Some developments in vertex operator algebra theory, old and new
aa r X i v : . [ m a t h . QA ] J un Some developments in vertex operatoralgebra theory, old and new
James Lepowsky ∗ Abstract
In this exposition, I discuss several developments in the theory of ver-tex operator algebras, and I include motivation for the definition.
This is a more detailed version of the talk I gave at the Conference on LieAlgebras, Vertex Operator Algebras and Their Applications. I thank Yi-ZhiHuang and Kailash Misra very much for organizing this conference.I would like to motivate the concept of vertex (operator) algebra, includ-ing the definition; to discuss some of the main sources of the theory, includingLie algebras and partition identities, the “monstrous moonshine” problem, andstring theory and conformal field theory; and to mention a selection of develop-ments. (This writeup is not intended to be a comprehensive survey.) Some ofthis talk is drawn from the introductory material in [FLM3]–[FLM5], [DL2] and[LL], as well as from earlier expositions such as [LW5], [L1], [L2], [HL7], [HL8]and [L4].The mathematical notion of “vertex algebra” was introduced by R. Borcherds[B1]. A variant of it, a notion of “vertex operator algebra,” was introduced in[FLM5]. These notions are algebraic formulations of concepts that had beendeveloped by many string theorists, conformal field theorists and quantum fieldtheorists, and formalized in [BPZ] as certain “operator algebras,” later called“chiral algebras” in physics.Vertex (operator) algebra theory is inherently “nonclassical,” in the samespirit in which string theory in physics is nonclassical and also in the samespirit in which the sporadic finite simple groups in mathematics are nonclassi-cal. String theory, its initial version having been introduced in the late 1960s,is based on the premise that elementary particles manifest themselves as “vi-brational modes” of fundamental strings, rather than points, moving throughspace according to quantum-field-theoretic principles. A string sweeps out atwo-dimensional “world-sheet” in space-time, and it is fruitful to focus on thecase in which the surface is a Riemann surface locally parametrized by a complexcoordinate. The resulting two-dimensional conformal (quantum) field theory hasbeen studied extensively. ∗ The author is partially supported by NSF grant DMS–0401302. M , had been predictedby B. Fischer and R. Griess and was constructed by Griess [G] as a symmetrygroup (of order about 10 ) of a remarkable new commutative but very, veryhighly nonassociative, seemingly ad-hoc, algebra B of dimension 196,883. The“structure constants” of the Griess algebra B were “forced” by expected prop-erties of the conjectured-to-exist Monster. It was proved by J. Tits that M isactually the full symmetry group of B .A bit earlier (1978–79), J. McKay, J. Thompson, J. Conway and S. Norton(see especially Conway-Norton [CN]) had discovered astounding “numerology”culminating in the “monstrous moonshine” conjectures relating the not-yet-proved-to-exist Monster M to modular functions in number theory, namely:There should exist a (natural) infinite-dimensional Z -graded module for M (i.e., representation of M ) V = M n = − , , , , ,... V n such that X n = − , , , , ,... (dim V n ) q n = J ( q ) , (1)where J ( q ) = q − + 0 + 196884 q + higher-order terms , (2)the classical modular function with its constant term set to 0. J ( q ) is the suit-ably normalized generator of the field of SL (2 , Z )-modular invariant functionson the upper half-plane, with q = e πiτ , τ in the upper half-plane. (Note:196884=196883+1.) The existence of such a structure V was conjectured byMcKay and Thompson.More generally, as Conway and Norton conjectured, for every g ∈ M (notjust g = 1), the generating function X n = − , , , , ,... (tr g | V n ) q n (3)should be the analogous “Hauptmodul” for a suitable discrete subgroup of SL (2 , R ), a subgroup having a fundamental “genus-zero property,” so that itsassociated field of modular-invariant functions has a single generator (a Haupt-modul). (The left-hand side of (1) is the graded dimension of the graded vectorspace V , and (3) is the graded trace of the action of g on the graded space V ; the graded dimension is of course the graded trace of the identity element g = 1.) The Conway-Norton conjecture subsumed a remarkable coincidencethat had been noticed earlier—that the 15 primes giving rise to the genus-zeroproperty (see A. Ogg [O]) are precisely the primes dividing the order of the(conjectured-to-exist) Monster. 2roving these conjectures would give a remarkable connection between clas-sical number theory and “nonclassical” sporadic group theory. The existence ofa structure V was soon essentially (but nonconstructively) proved by Thomp-son, A. O. L. Atkin, P. Fong and S. Smith. After Griess constructed M , withI. Frenkel and A. Meurman [FLM2] we (constructively) proved the McKay-Thompson conjecture, that there should exist a natural (whatever that wasgoing to mean) infinite-dimensional Z -graded M -module V whose graded dimen-sion is J ( q ), as in (1). The graded traces of some, but not all, of the elementsof the Monster—the elements of an important subgroup of M , namely, a certaininvolution centralizer involving the largest Conway sporadic group Co —wereconsequences of the construction, and these graded traces were indeed (suit-ably) modular functions [FLM2]. We called this V “the moonshine module V ♮ ”because of its naturality (although the contruction of V ♮ is not short). See J.Tits ([Ti1], [Ti2]) for discussions of the construction of the Monster and of themoonshine module.The construction [FLM2] heavily used a number of different types of “vertexoperators,” all of them recently constructed, or newly constructed as steps in[FLM2], along with their algebraic structure and relations. These were neededfor the construction of the structure V ♮ itself, including a natural “algebra ofvertex operators” acting on it. They were also needed for the construction of anatural infinite-dimensional “affinization” of the Griess algebra B acting on V ♮ .This “affinization,” which was part of the new algebra of vertex operators, isanalogous to, but more subtle than, the notion of affine Lie algebra, an exampleof which is discussed below. More precisely, the vertex operators were needed fora “commutative affinization” of a certain natural 196884-dimensional enlarge-ment B of B , with an identity element (rather than a “zero” element) adjoinedto B . This enlargement B naturally incorporated the Virasoro algebra—thecentral extension of the Lie algebra of formal vector fields on the circle—actingon V ♮ ; for us, the Virasoro algebra arose not because of its role as a fundamen-tal “symmetry algebra” in string theory but rather because of the fact that anatural identity element for the algebra was “forced” on us by our construction.The vertex operators were also needed for a natural “lifting” of Griess’s actionof M from the finite-dimensional space B to the infinite-dimensional structure V ♮ , including its algebra of vertex operators and its copy of the affinization of B . Thus the Monster was now realized as the symmetry group of a certain ex-plicit “algebra of vertex operators” based on an infinite-dimensional Z -gradedstructure whose graded dimension is the modular function J ( q ).Griess’s construction of B and of M acting on B was a crucial guide for us,although we did not start by using his construction; rather, we recovered it,as a finite-dimensional “slice” of a new infinite-dimensional construction usingvertex operator considerations. In fact, our presentation of the Griess algebra(see [FLM1], [FLM2], [FLM5]) was short and entirely canonical, and involvedno choices or guesses of signs or structure constants; one “reads” this algebraas in [FLM1] from canonical untwisted and twisted vertex operator structuresnewly constructed starting from the Leech lattice (mentioned below). As pre-sented in [FLM1], the 196884-dimensional algebra is simply the direct sum of3he weight-two subspace of the canonical involution-fixed subspace of the un-twisted Leech lattice vertex operator structure with the analogous subspace ofa canonically twisted vertex operator structure; the commutative nonassocia-tive algebra structure and natural “associative” symmetric bilinear form struc-ture on B are essentially described “in words.” The initally strange-seeming finite-dimensional Griess algebra was now embedded in a natural new infinite-dimensional space on which a certain algebra of vertex operators acts, via a newkind of “generalized commutation relation” (relations of this type are discussedbelow); such relations are what gave the commutative affinization mentionedabove. At the same time, the Monster, a finite group, took on a new ap-pearance by now being understood in terms of a natural infinite-dimensional structure. The very-highly-nonassociative Griess algebra, or rather, from ourviewpoint, the natural modification of the Griess algebra, with an identity el-ement adjoined, coming from a “forced” copy the Virasoro algebra, becamesimply the conformal-weight-two subspace of an algebra of vertex operators of acertain “shape.” The word “simply” refers to the ease of defining a commutativenonassociative algebra with an associative symmetric bilinear form (generalizingthe Griess algebra with identity element adjoined, for the special case of V ♮ )in the new general context of algebras of vertex operators of “shape” similar tothat of V ♮ (as was explained in [FLM2] and [FLM5]); the actual constructionof the particular algebra V ♮ remains complex. In any case, the largest sporadic finite simple group, the Monster, was “really” infinite-dimensional .In the expansion (2), the constant term of J ( q ) is zero, and this choice ofconstant term, which is not uniquely determined by number-theoretic principles,is not traditional in number theory. It turned out that the vanishing of the con-stant term in (2) was canonically “forced” by the requirement that the Monstershould act naturally on V ♮ and on an associated algebra of vertex operators.This vanishing of the degree-zero subspace of V ♮ is actually analogous in a cer-tain strong sense to the absence of vectors in the Leech lattice of square-lengthtwo; the Leech lattice is a distinguished rank-24 even unimodular (self-dual)lattice with no vectors of square-length two. In addition, this vanishing of thedegree-zero subspace of V ♮ and the absence of square-length-two elements of theLeech lattice are in turn analogous to the absence of code-words of weight 4 inthe Golay error-correcting code, a distinguished self-dual binary linear code ona 24-element set, with the lengths of all code-words divisible by 4. In fact, theGolay code was used in the original construction of the Leech lattice, and theLeech lattice was used in the construction of V ♮ . This was actually to be ex-pected (if V ♮ existed) because it was well known that the automorphism groupsof both the Golay code and the Leech lattice are (essentially) sporadic finitesimple groups; the automorphism group of the Golay code is the Mathieu group M and the automorphism group of the Leech lattice is a double cover of theConway group Co mentioned above, and both of these sporadic groups werewell known to be involved in the Monster (if it existed) in a fundamental way.The work [FLM2], [FLM5] revealed, and exploited, a new hierarchy, namely:error-correcting codes, lattices, and vertex-operator-theoretic structures. TheGolay code is actually unique subject to its distinguishing properties mentioned4bove (proved by V. Pless [Pl]) and the Leech lattice is unique subject to itsdistinguishing properties mentioned above (proved by Conway [Co] and others).Is V ♮ unique? If so, unique subject to what? The answer to this question can beviewed as serving as a motivation of the very notion of vertex operator algebra.But this uniqueness is an unsolved problem; more on this below.After [FLM2] appeared, what has been called the “first string theory revolu-tion” started in the summer of 1984, stimulated by the work [GS] of M. Greenand J. Schwarz. In this suddenly-active period in string theory, the new struc-ture V ♮ came to be viewed in retrospect by string theorists as an inherently string-theoretic structure: the “chiral algebra” underlying the Z -orbifold con-formal field theory based on the Leech lattice. The string-theoretic geometryis this: One takes the torus that is the quotient of 24-dimensional Euclideanspace modulo the Leech lattice, and then one takes the quotient of this mani-fold by the “negation” involution x
7→ − x , giving rise to an orbit space calledan “orbifold”—a manifold with, in this case, a “conical” singularity. Thenone takes the “conformal field theory” (presuming that it exists mathemati-cally) based on this orbifold, and from this one forms a “string theory” in two-dimensional space-time by compactifying a 26-dimensional “bosonic string” onthis 24-dimensional orbifold. The string vibrates in a 26-dimensional space, 24dimensions of which are curled into this 24-dimensional orbifold, and space-timeis thus 2-dimensional in this “toy-model” string theory. Such an adjunction ofa two-dimensional structure is a natural and standard procedure in string the-ory; 26 is the “critical dimension” in bosonic string theory. So in retrospect,the mathematical construction [FLM2] was essentially the construction of anorbifold string theory (actually, the first example of a theory of a string prop-agating on an orbifold that is not a torus). As discussed in the Introductionof [FLM5], some of the basic string-theoretic papers on these aspects of orb-ifold string theory are [DHVW1], [DHVW2], [Ha], [DFMS], [HV], [NSV], [M],[DGH] and [DVVV]. The idea of “orbifolding” (as string theorists were to callit) came, in the development of the work [FLM2], [FLM5], from the construc-tion of general twisted vertex operators and their algebraic relations, includingrelations involving what sometimes came to be called “intertwining operators”among “twisted sectors,” treated in detail in [FLM5] and related works discussedthere. The construction in [FLM2] also came to be viewed as a conformal-field-theoretic structure in the sense of [BPZ], which appeared around the same timeas [FLM2]. (These ideas are all discussed in [FLM5].)As I mentioned at the beginning, in [B1] Borcherds introduced the axiomaticnotion of vertex algebra . This naturally extended the relations [FLM2] for thevertex operators for V ♮ and also other known mathematical and physical fea-tures of known vertex operators, and these axioms turned out to be essentiallyequivalent to Belavin-Polyakov-Zamolodchikov’s physical axioms [BPZ] for thebasic “algebras” of vertex operators underlying conformal field theory. In [B1]Borcherds asserted that V ♮ admits a vertex algebra structure, generated by thealgebraic structure constructed in [FLM2], on which M (still) acts as a symme-try group. This assertion was proved in [FLM5], by an (elaborate) extensionof the proof of the results announced in [FLM2] (rather than by a direct use of5he results announced in [FLM2]). In [FLM5], Borcherds’s definition of vertexalgebra was modified, giving the variant notion of vertex operator algebra . Themain modification was the introduction of what we called the “Jacobi identity,”discussed below. (Actually, we had first thought of this identity as the “masterformula” for reasons mentioned below, but then we decided to emphasize itsanalogy with the Jacobi identity in the definition of the notion of Lie algebra.)Another modification was the emphasis of the viewpoint that the elements ofthe algebra essentially “are” vertex operators. Also, the definition of “vertexoperator algebra” in [FLM5] included two natural grading-restriction conditionsand the presence of a copy of the Virasoro algebra, because these features natu-rally arose in the construction of V ♮ . The term “vertex algebra” generally refersto any notion equivalent to Borcherds’s notion in [B1], and the term “vertex op-erator algebra” generally refers to any notion equivalent to the notion in [FLM5]and also in the sequel [FHL]—that is, including the two grading restrictions—even though, from the strictly logical point of view, the notion in [FLM5] and[FHL] does not have any more “operators” in it than does the notion in [B1](except in the notation).Then in [B2], Borcherds used all this and new ideas, including his results ongeneralized Kac-Moody algebras, also called Borcherds algebras, together withcertain ideas from string theory, including the “physical space” of a bosonicstring along with the “no-ghost theorem” of R. Brower, P. Goddard and C. Thorn[Br], [GT], to prove the remaining Conway-Norton conjectures for the structure V ♮ . What had remained to prove was that the formal series P (tr g | V ♮n ) q n ((3) above, but now, rather, (3) for the known structure V ♮ instead of for astill-unknown space V ) for the Monster elements g (or conjugacy classes) nottreated in [FLM2], [FLM5]—that is, the conjugacy classes outside the involu-tion centralizer—were indeed the desired Hauptmoduls; the methods of [FLM2],[FLM5] did not handle these conjugacy classes. He accomplished this by con-structing a copy of his “Monster Lie algebra” from the “physical space” associ-ated with V ♮ , enlarged to a central-charge-26 vertex algebra closely related tothe 26-dimensional bosonic-string structure mentioned above. He transportedthe known action of the Monster from V ♮ to this copy of the Monster Lie alge-bra, and by using his twisted denominator formula for this Lie algebra he provedcertain recursion formulas for the coefficients of the formal series P (tr g | V ♮n ) q n (that is, (3) for the known structure V ♮ ), for all Monster elements g . Thisentailed a generalization to Borcherds algebras of the work [GL], which hadgeneralized B. Kostant’s homology theorem [Ko]. The resulting recursion for-mulas for P (tr g | V ♮n ) q n agreed with the “replication formulas” in [CN], satisfiedby the coefficients of the Hauptmoduls listed in [CN]. By numerically verifyingthat the first few terms of some of the formal series P (tr g | V ♮n ) q n (for g rangingthrough certain elements of the involution centralizer, whose action on V ♮ hadbeen constructed in [FLM5]) agreed with the corresponding coefficients of thecorresponding Hauptmoduls listed in [CN], he succeeded in concluding that allthe graded traces P (tr g | V ♮n ) q n for V ♮ must coincide with the formal series forthe Hauptmoduls listed in [CN]. 6his remarkable work of Borcherds has been further illuminated in a num-ber of ways. In [Ju], E. Jurisich simplified Borcherds’s argument (proving thereplication formulas for the structure V ♮ ) by exploiting a certain “large” free Liealgebra inside the Monster Lie algebra; this simplification is further discussed in[JLW]. Rather than a “Borel subalgebra” of the Monster Lie algebra, a certainnatural “parabolic” subalgebra was used, allowing the simplification.Moreover, in [CG], C. Cummins and T. Gannon discovered a conceptualproof that the replication formulas lead to the genus-zero property. In particu-lar, the numerical checking in [B2] using the first few terms of the formal seriesconstructed in [FLM2], [FLM5] can essentially be bypassed. Once one provesthe replication formulas for the action of the Monster on V ♮ [B2] (or with theshorter argument in [Ju] (or [JLW])), then by [CG] one knows that the “McKay-Thompson series” for all the Monster elements acting on the structure V ♮ havethe genus-zero property. Also, the fact [FLM2], [FLM5] that the graded dimen-sion of V ♮ is the modular function J ( q ), given certain established properties ofthe vertex operator algebra V ♮ , follows alternatively from a major theorem ofY. Zhu [Z] on the modular transformation properties of the graded dimensionsof modules for suitable vertex operator algebras.The original McKay-Thompson-Conway-Norton conjectures are conceptu-ally proved. But there is also much, much more to monstrous moonshine (someof it mentioned below). See in particular Gannon’s treatments in [Ga1] and[Ga2], which include references to many works and surveys.As it turned out, then, the numerology of “monstrous moonshine” is muchmore than an astonishing relation between finite group theory and number the-ory; its underlying theme is the new theory of vertex (operator) algebras, itselfthe foundational structure for conformal field theory, which is in turn the foun-dational structure underlying string theory.It’s in this sense that (as I said at the beginning) vertex operator algebratheory is inherently “nonclassical” in the same way in which sporadic grouptheory and string theory are “nonclassical” in their respective domains.In order to motivate the precise definition of vertex (operator) algebra, whichI’ll give later, I’ll first repeat that there is a vertex operator algebra (namely, V ♮ )whose symmetry group is the Monster M and which implements the McKay-Thompson-Conway-Norton conjectures relating M to modular functions includ-ing J ( q ). But in fact, this vertex operator algebra V ♮ has the following threesimply-stated properties— properties that have nothing at all to do with the Mon-ster :(1) V ♮ , which is an irreducible module for itself (proved in [FLM5]), isits only irreducible module, up to equivalence. C. Dong [D] proved this, andC. Dong-H. Li-G. Mason [DLM] proved the stronger result that every modulefor the vertex operator algebra V ♮ is completely reducible and is in particular adirect sum of copies of itself. Thus the vertex operator algebra V ♮ has no morerepresentation theory than does a field! (I mean a field in the sense of mathemat-ics, not physics. Given a field, every one of its modules—called vector spaces,of course—is completely reducible and is a direct sum of copies of itself.)72) dim V ♮ = 0. This corresponds to the zero constant term of J ( q ); whilethe constant term of the classical modular function is essentially arbitrary, andis chosen to have certain values for certain classical number-theoretic purposes,the constant term must be chosen to be zero for the purposes of moonshine andthe moonshine module vertex operator algebra.(3) The central charge of the canonical Virasoro algebra in V ♮ is 24. “24”is the “same 24” so basic in number theory, modular function theory, etc. Asmentioned above, this occurrence of 24 is also natural from the point of view ofstring theory. These three properties are actually “smallness” properties in the sense ofconformal field theory and string theory. These properties allow one to say that V ♮ essentially defines the smallest possible nontrivial string theory (cf. [Ha],[Na] and the Introduction in [FLM5]). (These “smallness” properties essen-tially amount to: “no nontrivial representation theory,” “no nontrivial gaugegroup,” i.e., “no continuous symmetry,” and “no nontrivial monodromy”; thislast condition actually refers to both the first and third “smallness” properties.)Conversely, conjecturally [FLM5], V ♮ is the unique vertex operator algebrawith these three “smallness” properties (up to isomorphism). This conjectureturns out to be very hard to prove (without additional strong hypotheses);in any case, it remains unproved. It would be the conformal-field-theoreticanalogue of the uniqueness of the Leech lattice in sphere-packing theory andof the uniqueness of the Golay code in error-correcting code theory, mentionedabove. Proving this uniqueness conjecture can be thought of as the “zerothstep” in the program of classification of (reasonable classes of) conformal fieldtheories. M. Tuite [Tu] has related this conjecture to the genus-zero property inthe formulation of monstrous moonshine. With additional (strong) hypothesesassumed, uniqueness results have been proved by Dong-Griess-C. H. Lam [DGL]and by Lam-H. Yamauchi [LY].Up to this conjecture, then, we have the following remarkable characteriza-tion of the largest sporadic finite simple group:
The Monster is the automor-phism group of the smallest nontrival string theory that nature allows, or moreprecisely, the automorphism group of the vertex operator algebra with the canon-ical “smallness” properties. (As I mentioned above, space-time is 2-dimensionalfor this “toy-model” string theory. Bosonic 26-dimensional space-time is “com-pactified” on 24 dimensions, using the orbifold construction V ♮ ; again cf. [Ha],[Na] and the Introduction in [FLM5].) Note that (up to the conjecture) the“smallness” properties characterize the vertex operator algebra, but in order toactually construct it and to construct its automorphism group one needs thework in [FLM5] or the equivalent.This definition of the Monster in terms of “smallness” properties of a ver-tex operator algebra provides a remarkable motivation for the definition of theprecise notion of vertex (operator) algebra. The discovery of string theory (asa mathematical, even if not necessarily physical) structure sooner or later mustlead naturally to the question of whether this “smallest” possible nontrivial ver-tex operator algebra V ♮ exists, and the question of what its symmetry group(which turns out to be the largest sproradic finite simple group) is. And on the8ther hand, the classification of the the finite simple groups— a mathematicalproblem of the absolutely purest possible sort —leads naturally to the questionof what natural structure the largest sporadic group is the symmetry group of;the answer entails the development of string theory and vertex operator alge-bra theory (and involves modular function theory and monstrous moonshine aswell). The Monster, a singularly exceptional structure—in the same spirit thatthe Lie algebra E is “exceptional,” though M is far more “exceptional” than E —helped lead to, and helps shape, the very general theory of vertex operatoralgebras. (The exceptional nature of structures such as E , the Golay code andthe Leech lattice in fact played crucial roles in the construction of V ♮ , as isexplained in detail in [FLM3] and [FLM5].)Incidentally, whatever the ultimate role of string theory turns out to be inphysics, string theory is here to stay; string theory has been “experimentallytested” very successfully—in mathematics (whether string theory is done byphysicists or mathematicians or both), and in many, many ways, going farbeyond what I have been discussing.The results in [FLM5] include that V ♮ is defined over the field of real num-bers, and in fact over the field of rational numbers, in such a way that theMonster preserves the real and in fact rational structure, and that the Monsterpreserves a rational-valued positive-definite symmetric bilinear form on this ra-tional structure. More recent proofs that V ♮ is a vertex operator algebra havebeen found—by L. Dolan-P. Goddard-P. Montague [DGM], by Y.-Z. Huang[Hua3] and by M. Miyamoto [Mi]. (The proof in [FLM5] is perhaps still theshortest; any complete proof must include the full construction itself.) Huang’sproof of (the hard part of) the vertex-operator-algebra property of V ♮ uses thetensor product theory for modules for a (suitable) vertex operator algebra; I’llmention this later. Y. Kawahigashi and R. Longo [KLo] have interpreted the“orbifold” construction of V ♮ in terms of algebraic quantum field theory, specif-ically, in terms of local conformal nets of von Neumann algebras on the circle.The Monster is not the only sporadic finite simple group to which a vertex-operator-algebraic structure has been attached. G. H¨ohn [Ho] has constructeda vertex-operator-superalgebraic structure for the Baby Monster, which is in-volved in the Monster. Also, J. Duncan [Du1] has done so for the Conwaygroup Co , and has proved the uniqueness of the structure. Evidence for theexistence of such a structure was given in [FLM3]. See also Borcherds-A. Ryba[BR]. In a remarkable development, Duncan ([Du2], [Du3]) has constructed twovertex-algebraic structures for a sporadic group not involved in the Monster,namely, the Rudvalis group, yielding moonshine-type phenomena, including agenus-zero property. This supports the hope, expressed in [FLM5], that all thesporadic groups (as well as all the other finite simple groups) can eventually bedescribed in vertex-algebraic terms.So, exactly what is a vertex operator algebra? And what are vertex opera-tors? First of all, with what is now understood, vertex operators are (or rathercorrespond to) elements of vertex operator algebras , by analogy with how (forexample) vectors are elements of (abstract) vector spaces ; the notion of vectorspace is of course in turn defined by an axiom system. But before abstract9ector spaces had been formalized, vectors already “were” something (little ar-rows, etc.), and this of course helped motivate the eventual axiom system forthe notion of vector space. Here is (an oversimplified version of) what vertexoperators “already were”:In string theory and conformal field theory, when two (closed) strings inter-act at a “vertex,” one has a standard picture that looks like a “pair of pants,”conformally equivalent to a three-punctured Riemann sphere (after a suitableinterpretation). Such a picture is the string-theoretic analogue of a simple Feyn-man diagram that looks like the letter “Y”—a schematic diagram for two incom-ing particles interacting at a “vertex” and producing one outgoing particle. Inthe “pair of pants,” the singularity of the vertex in traditional (point-particle)quantum field theory is replaced by a smooth Riemann surface. This allowsstring theory to avoid the “ultraviolet divergences” in point-particle quantumfield theory. In string theory and conformal field theory, such “Riemann-surfacevertex diagrams” get “represented” by “vertex operators” acting on suitableinfinite-dimensional vector spaces; vertex operators “describe” the particle (orrather, string) interactions in a given conformal field theory model. Geomet-ric relations among Riemann-surface “diagrams” are reflected by algebraic andanalytic relations among vertex operators.I’ll next give a concrete example of a vertex operator, as it arose in mathe-matics:For certain mathematical reasons, with R. Wilson [LW1] we focused on the“philosophical” problem of trying to construct the affine Kac-Moody Lie algebra A (1)1 = d sl (2) = sl (2) ⊗ C [ t, t − ] ⊕ C c (4)(the “affinization” of the Lie algebra sl (2) of 2-by-2 matrices of trace 0), withLie brackets given by[ a ⊗ t m , b ⊗ t n ] = [ a, b ] ⊗ t m + n + tr( ab ) mδ m + n, c for a, b ∈ sl (2) and m, n ∈ Z and [ c, d sl (2)] = 0 , as some kind of “concrete” Lie algebra of (as-yet unknown) operators on somekind of “concrete” space.In fact, our main reasons for formulating and trying to solve this problemwere that we already knew some pieces of evidence, stemming from joint workwith S. Milne [LM] and with A. Feingold [FL], that such a construction mighteventually shed light on the classical Rogers-Ramanujan combinatorial iden-tities. One of these two identities states that the number of partitions of anonnegative integer n into parts congruent to 1 or 4 mod 5 equals the numberof partitions of n into parts whose successive differences are at least 2, and theother of these identities states that the number of partitions of a nonnegativeinteger n into parts congruent to 2 or 3 mod 5 equals the number of partitionsof n into parts whose successive differences are at least 2 and such that the10mallest part is at least 2. These two theorems have a long and interesting his-tory and are highly nontrivial; cf. [A1]. When these two identities are writtenin their original, classical, generating-function form (cf. [A1]), each of themasserts the equality of two formal power series ( q -series)—one of them a formalinfinite product in q and the other a formal infinite sum. The work [LM], whichused the Weyl-Kac character formula [Ka], showed that the product sides of thetwo Rogers-Ramanujan identities had something interesting to do with stan-dard (= integrable highest weight) d sl (2)-modules of levels 1 and 3; the “level”is the scalar by which the central element c in (4) acts. It seemed natural totry to “construct” these standard modules somehow, starting with the level 1standard modules (the “basic” modules). The hope was to try to “discover” thesum sides of the Rogers-Ramanujan identities, somehow, in the level 3 standardmodules. The Rogers-Ramanujan identities had been proved many times, butthe question now was: What do the sum sides of the Rogers-Ramanujan identi-ties “count,” in this new context? (Classically, they count partitions satisfyingthe difference-two condition.)Here is the result of the work [LW1] (expressed in different, but equivalent,notation): Consider the (commutative associative) algebra S = C [ y , y , y , . . . ]of polynomials in the formal variables y n , n = , , . . . . Form the expression Y ( x ) = exp X n = , , ,... y n n x n exp − X n = , , ,... ∂∂y n x − n , where “exp” is the formal exponential series and x is another formal variablecommuting with the y n ’s. The y n ’s (understood as multiplication operators on S ) can be thought of as “creation operators” and the ∂∂y n ’s as “annihilationoperators,” acting on the “Fock space” S , using some terminology from quan-tum field theory. Together with the identity operator on the space S , they spanan (infinite-dimensional) Heisenberg Lie algebra acting on S ; the commutatorsamong these operators are the classical Heisenberg commutation relations, oninfinitely many generators. The operator Y ( x ) is a well-defined formal differ-ential operator in infinitely many formal variables, including the extra variable x . Viewing Y ( x ) as a generating function with respect to the formal variable x ,we write Y ( x ) = X j ∈ Z A j x − j , thus giving a family of (well-defined) linear operators A j , j ∈ Z , acting on S . Each A j can be computed, as a certain formal differential operator, in theform of an infinite sum of products of multiplication operators with partialdifferentiation operators, multiplied in this order; this infinite sum actually actsas a finite sum when applied to any given element of the space S . The explicitexpression for each A j is in fact complicated, and while one can write it down11xplicitly, one does not want to have to do this, although in our original work wedid in fact find these explicit formal differential operators A j “directly”; it wasonly after the fact that we realized that if we added up all of these complicatedoperators A j and thus formed their generating function as above, then all ofthese operators A j could be described by the single product of exponentials Y ( x ), which looked much simpler than any of the individual operators A j .The main point of this was: Theorem 1 [LW1] The operators , y n , ∂∂y n (cid:16) n = 12 , , , . . . (cid:17) and A j (cid:16) j ∈ Z (cid:17) (1 is the identity operator on the space S ) span a Lie algebra of operators actingon S , that is, the commutator of any two of these operators is a (finite) linearcombination of these operators, and this Lie algebra is a copy of the affine Liealgebra d sl (2) . This operator Y ( x ) turned out to be a variant of the vertex operators thathad arisen in string theory, as H. Garland pointed out (although that is not howwe had found it). It turned out that vertex operators and symmetry are closelyrelated. Instances of this had already been discovered in physics, including theworks [H] and [BHN].With what is now known, this operator Y ( x ) is an example of a “twistedvertex operator” (a vertex operator appearing in a twisted module for a vertexoperator algebra). This particular vertex operator construction happens to beamong the (many) ingredients playing crucial roles in the construction of V ♮ ,as are the untwisted vertex operator constructions of I. Frenkel-V. Kac [FK]and G. Segal [S1] as well. These and other vertex operator constructions alsoenter into a variety of other, very different, mathematical problems. For in-stance, the operator Y ( x ) above was interpreted by E. Date, M. Kashiwara andT. Miwa [DKM] to be precisely the infinitesimal B¨acklund transformation forthe Korteweg-de Vries hierarchy of differential equations in soliton theory; inHirota’s bilinear formalism, Y ( x ) generates the multi-soliton solutions.In Theorem 1, the d sl (2)-module that is constructed using the twisted vertexoperator Y ( x ) is one of the two basic (level 1 standard) d sl (2)-modules. Theother one of the two basic modules is constructed in exactly the same way, butwith each operator A j in Theorem 1 replaced by − A j (or equivalently, with thegenerating-function Y ( x ) of the operators A j replaced by its negative − Y ( x )).Further work with Wilson led to the construction of structures we called“ Z -algebras” [LW2]–[LW4], which provided a vertex-operator-theoretic inter-pretation and proof the Rogers-Ramanujan identities (mentioned above), in thefollowing way (very briefly):The (higher-level) standard d sl (2)-modules of level k > k copies of basic modules, but it was an openproblem to construct these higher-level modules “concretely,” by exhibiting12atural bases of them. The work [LM] led to a natural conjecture that theRogers-Ramanujan identities should “take place” inside the level 3 standard d sl (2)-modules, with a structure that looks like a copy of the basic d sl (2)-modulesomehow “factored out,” as a tensor factor. It turned out that the HeisenbergLie subalgebra of d sl (2) entering into Theorem 1 acts completely reducibly oneach standard module L , and that the “vacuum space” Ω L in L for the actionof this Heisenberg subalgebra (the subspace of L annihilated by all the “annihi-lation operators” mentioned above) has an easily-computed graded dimension.This vacuum space Ω L implemented the desired “factoring out,” and in casethe level of L is 3, the q -series that are the graded dimensions of the spacesΩ L are exactly the product sides of the two Rogers-Ramanujan identities. Thenext problem appeared to be to find a basis of the vacuum space Ω L (for alevel 3 standard module) that would exhibit the graded dimension of Ω L as the sum side of the corresponding Rogers-Ramanujan identity—the classical q -seriesgenerating function of the number of partitions of n satisfying one of the twodifference-two conditions mentioned above.Using the type of structure involved in Theorem 1, in [LW2]–[LW4] we even-tually constructed certain operators that commute with the action of the Heisen-berg Lie subalgebra of d sl (2), acting on suitable modules, and we called theseoperators “ Z -operators” (“ Z ” referring to the centralizing of this subalgebra).A typical Z -operator is a generating function of the shape Z ( x ) = X j ∈ Z Z j x − j , where each Z j is an operator of “degree” j that (because of the centralizingproperty) preserves Ω L . The goal seemed to be to prove that the vacuum spaceΩ L , for a level 3 standard module, has a basis of the form Z j Z j · · · Z j n , j i < , (5)with j ≤ j − , j ≤ j − , . . . , j n − ≤ j n − , (6)and with j n ≤ − j i play the role of the negatives of the parts in apartition of a nonnegative integer.) This difference-two condition on the indicesof such a basis monomial would exhibit the graded dimension of Ω L as the sumside of a Rogers-Ramanujan identity, and this would solve the problem. Butachieving this would require the “straightening” of monomials (5) to obtain theinequalities (6) on the indices. Such “straightening” could be accomplished ifthere were good enough algebraic relations involving the generating function Z ( x ), such as perhaps a commutator formula for [ Z ( x ) , Z ( x )].But there is no such commutator formula. Instead, what turned out tobe possible was the construction of “generalized commutation relations” of theshape A ( x , x ) Z ( x ) Z ( x ) − B ( x , x ) Z ( x ) Z ( x ) = C ( x , x ) , (7)13here A ( x , x ) and B ( x , x ) are suitable formal expansions of suitable formalalgebraic functions, and C ( x , x ) is some operator that is “simpler than” both Z ( x ) Z ( x ) and Z ( x ) Z ( x ). The term “generalized” refers to the presence ofthe formal algebraic functions A and B ; in the case when these are 1, thenone of course has ordinary commutation relations. Also, there are “generalizedanticommutation relations,” of a generally still-more-complicated shape, thatinvolve arbitrary numbers of generating functions Z ( x i ) in general. Each ofthese generalized commutation and anticommutation relations can be viewedas the generating function of an infinite family of relations among monomialsin the operators Z j ; each such relation among such monomials is of the follow-ing form: A (well-defined) formal infinite linear combination of monomials inthe Z j ’s, with coefficients coming from the coefficients of the formal algebraicfunctions such as A and B , is equated with a “simpler” expression involving the Z j ’s. (Again, if the formal algebraic functions A and B are 1 in (7), then eachsuch relation among the Z j ’s is a commutation relation of the following form:[ Z j , Z j ] equals a simpler expression such as perhaps a multiple of a single Z j or a scalar.) These various types of generalized commutation and anticommu-tation relations entering into the solution of the present problem are detailed in[LW2]–[LW4]. What they were used for was to “straighten” monomials (5) toobtain the difference-two condition (6) on the subscripts, in the case of the thelevel 3 standard d sl (2)-modules. This showed that the monomials (5) satisfyingthe difference-two condition span the vacuum space Ω L . Much more work wasneeded to prove that these difference-two monomials are also linearly indepen-dent , thus proving that they form a basis. Once these difference-two monomialswere proved to form a basis, this vertex-operator interpretation and proof of theRogers-Ramanujan identities was complete.I had mentioned generalized commutation relations when I was discussinghow certain such relations among certain vertex operators gave a natural “com-mutative affinization” of the (enlargement of) the Griess algebra acting on V ♮ ,in [FLM2], [FLM5]. In that situation, the formal algebraic functions A and B in (7) are particularly (but deceptively!) simple: A ( x , x ) = B ( x , x ) = x − x . (8)That is, the relations yielding the commutative affinization of the 196884-dimensional enlargement B of the Griess algebra in V ♮ are of the shape( x − x )[ Z ( x ) , Z ( x )] = simpler , (9)where in this formula, the generating functions Z i ( x j ) now refer to the rele-vant vertex operators in [FLM2], [FLM5]. These relations, which we called“cross-bracket” relations in [FLM2], [FLM5], exhibit the desired commutativeaffinization, once one has constructed the space V ♮ .The process discussed above for the level 3 standard d sl (2)-modules was ex-tended to all the standard d sl (2)-modules in [LW2]–[LW4], and the result was acorresponding vertex-algebraic interpretation of a known family of generaliza-14ions of the Rogers-Ramanujan identities, which had been discovered by B. Gor-don, G. Andrews and D. Bressoud. Many of these identities are treated in [A1].The case of the level 2 standard d sl (2)-modules actually gave certain infinite-dimensional Clifford algebras, and correspondingly, a “difference-one” condition,familiar in a natural basis of an exterior algebra, rather than a “difference-twocondition.” It was for this reason that we thought of the general phenomenon,arising now for all the standard d sl (2)-modules, as the emergence of a new “gen-eralized Pauli exclusion principle” for a natural family of operators generalizingclassical Clifford-algebra operators that are familiar in quantum mechanics andquantum field theory and producing fermionic particles; starting at level 3,fermionic (“difference-one”) statistics changed into “difference-two statistics,”and for the levels greater than 3, “difference-two-at-a-distance statistics,” re-flecting the sum sides of the Gordon-Andrews-Bressoud identities. (See [LW2]–[LW4].)However, we were unable to prove the linear independence of the relevantmonomials, analogous to those (for the level-3 case) in (5), (6), for the levelsgreater than 3. This problem was solved by A. Meurman and M. Primc [MP1],who thus provided a vertex-algebraic proof of the Gordon-Andrews-Bressoudidentities beyond the case of the Rogers-Ramanujan identities. Also, C. Husu[Hus1] discovered an elegant “multi-Jacobi-identity” interpretation and proof ofthe complicated generalized anticommutation relations in [LW2], [LW4].The Z -algebra viewpoint in [LW2]–[LW4] was used by K. Misra ([Mis1]–[Mis4]), M. Mandia [Ma], C. Xie [X], S. Capparelli [Ca1] and M. Bos-K. Misra[BosMis] to construct difference-two-type bases for the vacuum spaces Ω L forcertain standard modules for a range of affine Lie algebras, giving still more in-terpretations and proofs of the Rogers-Ramanujan and Gordon-Andrews-Bressoudidentities, and Bos [Bos] proved that in fact the complete list of such occurrencesof the Rogers-Ramanujan identities consists of certain low-level standard mod-ules for the affine Kac-Moody algebras A (1)1 , A (1)2 , A (2)2 , A (2)7 , C (1)3 , F (1)4 and G (1)2 ; all these cases are covered by the papers mentioned.One goal was to discover new identities with these ideas. In his Ph.D. the-sis research at Rutgers, S. Capparelli set out to construct Z -algebra bases ofthe vacuum spaces Ω L for the standard A (2)2 -modules of level 3, and he suc-ceeded in obtaining a construction, and a pair of partition identities, but onlyas a conjecture, because he had not yet completed his Z -algebra proof of thelinear independence of his spanning sets of Z -operator monomials in the vac-uum spaces. (He completed this in [Ca2], and M. Tamba-C. Xie [TX] did so aswell.) Meanwhile, in a talk at the Rademacher Centenary Conference in 1992, Imentioned Capparelli’s still-conjectured identities, or rather, one of them, in asurvey of the research program. After confirming that the identity was indeednew, Andrews proved it, before the conference had ended (see [A2]), and soonafterward, K. Alladi, Andrews and Gordon formulated and proved a refinementand generalizations of it [AAG].A central goal of these ideas was to find interesting new structure. The Z -algebra constructions mentioned so far are based on a certain twisting, the15ame twisting as in [LW1]. In joint work with Primc [LP], we developed Z -algebras in the untwisted case, and using these and related ideas in this analo-gous but still surprisingly different setting, we constructed combinatorial basesexhibiting “difference-two-at-a-distance generalized fermionic statistics,” andcorresponding (“fermionic”) character formulas, for the higher-level standard d sl (2)-modules. In a sequel [MP3] to [MP1], Meurman and Primc discovered andexploited new structure related to [LP].A. Zamolodchikov-and V. Fateev introduced “nonlocal parafermion cur-rents” in [ZF1] and a twisted analogue in [ZF2], and these conformal-field-theoretic constructions turned out to be essentially equivalent to the untwistedand twisted Z -algebra constructions in [LP] and [LW2], respectively; see [DL1],[DL2] for the untwisted case, and [Hus1] for the twisted case. The “nonlocality”in Zamolodchikov-Fateev’s terminology refers to the fact that in the notationabove, the formal algebraic functions A ( x , x ) and B ( x , x ) in (7) are (for-mally) multiple-valued, and the term “parafermion” refers to the generalizationof fermionic statistics mentioned above, or more precisely, to the reflection ofsuch statistics in the form of the formal algebraic functions A and B . In the ter-minology of [DL2], the Z -algebra structures generate certain examples of “gener-alized vertex algebras” and “abelian intertwining algebras,” whose main axiomis a “generalized Jacobi identity,” incorporating the relevant formally-multiple-valued formal algebraic functions. Notions essentially equivalent to generalizedvertex algebras were also introduced by A. Feingold-I. Frenkel-J. Ries [FFR] andG. Mossberg [Mos]. The introductory material in [DL2] includes discussions ofthese developments.One of the fundamental methods classically used to study and prove theRogers-Ramanujan identities was the Rogers-Ramanujan recursion, as explainedin [A1]. But this recursion did not arise in any of the vertex-operator work onpartition identities that I have mentioned. More recently, in joint work workwith Capparelli and A. Milas ([CapLM1], [CapLM2]), using vertex operatoralgebra theory in the context of “principal subspaces” ([FS1], [FS2]) of standardmodules, we have been able to incorporate these into the theory, as well as themore general Rogers-Selberg recursions, satisfied by the sum sides of the Gordon-Andrews identities. This leads to new questions, being addressed in work ofC. Calinescu [Cal1], [Cal2] and joint work with Calinescu and Milas [CalLM].The main theme here is to use intertwining operators (which I’ll mention below)among modules for vertex (operator) algebras to find new structure. In fact,perhaps the main theme of all the work I’ve mentioned on partition identities isto use known and sometimes new identities as clues to look for interesting newstructure.Now I’ll give the definition of the notion of vertex operator algebra. Theconsiderations I’ve mentioned (and related additional ones) led to the followingvariant ([FLM5] and [FHL]) of Borcherds’s definition of the notion of vertexalgebra; in this definition we use commuting independent formal variables x , x , x , etc.: 16 efinition 1 A vertex operator algebra consists of a Z -graded vector space V = M n ∈ Z V ( n ) (this grading, by conformal weights , is “shifted” from the grading we have al-ready been using on V ♮ ; that grading is adapted to the modularity propertiesof the generating functions, including the J -function, that we have mentioned,while the present grading is adapted to the action of the Virasoro algebra onthe vertex operator algebra, as we mention below) such thatdim V ( n ) < ∞ for n ∈ Z ,V ( n ) = 0 for n sufficiently negative , equipped with a linear map (the “state-field correspondence” ) V → (End V )[[ x, x − ]] v Y ( v, x ) = X n ∈ Z v n x − n − (where v n ∈ End V , the algebra of linear operators on V ), Y ( v, x ) denotingthe vertex operator associated with v (the letter “ Y ” happens to look like thevertex Feynman diagram that we mentioned above), and equipped also with twodistinguished homogeneous vectors ∈ V (0) (the vacuum vector ) and ω ∈ V (2) (the conformal vector ). The axioms are: For u, v ∈ V , u n v = 0 for n sufficiently large(the truncation condition ); Y ( , x ) = the identity operator on V ;the creation property : Y ( v, x ) has no pole in x and its constant term is the vector v (which implies that the state-field correspondence is one-to-one); the Virasoroalgebra relations :[ L ( m ) , L ( n )] = ( m − n ) L ( m + n ) + m − m δ n + m, c for m, n ∈ Z , where c ∈ C (the central charge ) and where Y ( ω, x ) = X n ∈ Z L ( n ) x − n − (i.e., the conformal vector generates the Virasoro algebra); L (0) v = nv n ∈ Z and v ∈ V ( n ) ; ddx Y ( v, x ) = Y ( L ( − v, x );and finally, the main axiom (the vast bulk of the definition of vertex operatoralgebra), the Jacobi identity : Writing δ ( x ) = X n ∈ Z x n , (10)the “formal delta function” (this really is a natural analogue of the Dirac deltafunction), we have: x − δ (cid:18) x − x x (cid:19) Y ( u, x ) Y ( v, x ) − x − δ (cid:18) − x + x x (cid:19) Y ( v, x ) Y ( u, x )= x − δ (cid:18) x − x x (cid:19) Y ( Y ( u, x ) v, x ) , (11)where each binomial, such as for example ( x − x ) − n , occurring in the ex-pansions here is understood to be expanded in nonnegative integral powers ofthe second variable; the truncation condition insures that all the expressionshere are well defined. The identity (11) is actually the generating function ofan infinite list of identities among operators on V , obtained by equating thecoefficients of all the monomials in x , x and x .The “physical” terms mentioned in this definition reflect the well-understoodrelation between this algebraic notion and the notion of “chiral algebra” inconformal field theory. The “language” of this definition, in particular, of theJacobi identity, is “formal calculus,” which is discussed in detail in [FLM5],[FHL] and [LL], for instance. We first learned about the viewpoint of the formaldelta function (10) from Garland; see also [DKM].Formal calculus is completely different from the algebra of formal powerseries. Formal power series (with the powers of the formal variable(s) all non-negative), or formal Laurent series with the powers of the formal variable(s)truncated from below, form rings. They can be multiplied. But the formalseries in the “formal calculus” of vertex operator algebra theory are “doublyinfinite”; the powers of the formal variables in a given formal series can be arbi-trary negative or positive integers, as is the case in (10) and (11), and sometimeseven more generally, the powers of the formal variables in a given formal seriesmust be allowed to be arbitrary rational or even complex numbers. These gen-eral kinds of “doubly infinite formal Laurent series” and so on certainly do not form rings. Yet there is a highly-developed algebra (“formal calculus”) for suchformal series, and indeed, one needs this in vertex (operator) algebra theory andin particular in the representation theory (for instance, in the tensor product18heory for module categories for a suitable vertex operator algebra; I’ll mentionthis later). A typical pattern in the theory is: First one does formal calculus,and then one (suitably and systematically) specializes the formal variables tocomplex variables and proves (analytic) convergence, for those results that needto be formulated analytically.Now that we have the precise definition of the notion of vertex operatoralgebra, we have a canonical definition of the Monster without reference to finitegroup theory: It is the symmetry group of the (conjecturally) unique vertexoperator algebra (a structure satisfying the Jacobi identity and the “relativelyminor” axioms) having the three “smallness” properties mentioned above. Thusthe Jacobi identity plus a few words determine the largest sporadic finite simplegroup.Notice that there is a certain resemblance between the “shape” of the Ja-cobi identity (11) and the (schematic) generalized commutation relation (7);the three-variable formal-delta-function expressions in the left-hand side of (11)are analogous to the formal expressions A ( x , x ) and B ( x , x ) in (7). Thisis actually more than a coincidence. In [B1], Borcherds had discovered identi-ties equivalent, in retrospect, to certain consequences of (11). These includedin particular a formula for the commutator [ Y ( u, x ) , Y ( v, x )], which can beobtained from (11) by taking the (formal) residue with respect to the formalvariable x (that is, the coefficient of x − ) of each of the two sides of (11) andequating the results, as well as a formula for Y ( Y ( u, x ) v, x ), which can beobtained by taking the residue with respect to the formal variable x of each ofthe two sides of (11) and equating the results. In [B1], these and other identi-ties for a vertex algebra were actually expressed in component form (that is, interms of the component operators v n of the vertex operators Y ( v, x )) instead ofgenerating-function form, but they could be recast in generating-function form.Also, Borcherds used the second one of these two identities, together with cer-tain other axioms, not including his commutator formula, is his definition of“vertex algebra.” When expressed, in retrospect, using the formal variables of(11), each of these two identities involves only two of the three formal variables x , x , x (the third variable being eliminated by the residue procedure).Now, the identity of the shape (9) that was needed for the commutativeaffinization, using “cross-brackets,” in [FLM2], [FLM5] suggested the formula-tion of a family of generalized commutation relations, with the left-hand sidesof the form ( x − x ) n Z ( x ) Z ( x ) − ( − x + x ) n Z ( x ) Z ( x ) (12)for all n ∈ Z (where the binomial expansion convention mentioned just after (11)is being used here), so that the case n = 0 would give a commutator formulaand the case n = 1 would give an expression for the cross-bracket of type (9). In[FLM5] we were in fact able to do this, and the most natural way to formulatethe resulting infinite family of identities was to put all these identities into asingle generating function, using a new formal variable x ; that is, there was19ow a formula for X n ∈ Z x − n − ( x − x ) n Z ( x ) Z ( x ) − X n ∈ Z x − n − ( − x − x ) n Z ( x ) Z ( x ) (13)(where the operators called Z i here are now the vertex operators entering intothe construction of V ♮ ). But (13) is exactly the left-hand side of (11) (with theoperators Z i now playing the roles of the vertex operators in (11)). In otherwords, (11) is the generating function of an infinite family of generalized com-mutation relations. Also, we were able to prove in [FLM5] that this generatingfunction actually equaled the right-hand side of (11), for all the vertex operatorsinvolved in V ♮ . Thus for us, in [FLM5] the idea for formulating, and proving, theJacobi identity arose from the axioms and properties that Borcherds wrote in[B1] together with the generalized-commutation-relation idea. Notice that thethree expressions in (11) are analogous, and even quite similar, to one another.In fact in [FHL], an explicit three-variable symmetry of the Jacobi identity wasformulated and analyzed. If one takes the residue with respect to any of thethree formal variables in (11), the three-variable symmetry is “broken,” andcorrespondingly, with (11) not yet known to us, it seemed natural to try to con-struct the identity (11), completing this natural symmetry. The identity (11)exhibits “all” the information. It was for these reasons that we were initiallythinking of (11) as the “master formula.” (As we said in the Introduction of[FLM5], Borcherds informed us that he too found this identity, and in fact itis implicit in [B1]. It is also implicit in the work of physicists, as we have beendiscussing.)But we decided instead in [FLM5] to call (11) the Jacobi identity because itis analogous to the Jacobi identity in the definition of Lie algebra: For u and v in a Lie algebra, (ad u )(ad v ) − (ad v )(ad u ) = ad((ad u ) v ) , where ad u is the operation of left-bracketing with the element u . While thevariables in (11) must be understood as formal and not complex variables (forinstance, note that in the formal delta function (10) itself, the formal variablecannot be specialized to a complex variable; the resulting doubly-infinite formalseries converges nowhere), it is in fact possible (and very important) to special-ize the formal variables to complex variables in certain systematic, and subtle,ways. This process is crucial, in particular, for making the connection betweenthe notion of vertex operator algebra and the notion of chiral algebra in confor-mal field theory, and it is also crucial for mathematical reasons. In fact, becauseof the fundamental complex-analytic geometry implicit in (11), suitably inter-preted, we also called (11) the “Jacobi-Cauchy identity” in [FLM5], for reasonsexplained in detail in the Appendix of [FLM5].The notion of vertex operator algebra, then, is indeed deeply analogous to thenotion of Lie algebra, and is actually the “one-complex-dimensional analogue”of the notion of Lie algebra (which is the corresponding “one-real-dimensional”notion, in this sense); this statement can be made precise using the language and20iewpoint of operads, which I will mention below in connection with Huang’swork incorporating the geometry underlying the Virasoro algebra into the struc-ture.In addition to being the generating function of an infinite family of generalized-commutation-relation identities, (11), as we mentioned above, is the generatingfunction of an infinite family of (generally highly-nontrivial) identities for thecomponent operators v n of the vertex operators in a vertex operator algebra V , one identity for each monomial in the three formal variables x , x and x . Each identity in this infinite list involves binomial coefficients, coming fromthe three formal delta-function expressions. It is the generating function formof these identities (namely, the Jacobi identity) that is the natural analogueof the Jacobi identity in the definition of Lie algebra. Even more basically, Y ( u, x ) is itself the generating function of the infinite family of operators u n acting on V , as we saw in the concrete example of the (twisted) vertex oper-ator Y ( x ) above (acting on the space S , in that case; as we emphasized, thegenerating function is much easier to work with than the individual operators A j ). Moreover, the single “ x -parametrized product operation” taking the or-dered pair ( u, v ), u, v ∈ V , to the generating function Y ( u, x ) v can certainly bethought of as specifying an infinite family of nonassociative product operations u n v on V , for n ∈ Z , corresponding to the powers of x . In very special cases,the “component identities” of the Jacobi identity include the relations definingaffine Lie algebras; the Virasoro algebra; the infinite-dimensional “affinization”of the (modified) Griess algebra B ; and a vast array of other remarkable alge-braic structures. In “formal calculus,” it is generally much more natural, andmuch easier, to work with formal delta functions, and with generating func-tions in general, rather than with individual components of vertex operatorsand individual relations among them. Generating functions of otherwise verycomplicated objects, such as nonassociative product operations, or operators ona space, or identities among such operators, pervade vertex operator algebratheory and allow one to work efficiently.There are many generalizations and analogues of (11), including twistedJacobi identities, as in [FLM5]; generalized Jacobi identities for abelian inter-twining algebras, etc., as in [DL1], [DL2], [FFR] and [Mos]; Huang’s much moresubtle identity for nonabelian intertwining algebras [Hua5]; multi-Jacobi iden-tities, as in [Hus1], [Hus2]; and “logarithmic analogues” of the Jacobi identity,both untwisted [L3] and twisted [DoyLM], which serve to “explain” and gener-alize work of S. Bloch [Blo] on the appearance of certain values of the Riemannzeta function that arose in certain vertex operator computations.We have emphasized that the notion of vertex operator algebra is actuallythe “one-complex-dimensional analogue” of the notion of Lie algebra. But atthe same time that it is the “one-complex-dimensional analogue” of the notionof Lie algebra, the notion of vertex operator algebra is also the “one-complex-dimensional analogue” of the notion of commutative associative algebra (whichagain is the corresponding “one-real-dimensional” notion). Again, operad lan-guage can be used to make this precise, as we comment below. The remarkable and paradoxical-sounding fact that the notion of vertex oper- tor algebra can be, and is, the “one-complex-dimensional analogue” of BOTH the notion of Lie algebra
AND the notion of commutative associative algebralies behind much of the richness of the whole theory, and of string theory andconformal field theory.
When mathematicians realized a long time ago thatcomplex analysis was qualitatively entirely different from real analysis (becauseof the uniqueness of analytic continuation, etc., etc.), a whole new point of viewbecame possible. In vertex operator algebra theory and string theory, there isagain a fundamental passage from “real” to “complex,” this time leading fromthe concepts of both
Lie algebra and commutative associative algebra to the con-cept of vertex operator algebra and to its theory, and also leading from pointparticle theory to string theory.This analogy with the notion of commutative associative algebra comesfrom the “commutativity” and “associativity” properties of the vertex oper-ators Y ( v, x ) in a vertex operator algebra, detailed in [FLM5] and [FHL], anddiscussed in many places, including the book [LL]. These properties are rootedin conformal-field-theoretic properties of vertex operators, as in [BPZ]; see [Go].In fact, the Jacobi identity (11) follows from the commutativity property, in thepresence of certain “minor” axioms; see [FLM5], [FHL], [Go]. The term “com-mutativity” actually refers to a certain “commutativity of left-multiplicationoperations,” which is why it can (and in fact does) imply associativity and theJacobi identity.It is natural to ask: Can the Jacobi identity axiom (11) in the definition ofvertex (operator) algebra be simplified?
As we have discussed, the Jacobi identityis actually the generating function of an infinite list of generally highly-nontrivialidentities, and one needs many of these individual component identities in work-ing with the theory . But is there some “simpler” condition that in fact impliesthe Jacobi identity (in the presence of the “minor” axioms in the definition ofvertex operator algebra)?In fact there is, and this simpler condition, which is related to the “com-mutativity” property, does indeed look much simpler than the Jacobi identityaxiom, but it turns out that the apparent simplicity is deceptive.This simple-looking replacement axiom is:For all u, v ∈ V , where V is a structure satisfying all the conditions inDefinition 1 except the Jacobi identity (11), there exists a nonnegative integer k such that ( x − x ) k [ Y ( u, x ) , Y ( v, x )] = 0 . (14)This “weak commutivity” condition, and also, more significantly, the theoremthat it implies the Jacobi identity (in the presence of “minor” axioms), werediscovered in [DL2], where (14) was actually a special case of a much more gen-eral condition, namely, the analogous assertion for generalized vertex algebrasand abelian intertwining algebras, mentioned above. In that greater generality,formal algebraic functions schematically called A and B in (7) replace the ex-pression ( x − x ) k in (14). All this is treated in [DL2], in the full generality. Thespecial case (14) is discussed in the Introduction of [DL2], formula (1.4). Theproof [DL2] that (14) implies the Jacobi identity, and that the generalizations22f (14) imply the corresponding generalized Jacobi identities, for generalizedvertex algebras and abelian intertwining algebras, are continuations of the ideathat “commutativity” implies the Jacobi identity. In [DL2], we were workingwith graded structures; just as a vertex operator algebra is Z -graded, a gen-eralized vertex algebra or abelian intertwining algebra is graded, too (actually Q -graded), and this grading was useful in the proof [DL2] that weak commu-tativity and its generalizations imply the corresponding (generalized) Jacobiidentities. Soon after [DL2], H. Li was able to remove the grading hypothesis,and in particular, he was able to prove that for a vertex algebra (without grad-ing), weak commutativity (14) implies the Jacobi identity, in the presence of“minor” axioms. This and a number of related results are covered in [LL].Notice that the condition (14) is reminiscent of the generalized commutationrelations discussed above, such as (7) and in particular, (9); in fact, (14) is ofcourse an example of a generalized commutation relation.The fact that such a simple-looking generalized commutation relation as (14)can serve as an axiom replacing the Jacobi identity in the definition of vertexoperator algebra is not as useful as it might seem. In fact, starting in [DL2]itself, we chose not to take (14) as an “official” replacement axiom, in spite of thefact that we proved, and stated, there that it could be taken as a replacementaxiom. There are essentially three reasons why we have chosen not to take(14) as an “official” axiom replacing the Jacobi identity: First, one needs “all”the information in the Jacobi identity (and in the relevant generalized Jacobiidentities, in the context of generalizations of the notion of vertex operatoralgebra). Second, if one wants to prove that a certain structure V is indeed avertex (operator) algebra, it is essentially just as hard to prove the condition(14) as it is to prove the Jacobi identity, for all u, v ∈ V . (In other words,the proof that (14) implies the Jacobi identity is quite short, and in particularis much simpler than the proof that either (14) or the Jacobi identity holdsfor all u, v ∈ V , for interesting examples of vertex (operator) algebras.) Andthird, (14) fails as a replacement axiom for the notion of module for a vertex(operator) algebra. In general, one can think of a module for an algebraicstructure as a space on which the algebra acts linearly, such that all the axioms(in the definition of algebra) that make sense hold; this principle is compatiblewith the standard definition of “module” for a Lie algebra and the standarddefinition of “module” for an associative algebra, for instance. This principleindeed motivates the standard definition of “module” for a vertex (operator)algebra, and using (14) in place of the Jacobi identity would not lead to thecorrect notion of module. Analogous comments hold in generalized settingssuch as abelian intertwining algebras, and also, twisted modules. For instance,in [FLM5], certain twisted Jacobi identities were proved, and these identitiesendowed certain “twisted” spaces with what came to be called twisted modulestructure (although the term “twisted module” was not used on [FLM5]); thistwisted module structure was necessary for the construction of V ♮ . (I mentionedabove that the vertex operator Y ( x ) entering into Theorem 1 is an example ofa “twisted vertex operator,” that is, a vertex operator appearing in a twistedmodule.) 23s we have been suggesting, in vertex operator algebra theory it is notori-ously difficult to construct nontrival examples of vertex operator algebras, evenexamples that are much simpler than V ♮ , and of course one cannot do the the-ory without examples; in fact, the theory is so rich because the examples are sorich. How can one efficiently construct families of examples of vertex operatoralgebras and their modules?
In classical algebraic subjects like group theory, Lie algebra theory, etc., oneof course has many interesting examples available from the beginning, guidingthe development of the general theory. In vertex operator algebra theory, thereare no essentially nontrivial examples that are easy to construct and prove theaxioms for. (The only “easy” examples of vertex algebras are commutative asso-ciative algebras equipped with derivations; see [B1].) This is yet another reasonwhy vertex operator algebra theory is inherently “nonclassical.” In “classical”mathematics, there simply were no nontrivial examples of vertex operator al-gebras “lying around waiting to be axiomatized,” in contrast with, say, vectorspaces, groups, Lie algebras, etc., etc.A conceptually elegant, extremely general and extremely convenient-to-usesolution of the problem of constructing families of examples of vertex operatoralgebras, and also modules for them, was developed by H. Li ([Li1], [Li2]), gener-alizing earlier constructions of examples of vertex operator algebras, including,among others, constructions of B. Feigin-E. Frenkel [FF] and I. Frenkel-Y. Zhu[FZ] (and constructions in [FLM5]). Briefly, Li formulated a subtle notion of representation of, as opposed to module for, a vertex operator algebra, by de-veloping the theory of a “vertex-algebraic analogue” of the notion of the usualendomorphism algebra End W of a vector space W and by defining a represen-tation of a vertex algebra V on W to be a (suitable kind of) homomorphism from V to this endomorphism-algebra structure on W . This endomorphism-algebrastructure has roots in conformal field theory. It has also been exploited in workof B. Lian and G. Zuckerman [LZ1], [LZ2]. Li’s analysis of this point of viewculminated in relatively-easy-to-implement sufficient conditions for constructingfamilies of vertex operator algebras and their modules. An enhanced treatmentof this work of Li’s is the main goal of the book [LL], which also highlightsgeneral theorems of E. Frenkel-V. Kac-A. Radul-W. Wang [FKRW], Meurman-Primc [MP2] and X. Xu [Xu] useful for constructing families of examples ofvertex (operator) algebras, including those based on the Virasoro algebra, thosebased on Heisenberg Lie algebras, those based on affine Lie algebras, and thosebased on lattices. (It happens, though, that such theorems do not serve to sim-plify the construction of the vertex operator algebra V ♮ , even though all thekinds of vertex operator algebras just mentioned do enter into the constructionof V ♮ .)The analogy between the notion of vertex operator algebra and the notionof commutative associative algebra is in fact directly related to the conformal-field-theory viewpoint. This analogy was pointed out by I. Frenkel [F], in theinitiation of a program to construct (geometric) conformal field theory usingvertex operator algebras. In [Hua1], [Hua4], Huang introduced a precise, and24eep, analytic-geometric notion of “geometric vertex operator algebra,” andestablished that it is equivalent to the (algebraic) notion of vertex operatoralgebra. Formulating and proving the geometric aspects of the action of theVirasoro algebra was the hard part, involving differential geometry and analysison infinite-dimensional moduli spaces; the infinite-dimensionality comes fromthe arbitrariness of analytic local coordinates at punctures on a Riemann sphere.The sewing of multipunctured Riemann spheres, with analytic local coordinatesvanishing at the punctures (both the “incoming” and “outgoing” punctures), isreflected in the structure of vertex-operator-algebraic operations. In particular,the formal variables are systematically specialized to complex variables, and onedoes extensive analysis in addition to extensive algebra.An interpretation—actually, restatement—of these constructions and the-orems of Huang, including some discussion of the principle mentioned abovethat the notion of vertex operator algebra is a natural “complexification” of thenotion of both Lie algebra and of commutative associative algebra, appears in[HL1], [HL2], in the language of operads. What is being “complexified” is the(one-real-dimensional) operads underlying both the notion of Lie algebra andthe notion of commutative associative algebra; the new analytic partial operadunderlying the notion of vertex operator algebra [Hua4] (cf. [HL1], [HL2]) isone-complex-dimensional. In fact, this one-complex-dimensional partial operad“dictates” the algebra of vertex operator algebra theory. More precisely, whenone takes into account the arbitrary local coordinates vanishing at the punc-tures, one really has an infinite-dimensional structure. The algebraic operationsare “mediated” by the infinite-dimensional analytic geometry of this partial op-erad, in a precise way; a vertex operator algebra becomes a representation ofthis analytic partial operad. Again, we are seeing a mathematical reflection ofa passage from point particle theory to string theory.In an extensive series of works, K. Barron ([Ba1]–[Ba6]) has carried out asophisticated super-geometric (super-conformal-field-theoretic) analogue of thiswork of Huang’s, using super-Riemann spheres with general superconformal lo-cal coordinates vanishing at the “incoming” and “outgoing” punctures, and us-ing vertex operator superalgebras endowed with the appropriate super-geometricstructure.As we have mentioned a number of times, in vertex operator algebra theory,it is crucial to make precise distinctions between formal variables and complexvariables. This distinction is particularly dramatic in this work of Huang andBarron. First they had to carry out elaborate formal algebra, and then theyhad to systematically specialize the (infinitely many) formal variables to com-plex variables to obtain the desired results. In classical mathematics, one is usedto being (appropriately!) “careless” about the distinction between formal andcomplex variables; for instance, one routinely writes formal expressions suchas P n ≥ x n without saying much about whether the variable x is supposed tobe formal or complex. In this simple example, it is so clearly understood thatthis geometric series converges for certain complex numbers x and not othersthat the notation P n ≥ x n can easily be used, in the same discussion, for ei-ther the formal sum or the convergent series, depending on what is being said.25owever, in vertex operator algebra theory, where one also needs both formaland complex variables, the formal algebra (and “formal calculus,” mentionedabove in connection with (11), in which the three formal variables cannot allbe specialized to complex variables) becomes so subtle and elaborate that it isnecessary to be very explicit about the distinction between the two kinds of vari-ables. Correspondingly, in Huang’s and Barron’s work, there are many formaltheorems, which cannot be initially and directly formulated in terms of complexvariables, and these theorems are then systematically applied to give analyticconsequences. One particular “formal” theorem that was proved by Huang andextended to the superalgebraic setting by Barron has been considerably gener-alized in [BHL], to a factorization theorem for formal exponentials, in a settinginvolving arbitrary infinitie-dimensional Lie algebras. This theorem reflects andgeneralizes the formal algebra required for the sewing of Riemann spheres orsuper-Riemann spheres with general coordinates vanishing at punctures thatare sewn together.Huang’s work on geometric vertex operator algebras is a major step of manyin a program to use vertex operator algebra theory and its representation the-ory to construct conformal field theories in the precise sense of G. Segal’s andM. Kontsevich’s definition of the (mathematical) notion of conformal field theory(see [S2]–[S4]); this definition was clarified by P. Hu-I. Kriz ([HuK1], [HuK2]).Geometric vertex operator algebras amount to the holomorphic, genus-zero partof the construction of conformal field theories. The term “genus-zero” now refersto the Riemann spheres mentioned above; it does not refer to the “genus-zeroproperty” of the discrete subgroups of SL (2 , R ) discussed earlier in connectionwith moonshine. This use of “genus zero” corresponds to conformal field theoryat “tree level,” in physics terminology.A major achievement of this program so far is Huang’s proof ([Hua7]–[Hua11]), in a general setting, of the Verlinde conjecture and his solution ofthe problem of constructing modular tensor categories from the representationtheory of vertex operator algebras. E. Verlinde conjectured [Ve] that certainmatrices formed by numbers called the “fusion rules” in a “rational” conformalfield theory are diagonalized by the matrix given by a certain natural action ofthe fundamental modular transformation τ
7→ − /τ on “characters.” A greatdeal of progress has been achieved in interpreting and proving Verlinde’s (phys-ical) conjecture and the related “Verlinde formula” in mathematical settings,in the case of the Wess-Zumino-Novikov-Witten models in conformal field the-ory, which are based on affine Lie algebras. On the other hand, G. Moore andN. Seiberg [MSe1], [MSe2] showed, on a physical level of rigor, that the gen-eral form of the Verlinde conjecture is a consequence of the axioms for rationalconformal field theories, thereby providing a conceptual understanding of theconjecture. In the process, they formulated a conformal-field-theoretic analogue,later termed “modular tensor category” by I. Frenkel, of the classical notion oftensor category for representations of (i.e., modules for) a group or a Lie alge-bra. A modular tensor category is in particular a braided tensor category thatis also rigid and “nondegenerate.”Now, there is a general tensor product theory for modules for a suitably26eneral vertex operator algebra, a theory based on intertwining operators [FHL]among modules (the dimensions of spaces of intertwining operators are the “fu-sion rules” mentioned above): The tensor product functors and appropriatestructure were constructed in [HL4]–[HL6], and in [Hua2] Huang proved a gen-eral operator-product-expansion theorem for intertwining operators, enablinghim to construct the natural associativity isomorphisms between suitable tensorproducts of triples of modules. (The existence of such an operator product ex-pansion was a key assumption —not theorem —in [MSe2].) The resulting braidedtensor category structure was enhanced in [HL3] to what we called “vertex tensorcategory” structure. This structure is much richer than braided tensor categorystructure. Vertex tensor category structure is “mediated” by the analytic partialoperad [Hua4], [HL1], [HL2], based on multipunctured Riemann spheres witharbitrary analytic local coordinates vanishing at the punctures, by analogy withhow the structure of ordinary, classical, braided tensor categories is “mediated”by operadic structure in one real dimension. That is, not only is the concept ofvertex operator algebra itself “based” on the one-complex-dimensional operadicstructure discussed above, but so is a tensor category theory [HL3] of modulesfor a (suitable) general vertex operator algebra. In particular, in place of a sin-gle tensor product functor, there is a natural family of tensor product functors,indexed by a power of the determinant line bundle over the moduli space ofthree-punctured Riemann spheres with analytic local coordinates vanishing atthe punctures, and the natural associativity isomorphisms among triple tensorproducts, and the coherence, are controlled by this geometric structure; this isexplained in [HL3]. In this tensor product theory, the underlying vector spaceof the tensor product of suitable modules for a vertex operator algebra is not the tensor product vector space of the modules. Instead, intertwining operatorsamong triples of modules form the starting point for a family of analytically-defined tensor product functors, and it is a subtle matter to construct the tensorproduct spaces.The work [HL3]–[HL6] and [Hua2] was originally inspired by the work ofD. Kazhdan and G. Lusztig, starting in [KLu1]–[KLu3], constructing a tensorproduct theory for certain categories of modules of a fixed non-positive-integrallevel for an affine Lie algebra. However, while the theory of [HL3]–[HL6] and[Hua2] applies to the module categories of many families of vertex operator al-gebras, this theory does not include [KLu1]–[KLu3] as a special case, becausethe modules considered by Kazhdan-Lusztig are not semisimple. But recently,in joint work with L. Zhang [HLZ], we have generalized the tensor product the-ory [HL4]–[HL6], [Hua2] to “logarithmic” tensor product theory, which indeedaccommodates suitable non-semisimple module categories. It turned out thethe work necessary for this generalization was considerable. In particular, thealready-intricate formal calculus necessary for [HL3]–[HL6] and [Hua2] had tobe extended to “logarithmic formal calculus,” and many of the arguments in[HL4]–[HL6], [Hua2] had to be replaced by new ones, for the generalization. Us-ing the work [HLZ], Zhang has succeeded [Zha] in recovering the braided tensorcategory of Kazhdan-Lusztig as a special case of the theory, and of endowing itwith vertex tensor category structure. 27o return to Huang’s work announced in [Hua7]— Under very general andnatural conditions on a simple vertex operator algebra V , Huang proved theVerlinde conjecture, and using this result, he proved the rigidity and in factmodularity of the braided tensor category constructed in [HL4]–[HL6], [Hua2].Zhu’s theorem [Z], which I mentioned above, on modular transformation prop-erties of the graded dimensions of modules, is necessary in the formulation ofthe Verlinde conjecture in this general setting, and Zhu’s method of proof playsan important role in the development of the genus-one theory. In Huang’sproof of the Verlinde conjecture, a crucial step was to prove the modular in-variance of spaces of multipoint correlation functions involving compositions of(multivalued) intertwining operators (as opposed to single-valued vertex opera-tors). For this, Huang had to develop a new, analytically-based method [Hua9],since Zhu’s method (mainly, his use of the commutativity property of vertexoperators) cannot be generalized to this multivalued setting. (This situationis actually somewhat analogous to the situation discussed above in connectionwith generalized commutation relations (7), where it was impossible to con-struct ordinary commutation relations, which would have been easier to use,if they had existed. In Huang’s proof of the Verlinde conjecture, however, thesituation is even more subtle.) Huang established natural duality and mod-ular invariance properties for genus-zero and genus-one multipoint correlationfunctions constructed from intertwining operators for a vertex operator algebrasatisfying general hypotheses, and as I have mentioned, the multiple-valuednessof the multipoint correlation functions led to considerable subtleties that had tobe handled analytically and geometrically, rather than just algebraically. Then,with these results having been established, the strategy of Huang’s proof of theVerlinde conjecture reflected the pattern of [MSe1], [MSe2]; he used these re-sults to establish two formulas that Moore and Seiberg had derived from strongassumptions, namely, the axioms for rational conformal field theory, which ofcourse cannot be assumed here. It is much harder to (mathematically) prove theaxioms in [S2]–[S4] or the axioms in [MSe2] for conformal field theory than it isto prove the Verlinde conjecture; in Huang’s work, the truth of Verlinde conjec-ture was needed to prove the desired properties of the tensor category. In turn,his proof of the general Verlinde conjecture required a large amount of (existingand new) representation theory of vertex operator algebras. The hypotheses ofthe theorems entering into Huang’s work are very general, natural and purelyalgebraic, and have been verified in a wide range of important examples, whilethe theory itself is heavily analytic and geometric as well as algebraic.In another important direction, a great deal of extensive and deep workhas been done in conformal field theory using algebraic geometry. I will onlymention the books of A. Beilinson and V. Drinfeld [BD], and of E. Frenkel andD. Ben-Zvi [FB]. The review [Hua6] of [FB] includes an interesting discussionof the relation between the algebro-geometric viewpoint and the viewpoint ofthe representation theory of vertex operator algebras.I would like to mention a certain inspiring article by M. Atiyah [A], inwhich among many other things he compares geometry and algebra; discussesthe interaction between mathematics and physics; and comments on string the-28ry and also on finite simple groups, and in particular the Monster and itsconnections with elliptic modular functions, theoretical physics and quantumfield theory—some of which we’ve actually been discussing. Among his manystimulating comments are these, about “the dichotomy between geometry andalgebra”: “Geometry is, of course, about space... Algebra, on the other hand...isconcerned essentially with time. Whatever kind of algebra you are doing, a se-quence of operations is performed one after the other, and ‘one after the other’means you have got to have time... Algebra is concerned with manipulation in time , and geometry is concerned with space . These are two orthogonal aspects ofthe world, and they represent two different points of view in mathematics. Thusthe argument or dialogue between mathematicians in the past about the relativeimportance of geometry and algebra represents something very fundamental.”In the spirit of what we’ve been discussing: While a string sweeps out a two-dimensional (or, as we’ve been mentioning, one-complex-dimensional) “world-sheet” in space-time, a point particle of course sweeps out a one-real-dimensional“world-line” in space-time, with time playing the role of the “one real dimen-sion,” and this “one real dimension” is related in spirit to the “one real dimen-sion” of the classical operads that I’ve briefly referred to— the classical operads“mediating” the notion of associative algebra and also the notion of Lie algebra(and indeed, any “classical” algebraic notion), and in addition “mediating” theclassical notion of braided tensor category. The “sequence of operations per-formed one after the other” is related (not perfectly, but at least in spirit) tothe ordering (“time-ordering”) of the real line. But as we have emphasized, the“algebra” of vertex operator algebra theory and also of its representation theory(vertex tensor categories, etc.) is “mediated” by an (essentially) one- complex -dimensional (analytic partial) operad (or more precisely, as we have mentioned,the infinite-dimensional analytic structure built on this). When one needs tocompose vertex operators, or more generally, intertwining operators, after theformal variables are specialized to complex variables, one must choose not merelya (time-)ordered sequencing of them, but instead, a suitable complex number, ormore generally, an analytic local coordinate as well, for each of the vertex oper-ators. This process, very familiar in string theory and conformal field theory, isa reflection of how the one-complex-dimensional operadic structure “mediates”the algebraic operations in vertex operator algebra theory. Correspondingly, “al-gebraic” operations in this theory are not instrinsically “time-ordered”; they areinstead controlled intrinsically by the one-complex-dimensional operadic struc-ture. The “algebra” becomes intrinsically geometric. “Time,” or more precisely,as we discussed above, the one-real-dimensional world-line, is being replaced bya one-complex-dimensional world-sheet. This is the case, too, for the vertextensor category structure on suitable module categories. In vertex operatoralgebra theory, “algebra” is more concerned with one-complex-dimensional ge-ometry than with one-real-dimensional time.As we have discussed, the Monster is indeed (very deeply) related to stringtheory. I mentioned above that one of the distinguishing features of the moon-shine module vertex operator algebra V ♮ is that its representation theory istrivial. This means in particular that the braided tensor category attached to29t is also trivial. 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