KEK Report 89-22 January 1990 H Proceedings of the Workshop Topology, Field Theory and Superstrings" KEK, Tsukuba, Japan November 6-10, 1989 edited by Makoto KOBAYASHI and Shin'ichi NOJIRI NATIONAL LABORATORY FOR HIGH ENERGY PHYSICS © National Laboratory Tor High Energy Physics, 1990 KEK Reports are available from: Technical Information & Library National Laboratory for High Energy Physics 1-1 Oho, Tsukuba-shi Ibaraki-ken, 305 JAPAN Phone: 0298-64-1171 Telex: 3652-534 (Domestic) (0)3652-534 (International) Fax: 0298-64-4604 Cable: KEKOHO FOREWORD The workshop "Topology, Field Theory and Superstrings" was held at KEK on November 6-10. This workshop was devoted to various recent developments in field theories and string theories, e.g. the deformations of conformal field theories, exactly solvable models, link theory, quantum gravity, topological field theories, superconductivity, etc. More than 50 talks, including several introduc­ tory lectures, were given. Many of 130 participants were young students. We hope that they generate the new currents in physics toward 1990's. This workshop was supported by the Grant-in Aid for scientific research on Priority Areas from the Ministry of Education, Science and Culture. We wish to thank the speakers and participants for their efforts which made this workshop a successful one. Makoto Kobayashi Shin'ichi Nojiri CONTENTS Conservation Laws in a Perturbed k=l SU(2) Wess-Zumino-Witten Model K-I. Kobayashi, T. Uematsu i On the Integrals of Motion away from criticality J. Kodaira, Y. Sasai and H. Sato < g Toda Field Theory R. Sasaki 95 Algebraic Aspects of the Deformation of Conforroal Field Theory M. Fukuma 27 On q-analog Racah-Wigner Algebra, Yang-Baxter Relations and Witten's Quadratic Relation M. Nomura 35 Link Polynomials, Linking number and Exactly Solvable Model T. Deguchi 45 Exact Solution of the Fully Anisotropic Kondo Problem Y. Quano 77 Comments on Super Schwarzian Derivatives S. Matsuda 87 Lie Algebra Cohomology and N=2 Superconformal Field Theory S. Hosono and A. Tsuchiya 109 Parafermions as Z^ orbifold and level 3 SU(3) Kac-Moody Algebra A. Fujitsu 123 c=3d Algebra S. Odake ^33 N = 2 Superconfoimal Symmetry based on N = 1 Snpersymmetric Non-Compact Group Current Algebra S- Nojiri 143 Spontaneous Deformation of Maximally Symmetric Calabi-Yau Manifolds and A Realistic Four-Generation Model T. Matsuoka I53 Torus-Orbifold Equivalence in Compactified String Theories M. Sakamoto }gg Scattering Amplitude of String on Orbifold S. Nima ^79 Operator Formalism on Higher Genus Riemann Surfaces S. Ojima 291 Renormalization Invariant Effective Action of the cr-Model String Theory Y.X. Cheng 207 Symmetry of Cutoff Theory and Symmetry of String Theory K. Sakai 219 Canonical Quantization of Witten's String Field Theory in Mid-point Time For­ malism M. Maeno 229 Non-Polynomial Closed String Field Theory K. Suehiro 241 — ii — Wormhole Solutions in the Skyime Model A. Iwazaki 253 (2+l)-Dimensional Gravity with the Cosmological Constant Y. Fujiwara and J. Soda 261 Linearized Analysis of (2+l)-dimensional Einstein-Maxwell Theory J • Soda 273 BRS Current and Related Anomalies in Two-dimensional Gravity and String Theories H. Suzuki 281 BRST Quantization of Chern-Simons Gauge Theory H. Imai 283 Super Wess-Zumino-Witten Models from Super Chern-Simons Theories N. Sakai and Y. Tanii 293 Topological Quantum Field Theories In Higher Dimensions I. Oda 305 Action Principle for Chiral Bosons K. Harada 323 Path-Integral Quantization of a Particle Coupled with Chern-Simons Gauge Field C. Itoi 33! Effective Actions of ID and 2D Heisenberg Anliferromagnels in CPl Represen­ tation H. Mukaida 341 Nonlinear Sigma Model and Quantum Antiferromagnets H. Yamamoto 351 — 111- "String Amplitudes": What Can We Do about the Divergent Integrals? K. Amano oyy Necessity of Finite Size Terms in WZW model -coherent-state path integral approach- S. Iso, C. Itoi and H. Mukaida ooq Conservation Laws in a Perturbed k=l 1. Introduction and Motivation SU(2) Wess-Zumino-Witten Model* In the last several years, there has been much interest in two-dimensional conformai field theories(CFT's) [1,2] in connection with the problems of string compactifications and also statistical models of critical phenomena. KEN-ICHIRO KOBAYASHI Conformally invariant field theories in 2-dimensions possess infinite dimen­ Lyman Laboratory of Physics, Harvard University sional conformai algebras known as Virasoro algebras which enable us to deter­ Cambridge, MA 02138, USA mine n-point correlation functions. This situation is quite in contrast to the conforrnal field theories in higher dimensions. The essential ingredient for the solvability here, in the 2-dimensional case, is the existence of the infinite number and of the conservation laws, which is more general criteria for the solvable models. Two-dimensional conformai field theories describing the critical points of sta­ tistical systems correspond to the renormalization-group fixed-points in a larger TSUNEO UEMATSU set of 2-dimensional field theories which are generally scale non-invariant. Now it has become increasingly interesting to study 2-dimensional field theories away Department of Physics, College of Liberal Arts and Sciences from critical points or off-critical behavior of conformai field theories [3,4,S,22]. Kyoto University, Kyoto 006, Japan It has also been suggested that integrable lattice models carry an infinite dimen­ sional algebraic structure characteristic to CFT even away fxcm critical points ABSTRACT [61. Since an infinite number of conserved quantities are crucial for solving the We study conservation laws in the SU(2) Wess-Zumino-WiUen (WZW) model theories exactly, as mentioned above, there has recently been numerous works on with the level k = 1 perturbed by a certain relevant operator. The perturbed the conservation laws in the conformai field theories away from criticality which system is a special case of the sine-Gordon theory to the lowest order in the are achieved by applying some relevant perturbations on the original theories perturbation theory, and it turns out that there exist extra conserved currents [7,15,16,17]. In fact, it has been known that there exist higher integrals of motion due to the SU(2) symmetry in the original WZW model. in perturbed conformai field theories such as minimal models of CFT and W- algebras [8,9]. On the other hand, a well-known type of solvable models apart from CFT's • Talk presented by T. Uemitaii at KEK Worbhop on Topology, Field Theory and is soL'ton theory. In ref.[10], Sasaki and Yamanaka investigated the higher inte­ Snperatrlng Theory, Nov.6lh - 10th, 1989. »• Fellow of Nishina Memorial Foundation grals of motion in the quantum sine-Gordon system. Their prescription to get - 1 - -2- conserved quantities is writing down mutually commuting 'Polynomial'functions 2 S = ±Jd zdVdV (2) of the Virasoro generators. Recently Eguchi and Yang [11] have studied the deformation of the Virasoro minimum models perturbed by the (1,3) operator where v{z, z) = \{4(z) + fc)]. which leads to the sine-Gordon theory and clarified the connection between the This system is invariant under the transformation: f —» <p + 2irr. At the result of ref. [10] and those of Zaraolodchikov's [7,8,9]. They also argued that self-dual point r = l/\/2 there exists affine 51/(2) xSC(2) symmetry. The the perturbed CFT's based on the coset construction are described by the Toda SE/(2)xS!7(2) generators in our convention are given in terms of if> and $ as: field theory [23]. In this talk, based on our work [18], we discuss the conservation law in the J+(z)=: e,V5«*> : J+{z) =: e^*" : SU(2) Wess-Zumino-Witten (WZW) model [12,13,14] with the level k = 1 and J-(j)=:e-^*(-"': J-(z) =: e-'^W: (3) therefore with the central charge c = L perturbed by a certain relevant operator. 3 3 J [z) = -Ld4,{z) J {z) = -j=dfc) We will show that for a special value of the /? in the sine-Gordon theory, we get extra integrals of motion in addition to those obtained by Sasaki and Yamanaka where : : means the usual normal-ordered product. We note that they are suitably [10], 2 single-valued under ^(^) —• 0(0) + 2irr at r = \/y/2. The operators J*, J satisfy the following operator algebras: 2. SU(2) WZW Model 1 2J3(u)) + J (*)'-(»)~ 77-^72 2 + Now let us consider the SU(2) WZW model with the level * for which the (1 — tu) (z — w) central charge is given by c = jjj^?. The operator product expansion (OPE) reads (z —iu)* (z-w)' z — w where we have used the following formula: •/'(») , dwJ'(w) WH~^ + ^ (l) ofl (z - w) : e""**') :: e''*W : = (z - iu) : e'«*W+>0«»). (5) (z — w)2 z — w Here we normalize the holomorphic scalar field 0 as: where T[z) and J"(z) (a = 1,2,3) denote the energy-momentum tensor and the <0(sMui)>=-ln(z-iii) (6) SU(2) currents, respectively. The k = 1 i.e. c = 1 case is realized by a free boson <ji compactified on a and we take a similar normalization for 0. circle of radius r = 4- [19,20]. The action for this system is given by - 3- -4 - 3. Perturbation and Sine-Gordon Equation and we obtain dA = \(AQ)-i + XdB (12) We now add to the action S of this theory a relevant perturbation term as t t follows: where S - X^ / $(vi,w)d~w$(v>,w)d2 . (7) B = E 7Ti»l'(-1)*~lfl'*'",<A*)-»(*.*) • £J (*-l)! In order to study higher-order integrals of motion we shall follow the argument by Zamolodchikov based on perturbation theory(7,8,9]. Correlation functions in Therefore, if (A$)_i is a total derivative of the form OX' then A is a conserved the presence of perturbation are given by charge desity(current): x X 2 d, (8) 8tA = \6,X , (13) < >»= TX £ /•••/ -j < *M •••*(»«) >° <<« ••• y* withX = X' + B, since its integral gives a conserved charge (integral of motion): where Zx = £» „ J-~f%< *(»i) • • • *(y„) >o d'lft • • • dhM.