Изомонодромные деформации и квантовая теория поля Гавриленко Павел Георгиевич

brief description

Гавриленко, Павел Георгиевич.
Изомонодромные деформации и квантовая теория поля = Isomonodromic deformations and quantum field theory : Isomonodromic deformations and quantum field theory : диссертация ... кандидата физико-математических наук : 01.01.00 / Гавриленко Павел Георгиевич ; [Место защиты: Нац. исслед. ун-т "Высшая школа экономики"]. - Москва, 2018. - 211 с. : ил.
Оглавление диссертациикандидат наук Гавриленко Павел Георгиевич
Contents
1 Introduction
1.1 Basic concepts
1.1.1 Conformal field theory
1.1.2 Isomonodromic deformations
1.1.3 Isomonodromy-CFT correspondence
1.1.4 Twist fields
1.2 Outline
1.2.1 List of the key results
1.2.2 Organization of the thesis
2 Isomonodromic T-functions and WN conformal blocks
2.1 Introduction
2.2 Isomonodromic deformations and moduli spaces of flat connections
2.2.1 Schlesinger system
2.2.2 Moduli spaces of flat connections
2.2.3 Pants decomposition of M4
2.2.4 Pants decomposition for M®
2.3 Iterative solution of the Schlesinger system
2.3.1 si2 case
2.3.2 si3 case
2.4 Remarks on W3 conformal blocks
2.4.1 General conformal block
2.4.2 Degenerate field
2.5 Conclusions
3 Free fermions, W-algebras and isomonodromic deformations
3.1 Introduction
3.2 Abelian U(1) theory
3.2.1 Fermions and vertex operators
3.2.2 Matrix elements and Nekrasov functions
3.2.3 Riemann-Hilbert problem
3.2.4 Remarks
3.3 Non-Abelian U(N) theory
3.3.1 Nekrasov functions
3.3.2 N-component free fermions
3.3.3 Level one Kac-Moody and W-algebras
3.3.4 Free fermions and representations of W-algebras
3.4 Vertex operators and Riemann-Hilbert problem
3.4.1 Vertex operators and monodromies
3.4.2 Generalized Hirota relations
3.4.3 Riemann-Hilbert problem: hypergeometric example
3.5 Isomonodromic tau-functions and Fredholm determinants
3.5.1 Isomonodromic tau-function
3.5.2 Fredholm determinant
3.6 Conclusion
4 Fredholm determinant and Nekrasov sum representations of isomon-odromic tau functions
4.1 Introduction
4.1.1 Motivation and some results
4.1.2 Notation
4.1.3 Outline of the chapter
4.1.4 Perspectives
4.2 Tau functions as Fredholm determinants
4.2.1 Riemann-Hilbert setup
4.2.2 Auxiliary 3-point RHPs
4.2.3 Plemelj operators
4.2.4 Tau function
4.2.5 Example: 4-point tau function
4.3 Fourier basis and combinatorics
4.3.1 Structure of matrix elements
4.3.2 Combinatorics of determinant expansion
4.4 Rank two case
4.4.1 Gauss and Cauchy in rank
4.4.2 Hypergeometric kernel
4.5 Relation to Nekrasov functions
5 Exact conformai blocks for the W-algebras, twist fields and isomon-odromic deformations
5.1 Introduction
5.2 Twist fields and branched covers
5.2.1 Definition
5.2.2 Correlators with the current
5.2.3 Stress-tensor and projective connection
5.3 W-charges for the twist fields
5.3.1 Conformal dimensions for quasi-permutation operators
5.3.2 Quasipermutation matrices
5.3.3 W3 current
5.3.4 Higher W-currents
5.4 Conformal blocks and T-functions
5.4.1 Seiberg-Witten integrable system
5.4.2 Quadratic form of r-charges
5.4.3 Bergman t-function
5.5 Isomonodromic t-function
5.6 Examples
5.7 Conclusions
5.8 Diagram technique
5.9 W4(z) and the primary field
5.10 Degenerate period matrix
6 Twist-field representations of W-algebras, exact conformal blocks
and character identities
6.1 Abstract
6.2 Introduction
6.3 W-algebras and KM algebras at level one
6.3.1 Boson-fermion construction for GL(N)
6.3.2 Real fermions for D- and B- series
6.4 Twist-field representations from twisted fermions
6.4.1 Fermions and W-algebras
6.4.2 Twist fields and Cartan's normalizers
6.4.3 Twist fields and bosonization for gi(N)
6.4.4 Twist fields and bosonization for so(n)
6.5 Characters for the twisted modules
6.5.1 gi(N) twist fields
6.5.2 so(2N) twist fields, K' =
6.5.3 so(2N) twist fields, K' >
6.5.4 so(2N + 1) twist fields
6.5.5 Character identities
6.5.6 Twist representations and modules of W-algebras
6.6 Characters from lattice algebras constructions
6.6.1 Twisted representation of g
6.6.2 Calculation of characters
6.6.3 Characters from principal specialization of the Weyl-Kac formula177
6.7 Exact conformal blocks of W(so(2N)) twist fields
6.7.1 Global construction
6.7.2 Curve with holomorphic involution
6.7.3 Computation of conformal block
6.7.4 Relation between W(so(2N)) and W(gl(N)) blocks
6.8 Conclusion
6.9 Identities for lattice O-functions
6.9.1 First identity for AN-1 and DN O-functions
6.9.2 Product formula for AN-1 O-functions
6.9.3 An identity for DN and BN O-functions
6.10 Exotic bosonizations
6.10.1 NS x R
6.10.2 R x R
6.10.3 l twisted charged fermions
6.10.4 l charged fermions - standard bosonization
Bibliography