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Proceedings of the Workshop on New Aspects of Quantum Field Theories INS-T—513 JP9212001 INSTITUTE FOR NUCLEAR STUDY INS - T - 5i3| UNIVERSITY OF TOKYO September 1992 Tanashi, Tokyo 188 Japan Proceedings of the Workshop on New Aspects of Quantum Field Theories Institute for Nuclear Study (INS) University of Tokyo February 17-19, 1992 Edited by S. Nojiri PREFACE The workshop on New Aspect of Quantum Field Theory was held at INS on Feb. 17-19, 1992. This workshop was devoted to recent new developments in quantum field theories. More than 100 people, most of whom were young students, participated in the workshop. About 30 talks about latest results were given. Shin'ichi Nojiri Organizer: H. Ikemori (Nagoya Univ.), A. Kato (Tokyo Univ.), H. Kunitomo (Osaka Univ.), K. Nakanishi (Kyoto Univ.), S. Nojiri (INS). T. Mizoguchi (YITP) CONTENTS W-Geometries 1 Jean-Loup GERVAIS and Yutaka MATSUO Fusion Rules for the Vertex Operators of sl(2) with Functional Level 6 Hidetoshi AWATA and Yasuhiko YAMADA WZW Model on Higher Genus Riemann Surfaces 16 Hitoshi KONNO Topological Gauged WZW Model and 2D Gravity 27 Toshio NAKATSU and Yuji SUGAWARA Sklyanin Algebra and Solvable Lattice Models 29 Yas-Hiro QUANO The Geometry of Poisson Manifolds 45 Patrick CREHAN Physical States of the Massless QCD2 with g = 00 54 Hisamitsu MUKAIDA Dynamical Chiral Symmetry Breaking in QCD 65 Hirofumi YAMADA Strings from Black Hole 77 Ichiro ODA An Exact Cosmic String Solution as the Dislocation 85 Takehiro T. FUJISHIRO, Mitsuo J. HAYASHI and Shoji TAKESHITA Vacuum Polarization arround Black Holes with Quantum Hair 95 Kivoshi SHIRAISHI The Ashtekar's Variables for TV = 2 Supergravity 102 Hiroshi KVNITOMO and Takashi SANO Disk Amplitude in Two-Dimensional Open String Theories 107 Yoshiaki TANII and Shun-ichi YAMAGUCHI Coupling of Tachyons and Discrete States in c = 1 2-D Gravity 118 Yoichiw MATSUMURA, Norisuke SAKAI and Yoshiaki TANII Schwinger-Dyson Loop Equations as the Wi+oo-like Constraints for Hermitian Multi-Matrix Chain Model at Finite N 128 Yi-Xin CHENG Quantum Analysis of Jackiw and Teitelboim's Model for 1 + 1 D Gravity and Topological Gauge Theory 145 Haruhiko TERAO Stochastic Second Quantization of String 158 Masahiro Kanenaga. Kazuhiko NAKADA, Mikio NOMIKI and Ichiro OHBA Reexamination of Renormalizability Problem in Quantum Gravity 167 Shoichi ICHINOSE Non-perturbative Aspect of N = ^ Superstring in Zero Dimension 184 Shin'ichi NOJIRI W-GEOMETRIES Jean-Loup GERVAIS1 and Yutaka MATSUO2 Abstract; It is shown that, classically, the W-algebras are directly related to the extrinsic geometry of the embedding of two-dimensional manifolds with chiral parametrisation (W-surfaces) into higher dimensional Kahler manifolds. We study the local and the global geometries of such embeddings, and connect them to Toda equations. In this report, we briefly summarize geometrical aspects of the W-algebras [1] based on our works [2]. W-symmetries exhibit strikingly novel features. First, they are basically non-linear algebras. W-algebras generalise the dif- feomorphisms of the circle by including derivatives of degree higher than one. Going beyond linear approximation (tangent space) is a highly non-trivial step. Taking higher order derivatives changes the shape of the world-sheet in the target-space, thus W-geometry should be related to the extrinsic geometry of the embedding. In the present report, we show that 2D surfaces E with chiral parametri- sations, embedded into CPn (and Cn), are the geometrical framework where the features just recalled are naturally realised, at the classical level. The extrinsic geometry of the surface is neatly described by string version of the classical Frenet-Serret formula. Toda fields appear naturally as gauge fields of W-symmetry and we explain explicitly how local and global aspects of W-geometry are described in terms of them. Local W-geometry; Our basic strategy is to study embeddings of two- dimensional surfaces E given by equations of the form XA = fA(z),A = l,---,n, X* = fA(z),A = l,---,n, (1) n in a 2n-dimensional target manifold C with a flat metric, GAg = GBA = &AB- One may either work with Minkowski coordinates where z — IT + r, and z = a — T, are light-cone coordinates, or perform a Wick rotation and work with the complex z variable (isothermal coordinates). The first description makes it clear that fA and f* are independent functions, so that we are not dealing with a single complex curve as is often done in this context. In the present section, we shall be concerned with regular points of E where Taylor expansion gives a local frame, so that a discussion of the Frenet type applies. 'Laboratoire de Physique Theorique, Ecole Normale Superieure, 24 rue Lhomond, 75231 Paris CEDEX 05, France 2Dept. of Physics, Univ. of Tokyo, Bunkyo-ku, Hongo 7-3-1, Tokyo, Japan i In this geometric setting, W-transformations are defined to be of the form SwfA = T^wfA where T>w is an- lth order differential operators in z. As usual, the first order derivative corresponds to the Virasoro algebra and operators of the form et(z)dl (I = 0,2, • • •, n — 1) correspond to generators, W\,Wz, • • •, Wn respectively. The commutation relation between n—lth order differential operators can not be expressed as n — lth order operator. Hence we do not have Lie algebra of ordinary sense. Derivatives higher than n— lth order, however, may be eliminated using the fact that the functions fA are automatically solutions of the following differential equation f1 ••• r f An important point is that coupling constants depend on the embedding itself. This is a well-known feature of W-algebra. W-transformations in general mix the tangent and the normals at a given point of E. Thus they move the surface and the distinction between intrinsic and extrinsic geometry is not W-invariant. Our first result is that Toda fields naturally arise from the Gauss-Codazzi equations of this embedding. These equations are integrability conditions for derivatives of the tangent and of the normals to the surface. The latter are introduced by extending Frenet-Serre formula as follows. Define (3) 9tl A,= (4) 9i\ d and 8 are short hands for d/dz and d/dz respectively. ^ stands for (d)x. Vectors in C1 are denoted by boldface letters. It is straightforward to check that a set of In orthonormal vectors is given by (upper indices in between parenthesis denote derivatives) (5) 9i^zi 5oo-l 9\\ 9ia ea = 0=1,- (6) 9aa-l where determinants are to be computed for each component of the last line. A B A B Indeed, if we denote by (X, Y) the inner product 6AB(X Y + Y X ) in M, we have (ea,e6) = (ea,e6) = 0, (ea,ei,) = 6af>. et and §! are tangents to the surface, while the other vectors are clearly normals. The string version of the Frenet-Serret equation is described in terms of ea's and Toda fields fc, dea - w*aeb, dea = U;aeb, (7) 1 where <j>0 = 0 and other components of w are zero. Remarkably, one sees that the right member is just the Toda Lax pair[4]. Toda equations are equiv- alent to the zero-curvature condition on w. In the language of Riemannian geometry[5], w^ and w^ o > 1 give the second and third fundamental forms respectively. ea satisfy similar equations. Geometrically, since the ea and ea are 2n-beins, W-transformations appear as particular transformations of the local Lorentz group, which leave the form of the Toda Lax pair invariant. This allows us to interprete Toda fields as the gauge fields of the W-symmetry. Going to the second derivatives, one deduces from the above that 2 dd(f>a + e *«-v«+i-<t>a-i _ Q^ (a = 1, • • • ,ra — 1), (9) d8<j>n = 0, (10) which are the Gauss-Codazzi equations[5] for the embedding considered3. The first dequations are j4n_!-type Toda equation. The addtional field <f>n satisfies free field equation. It implies that we have chiral facorization of An = U(z)U(z). In order to get An_x Toda equation exactly, we need to change our target space Cn by a projective space CPn~1. We describe CPn~l by homogeneous coordinates [X1, • • •, Xn] with identification, In terms of the embedding functions, it is described by U(l) gauge invariance, 3Ideas, that are somewhat related to ours, have already been put forward in ref.[6] It is shown in [2] that this gauge symmetry can be used to remove the <j>n degree of freedom through identification, p(z) = (U(z))~l/n. Global W-geometry; The Grassmannian manifold Gn+k,k is the set of k) x k matrices W with the equivalence relation W ~ aW, where a is a k x k matrix. A chiral embedding in Gn+kik is thus defined by /u(Z) ... r+«-\z) W(z) = (12) P'k{z) ••• fn+k-"(z) T Following Ref.[7], we introduce the Kahler potential Kk = ln(det WW ), that is, .(13) The difference is that, in our case, / will not the complex conjugate of /. From the embedding E —> CPn, we can canonically construct the kth associated fc+1 n embedding E -» Grn+1<k+1 <-• P(A C ). We let /'•' = &-Vf'(z) in Eq.12, for s = 1, • • -, k + 1, I = 0, • • •, n. The induced metric on the corresponding Riemann surface is simply g^ = dd\nrk+l (14) so that the Toda field appears naturally. The infinitesimal Plucker formula is derived by computing the curvature (15) and this gives "•zl — ~9zl + Z9zt — 9zi The global Plucker formula contains the k-th instanton number dk = ± f dzdzg?}, (17) and follows from Gauss-Bonnet theorem for g\i\ ± f (18) 2TT JE A 01, is defined as a sum of the k-th ramification indices.
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