Table Of ContentResearch Reports in Physics
Research Reports in Physics
Nuclear Structure of the Zirconium Region
Editors: J. Eberth, R. A Meyer, and K. Sistemich
Ecodynamics Contributions to Theoretical Ecology
Editors: W Wolff, C.-J. Soeder, and F. R. Drepper
Nonlinear Waves 1 Dynamics and Evolution
Editors: A V. Gaponov-Grekhov, M. I. Rabinovich,
and J. Engelbrecht
Nonlinear Waves 2 Dynamics and Evolution
Editors: A V. Gaponov-Grekhov, M.1. Rabinovich,
and J. Engelbrecht
Nuclear Astrophysics
Editors: M. Lozano, M. I. Gallardo, and J. M. Arias
Optimized LCAO Method and the Electronic Structure
of Extended Systems By H. Eschrig
Nonlinear Waves in Active Media
Editor: J. Engelbrecht
Problems of Modern Quantum Field Theory
Editors: AA Belavin, AU. Klimyk, and AB. Zamolodchikov
Fluctuational Superconductivity of Magnetic Systems
By MA Savchenko and AV. Stefanovich
A.A. Belavin
A.U. Klimyk A.B. Zamolodchikov (Eds.)
Problems of
Modern Quantum Field Theory
Invited Lectures of the Spring School
held in Alushta USSR, April 24 - May 5,1989
Springer-Verlag Berlin Heidelberg New York
London Paris Tokyo Hong Kong
Professor Aleksandr A. Belavin
Professor Aleksandr B. Zamolodchikov
Landau Institute for Theoretical Physics, ul. Kosygina 2,
SU-117334 Moscow, USSR
Professor Anatolii U. Klimyk
Institute for Theoretical Physics, Ukrainian SSR Academy of Sciences,
ul. Metrologicheskaya 14b, SU-252130 Kiev 130, USSR
ISBN-13: 978-3-540-51833-4 e-ISBN-13: 978-3-642-84000-5
001: 10.1007/978-3-642-84000-5
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© Springer-Verlag Berlin Heidelberg 1989
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a
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2157/3150-543210 - Printed on acid-free paper
Preface
This volume is the compilation of invited lectures presented at the Spring
School "Problems of Modern Quantum Field Theory" held in Alushta (USSR)
April 24-May 5 1989, organized by the Institute for Theoretical Physics (Kiev)
and Landau Institute for Theoretical Physics (Moscow). Approximately one
hundred physicists and mathematicians attended lectures on aspects of mod
ern quantum field theory: Conformal Field Theory, Geometrical Quantization,
Quantum Groups and Knizhnik-Zamolodchikov Equations, Non-Archimedian
Strings, Calculations on Riemannian Surfaces. A number of experts active in
research in these areas were present and they shared their ideas in both formal
lectures and informal conversations.
V. Drinfeld discusses the relation between quasi-Hopf algebras, conformal
field theory, and knot invariants. The author sketches a new proof of Konno's
theorem on the equivalence of the braid group representations corresponding
to R-matrices and the Knizhnik-Zamolodchikov equation.
The main ideas of quantum analogs of simple Lie superalgebras and their
dual objects - algebras of functions on the quantum supergroup - are introduced
in the paper by P.P. Kulish. He proposes the universal R-matrix for simplest
superalgebra osp(2/1) and discusses the elements of a representation theory.
In the paper by A. Alekseev and S. Shatashvili the correspondence between
geometrical quantization and conformal field theory is established. It allows one
to develop a Lagrange approach to two-dimensional conformal field theory. The
authors also discuss the relation to finite R-matrices.
A. Marshakov considers the problems of bosonization of conformal field
theories. He focuses attention on a "primary" class of two-dimensional confor
mal field theories - Wess-Zumino-Witten (WZW) models. Their bosonization
is based on the representation of the Kac-Moody current algebra in terms of
some ,sf-systems (bosonic first-order systems) and scalar fields, taking on val
ues in some tori. The presence of the ,sf-systems in the proposed scheme is
the origin of the nontrivial formulas for multiloop correlation functions. The
author also deals with the bosonization of different rational conformal theories
by means of the coset method from WZW models.
L. Chekhov, A. Mironov, and A. Zabrodin have devoted their contribution
to the higher genera calculations in the theory of p-adic string world sheets as
=
a coset space F TIG, where T is the Bruhat-Tits tree for the p-adic linear
v
group GL(2,Qp) and G PGL(2,Qp) is some Schottky group. The boundary
of this world sheet corresponds to a p-adic Mamford curve of finite genus.
The string dynamics are governed by the local gaussian action on the coset
space F. The tachyon amplitudes expressed in terms of a p-adic 8-function
are proposed for the Mamford curve of arbitrary genus. These are compared
with the corresponding usual archimedian amplitudes. The sum over moduli
space of the algebraic curve is proposed to be expressed in arithmetic surface
terms.
The functional integral over fields on a sphere with discs cut out as a
function of boundary conditions is determined up to a multiplicative constant
expressed in terms of statsums on spheres by sewing holes with discs. A. Losev
calculates multiloop amplitudes obtained by cutting and sewing methods in
Schottky parametrization which are then experessed in terms of 8-functions.
P. Grinevich and A. Orlov present in their paper the explicit form of the
Wilson construction, which allows one to interpret the determinant of the {)
operator as a Wilson r-function defined on the fiber of j-differentials. They
construct the higher Kadomtsev-Petviashvili (KP) equations corresponding to
the variation of complex structure of Riemannian surfaces.
In the paper by P.I. Holod and S. Pakuliak: a new supergeneralization of the
Kadomtsev-Petviashvili equation is proposed. The bi-Hamiltonian Korteweg-de
Vries (KdV) superequation follows from the KP superequation after reduction.
The finite-gap integration of this KdV superequation leads to super-Riemannian
hyperelliptic surfaces. Some features of finite-gap solutions of supersymmetric
and bi-Hamiltonian KdV superequations are discussed in this paper.
A. Zabrodin and A. Mironov discuss the finite size effects in conformal
theories. The long wavelength asymptotics of the vacuum average of some
nonlocal operators are determined by means of these effects.
The solution of the two-dimensional Ising model on a strip with arbitrary
magnetic fields applied to the boundaries is obtained in the paper by A.I.
Bugrij and V. Shadura. The authors calculate the finite size corrections in two
dimensional conformal field theory with a central charge c = 1/2.
In the paper by D.V. Volkov, D.P. Sorokin, and V.I. Tkach the problems of
a field-theoretic description of the objects with anomaly statistics and fractional
spins are discussed. The authors propose a description of such objects with
spins 1/4 and 3/4 in dimensions d =( 2 + 1), (3 + 1) space-time on the basis of
the infinite representations of SL(2,JR) and SL(2,O::) groups with weights 1/4
and 3/4.
Yu.A. Sitenko reveals the supersymmetric structure of the equation of two
dimensional electron motion in an external static magnetic field of arbitrary
configuration. This paper considers the zero modes of the Dirac operator on
compact and noncom pact surfaces. The author discusses the equivalence of the
square integrability condition for zero modes on a noncompact surface to the
nonlocal boundary condition.
VI
Although all lectures presented at the school could not be included, we
hope that this volume communicates the wealth of ideas that exist in modern
quantum field theories.
The editors would like to express their gratitude to S. Pakuliak and V.
Shadura, whose enormous organizational efforts enabled the high level of
discussion at the school. Vladimir Gezha of the Central Committee of the
Ukrainian LKSM who assisted in organizing the school in Alushta at the guest
house "Yunost", and the cooperative "Effect" whose financial support resolved
many problems, including the preparation of this volume, also deserve heart
felt gratitude. Finally, we would like to thank Elena Kuprievich who kindly
consented to prepare these materials.
Moscow, Kiev A.A. Belavin
August 1989 A.U. Klimyk
A.B. Zamolodchikov
VII
Contents
Quasi-Hopf Algebras and Knizhnik-Zamolodchikov Equations
By V.G. Drinfeld ...................................... 1
Quantum Lie Superalgebras and Supergroups
By P.P. Kulish ........................................ 14
From Geometric Quantization to Conformal Field Theory
By A. Alekseev and S. Shatashvili .......................... 22
Bosonization of Wess-Zumino-Witten Models and Related Conformal
Theories
By A.V. Marshakov .................................... 43
The Cutting and Sewing Method in String Theory
By A. Losev ......................................... 58
P-Adic String World Sheets: Higher Genera
By L.O. Chekhov, A.D. Mironov, and A.V. Zabrodin ............. 76
a
Vrrasoro Action on Riemann Surfaces, Grassmannians, det J and Segal-
Wilson r-Function
By P.G. Grinevich and A.Yu. Orlov ......................... 86
On the Superextension of the Kadomtsev-Petviashvili Equation and Finite
Gap Solutions of Korteweg-de Vries Superequations
By P.I. Holod and S.Z. Pakuliak ............................ 107
Finite Size Effects in Conformal Field Theories and Non-local Operators
in One-Dimensional Quantum Systems
By A.D. Mironov and A. V. Zabrodin ........................ 117
The Solution of the Two-Dimensional Ising Model with Magnetic Fields
Applied to the Boundaries
By A.I. Bugrij and V.N. Shadura ........................... 122
On the Relativistic Field Theories with Fractional Statistics and Spin in
D=(2+ 1), (3+ 1)
By D.V. Volkov, D.P. Sorokin, and V.I. Tkach .................. 132
IX
Electron on a Surface in an External Magnetic Field: Hidden
Supersymmetry, Zero Modes and Boundary Conditions
By Yu.A. Sitenko ...................................... 146
Index of Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 157
x
Quasi-Hopf Algebras and Knizhnik-Zamolodchikov Equations
V.G. Drinfeld
Physico-Technical Institute of Low Temperatures (FI1NT),
SU-310164 Kharkov, USSR
This paper is a brief exposition of [6]. In §1 we remind the
notion of quasitriangular Hopf algebra which is an abstract version
of the notion of R-matrix. In §2 the notion of quasitriangular quasi
Hopf algebra is introduced (coassociativity is replaced by a weaker
axiom). In §3 we construct a class of quasitriangular quasi-Hopf
algebras using the differential equations for n-point functions in
the wzw theory introduced by v.G.Knizhnik and A.B.Zamolodchikov.
Theorem 1 asserts that wi thin perturbation theory with respect to
Planck's constant essentially all quasitriangular quazi-Hopf algebras
belong to this class. A natural proof of Kohno' s theorem on the
equivalence of two kinds of braid group representations is given. In
§4 we discuss applications to knot invariants. In §5 the classical
limit of various quantum notions is discussed.
1. Hopf algebras.
We remind that a Hopf algebra is an associative algebra A with a
homomorphism ~:A~ A8A called comultiplication. These data must
satisfy certain conditions. The most important of them is the
coassociativity axiom (id8~) o~=(~8id) o~ ( both sides of this equality
are mappings A~ A8A8A ). There are also axioms concerning the unit,
the counit and the antipode.
One of the reasons for the notion of Hopf algebra being useful
is the possibility to define the tensor product of representations.
If A is an arbitrary associative algebra and representations of A in
vector spaces V, and V 2 are given then A8A acts on V,8 V 2 but, in
general, there is no natural action of A on V,8 V2 • Now suppose that
A is a Hopf algebra. Then we can use ~:A ~ A8A to define a
representation of A in V 8 V • The coassociativity axiom means that
, 2
the tensor product of representations of A is obviously, associative.
The other Hopf algebra axiom can be used to define the unit
representation and contragredient representation.
Research Reports in Physics Problems or Modern Quantum Field Theory
Editors: A.A. Belavin· A.V. Klimyk. A.B. Zamolodchikov
© Spcinger-Verlag Berlin, Heidelberg 1989
