On effective field theories describing (2, 2) vacua of the heterotic string

Abstract

Classical vacua of the heterotic string corresponding to c = 9, N = (2,2) superconformal theories on the world sheet yield low-energy effective field theories with N = 1 space-time supersymmetry in four dimensions, gauge group E6 ⊗ E8, several families of 27 and 27 matter fields, and moduli fields. String theory relates matter fields to moduli; in this article we relate the kinetic terms in the effective lagrangian for both moduli and matter fields to the 273 and 273 Yukawa couplings. Geometrically, we recover the result (obtained previously via the type II superstring and N = 2 supergravity) that moduli space is a direct product of two Kähler manifolds of restricted type, spanned by the moduli related respectively to the 27 and 27 mattèr fields. The holomorphic functions of the moduli generating the two restricted Kähler metrics also determine the Yukawa couplings of the matter fields. We derive explicit formulae for the metric for the matter fields in terms of the metric for the corresponding moduli; the two metrics are not identical to each other. The precise relation between moduli and matter metrics takes a slightly different form on subspaces of the moduli space where the unbroken gauge symmetry is enhanced beyond E6 ⊗ E8; this phenomenon is illustrated using the examples of (2, 2) orbifolds and tensor products of minimal N = 2 theories.