Heterotic/heterotic duality in D = 6,4

Abstract

We consider E 8 × E 8 heterotic compactifications on K3 and K3 × T 2 . The idea of heterotic/heterotic duality in D = 6 has difficulties for generic compactifications since for large dilaton values some gauge groups acquire negative kinetic terms. Recently Duff, Minasian and Witten (DMW) suggested a solution to this problem which only works if the compactification is performed assuming the presence of symmetric gauge embeddings on both E 8 's . We consider an alternative in which asymmetric embeddings are possible and the wrong sign of kinetic terms for large dilaton value is a signal of spontaneous symmetry breaking. Upon further toroidal compactification to D = 4, we find that the duals in the DMW case correspond to N = 2 models in which the β-function of the different group factors verify β α = 12 whereas the asymmetric solutions that we propose have β α = 24. We check the consistency of these dualities by studying the different large T, S limits of the gauge kinetic function. Dual N = 1, D = 4 models can also be obtained by the operation of appropriate freely acting twists, as shown in specific examples.

Instanton numbers and exchange symmetries in N = 2 dual string pairs

“…At the end, let us briefly consider different Calabi-Yau spaces and also comment on the relation to the heterotic/heterotic duality in six dimensions [7,8], with the 6-dimensional heterotic string compactified on K 3 . (The decompactification limit from D = 4 to D = 6 is obtained by sending T → ∞ with U finite; as discussed in [20,7], the D = 6 heterotic/heterotic duality becomes an exchange symmetry of S with T in D = 4.)…”

“…Upon compactification to D = 4 on T 2 one arrives at the heterotic string with gauge group U(1) 4 , which is the dual to the considered type II string on the CY W P 1,1,2,8,12 (24). Semiclassically, at special points in the hypermultiplet moduli space, this gauge group can be enhanced to a non-Abelian gauge group, inherited from the E 7 × E ′ 7 with N = 2 β-function coefficient b α = 12(1 +ṽ α vα ) = 12 [8].…”

“…Next, consider embedding the SU(2) bundles in an asymmetric way into the two E 8 's [39,8]: s = 14, s ′ = 10. This corresponds to the elliptic fibration over F 2 [11,37].…”

“…We will also discuss the action of the S − T exchange symmetry in this context. At the end, we will comment on the relation of the four-dimensional exchange symmetry to the six-dimensional heterotic/heterotic duality symmetry [20,7,8] in this class of models.…”

The dilaton potential from

“…It has been found in orientifold models that the gauge couplings at tree level depend not only on S but also on the blowing-up modes M: f = S + kM and, depending on the dimension of the brane that hosts the gauge group, the gauge coupling may not depend on S at all but take the form f = T + kM [22]. The effective action after gaugino condensation will then be much richer than in the pure heterotic case because of this dependence [15,16].…”