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Geometrie Aspects of Quantum Field Theory Graeme Segal Mathematical Institute, 24-29 St. Giles, Oxford OX1 3LB, England Quantumfieldtheory has been the basic tool of particle physics for more than half a century, but unlike earlier such tools it has not been accompanied by a satisfying mathematical theory. Recently this has begun to change. One reason is that the ideas of quantum field theory have turned out to shed light on purely mathematical questions. These applications are my subject today. So far, nevertheless, the field theory has played either a heuristic or an explanatory role in the mathematics, and the actual theorems can, and often must, be proved by other means. I hope that this will be less true as the mathematics of field theory becomes better developed. Meanwhile I shall just indicate some areas where field theory and geometry have come together, trying to illustrate the point of view rather than formulate theorems. For the most part I shall be summarizing other people's work, predominantly Witten's. §1. The Framework In his address to the Berkeley ICM Witten described d + 1 dimensional quantum field theory as follows. One considers "fields" defined on some class of oriented rf -h 1 dimensional manifolds M. A "field" might mean a map from M to some auxiliary manifold X, or a section of some natural fibre bundle on M, or even an equivalence class of such sections. In any case one has a space F(M) of fields for each compact manifold M with boundary. Afield/ e F(M) has a boundary value f\dM which belongs to some space F0(dM) of fields on the boundary. We also suppose given an "action" functional S : F(M) -> R, defined uniformly for all M. Then field theory is the study of the functions XPM on F0(dM) of the form »V(/o)=[ e-™af, (1.1) JF(Mi/o) where F(M;f0) = {fe F(M) \f\dM = f0}. More generally, if the boundary dM = £0 II £t consists of an incoming part £0 and an outgoing part Z\ we are interested in operators WM : HEQ -• HSi9 where HZi is a space of functions on F0(Lt)9 on the form Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990 © The Mathematical Society of Japan, 1991 1388 Graeme Segal KM{fo,fi)+ifoWo, (**V)(/i) = F0(£0) where e~™®f. ^MC/OJ/I) = (1.2) F(M; fo.fi) (A boundary component is 'outgoing' or 'incoming' according as its orientation agrees or not with that of M; and E0 denotes E0 with reversed orientation.) The preceding formulae are only schematic, and so far it has proved impossible to develop an integration theory of the type needed. But let us at least try to abstract the essential structure. It comprises (i) a vector space HE for each closed oriented d-dimensional manifold with whatever structure is appropriate; (ii) a bilinear pairing H^ x HE-* <C; (iii) an element *FM e HdM for each d + 1 dimensional manifold M with appropriate structure. The most obvious properties these data should have are (a) HziUz2 = HLi®Hz2 and ^MiUMj = y M i ® ^M 2 i (in particular Hz = (C when E = 0 , and so WM e <C when M is closed.) (b) if two components Ex and E2 of the boundary of M are sewn together by an orientation-reversing diffeomorphism to form a new manifold M such that dM = E1UZ2UdM then the map HdM-+HdM defined by the bilinear pairing takes WM to ÎP^. In particular, when dM = 0 and WM is regarded as an operator HE^ -> HSj9 property (b) asserts that t r a c e d ) =¥^e<C. I do not know how far an axiomatization of this kind is appropriate or helpful in traditional quantumfieldtheory, but with some especially simple kinds of theory it works well and is a useful tool in geometry, rather like a new kind of cohomology theory. I shall mention some limitations of the framework in §§ 5 and 6 below. A feature of each of the examples I shall describe is that either the phase space isfinitedimensional because of the presence of a large group of gauge symmetries, or else the path integral (1.1) reduces to a finite dimensional integral because the integrand is an exact differential form outside afinitedimensional submanifold of the space of fields (i.e. "the stationary phase calculation is exact"). One might take this to mean that genuine quantum field theory is not involved. A more optimistic moral, however, is that one can sometimes best studyfinitedimensional problems by the infinite dimensional methods offieldtheory. Geometrie Aspects of Quantum Field Theory 1389 § 2. Index Theory and the Elliptic Genus A particle moving in a Riemannian manifold X affords the simplest example of the path integral idea, and can be regarded as a 0 + 1 dimensional field theory. The action for a path y : [0, T] -• X is S(y) = i J J ||y'(r)ll2 dt. For a point P the vector space HP is L2(X), and to the 1-manifold [0, T] is associated the heat operator e~TA in Hp, where A is the Laplacian of X. The formula (1.2) is then the usual path integral representation of the heat kernel; and if we replace [0, T] by a circle ST of length T we have a formula for trace(e _7M ) as an integral over the loop space ££TX = Map(S r ; X). This path integral does not reduce to a finite dimensional integral. The position is different and more relevant if we replace the action S : 3?TX -• IR with the inhomogeneous differential form S = S + co, where œ is the 2-form on 3?TX which to two deformations <!;, q of a loop y assigns the number T co(y; & ri) = <«*), ij'(t)> A. (Here rç'(t) is the covariant derivative.) (*) Witten observed (see [2], [15]) that the action S corresponds to the 0 + 1 dimensional field theory for which HP is the mod 2 graded space of L 2 spinor fields on X, while the operator associated to [0, T] is the spinorial heat operator e~Tâ. The graded trace (or "supertrace") tr(e~TA) is now independent of T, and is the index of the Dirac operator on X. This is a topological invariant of X called its/-genus. On the other hand the top degree component of the differential form e~s on 1£TX is exact outside the finite dimensional manifold of point loops, so by Stokes's theorem the path integral can be reduced to an integral over X (identified with the point loops). The outcome is the Atiyah-Singer formula for the index of the Dirac operator. The elaboration of this idea was described by Bismut [8] at the Berkeley ICM. So far we have been dealing with well-known material. But we can go on to consider a 1 + 1 dimensional theory whose action is a differential form on the space F(E) of maps from a surface E to X. Then the vector space Hs associated to a circle S will be the space of L2 spinors on the loop space SâX. When E is a torus the path integral over F(E) is called the elliptic genus eE(X) oiX. (In fact there are a number of variants, applying in slightly different situations.) The 0 -h 1 dimensional result that the supertrace of e~TA was independent of T has the analogue that e£(X) depends only on the conformai structure of E, i.e. for each X it is a modular function on the upper ^-plane. The elliptic genus can be interpreted formally as the equivariant index of a version of the Dirac operator on the manifold X. As with the /4-genus the path integral defining eL(X) collapses to an integral over X, and this expresses it in terms of the characteristic numbers of X already familiar in algebraic topology. Nevertheless the elliptic genus has striking and unexpected properties, especially in connection with the topology of circle actions, and it stimulated the discovery of elliptic cohomology, a new theory whose true nature remains obscure. (An account of this subject can be found in [22]. Cf. also [1], [26]). * More accurately, we replace e S(y)@y by e s. The integral of an inhomogeneous form means the integral of its component of top degree. 1390 Graeme Segal § 3. Topological Field Theories Afieldtheory is topological if it is defined for smooth manifolds with no additional structure (apart from a question of fixing projective multipliers which I shall suppress in this talk.) The vector spaces HE must then be finite dimensional. A discussion of the formal properties can be found in [4]. (a) 1 + 1 dimensional Theories. These are completely described by giving a commutative ring A with a F together with a linear map 0 : A -> C such that the bilinear form (a, b)\-*9(ab) is non-degenerate. In fact A = Hsi, and the product A (g) A -> A is WM, where M is a disc with two holes. (b) 2 + 1 dimensional Theories. These are by far the most studied, and the structure is much richer: it appears to be roughly equivalent to a quantum group. A theory gives us an invariant for each closed 3-manifold, and a representation of the mapping class group of each closed surface. If we choose an element Ç e i/ s i xS i we get an invariant fa(K9 M) e C for each knot K in a 3-manifold M by defining ^(X,M) = <^,'PM_t/>, where U is a tubular neighbourhood of K. In Reshetikhin's, Turaev's, and Feigin's talks at this Congress we heard how the same output arises from a quantum group. The relation between the two approaches does not seem completely understood, but I shall say a little more about it in § 4 below. 2 + 1 dimensional theories are important because there is a supply of natural examples which lead to the knot invariants of Vaughan Jones and others. There is a theory for each compact Lie group G and choice of "level" k. The level is an element k e H4(BG; Z), i.e. an integer if G is simple and simply connected. Regardingfcas a characteristic class for G-bundles there corresponds to it a secondary Chern-Simons characteristic class Sk with values in R/2TCZ which is defined on the space F(M) of isomorphism classes of G-bundles with connection on a 3-manifold M. This, or rather iSk, is the action defining the theory [33]. But the theory can be constructed without mentioning path-integrals in the following way. The vector space HE associated to a surface E is the "quantization" of the symplectic manifold ME of flat G-bundles on E. (This is the symplectic quotient [5] of the space of all connections on E by the action of the gauge group.) The symplectic structure of JiE depends on the level: its class is the image offcunder the transgression HAr(BG) -> H2(JtE). To obtain a definite quantization one method is to (i) choose a complex structure on E, (ii) identify ME with the moduli space of stable holomorphic G-bundles on E by the Narasimhan-Seshadri theorem [23], thereby giving JiE a Kahler structure, (iii) represent the symplectic form as the curvature of a holomorphic line bundle L on JtE, and (iv) define HE as the space of holomorphic sections of L. One must show that HE is essentially independent of the complex structure chosen. Even after that one must construct the vectors WM associated to 3-manifolds. No natural way of doing this is known, though in general terms one can say that if Geometrie Aspects of Quantum Field Theory 1391 E = dM then the boundaries of flat G-bundles on M form a Lagrangian submanifold in JfE, and this should define a vector in the quantization HE. But the connection of the spaces Jt E with 3-manifolds was a great surprise, for they arise more obviously from 1 + 1 dimensional conformai theories, as we shall see below. The application of this theory to the study of knots and 3-manifolds is discussed elsewhere at this Congress, so here I shall just emphasize that it has led to many new results about the geometry of the spaces J4E, notably Verlinde's beautiful formula [30] for the dimension of the space HE. By applying quantum field theory to JtE in a slightly different way Witten has recently been led to conjecture a formula for the volume of Jt.'E in terms of ÇG(2g — 2), where g is the genus of E, and Us) = 27(dim V)~s, the sum being over the irreducible representations V of G. (c) 3 + 1 dimensional Theories. For each compact group G there is an important 3 + 1 dimensional theory [3, 13, 17] which assigns to a closed 4-manifold W its Donaldson invariant, i.e. (roughly) the number of "instantons" on. W. (An instanton is a solution of the self-dual Yang-Mills equations.) This theory was described in Floer's talk at this Congress. The vector space HM for a 3-manifold M is the Floer cohomology group defined by applying infinite dimensional Morse theory to the space F(M) of isomorphism classes of G-connections on M, the Morse function being the circle-valued Chern-Simons form already mentioned. A 4-manifold W with boundary M has a relative Donaldson invariant in HM. Unfortunately field theory, although strikingly exemplified here, has not so far helped much with the geometry, except insofar as it is a field-theoretic idea to study the instanton moduli spaces in terms of the space of all connections. I should say a word about Floer cohomology. The infinite dimensional manifolds F which arise in field theory are usually polarized, in the sense that their tangent spaces are roughly decomposed into positive- and negative-energy halves. (Cf. [26] § 4.) Floer's Morse function defines a decomposition of this kind, into the positive and negative eigenspaces of the Hessian. For such a manifold F one expects to be able to define "middle dimensional cohomology", by considering infinite dimensional cycles whose tangent spaces roughly fill the negative half of the tangent spaces to F. This idea goes back, of course, to Dirac's treatment of electrodynamics in terms of a sea of negative energy electrons. The same idea has been formalized by Feigin in his "semi-infinite" cohomology of Lie algebras [14]. Apart from the space of connections above, Floer cohomology has also been applied to the loop space of a symplectic manifold [12], and there too it arises as the state space of a field theory, the 1 + 1 dimensional topological <7-model of [32]. § 4. 1 + 1 Dimensional Conformai Field Theory Conformai field theory is akin to topological field theory in the sense that, up to isomorphism, a compact surface has only a finite dimensional space of conformai structures. Conformai theories can be axiomatized in the same way as topological ones [27, 28]. They have been much studied since the influential paper [7], partly 1392 Graeme Segai for their relevance to string theory, but also because of their role in at least three areas of mathematics: (i) the representation theory of loop groups and of Diff^1), (ii) the study of the moduli spaces of Riemann surfaces and holomorphic bundles, and (iii) the construction of the monster simple group and its representations. For the third of these areas I refer to [15]. A slightly more conventional approach to conformai field theory is summarized in [18]. A conformai theory consists of a vector space H naturally associated to the standard circle S1, together with an operator WE : H®m -> H®n for each Riemann surface E with m incoming and n outgoing parametrized boundary circles. Thus Diffus1) acts (projectively) on H, and so does the semigroup sé of surfaces which are topologically cylinders. (The composition-law is sewing end-to-end.) The semigroup sé has twice the dimension of Diff^S1), and is a complex manifold. One of the important ideas of the theory is that sé plays the role of a complexification of the group Diffus1): more precisely, the relation between them is the same as that between the unitary group Un and the semigroup {g e GL„((C) : \\g\\ < 1} of contraction operators. (Cf. [28] and also Neretin [24].) Let us recall that the loop group S£G of a compact group G has an interesting class of irreducible projective representations {HktV} - the positive energy representations - which are parametrized by their levelfce H*(BG; Z), which describes the projective multiplier of the representation, and an irreducible representation V of G. (The image offcin H2(J£G) is the class of the circle bundle defined by the central extension.) For a given levelfconly afiniteset of representations Fcan occur. An important fact about the representations Hkv is that they possess a canonical projective action of Diff^1) intertwining with that of «JSfG. This action extends to more accurately, is the boundary value of- an action of sé by trace-class operators i FA:H-^H. In fact WA is characterized by intertwining with the group GA of holomorphic maps A -» Gc, which acts on the source and target of WA via restriction to the two ends of A. The remarkable fact is that the irreducible representations of a given level constitute something very close to a conformai field theory. To state this precisely one needs the concept of a modular functor [27,28]. (In the literature modular functors are usually referred to as "conformai blocks" [7] or "solutions of the Knizhnik-Zamolodchikov equations". For the latter, see Varchenko's talk at this Congress.) A modular functor has a finite set 0 of labels. It assigns a finite dimensional vector space EE to each Riemann surface with boundary where each boundary circle is labelled with an element of #. The axioms are (i) EE = C when E is the Riemann sphere, (ii) EElUEi^EEi®EE2, (iii) E è = ® EEt<£ where (27, (j)) is obtained from È by cutting it along a simple closed curve and giving both new boundary circles the label <j>. For the application 0 is the set of irreducible representations of £t?G of a given level. There is a modular functor E such that when E is a Riemann surface with m Geometrie Aspects of Quantum Field Theory 1393 incoming and n outgoing boundary circles labelled with representations Ham and Hßi,..., Hßn there is an operator Hai,..., ¥E^.Hai®--®Ham^Hßi®"-®Hßn for each £, e EE which intertwines with the action of the group GE of holomorphic maps E -> Gç. (In fact EE can be defined as the space of such intertwining operators: then the point to establish is property (iii) above, which amounts to a version of the Peter-Weyl theorem for loop groups.) The first complete proof of this result is in [29], (Cf. Tsuchiya's talk at this Congress.) One of the advantages of the field-theoretic viewpoint in the representation theory of loop groups is to make plain the otherwise mysterious modularity properties of the characters: in field theory the values of the characters are naturally associated to complex tori. Witten realized [33] that the modular functor EE just described is essentially independent of the complex structure of E, and is the state space of the corresponding 2 + 1 dimensional topological theory based on the Chern-Simons action. More recently Kontsevich [21] has sketched an argument to show, still more surprisingly, that the concepts of modular functor and 2 + 1 dimensional topological theory are exactly equivalent. A "topological" modular functor is closely related to a quantum group, for the quantum deformation of G amounts essentially to a way of defining an exotic tensor product on the category of representations of G. For a modular functor E we can define Vi®EV2 = w ®EEtVuVi,w®W, where W runs through the irreducible representations of G, and E is a disc with two holes whose boundary components are labelled Vx, V2 (incoming) and W (outgoing). It is easy to relate the modular functor EE to the space HE = r(JtE, L) of holomorphic sections described in § 3. Let us decompose the closed surface E as E1 u E2 by a simple closed curve S. A holomorphic bundle on E is automatically trivial on Ex and E2, so it can be described by a clutching function on S, i.e. by an element of J^G^. The set of isomorphism classes of bundles on E - essentially the same as JtE - is therefore the double coset space GEi\^,G(C/GE2, and the space r(JtE, L) is the GEi-invariant part of H = F(^£G^jGE2, n*L), where 7i : JSfGc/G^ -> JtE. If we now take E2 to be a standard disc then H is the basic representation of JSfG of levelfc,constructed by the Borei-Weil method [25]. Finally, it is easy to see that when the boundary of Ex is labelled with H we have EE = EEi, and so EE s HG*> £ r(J/E; L). The preceding argument, which shows how representations of i?G define functions on the moduli space of G-bundles, also shows how representations of Dif^S1) give functions on the moduli space (€E of complex structures on a smooth surface E. For (€E behaves like a double coset space of the semigroup sé: if we write E = ExuE2, and choose fixed complex structures on E1 and E2, then E1 u A u E2 runs through an open set of <ßE as A runs through sé. The representation theory of 1394 Graeme Segal Diff^S1) allows us, for example, to identify and classify holomorphic line bundles on #£ much more simply than does conventional algebraic geometry. (Cf. [6,28].) § 5. Zamolodchikov's c-Theorem The point of view of this talk is successful with topological and conformai field theories, but so far it has never been taken seriously in a wider context. At present the only general definition of a field theory is the classical one in terms of the vacuum expectation values of a class of operators varying from theory to theory. This is not sufficiently manageable for one to be able to speak, for instance, of the "space 5" of all 1 + 1 dimensional theories". Nevertheless one of the most interesting recent developments has been the following result of Zamolodchikov [36], which is framed in terms of the space 0~. Whatever may be the definition of a theory, it will presumably be true that from any theory T one can derive a 1-parameter family of theories Tx (for X e R x ) simply by multiplying all lengths by X. The resulting flow on 0' is the renormalization group flow. Conformai theories are fixed points of this flow. Zamolodchikov's idea is to define a Riemannian metric on 0' and a smooth function c : 0~ -» R such that (i) the renormalization group flow is the gradient flow of c, and (ii) c(T) is equal to the central charge if Tis a conformai theory. The central charge of a conformai theory is the number describing the central extension of Diffi^S1) which acts on the state space of the theory. It is fairly straightforward to calculate the possible non-conformal infinitesimal deformations of a conformai theory, so Zamolodchikov's theorem suggests that one could in principle discover the global topology of the space 0~ by Morse theory. Vafa and others have made some steps in this direction. Zamolodchikov's argument is based on perturbation theory. I think it is a fascinating challenge to put it in a better mathematical setting. § 6. 1 + 1 Dimensional Quantum Gravity The most dramatic recent development in quantum field theory has been a breakthrough in 2-dimensional quantum gravity. It has suddenly appeared possible to perform integrals over the space of all metrics on a surface and get very explicit answers. The main success has come from the technique of random matrices [9,11, 19], and I cannot discuss it here. One very unexpected outcome is to link the theory with the classical completely integrable systems of non-linear partial differential equations such as the KdV equation. So far the situation has not really been assimilated mathematically, but the results seem to describe the algebraic topology of the space of metrics on a surface, or - equivalently - the moduli spaces of complex structures. From ordinary algebraic geometry one knows (see Morita's talk at this Congress) a ring of stable cohomology classes on the moduli spaces Jiq of surfaces of genus g ("stable" means that they are defined independently of g), and Witten has Geometrie Aspects of Quantum Field Theory 1395 claimed that the field-theoretic results are the integrals of these classes over the spaces J4.g, i.e. the characteristic numbers of J4r This has led him to the striking conjecture [34] that the generating function for the characteristic numbers is a certain specific solution of the KdV hierarchy. His lectures [35] give an excellent account of the present state of the subject. From the point of view of this talk it is interesting that quantum gravity does not fit directly into the framework of § 2 above, but nevertheless seems likely to be axiomatizable along related but more subtle lines. The crucial point that distinguishes the gauge-theory situations to which §2 applies from the gravitational ones to which it does not is simply that an automorphism of a bundle P on Mx u M2 is just a pair of automorphisms of P\M1 and P\M2, but a diffeomorphism of Mi u M2 cannot be broken into two diffeomorphisms. References 1. O. Alvarez, T. Killingback, M. Mangano, P. Windey: String theory and loop space index theorems. Comm. Math. Phys. Ill (1987) 1-10 2. M.F. Atiyah: Circular symmetry and stationary phase approximation. Proc. Conf. in honour of L. Schwartz, Astérisque 131 (1985) 43-59 3. M.F. Atiyah: New invariants of 3- and 4-dimensional manifolds. Amer. Math. Soc, Proc. Symp. Pure Math. 48 (1988) 285-99 4. M.F. Atiyah: Topological quantum field theories, Pubi. Math. I.H.E.S. Paris 68 (1989) 175-86 5. S. Axelrod, S. Della Pietra, E. Witten: Geometric quantization of Chern-Simons gauge theory. J. Diff. Geom. 33 (1991) 787-902 6. A.A. Beilinson, V.V. Schechtman: Determinant line bundles and Virasoro algebras. Comm. Math. Phys. 118 (1990) 651-701 7. A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov: Infinite conformai symmetry in two-dimensional quantumfieldtheory. Nucl. Phys. B241 (1984) 333-380 8. J.-M. Bismut: Index theorem and the heat equation, Proc. Internat. Cong. Math. Berkeley 1986, pp. 491-504 9. E. Brezin, V. A. Kazakov: Exactly solvablefieldtheories of closed strings. Phys. Lett. 236B (1990) 144-150 10. S.K. Donaldson, P. Kronheimer: Geometry of 4-manifolds. Oxford U.P. 1990 11. M. Douglas, S. Shenker: Strings in less than one dimension. Nucl. Phys. B 335 (1990) 635-654 12. A Floer: Morse theory forfixedpoints of symplectic diffeomorphisms. Bull. Amer. Math. Soc. 16(1987)279-281 13. A. Floer: An instanton invariant for 3-manifolds. Comm. Math. Phys. 118 (1988) 215240 14. LB. Frenkel, H. Garland, G. Zuckerman: Semi-infinite cohomology and string theory, Proc. Nat. Acad. Sci. USA. 83 (1986) 8442-6 15. LB. Frenkel, J. Lepowsky, A. Meurman: Vertex operator algebras and the monster. Academic Press, 1988 16. D. Friedan, P. Windey: Supersymmetric derivation of the Atiyah-Singer index theorem and the chiral anomaly. Nucl. Phys. B235 (1984) 395-416. 17. M. Furuta, D. Kotschick, S. Donaldson: Floer homology groups in Yang-Mills theory. (To appear) 18. K. Gawedzki: Conformai field theory. Seminaire Bourbaki 704 (1988) in Astérisque. 1396 Graeme Segal 19. D.J. Gross, A.A. Migdal: A nonperturbative treatment of two dimensional quantum gravity. Phys. Rev. Lett. 64 (1990) 127-130 20. N.J. Hitchin: Flat connections and geometric quantization. Comm. Math. Phys. 131 (1990) 347-380 21. M. Kontsevich: Rational conformai field theory and invariants of 3-dimensional manifolds (Marseilles preprint, to appear) 22. P.S. Landweber (ed.): Elliptic curves and modular forms in algebraic topology, Proceedings, Princeton 1986. (Lecture Notes in Mathematics, vol. 1326.) Springer, Berlin Heidelberg New York 1986 23. M.S. Narasimhan, CS. Seshadri: Stable and unitary bundles on a compact Riernann surface. Ann. Math. 82 (1965) 540-67 24. Yu. A. Neretin: A complex semigroup containing the group of diffeomorphisms of a circle. Funkts. Anal. Prilozh. 21 (1987) 82-83 25. A. Pressley, G. Segal: Loop groups. Oxford, U.P. 1986 26. G. Segal: Elliptic cohomology, Seminaire Bourbaki 695 (1988). In Astérisque 161-162 (1988) 187-201 27. G. Segal: Two dimensional conformai field theories and modular functors. IXth Proc. Internat. Cong. Math. Phys. Swansea 1988 (B. Simon, A. Truman, I.M. Davies, (eds).) Adam Hilger, 1989, pp. 22-37 28. G. Segal: The definition of conformai field theory. (To appear) 29. A. Tsuchiya, K. Ueno, Y. Yamada: Conformai field theory on the universal family of stable curves with gauge symmetry. In Conformai field theory and solvable lattice models. Adv. Stud. Pure Math. 16 (1988) 297-372 30. E. Verlinde: Fusion rules and modular transformations in two dimensional conformai field theory. Nucl. Phys. B 300 (1988) 360-376 31. E. Witten: Topological quantumfieldtheory. Comm. Math. Phys. 117 (1988) 353-86 32. E. Witten: Topological <7-models. Comm. Math. Phys. 118 (1988) 411-449 33. E. Witten: Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121 (1989) 351-99 34. E. Witten: On the structure of the topological phase of two dimensional gravity. IAS Preprint IAS SNS HEP 89/66 35. E. Witten: Two-dimensional gravity and intersection theory on moduli space- Lectures, Harvard 1990 (To appear) 36. A.B. Zamolodchikov: Renormalization group and perturbation theory nearfixedpoints in two-dimensional field theory. Yad. Fiz. 46 (1987) 1819-1831