansionsδαβ+ Jαβ (w) + O(z − w)z−wψα (z)ψβ (w) = reg.ψ̃α (z)ψ̃β (w) = reg.ψ̃α (z)ψβ (w) =(3.52)Similarly to (3.3) it is also possible and useful to introduce the bosonization formulasfor these fermionic fields!!X Jα,nX Jα,nψ̃α (z) = exp −exp −eQα z Jα,0 α (J 0 )nnnznzn<0n>0!!(3.53)X Jα,nX Jα,nψα (z) = expexpe−Qα z −Jα,0 α (J 0 )nnnznzn<0n>0Here Jα,n form the Heisenberg algebra[Jα,n , Jβ,m ] = nδαβ δm+n,0 ,46[J0,α , Qβ ] = δαβ(3.54)3.3. Non-Abelian U (N ) theoryQα−1and α (J 0 ) = β=1(−1)J0,β , we may also note that α (x + y) = α (x)β (y).

Theseextra sign factors do the same as the Jordan-Wigner transformation: they convertcommuting objects into the anticommuting ones.A standard representation of this algebra Hσ is constructed from the vacuumvector |σi, with the charges J 0 |σi = σ|σi and killed by all positive modesσψα,r>0|σi = 0 ,σψ̃α,r>0|σi = 0 .(3.55)Basis vectors of this representation can be given by|qα,j ||pα,i |NYYYψα,−qα,j  |σiψ̃α,−pα,i|{pα,i }, {qα,i }, σi =α=1(3.56)j=1i=1The letters pα,i and qα,i , at least in the case when #pα = #qα , should be interpreted asFrobenius coordinates of the N -tuple of the Young diagrams. It will be also convenientin what follows to use the vacuum-shifting operators PαnPα0 = 1,σσσPαn<0 = ψα,n+. . .

ψα,−1ψ1 |σi ,α,n+ 3222σσσPαn>0 = ψ̃α,−n+. . . ψ̃α,−1 |σi1 ψ̃α,−n+ 322(3.57)2and the corresponding states|n, σi =NYPαnα |σi .(3.58)α=1in particular for the vectors n = ±1β with components nα = ±δαβ .Level one Kac-Moody and W-algebrasConsider the W-algebras for g = sl(N ) series, possibly extended to gl(N ) where weshall call it WN ⊕ H.

Their generators in current representation can be identifiedwith the symmetric functions of the normally ordered currents J(z) ∈ h ⊂ g with thevalues in Cartan subalgebra, or equivalently, up to a coefficient, as certain “Casimir\) ). The Virasoro central charge at levelelements” in the universal enveloping U (sl(N1k = 1 isk dim gN2 − 1(3.59)==N −1c=k + CV1+N\) ) this current algebra has nice representation in termsWhen embedded to U (gl(N1of the multi-component free holomorphic fermionic fieldsJαβ (z) =: ψ̃α (z)ψβ (z) :,α, β = 1, .

. . , N(3.60)The WN -algebra can be defined in terms of invariant Casimirpolynomials of the¸currents, commuting with the screening charges Qαβ = Jαβ (z) (it is enough torequire commutativity only with those, corresponding to the positive simple roots).473. Free fermions, W-algebras and isomonodromic deformationsThen the W-generators turn to be just the symmetric polynomials of the diagonalCartan currents Jα = Jαα (z) ∈ h, i.e.XWn (z) =: Jα1 (z)Jα2 (z) . . . Jαn (z) : ,n = 1, . .

. , N(3.61)α1 <α2 <...<αn\) ) and WN ⊕ H in Hσ . For thisOne can consider the representations of U (gl(N1purpose it is convenient to introduce the generating functionsψασ (z) =Xσψα,r1r∈Z+ 21z r+ 2 +σα,ψ̃ασ (z) =Xσψ̃α,r1r∈Z+ 21z r+ 2 −σα.(3.62)where shifts of the powers of the coordinate z come naturally, e.g. from the bosonization formulas (3.53). For these fields instead of (3.52) one getsψ̃ασ (z)ψβσ (w) = δαβz σα w−σα+ : ψ̃ασ (z)ψβσ (w) : =z−wδαβσα+ δαβ + : ψ̃ασ (w)ψβσ (w) : + O(z − w) ,=z−ww(3.63)σthen it is clear that the modes of Jαβ(w) from (3.52) acquire in this representationthe formXσσσJαβ,n= δαβ δn,0 σα +: ψ̃α,n−pψβ,p:(3.64)1p∈Z+ 2As in the U (1) case (see (3.42)) in the fermionic realization of WN ⊕ H, then one canchoose the set of generators in a form of the fermionic bilinears:∞Xψ̃ασ (zαttN X tk−1+ )ψασ (z − ) =+Ukσ (z) .22t(k−1)!k=1(3.65)The l.h.s.

of this formula givesσαt σt1 1 + 2zt+ )ψα (z − ) =+22t 1 − 2zt1tX 1 + t p+ 2 +σα1X 1σσz2z+: ψ̃m−p,αψα,p: .tt m∈Z z m (1 + 2zt )m+11−2z1ψ̃ασ (zp∈Z+ 248(3.66)3.3. Non-Abelian U (N ) theoryIntroducing two collections of polynomials1+1−x2x2x(1 + x2 )m+1p=1+1∞Xuk (p)k=01+1−x2x2p+ 12xk,(k − 1)!(3.68)∞Xxk,=vk,m (p)(k−1)!k=1the generators in the r.h.s. of (3.65) explicitly become!σ=Uk,mXδm,0 uk (σα ) +αXσσ: .ψp,αvk,m (p + σα ) : ψ̃α,m−p(3.69)p∈ZσThis set of generators of WN ⊕ H contains commuting zero modes Uk,0which wereshown to play an important role in the study of the extended Seiberg-Witten theoryand AGT correspondence [LMN, MN, Mint, FLitv12].