Lattices and strings

Abstract

Self-dual lattices can be used to construct simple conformal field theories for “compactified” degrees of freedom in string theory. While these are by no means the most general conformal field theories that can be used, one does obtain in this way a large number of bosonic, heterotic and type-II theories that display nearly all interesting features of more general string theories. We present a pedagogical introduction to the construction and properties of these lattice theories, indicating generalizations whenever possible. While lattice constructions of bosonic strings are merely torus compactifications, heterotic and type-II strings provide more interesting possibilities, because one can include bosonized NSR-fermions and ghosts on the lattice. This covariant lattice construction of fermionic strings is explained in detail. We include a discussion of two subjects that are not limited to lattices, but originated from lattice constructions historically, namely the relation between exceptional algebras and space-time supersymmetry in string theory, and the relation between anomaly cancellation, modular invariance and elliptic genera. Three more mathematical topics are discussed in appendices, namely properties of self-dual lattices, Weyl groups and shift vectors, and modular functions such as multi-loop ϑ-functions for Lie algebra conjugacy classes as well as for spin structures.