Уравнения Пенлеве и теория представлений Щечкин Антон Игоревич

brief description

Щечкин, Антон Игоревич.Уравнения Пенлеве и теория представлений = Painlev´e equations and representation theory = Painlev´e equations and representation theory : диссертация ... кандидата физико-математических наук : 01.01.03 / Anton Shchechkin; [Место защиты: Нац. исслед. ун-т "Высш. шк. экономики"]. - Москва, 2020. - 118 с. : ил.

Оглавление диссертациикандидат наук Щечкин Антон Игоревич
Contents
Introduction
1 Painleve equations
1.1 Isomonodromic problem and Painleve VI(D^) equation
1.2 Tau-forms of Painleve equations
1.2.1 Painleve equations as non-autonomous Hamiltonian systems
1.2.2 Asymptotic parametrization of Painleve tau functions
1.2.3 Toda-like and Okamoto-like forms of Painleve III(D^1))
1.3 q-deformation: Painleve A^ and A^ equations
(1)'
1.3.1 Sakai's approach to Painleve A7 ; equation
1.3.2 Tau function variables in q-deformed case
1.3.3 Other Toda-like equations
2 Representation theory of vertex algebras
2.1 Verma modules of Virasoro and super Virasoro algebra
2.2 Vir © Vir embedding into U(F © NSR) and corresponding module decompositions
2.3 Conformal blocks of Virasoro and super Virasoro algebras
2.4 Vir © Vir decomposition of chain vectors and vertex operators
2.5 Matrix elements (P',n|$a(1)|P/,n/) in NS sector
3 Correspondence
3.1 Power series representation for the tau function
3.2 Proof for Painleve III equation
3.2.1 Representational explanation of algebraic solution
3.2.2 NS sector
3.2.3 R sector
3.3 Proof for Painleve VI tau function
4 Nekrasov functions and blowup relations
4.1 Nekrasov partition functions
4.1.1 Instanton part of 5d Nekrasov partition function
4.1.2 Classical and 1-loop part of Nekrasov partition functions
4.1.3 Four-dimensional limit
4.2 Blowup relations
4.2.1 Nakajima-Yoshioka blowup relations
4.2.2 Blowup relations on C2/Z2: general approach
4.2.3 Blowup relations on C2/Z2: explicit relations
4.2.4 Application: relations on Nekrasov partition functions, modified by Chern-Simons theory
5 q-deformation of the Isomonodromy/CFT correspondence
5.1 Tau functions of Painleve A^1) and A^1) equations
5.2 Convergence of tau functions
5.3 q-deformation of Painleve III(D^1)) algebraic solution
6 c = —2 tau functions
6.1 Dual formulation of Nakajima-Yoshioka blowup relations
6.1.1 c = —2 tau functions
6.1.2 Painleve equations from c = —2 tau functions
6.1.3 Algebraic solution and q-deformed c = —2 tau functions
6.1.4 TS/ST
6.1.5 Continuous limit and KZ equation
6.2 Folding
6.2.1 Cluster dynamics
6.2.2 Limit
Conclusion
A Some special functions
A.1 q-Pochhammer symbols and related q-special functions
A.2 Multiple gamma functions
A.3 Continuous limit
References