Problems of Modern Quantum Field Theory: Invited Lectures of the Spring School held in Alushta USSR, April 24 – May 5, 1989 PDF
Preview Problems of Modern Quantum Field Theory: Invited Lectures of the Spring School held in Alushta USSR, April 24 – May 5, 1989
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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 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is con cerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad-casting, re production on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9,1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1989 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 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
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