Horizontal and Vertical Five-Branes in Heterotic/F-Theory Duality

We consider the heterotic string on an elliptic Calabi-Yau three-fold with five-branes wrapping curves in the base ('horizontal' curves) of the Calabi-Yau as well as some elliptic fibers ('vertical' curves). We show that in this generalized set-up, where the association of the heterotic side with the $F$-theory side is changed relative to the purely vertical situation, the number of five-branes wrapping the elliptic fibers still matches the corresponding number of $F$-theory three-branes.Comment: 21 pages harvmac, reference adde

Chiral matter and transitions in heterotic string models

In the framework of N=1 supersymmetric string models given by the heterotic string on an elliptic Calabi-Yau \pi :Z\ra B together with a SU(n) bundle we compute the chiral matter content of the massless spectrum. For this purpose the net generation number, i.e. half the third Chern class, is computed from data related to the heterotic vector bundle in the spectral cover description; a non-technical introduction to that method is supplied. This invariant is, in the class of bundles considered, shown to be related to a discrete modulus which is the heterotic analogue of the $F$-theory four-flux. We consider also the relevant matter which is supported along certain curves in the base $B$ and derive the net generation number again from the independent matter-related computation. We then illustrate these considerations with two applications. First we show that the construction leads to numerous 3 generation models of unbroken gauge group $SU(5), SO(10)$ or $E_6$. Secondly we discuss the closely related issue of the heterotic 5-brane/instanton transition resp. the F-theoretic 3-brane/instanton transition. The extra chiral matter in these transitions is related to the Hecke transform of the direct sum of the original bundle and the dissolved 5-brane along the intersection of their spectral covers. Finally we point to the corresponding $F$-theory interpretation of chiral matter from the intersection of 7-branes where the influence of four-flux on the twisting along the intersection curve plays a crucial role.Comment: 13 pages, Latex, reference adde

Anomalous D-Brane Charge in F-Theory Compactifications

Tadpole cancellation in F-theory on an elliptic Calabi-Yau fourfold $X\to B_3$ demands some spacetime-filling three-branes (points in $B_3$). If moved to the discriminant surface, which supports the gauge group, and dissolved into a finite size instanton, the second Chern class of the corresponding bundle $E$ is expected to give a compensating contribution. However the dependence of D-brane charge on the geometry of $W$ and on the embedding $i: W\to B_3$ gives a correction to $c_2(E)$. We show how this is reconciled by considering the torsion sheaf $i_*E$ and discuss some integrality issues related to global properties of $X$ as well as the moduli space of this object.Comment: 11 pages, harvmac, reference adde

A Class of N=1 Dual String Pairs and its Modular Superpotential

We compare the N=1 F-theory compactification of Donagi, Grassi and Witten with modular superpotential - and some closely related models - to dual heterotic models. We read of the F-theory spectrum from the cohomology of the fourfold and discuss on the heterotic side the gauge bundle moduli sector (including the spectral surface) and the necessary fivebranes. Then we consider the N=1 superpotential and show how a heterotic superpotential matching the F-theory computation is built up by worldsheet instantons. Finally we discuss how the original modular superpotential should be corrected by an additional modular correction factor, which on the F-theory side matches nicely with a `curve counting function' for the del Pezzo surface. On the heterotic side we derive the same factor demanding correct T-duality transformation properties of the superpotential.Comment: 18 pages, Late