Диссертация (Isomonodromic deformations and quantum field theory), страница 2

. . . . . . . . . . . . . . . . . . . . . . . . 1896.9 Identities for lattice Θ-functions . . . . . . . . . . . . . . . . . . . . 1906.9.1 First identity for AN −1 and DN Θ-functions . . . . . . . . . . 1906.9.2 Product formula for AN −1 Θ-functions . . .

. . . . . . . . . . 1926.9.3 An identity for DN and BN Θ-functions . . . . . . . . . . . . 1936.10 Exotic bosonizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.10.1 N S × R . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 1936.10.2 R × R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1956.10.3 l twisted charged fermions . . . . . . . . . . . . . . . . . . . . 1966.10.4 l charged fermions – standard bosonization . . . . . . . . . . . 198Bibliography199v1IntroductionIn my thesis I present a correspondence between isomonodromic deformations ofhigher-rank Fuchsian linear systems and conformal field theory with higher-spin symmetry, or W-symmetry.

The correspondence that I describe is a generalization tohigher rank of the one found by Gamayun, Iorgov and Lisovyy in [GIL12]. This generalization is first found numerically and then proved in the free-fermionic framework byan explicit construction of the twist-fields that are at the same time monodromy fieldsand W-primary fields. Next I use this construction to give the Fredholm-determinantrepresentation of the general isomonodromic tau-function. The determinantal representation found in this way can also be proven without using any field theory by acareful analysis of derivatives of the determinant.Another part of thesis deals with the special case that the monodromy groupis given by quasi-permutation matrices I present a construction of the W-primaryfields in terms of twisted bosons and give an expression of their conformal blocks interms of algebro-geometric objects associated with branched covers of the complexsphere.

Such correlation functions are related to exact isomonodromic tau-functionsintroduced by Korotkin. I also present the interpretation of such fields in terms of freefermions and a computation of the characters of related W-algebra representations. Inthis part of the investigations also W-algebras of the orthogonal series are considered.Basic conceptsIn this section I try to give a self-contained overview of basic objects considered inthis thesis. My goal is to make this into an introduction for non-experts.Conformal field theoryBy a conformal field theory (CFT) is meant by default a two-dimensional quantumfield theory with conformal symmetry, i.e., the symmetry that preserves angles andmultiplies metrics by a scalar factor: gµν 7→ λgµν . A remarkable feature of thetwo-dimensional case is the fact that Lie algebra of local conformal transformationsbecomes infinite-dimensional and is generated by the holomorphic functions f : C →C, z 7→ f (z).

In the infinitesimal form such transformations may be rewritten asz 7→ z + (z) + O(2 )1(1.1)1. IntroductionThe Lie algebra of the corresponding vector fields (z)∂z has as a natural basis the{`n = −z n+1 ∂z }. In this basis its commutations relations acquire the following form:[`n , `m ] = (n − m)`n+m(1.2)This Lie algebra is called Witt algebra, or Virasoro algebra with zero central extension.All local fields in a conformal theory transform under conformal transformationsin some non-trivial way. It happens though that one may always choose a basis in thespace of fields which is formed by elements that transform as differential forms of the¯¯ are calledkind φ∆,∆¯ (z, z̄)dz ∆ dz̄ ∆ . Such fields are called primary fields, and (∆, ∆)¯ is important, we will alwaystheir dimensions.

Though in actual physical models ∆consider only the holomorphic part. Infinitesimal transformation of the primary field,or, in other words, the action of the Lie derivative, is given by the formulaδ(z) φ∆ (z) = (0 (z)∆ + ∂z ) φ∆ (z)(1.3)By Noether’s theorem, any symmetry in a quantum theory gives rise to conservedcharges. Conformal transformations in CFT give rise to charges that are encoded bya single energy-momentum tensor T (z). The quantum version of Noether’s theorem isformed by the Ward identities that relate infinitesimal transformations of fields withthe action of conserved charges. In CFT they read as˛dwR T (w)φ(z)δ(z) φ(z) =(1.4)2πizIn this formula the integral goes around a small circle around w = z, and the symbolR means radial ordering of the operatorsRφ(z)ψ(w) = φ(z)ψ(w),Rφ(z)ψ(w) = (−1)pψ ·pφ ψ(w)φ(z),|z| > |w||z| < |w|(1.5)Here pφ is fermionic parity.

If we work in the path integral formulation we may justskip this notation: any product is already radially ordered.A very important concept in CFT is the so-called operator product expansion, i.e.,the expansion of the radially ordered product of two fields in the neighbouring points:NX(AB)n (w)A(z)B(w) =(z − w)n+1n=−∞(1.6)Radial ordering will be usually omitted in all OPE expansions due to historical reasons, though it is important.The singular part of the OPE contains all information about the commutationrelations between modes of the operators 1 :˛1[An , B(w)] =dzz n+∆−1 A(z)B(w)(1.7)2πiw1This can be easily deduced using properties of radial ordering and doing manipulations withcontour integrals21.1. Basic concepts¸ n+∆−11where An = 2πizA(z)dz.For example, one can write down an OPE of the primary field with the energymomentum tensor:T (z)φ∆ (w) =∆φ∆ (w) ∂φ∆ (w)+ reg.+(z − w)2z−w(1.8)An analogous OPE for the energy-momentum tensor itself has the formT (z)T (w) =c/22T (w)∂T (w)+ regular(= reg.)++42(z − w)(z − w)z−w(1.9)One can introduce the components of the Laurent expansion of the energy-momentumtensorX LnT (z) =(1.10)z n+2n∈Zand then rewrite the above OPEs in terms of these components:[Ln , φ∆ (z)] = (n + 1)∆z n φ∆ (z) + z n+1 φ∆ (z)(1.11)c 3(n − n) + (n − m)Ln+m(1.12)12The Lie algebra generated by the operators Ln is called the Virasoro algebra, and aswe see, it is a central extension of the algebra of vector fields by the element c calledcentral charge.

The value of the central charge is an important characteristic of CFT.[Ln , Lm ] =Free bosonic CFTOne of not the most elementary, but very important examples of CFT is a free bosonictheory with N elementary fields φα (z, z̄) – Gaussian fields with the following OPEs:φα (z)φβ (w) = −δαβ log |z − w|2 + reg.(1.13)It is also useful to introduce derivatives of the φα , the so-called U (1) currents Jα (z) =i∂φα (z) with conformal dimension (1, 0). The commutation relation of the modesof such currents are given by [Jα,n , Jβ,k ] = nδn+k,0 δαβ . The Lie algebra with thesecommutation relations is called the Heisenberg algebra.One can check that the energy-momentum tensorT (z) =NX: Jα (z)2 :(1.14)α=1actually generates the Virasoro algebra with c = N .

In this way we can get a realization of the complicated Virasoro algebra in terms of a simpler Heisenberg.The simplest primary fields, or vertex operators of the constructed Virasoro, canbe given explicitly by the exponentsiV~a (z) = : ePα3aα φα (z):(1.15)1. IntroductionOne can check that the following OPE with the energy-momentum tensor holds forsuch fields:∆(~a)Va (w) ∂V~a (w)T (z)V~a (w) =+ reg.+(1.16)(z − w)2z−wPIn this formula the conformal dimension is given by the formula ∆(~a) = 12 a2α .αIn this way we can construct some examples of primary fields, but not an arbitraryone: in our case we have a serious problem, conservation of the U(1) charge.

Namely,colliding two fields with charges ~a and ~b we get another field with charge ~a + ~b:~V~a (z)V~b (w) = (z − w)(~a,b) V~a+~b (w) + . . . ,(1.17)whereas in the general CFT any fields can appear in this OPE. However, we willpresent an almost free-field generalization of this construction in Chapter 3, whichis not restricted by this charge-conservation condition.The free-bosonic theory gives also an example of a theory with W-symmetry, thenon-linear higher spin symmetry. Generators of this symmetry are expressed viainitial bosonic fields as elementary symmetric polynomials (the energy-momentumtensor was a quadratic symmetric polynomial):NXWk (z) =: Jα1 (z) .