DAMTP-1999-143 REVIEW ARTICLE An Introduction to Conformal Field Theory Matthias R Gaberdiel z 9 Department of Applied Mathematics and Theoretical Physics, Silver Street, 9 Cambridge, CB3 9EW, UK and 9 1 FitzwilliamCollege, Cambridge, CB3 0DG,UK v o Abstract. A comprehensive introduction to two-dimensionalconformal (cid:12)eld theory N is given. 1 2 PACS numbers: 11.25.Hf v 6 5 1 Submitted to: Rep. Prog. Phys. 0 1 9 9 / h t - p e h : v i X r a Email: [email protected] z Conformal Field Theory 2 1. Introduction Conformal (cid:12)eld theorieshavebeen at thecentreof muchattentionduring the last (cid:12)fteen years since they are relevant for at least three di(cid:11)erent areas of modern theoretical physics: conformal (cid:12)eld theories provide toy models for genuinely interacting quantum (cid:12)eldtheories, theydescribetwo-dimensional criticalphenomena, and they playa central ro^le in string theory, at present the most promising candidate for a unifying theory of all forces. Conformal (cid:12)eld theories have also had a major impact on various aspects of modern mathematics,inparticularthetheoryof vertexoperator algebras and Borcherds algebras, (cid:12)nite groups, number theory and low-dimensional topology. From an abstract point of view, conformal (cid:12)eld theories are Euclidean quantum (cid:12)eld theories that are characterised by the property that their symmetry group contains, in addition to the Euclidean symmetries, local conformal transformations, i.e. transformations that preserve angles but not lengths. The local conformal symmetry is of special importance in two dimensions since the corresponding symmetry algebra is in(cid:12)nite-dimensional in this case. As a consequence, two-dimensional conformal (cid:12)eld theories have an in(cid:12)nite number of conserved quantities, and are completelysolvable by symmetry considerations alone. As a bona (cid:12)de quantum (cid:12)eld theory, the requirement of conformal invariance is very restrictive. In particular, since the theory is scale invariant, all particle-like excitations of the theory are necessarily massless. This might be seen as a strong argument against any possible physical relevanceof such theories. However,all particles of any (two-dimensional) quantum (cid:12)eld theory are approximately massless in the limit of high energy, and many structural features of quantum (cid:12)eld theoriesare believedto be unchanged in this approximation. Furthermore, it is possible to analyse deformations of conformal (cid:12)eld theories that describe integrable massive models [1,2]. Finally, it might be hoped that a good mathematical understanding of interactions in any model theory should have implications for realistic theories. The more recent interest in conformal (cid:12)eld theories has di(cid:11)erent origins. In the description of statistical mechanics in terms of Euclidean quantum (cid:12)eld theories, conformal (cid:12)eld theories describe systems at the critical point, where the correlation length diverges. One simple system where this occurs is the so-called Ising model. This model is formulated in terms of a two-dimensional lattice whose lattice sites represent atomsofan(in(cid:12)nite)two-dimensionalcrystal. Eachatomistakentohaveaspinvariable (cid:27) that can take the values 1, and the magnetic energy of the system is the sum over i (cid:6) pairs of adjacent atoms E = (cid:27) (cid:27) : (1) i j (ij) X If we consider the system at a (cid:12)nite temperature T, the thermal average behaves h(cid:1)(cid:1)(cid:1)i as i j (cid:27) (cid:27) (cid:27) (cid:27) exp j (cid:0) j ; (2) i j i j h i(cid:0)h i(cid:1)h i (cid:24) (cid:0) (cid:24) (cid:18) (cid:19) Conformal Field Theory 3 where i j 1 and (cid:24) is the so-called correlation length that is a function of the j (cid:0) j (cid:29) temperature T. Observable (magnetic) properties can be derived from such correlation functions, and are therefore directly a(cid:11)ected by the actual value of (cid:24). The system possesses a critical temperature, at which the correlation length (cid:24) diverges, and the exponential decay in (2) is replaced by a power law. The continuum theory that describes the correlation functions for distances that are large compared to the lattice spacing is then scale invariant. Every scale-invariant two-dimensional local quantum (cid:12)eld theory is actually conformally invariant [3], and the critical point of the Ising model is therefore described by a conformal (cid:12)eld theory [4]. (The conformal (cid:12)eld theory in question will be brie(cid:13)y described at the end of section 4.) TheIsingmodelisonly arather rough approximationto theactual physicalsystem. However, the continuum theory at the critical point | and in particular the di(cid:11)erent critical exponents that describe the power law behaviour of the correlation functions at the critical point | are believed to be fairly insensitive to the details of the chosen model; this is the idea of universality. Thus conformal (cid:12)eld theory is a very important method in the study of critical systems. The second main area in which conformal (cid:12)eld theory has played a major ro^le is string theory [5,6]. String theory is a generalised quantum (cid:12)eld theory in which the basic objects are not point particles (as in ordinary quantum (cid:12)eld theory) but one dimensional strings. These strings can either form closed loops (closed string theory), or they can have two end-points, in which case the theory is called open string theory. Strings interact by joining together and splitting into two; compared to the interaction of point particles where two particles come arbitrarily close together, the interaction of strings is more spread out, and thus many divergenciesof ordinary quantum (cid:12)eld theory are absent. Unlike point particles, a string has internal degrees of freedom that describe the di(cid:11)erent ways in which it can vibrate in the ambient space-time. These di(cid:11)erent vibrational modes are interpreted as the ‘particles’ of the theory | in particular, the whole particle spectrum of the theory is determined in terms of one fundamental object. The vibrations of the string are most easily described from the point of view of the so-called world-sheet, the two-dimensional surface that the string sweeps out as it propagates through space-time; in fact, as a theory on the world-sheet the vibrations of the string are described by a conformal (cid:12)eld theory. In closed string theory, the oscillations of the string can be decomposed into two waves which move in opposite directions around the loop. These two waves are essentially independent of each other, and the theory therefore factorises into two so- calledchiral conformal (cid:12)eld theories. Many properties of the local theory can be studied separately for the two chiral theories, and we shall therefore mainly analyse the chiral theory in this article. The main advantage of this approach is that the chiral theory can be studied using the powerful tools of complex analysis since its correlation functions are analytic functions. The chiral theories also play a crucial ro^le for conformal (cid:12)eld theories that are de(cid:12)ned on manifolds with boundaries, and that are relevant for the Conformal Field Theory 4 description of open string theory. All known consistent string theories can be obtained by compacti(cid:12)cation from a rather small number of theories. These include the (cid:12)ve di(cid:11)erent supersymmetric string theories in ten dimensions, as well as a number of non-supersymmetric theories that are de(cid:12)nedineithertenortwenty-sixdimensions. Therecentadvances instring theoryhave centeredaround the idea of duality, namelythat these theories are further related in the sense that the strong coupling regime of one theory is described by the weak coupling regimeof another. A crucialelementinthese developmentshas been therealisation that the solitonic objects that de(cid:12)ne the relevant degrees of freedom at strong coupling are Dirichlet-branes that have an alternative description in terms of open string theory [7]. In fact, the e(cid:11)ect of a Dirichlet brane is completely described by adding certain open string sectors (whose end-points are (cid:12)xed to lieon the world-volumeof thebrane) to the theory. The possible Dirichlet branes of a given string theory are then selected by the condition that the resulting theory of open and closed strings must be consistent. These consistency conditions contain (and may be equivalent to) the consistency conditions of conformal (cid:12)eld theory on a manifold with a boundary [8{10]. Much of the structure of the theory that we shall explain in this review article is directly relevant for an analysis of these questions, although we shall not discuss the actual consistency conditions (and their solutions) here. Any review article of a well-developed subject such as conformal (cid:12)eld theory will miss out important elements of the theory, and this article is no exception. We have chosen to present one coherent route through some section of the theory and we shall not discuss in any detail alternative view points on the subject. The approach that we have taken is in essence algebraic (although we shall touch upon some questions of analysis), andisinspiredbythework ofGoddard [11]as wellas themathematicaltheory of vertex operator algebras that was developed by Borcherds [12,13], Frenkel, Lepowsky & Meurman [14], Frenkel, Huang & Lepowsky [15], Zhu [16], Kac [17] and others. This algebraic approach will be fairly familiar to many physicists, but we have tried to give it a somewhat new slant by emphasising the fundamental ro^le of the amplitudes. We have also tried to explain some of the more recent developments in the mathematical theory of vertex operator algebras that have so far not been widely appreciated in the physics community, in particular, the work of Zhu. There exist in essence two other view points on the subject: a functional analytic approach inwhichtechniquesfromalgebraic quantum(cid:12)eldtheory[18]are employedand which has been pioneered by Wassermann [19] and Gabbiani and Fro(cid:127)hlich [20]; and a geometrical approach that is inspired by string theory (for example the work of Friedan & Shenker [21]) and that has been put on a solid mathematical foundation by Segal [22] (see also Huang [23,24]). We shall also miss out various recent developments of the theory, in particular the progress in understanding conformal (cid:12)eld theories on higher genus Riemann surfaces [25{29], and on surfaces with boundaries [30{35]. Conformal Field Theory 5 Finally, we should mention that a number of treatments of conformal (cid:12)eld theory arebynowavailable,inparticularthereviewarticlesofGinsparg [36]and Gawedzki[37], and the book by Di Francesco, Mathieu and S(cid:19)en(cid:19)echal [38]. We have attempted to be somewhat more general, and have put less emphasis on speci(cid:12)c well understood models such as the minimal models or the WZNW models (although they will be explained in due course). We have also been more in(cid:13)uenced by the mathematical theory of vertex operator algebras, although we have avoided to phrase the theory in this language. The paper is organised as follows. In section 2, we outline the general structure of the theory, and explain how the various ingredients that will be subsequently described (cid:12)t together. Section 3 is devoted to the study of meromorphic conformal (cid:12)eld theory; this is the part of the theory that describes in essence what is sometimes called the chiral algebra by physicists, or the vertex operator algebra by mathematicians. We also introduce the most important examples of conformal (cid:12)eld theories, and describe standard constructions such as the coset and orbifold construction. In section 4 we introducetheconceptofa representationof themeromorphicconformal(cid:12)eldtheory,and explain the ro^le of Zhu’s algebra in classifying (a certain class of) such representations. Section 5 deals with higher correlation functions and fusion rules. We explain Verlinde’s formula, and give a brief account of the polynomial relations of Moore & Seiberg and their relation to quantum groups. We also describe logarithmic conformal (cid:12)eld theories. We conclude in section 6 with a number of general open problems that deserve, in our opinion, more work. Finally, we have included an appendix that contains a brief summary about the di(cid:11)erent de(cid:12)nitions of rationality. 2. The General Structure of a Local Conformal Field Theory Let us begin by describing somewhat sketchily what the general structure of a local conformal (cid:12)eld theory is, and how the various structures that will be discussed in detail later (cid:12)t together. 2.1. The Space of States In essence, a two-dimensional conformal (cid:12)eld theory (like any other (cid:12)eld theory) is determined by its space of states and the collection of its correlation functions. The space of states is a vector space (that may or may not be a Hilbert space), and the HHH correlation functions are de(cid:12)ned for collections of vectors in some dense subspace FFF of . These correlation functions are de(cid:12)ned on a two-dimensional space-time, which HHH we shall always assume to be of Euclidean signature. We shall mainly be interested in the case where the space-time is a closed compact surface. These surfaces are classi(cid:12)ed (topologically) by their genus g which counts the number of ‘handles’; the simplest such surface is the sphere with g = 0, the surface with g = 1 is the torus, etc. In a (cid:12)rst step we shall therefore consider conformal (cid:12)eld theories that are de(cid:12)ned on the sphere; as we shall explain later, under certain conditions it is possible to associate to such a theory Conformal Field Theory 6 families of theories that are de(cid:12)ned on surfaces of arbitrary genus. This is important in the context of string theory where the perturbative expansion consists of a sum over all such theories (where the genus of the surface plays the ro^le of the loop order). One of the special features of conformal (cid:12)eld theory is the fact that the theory is naturally de(cid:12)ned on a Riemann surface (or complex curve), i.e. on a surface that possesses suitable complex coordinates. In the case of the sphere, the complex coordinates can be taken to be those of the complex plane that cover the sphere except for the point at in(cid:12)nity; complex coordinates around in(cid:12)nity are de(cid:12)ned by means of the coordinate function (cid:13)(z) = 1=z that maps a neighbourhood of in(cid:12)nity to a neighbourhood of 0. With this choice of complex coordinates, the sphere is usually referred to as the Riemann sphere, and this choice of complex coordinates is up to some suitable class of reparametrisations unique. The correlation functions of a conformal (cid:12)eld theory that is de(cid:12)ned on the sphere are thus of the form V( ;z ;z(cid:22) ) V( ;z ;z(cid:22) ) ; (3) 1 1 1 n n n h (cid:1)(cid:1)(cid:1) i where V( ;z) is the (cid:12)eld that is associated to the state , , and z and z(cid:22) i i i 2 FFF (cid:26) HHH are complex numbers (or in(cid:12)nity). These correlation functions are assumed to be local, i.e. independent of the order in which the (cid:12)elds appear in (3). One of the properties that makes two-dimensional conformal (cid:12)eld theories exactly solvable is the fact that the theory contains a large (in(cid:12)nite-dimensional) symmetry algebra with respect to which the states in fall into representations. This symmetry HHH algebra is directly related (in a way we shall describe below) to a certain preferred subspace of that is characterised by the property that the correlation functions 0 F FFF (3) of its states depend only on the complex parameter z, but not on its complex conjugate z(cid:22). More precisely, a state is in if for any collection of , 0 i 2FFF F 2FFF (cid:26)HHH the correlation functions V( ;z;z(cid:22))V( ;z ;z(cid:22) ) V( ;z ;z(cid:22) ) (4) 1 1 1 n n n h (cid:1)(cid:1)(cid:1) i do not depend on z(cid:22). The correlation functions that involve only states in are then 0 F analytic functions on the sphere. These correlation functions de(cid:12)ne the meromorphic (sub)theory [11] that will be the main focus of the next section. x Similarly, we can consider the subspace of states that consists of those states 0 F for which the correlation functions of the form (4) do not depend on z. These states de(cid:12)ne an (anti-)meromorphic conformal (cid:12)eld theory which can be analysed by the same methods as a meromorphic conformal (cid:12)eld theory. The two meromorphic conformal subtheories encode all the information about the symmetries of the theory, and for the most interesting class of theories, the so-called (cid:12)nite or rational theories, the whole theory can be reconstructed from them up to some (cid:12)nite ambiguity. In essence, this means that the whole theory is determined by symmetry considerations alone, and this is at the heart of the solvability of the theory. Our use of the term meromorphic conformal (cid:12)eld theory is di(cid:11)erent from that employed by, e.g., x Schellekens [39]. Conformal Field Theory 7 The correlation functions of the theory determine the operator product expansion (OPE) of the conformal (cid:12)elds which expresses the operator product of two (cid:12)elds in terms of a sum of single (cid:12)elds. If and are two arbitrary states in then the OPE 1 2 FFF of and is an expansion of the form 1 2 V( ;z ;z(cid:22) )V( ;z ;z(cid:22) ) 1 1 1 2 2 2 = (z z )(cid:1)i(z(cid:22) z(cid:22) )(cid:1)(cid:22)i V((cid:30)i ;z ;z(cid:22) )(z z )r(z(cid:22) z(cid:22) )s; (5) 1 (cid:0) 2 1 (cid:0) 2 r;s 2 2 1(cid:0) 2 1(cid:0) 2 i r;s 0 X X(cid:21) where (cid:1) and (cid:1)(cid:22) are real numbers, r;s IN and (cid:30)i . The actual form of this i i 2 r;s 2 FFF expansion can be read o(cid:11) from the correlation functions of the theory since the identity (5) has to hold in all correlation functions, i.e. V( ;z ;z(cid:22) )V( ;z ;z(cid:22) )V((cid:30) ;w ;w(cid:22) ) V((cid:30) ;w ;w(cid:22) ) 1 1 1 2 2 2 1 1 1 n n n (cid:1)(cid:1)(cid:1) D = (z z )(cid:1)i(z(cid:22) z(cid:22) )(cid:1)(cid:22)i (z z )r(z(cid:22)E z(cid:22) )s 1 2 1 2 1 2 1 2 (cid:0) (cid:0) (cid:0) (cid:0) i r;s 0 X X(cid:21) V((cid:30)i ;z ;z(cid:22) )V((cid:30) ;w ;w(cid:22) ) V((cid:30) ;w ;w(cid:22) ) (6) r;s 2 2 1 1 1 (cid:1)(cid:1)(cid:1) n n n D E for all (cid:30) . If both states and belong to the meromorphic subtheory , (6) j 1 2 0 2 FFF F only depends on z , and (cid:30)i also belongs to the meromorphic subtheory . The OPE i r;s F0 thereforede(cid:12)nesacertainproduct onthemeromorphic(cid:12)elds. Sincetheproductinvolves the complex parameters z in a non-trivial way, it does not directly de(cid:12)ne an algebra; i the resulting structure is usually called a vertex (operator) algebra in the mathematical literature [12,14], and we shall adopt this name here as well. By virtue of its de(cid:12)nition in terms of (6), the operator product expansion is associative, i.e. V( ;z ;z(cid:22) )V( ;z ;z(cid:22) ) V( ;z ;z(cid:22) ) = V( ;z ;z(cid:22) ) V( ;z ;z(cid:22) )V( ;z ;z(cid:22) ) ; (7) 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 (cid:16) (cid:17) (cid:16) (cid:17) where the brackets indicate which OPE is evaluated (cid:12)rst. If we consider the case where both and are meromorphic (cid:12)elds (i.e. in ), then the associativity of the OPE 1 2 0 F implies that the states in form a representation of the vertex operator algebra. The FFF same also holds for the vertex operator algebra associated to the anti-meromorphic (cid:12)elds, and we can thus decompose the whole space (or ) as FFF HHH = ; (8) (j;|(cid:22)) HHH HHH (j;|(cid:22)) M where each is an (indecomposable) representation of the two vertex operator (j;|(cid:22)) HHH algebras. Finite theories are characterised by the property that only (cid:12)nitely many indecomposable representations of the two vertex operator algebras occur in (8). 2.2. Modular Invariance The decomposition of the space of states in terms of representations of the two vertex operator algebras throws considerablelighton theproblemofwhether thetheoryiswell- de(cid:12)ned on higher Riemannsurfaces. One necessaryconstraint for this (whichis believed Conformal Field Theory 8 also to be su(cid:14)cient[40]) is that the vacuum correlator on the torus is independent of its parametrisation. Every two-dimensional torus can be described as the quotient space of IR2 C by the relations z z +w and z z +w , where w and w are not parallel. 1 2 1 2 ’ (cid:24) (cid:24) The complex structure of the torus is invariant under rotations and rescalings ofC, and therefore every torus is conformally equivalent to (i.e. has the same complex structure as) a torus for which the relations are z z +1 and z z +(cid:28), and (cid:28) is in the upper (cid:24) (cid:24) half plane of C. It is also easy to see that (cid:28), T((cid:28)) = (cid:28) +1 and S((cid:28)) = 1=(cid:28) describe (cid:0) conformally equivalenttori; the two maps T and S generate the group SL(2;Z)=Z2 that consists of matrices of the form a b A = where a;b;c;d Z; ad bc = 1; (9) c d 2 (cid:0) ! and the matrices A and A have the same action on (cid:28), (cid:0) a(cid:28) +b (cid:28) A(cid:28) = : (10) 7! c(cid:28) +d The parameter (cid:28) is sometimescalled the modular parameter of the torus, and the group SL(2;Z)=Z2 is called the modular group (of the torus). Given a conformal (cid:12)eld theory that is de(cid:12)ned on the Riemann sphere, the vacuum correlator on the torus can be determined as follows. First, we cut the torus along one of its non-trivial cycles; the resulting surface is a cylinder (or an annulus) whose shape depends on one complex parameter q. Since the annulus is a subset of the sphere, the conformal (cid:12)eldtheoryon the annulus isdeterminedintermsof the theoryon the sphere. In particular, the states that can propagate in the annulus are precisely the states of the theory as de(cid:12)ned on the sphere. In order to reobtain the torus from the annulus, we have to glue the two ends of the annulus together; in terms of conformal (cid:12)eld theory this means that we have to sum over a complete set of states. The vacuum correlator on the torus is therefore described by a trace over the whole space of states, the partition function of the theory, Tr ( (q;q(cid:22))) ; (11) HHH(j;|(cid:22)) O (j;|(cid:22)) X where (q;q(cid:22)) is the operator that describes the propagation of the states along the O annulus, (q;q(cid:22)) = qL0(cid:0)2c4 q(cid:22)L(cid:22)0(cid:0)2c(cid:22)4 : (12) O (cid:22) Here L and L are the scaling operators of the two vertex operator algebras and c and 0 0 c(cid:22) their central charges; this will be discussed in more detail in the following section. The propagator depends on the actual shape of the annulus that is described in terms of the complex parameter q. For a given torus that is described by (cid:28), there is a natural choice for how to cut the torus into an annulus, and the complex parameter q that is associated to this annulus is q = e2(cid:25)i(cid:28). Since the tori that are described by (cid:28) and A(cid:28) (whereA SL(2;Z)) are equivalent,thevacuum correlatoris onlywell-de(cid:12)nedprovided 2 that (11) is invariant under this transformation. This provides strong constraints on the spectrum of the theory. Conformal Field Theory 9 For most conformal (cid:12)eld theories (although not for all, see for example [41]) each of the spaces is a tensor product of an irreducible representation of the (j;|(cid:22)) j HHH H (cid:22) meromorphic vertex operator algebra and an irreducible representation of the anti- |(cid:22) H meromorphic vertex operator algebra. In this case, the vacuum correlator on the torus (11) takes the form (cid:31) (q)(cid:31)(cid:22) (q(cid:22)); (13) j |(cid:22) (j;|(cid:22)) X where (cid:31) is the character of the representation of the meromorphic vertex operator j j H algebra, (cid:31)j((cid:28)) = TrHj(qL0(cid:0)2c4) where q = e2(cid:25)i(cid:28) ; (14) and likewise for (cid:31)(cid:22) . One of the remarkable facts about many vertex operator algebras |(cid:22) (that has now been proven for a certain class of them [16], see also [42]) is the property that the characters transform into one another under modular transformations, (cid:31) ( 1=(cid:28)) = S (cid:31) ((cid:28)) and (cid:31) ((cid:28) +1) = T (cid:31) ((cid:28)); (15) j jk k j jk k (cid:0) k k X X where S and T are constant matrices, i.e. independent of (cid:28). In this case, writing (cid:22) = M ; (16) i|(cid:22) i |(cid:22) HHH H (cid:10)H i;|(cid:22) M (cid:22) where M IN denotes the multiplicitywith which the tensor product appears i|(cid:22) i |(cid:22) 2 H (cid:10)H in , the torus vacuum correlation function is well de(cid:12)ned provided that HHH S M S(cid:22) = T M T(cid:22) = M ; (17) il i|(cid:22) |(cid:22)k(cid:22) il i|(cid:22) |(cid:22)(cid:22)k lk(cid:22) i;|(cid:22) i;|(cid:22) X X and S(cid:22) and T(cid:22) are the matrices de(cid:12)ned as in (15) for the representations of the anti- meromorphic vertex operator algebra. This provides very powerful constraints for the multiplicity matrices M . In particular, in the case of a (cid:12)nite theory (for which each of i|(cid:22) thetwovertexoperatoralgebrashasonly(cid:12)nitelymanyirreduciblerepresentations)these conditions typically only allow for a (cid:12)nite number of solutions that can be classi(cid:12)ed; this has been done for the case of the so-called minimal models and the a(cid:14)ne theories with group SU(2) byCappelli, Itzyksonand Zuber [43,44] (for a modern proof involving some Galois theorysee [45]),and for thea(cid:14)netheories withgroup SU(3) and the N = 2 superconformal minimal models by Gannon [46,47]. This concludes our brief overview over the general structure of a local conformal (cid:12)eld theory. For the rest of the paper we shall mainly concentrate on the theory that is de(cid:12)ned on the sphere. Let us begin by analysing the meromorphic conformal subtheory in some detail. Conformal Field Theory 10 3. Meromorphic Conformal Field Theory In this section we shall describe in detail the structure of a meromorphic conformal (cid:12)eld theory; our exposition follows closely the work of Goddard [11] and Gaberdiel & Goddard [48], and we refer the reader for some of the mathematical details (that shall be ignored in the following) to these papers. 3.1. Amplitudes and Mo(cid:127)bius Covariance As we have explained above, a meromorphic conformal (cid:12)eld theory is determined in terms of its space of states , and the amplitudes involving arbitrary elements in a 0 i H dense subspace of . Indeed, for each state , there exists a vertex operator 0 0 0 F H 2 F V( ;z) that creates the state from the vacuum (in a sense that will be described in more detail shortly), and the amplitudes are the vacuum expectation values of the corresponding product of vertex operators, ( ;:::; ;z ;:::;z ) = V( ;z ) V( ;z ) : (18) 1 n 1 n 1 1 n n A h (cid:1)(cid:1)(cid:1) i Each vertex operator V( ;z) depends linearly on , and the amplitudes are meromor- phic functions that are de(cid:12)ned on the Riemann sphere P = C , i.e. they are [ f1g analytic except for possible poles at z = z , i = j. The operators are furthermore i j 6 assumed to be local in the sense that for z = (cid:16) 6 V( ;z)V((cid:30);(cid:16)) = " V((cid:30);(cid:16))V( ;z); (19) where " = 1 if both and (cid:30) are fermionic, and " = +1 otherwise. In formulating (19) (cid:0) we have assumed that and (cid:30) are states of de(cid:12)nite fermion number; more precisely, this means that decomposes as 0 F = B F ; (20) F0 F0 (cid:8)F0 where B and F is thesubspace of bosonic and fermionicstates, respectively,and that F0 F0 both and (cid:30) are either in B or in F. In the following we shall always only consider F0 F0 states of de(cid:12)nite fermion number. In terms of the amplitudes, the locality condition (19) is equivalentto the property that ( ;:::; ; ;:::; ;z ;:::;z ;z ;:::;z ) 1 i i+1 n 1 i i+1 n A = " ( ;:::; ; ;:::; ;z ;:::;z ;z ;:::;z ); (21) i;i+1 1 i+1 i n 1 i+1 i n A and " is de(cid:12)ned as above. Asthe amplitudesare essentiallyindependentof the order i;i+1 of the (cid:12)elds, we shall sometimes also write them as n ( ;:::; ;z ;:::;z ) = V( ;z ) : (22) 1 n 1 n i i A h i i=1 Y