ar X iv : h ep - t h / 99 08 06 7 v 2 2 N ov 1 99 9 ASPECTS OF HETEROTIC / M - THEORY DUALITY IN D = 3

We study the duality between M-theory compactified on Calabi-Yau fourfolds and the heterotic string compactified on Calabi-Yau threefolds times a circle. Our analysis is based on a comparison of the low energy effective actions in three dimensions. Non-perturbative N=1 vacua of the heterotic string in D=4 are of particular interest due to their phenomenological prospects [1]. A certain class of such vacua is best described by F-theory compactified on elliptic Calabi-Yau fourfolds Y4 [2, 3]. A closely related but technically simpler class of vacua is obtained by further compactifying the heterotic string on a circle S1 down to D=3; some of their non-perturbative properties are captured by M-theory compactified on Y4 [4]. In this short note we display the corresponding effective Lagrangians and their relation following a similar analysis carried out for the duality of type IIA on Y3 and the heterotic string on K3×T 2 in ref. [5]. A more detailed presentation of our results will be given in a forthcoming paper [6]. Our starting point is a generic effective Lagrangian of N=1, D=4 heterotic string vacua1 (using the conventions of ref. [7]) L = R 2 −G īj ∂mΦ̄ ∂Φ − 1 4 Refa(Φ)F a mnF amn + 1 4 Imfa(Φ)F a mnF̃ amn + . . . , (1) Presented by M. Haack. They can be constructed as compactifications on Y3 or more generally from appropriate (0,2) SCFTs. 1 2 where m,n = 0, . . . , 3. Φi are moduli fields including the (1,1) and (1,2) moduli of Y3 as well as moduli arising from the choice of the gauge bundle.2 G (4) īj is the Kähler metric which is determined by a Kähler potential K(4)(Φ, Φ̄) and the F a mn denote the field strengths of the (nonAbelian) gauge bosons Am. The S 1-reduction to D = 3 uses the Ansatz [8]: g mn = ( g (3) μν + eBμBν e Bμ eBν e 2σ )