Notion Of Vertex Operator Algebra Research Articles

Constructions of vertex operator coalgebras via vertex operator algebras

The notion of vertex operator coalgebra is presented which corresponds to the family of correlation functions modeling one string propagating in space–time splitting into n strings in conformal field theory. This notion is in some sense dual to the notion of vertex operator algebra. We prove that any vertex operator algebra equipped with a nondegenerate, Virasoro preserving, bilinear form gives rise to a corresponding vertex operator coalgebra. We then explicitly calculate the vertex operator coalgebra structure and unique bilinear form for the Heisenberg algebra case, which corresponds to considering free bosons in conformal field theory.

Factorization of Formal Exponentials and Uniformization

Let g be a Lie algebra over a field of characteristic zero equipped with a vector space decomposition g=g−⊕g+, and let s and t be commuting formal variables commuting with g. We prove that the map C: sg−[[s,t]]×tg+[[s,t]]→sg−[[s,t]]⊕tg+[[s,t]] defined by the Campbell–Baker–Hausdorff formula and given by esg−etg+=eC(sg−,tg+) for g±∈g±[[s,t]] is a bijection, as is well known when g is finite-dimensional over R or C, by geometry. It follows that there exist unique Ψ±∈g±[[s,t]] such that etg+esg−=esΨ−etΨ+ (also well known in the finite-dimensional geometric setting). We apply this to a Lie algebra g consisting of certain formal infinite series with coefficients in a Z-graded Lie algebra p, for instance, an affine Lie algebra, the Virasoro algebra, or a Grassmann envelope of the N=1 Neveu–Schwarz superalgebra. For p the Virasoro algebra, the result was first proved by Huang as a step in the construction of a geometric formulation of the notion of vertex operator algebra, and for p a Grassmann envelope of the Neveu–Schwarz superalgebra, it was first proved by Barron as a corresponding step in the construction of a supergeometric formulation of the notion of vertex operator superalgebra. In the special case of the Virasoro (resp., N=1 Neveu–Schwarz) algebra with zero central charge the result gives the precise expansion of the uniformizing function for a sphere (resp., supersphere) with tubes resulting from the sewing of two spheres (resp., superspheres) with tubes in two-dimensional genus-zero holomorphic conformal (resp., N=1 superconformal) field theory. The general result places such uniformization problems into a broad formal algebraic context.

GENUS-ZERO MODULAR FUNCTORS AND INTERTWINING OPERATOR ALGEBRAS

In [7] and [9], the author introduced the notion of intertwining operator algebra, a nonmeromorphic generalization of the notion of vertex operator algebra involving monodromies. The problem of constructing intertwining operator algebras from representations of suitable vertex operator algebras was solved implicitly earlier in [5]. In the present paper, we generalize the geometric and operadic formulation of the notion of vertex operator algebra given in [3, 4, 11, 12, 8] to the notion of intertwining operator algebra. We show that the category of intertwining operator algebras of central charge [Formula: see text] is isomorphic to the category of algebras over rational genus-zero modular functors (certain analytic partial operads) of central charge c satisfying a certain generalized meromorphicity property. This result is one main step in the construction of genus-zero conformal field theories from representations of vertex operator algebras announced in [7]. One byproduct of the proof of the present isomorphism theorem is a geometric construction of (framed) braid group representations from intertwining operator algebras and, in particular, from representations of suitable vertex operator algebras.

A supergeometric interpretation of vertex operator superalgebras

Conformal field theory (or more specifically, string theory) and related theories (cf. [BPZ], [FS], [V], and [S]) are the most promising attempts at developing a physical theory that combines all fundamental interactions of particles, including gravity. The geometry of this theory extends the use of Feynman diagrams, describing the interactions of point particles whose propagation in time sweeps out a line in space-time, to one-dimensional “particles” (strings) whose propagation in time sweeps out a two-dimensional surface. For genus zero holomorphic conformal field theory, algebraically, these interactions can be described by products of vertex operators or more precisely, by means of vertex operator algebras (cf. [Bo] and [FLM]). However, until 1990 a rigorous mathematical interpretation of the geometry and algebra involved in the “sewing” together of different particle interactions, incorporating the analysis of general analytic coordinates, had not been realized. In [H1] and [H2], motivated by the geometric notions arising in conformal field theory, Huang gives a precise geometric interpretation of the notion of vertex operator algebra by considering the geometric structure consisting of the moduli space of genus zero Riemann surfaces with punctures and local coordinates vanishing at the punctures, modulo conformal equivalence, together with the operation of sewing two such surfaces, defined by cutting discs around one puncture from each sphere and appropriately identifying the boundaries. Important aspects of this geometric structure are the concrete realization of the moduli space in terms of exponentials of a representation of the Virasoro algebra and a precise analysis of sewing using these resulting exponentials. Using this geometric structure, Huang then introduces the notion of geometric vertex operator algebra with central charge c ∈ C, and proves that the category of geometric vertex operator algebras is isomorphic to the category of vertex operator algebras. In [F], Friedan describes the extension of the physical model of conformal field theory to that of superconformal field theory and the notion of a superstring whose propagation in time sweeps out a supersurface. Whereas conformal field theory attempts to describe the interactions of bosons, superconformal field theory attempts to describe the interactions of boson-fermion pairs. This, in particular, requires an operator D such that D 2 = @ @z . Such an operator arises naturally in supergeometry. In [BMS], Beilinson, Manin and Schechtman study some aspects of superconformal symmetry, i.e., the Neveu-Schwarz algebra, from the viewpoint of algebraic geometry. In this work, we will take a differential geometric approach, extending Huang’s geometric interpretation of vertex operator algebras to a supergeometric interpretation of vertex operator superalgebras. Within the framework of supergeometry (cf. [D], [R] and [CR]) and motivated by superconformal field theory, we define the moduli space of super-Riemann surfaces with genus zero “body”, punctures, and local superconformal coordinates vanishing at the punctures, modulo superconformal equivalence. We announce the result that any local superconformal coordinates can be expressed in terms of exponentials of certain superderivations, and that these superderivations give a representation of the Neveu-Schwarz algebra with zero central charge. We define a