Algebraic constructions of code lattices in Narain conformal field theories

Abstract

We give new results on the structure and representations of the three lattices Λk,ΛkC,Λk* relevant to code conformal field theories realising Narain theories. In this construction, Λk* denotes the dual of the even lattice Λk and ΛkC is an even self dual intermediate lattice with (d, d) signature. We study the inclusion relations Λk⊂ΛkC⊂Λk* characterised by the discriminant group Λk*/Λk isomorphic to Zk. Explicitly constructions of these R(rd,rd) lattices are also provided first for rank r = d = 1 and then for higher dimensional Lie algebras with r = d > 1. Additional structural features and generalizations are also discussed.