Диссертация (Isomonodromic deformations and quantum field theory)
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National Research University Higher School of EconomicsFaculty of MathematicsPavlo GavrylenkoIsomonodromic deformations and quantum field theoryPhD thesisSupervisorAndrei MarshakovDr.Sc., professorMoscow – 2018Contents1 Introduction1.1 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 τ -functions and WN conformal blocks2.1 Introduction . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .2.2 Isomonodromic deformations and moduli spaces of flat connections2.2.1 Schlesinger system . . . . . . . . . . . . . . . . . . . . . . .2.2.2 Moduli spaces of flat connections . . . . . . . . . . . . . . .2.2.3 Pants decomposition of Mg4 . . . . . .
. . . . . . . . . . . .2.2.4 Pants decomposition for Mgn . . . . . . . . . . . . . . . . . .2.3 Iterative solution of the Schlesinger system . . . . . . . . . . . . . .2.3.1 sl2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.2 sl3 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 deformations3.1 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 . . . . . . .
. . . . . . . . . . . .iii............................................111789111113.............1515171819202122262831313234..................37373838414344454546475052525658606061CONTENTS3.6Conclusion . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .624 Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions654.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .654.1.1 Motivation and some results . . . . . . . . . . . . . . . . . . .654.1.2 Notation . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .704.1.3 Outline of the chapter . . . . . . . . . . . . . . . . . . . . . .704.1.4 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . .714.2 Tau functions as Fredholm determinants . . . . . . . . . . . . . . . .744.2.1 Riemann-Hilbert setup . . .
. . . . . . . . . . . . . . . . . . .744.2.2 Auxiliary 3-point RHPs . . . . . . . . . . . . . . . . . . . . .774.2.3 Plemelj operators . . . . . . . . . . . . . . . . . . . . . . . . .794.2.4 Tau function . . . . . . . . . . . . . . . . . . .
. . . . . . . . .844.2.5 Example: 4-point tau function . . . . . . . . . . . . . . . . . .874.3 Fourier basis and combinatorics . . . . . . . . . . . . . . . . . . . . .914.3.1 Structure of matrix elements . . . . . . . . . . . . . . . . . . .914.3.2 Combinatorics of determinant expansion . . . . .
. . . . . . .944.4 Rank two case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .984.4.1 Gauss and Cauchy in rank 2 . . . . . . . . . . . . . . . . . . .994.4.2 Hypergeometric kernel . . . . . . . . . . . . . . . . . . . . . . 1064.5 Relation to Nekrasov functions . . . .
. . . . . . . . . . . . . . . . . 1105 Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations1215.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.2 Twist fields and branched covers . . . . . . . . . . . . . . . . . . . . 1235.2.1 Definition . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 1235.2.2 Correlators with the current . . . . . . . . . . . . . . . . . . . 1265.2.3 Stress-tensor and projective connection . . . . . . . . . . . . . 1285.3 W-charges for the twist fields . . . . . . . . . . . . . . . . . . . . . . 1295.3.1 Conformal dimensions for quasi-permutation operators . . . .
1295.3.2 Quasipermutation matrices . . . . . . . . . . . . . . . . . . . . 1305.3.3 W3 current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.3.4 Higher W-currents . . . . . . . . . . . . . . . . . . . . . . . . 1325.4 Conformal blocks and τ -functions .
. . . . . . . . . . . . . . . . . . . 1345.4.1 Seiberg-Witten integrable system . . . . . . . . . . . . . . . . 1345.4.2 Quadratic form of r-charges . . . . . . . . . . . . . . . . . . . 1355.4.3 Bergman τ -function . . . . . . . . . . . . . . . . . . . . . . . . 1385.5 Isomonodromic τ -function . . . . . . . . .
. . . . . . . . . . . . . . . 1405.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.8 Diagram technique . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 1455.9 W4 (z) and the primary field . . . . . . . . . . . . . . . . . . . . . . . 1475.10 Degenerate period matrix . . . . . . . . . . . . . . . . . . . . . . . . 148ivCONTENTS6 Twist-field representations of W-algebras, exact conformal blocksand character identities1516.1 Abstract .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.3 W-algebras and KM algebras at level one . . . . . . . . . . . . . . . 1536.3.1 Boson-fermion construction for GL(N) . . . . . . . . . . . . . 1536.3.2 Real fermions for D- and B- series . . . . . . . . . . . . . . . 1556.4 Twist-field representations from twisted fermions . . . . . .
. . . . . 1576.4.1 Fermions and W-algebras . . . . . . . . . . . . . . . . . . . . 1576.4.2 Twist fields and Cartan’s normalizers . . . . . . . . . . . . . . 1586.4.3 Twist fields and bosonization for gl(N ) . . . . . . . . . . . . . 1616.4.4 Twist fields and bosonization for so(n) . . . . . . . . . .
. . . 1636.5 Characters for the twisted modules . . . . . . . . . . . . . . . . . . . 1646.5.1 gl(N ) twist fields . . . . . . . . . . . . . . . . . . . . . . . . . 1656.5.2 so(2N ) twist fields, K 0 = 0 . . . . . . . . . . . . . . . . . . . . 1666.5.3 so(2N ) twist fields, K 0 > 0 . . . . . . . . . . . . . . . . . .
. . 1666.5.4 so(2N + 1) twist fields . . . . . . . . . . . . . . . . . . . . . . 1676.5.5 Character identities . . . . . . . . . . . . . . . . . . . . . . . 1686.5.6 Twist representations and modules of W-algebras . . . . . . . 1716.6 Characters from lattice algebras constructions . . . . . . . . . . . .
. 1736.6.1 Twisted representation of bg1 . . . . . . . . . . . . . . . . . . . 1736.6.2 Calculation of characters . . . . . . . . . . . . . . . . . . . . . 1756.6.3 Characters from principal specialization of the Weyl-Kac formula1776.7 Exact conformal blocks of W (so(2N )) twist fields . . . . . . . . .
. . 1826.7.1 Global construction . . . . . . . . . . . . . . . . . . . . . . . . 1826.7.2 Curve with holomorphic involution . . . . . . . . . . . . . . . 1846.7.3 Computation of conformal block . . . . . . . . . . . . . . . . . 1866.7.4 Relation between W (so(2N )) and W (gl(N )) blocks . . . . . . 1886.8 Conclusion . . . . . . . . .
