How Vertex Algebras Are Almost Algebras: on a Theorem of Borcherds

1.1. Today, vertex algebras have become rather ubiquitous in mathematics. Among the subjects on which they bear direct influence, we can list: finite group theory, number theory (both through the classical theory of elliptic functions and the modern geometric Langlands theory), combinatorics, representation theory, and algebraic geometry. Before we discuss these relations, let us recall that despite their firm entrenchment within the world of pure mathematics, vertex algerbas (or rather their consituent elements vertex operators) actually first arose in the 1970s within the early string theory literature concerning dual resonance models [ref?]. Independently of this, [LW] Lepowsky and Wilson were interested in constructing representations of the affine Lie algebra ŝl2 using differential operators acting on the ring of infinite polynomials C[x1,x2, . . .]. The main difficulty in doing so was to construct representations of a certain infinite Heinsenberg algebra using rather complicated looking formulae. Howard Garland then noticed that these same formulae occurred within the string theory literature [S], and he christened the term "vertex operator" to the mathematical world. Shortly thereafter, Igor Frenkel and Victor Kac (and G. Segal independetly) made precise sense of Garland’s observation and constructed the most important representation of arbitrary affine Lie algebras, the so called basic representation, again using vertex operators to realize certain infinite Heinsenberg subalgebras. Thus within the world of representation theory of affine Lie algebras, vertex operators play a central role. What may come as a surprise is how pervasive vertex algebras have now become in mathematics.