NS5-branes and line bundles in heterotic/F-theory duality

We study F-theory duals of heterotic line bundle models on elliptically fibered Calabi-Yau threefolds. These models necessarily contain NS5-branes which are geometrized in the dual F-theory compactifications. We initiate a systematic study of the correspondence between various configurations of NS5-branes and the dual geometries in F-theory and perform several checks of the duality. Furthermore, we discuss the singular transitions between different configurations of NS5-branes.

Figure 1

Schematic depiction of the F-theory base after stable degeneration of a Hirzebruch base. Note this is a rendition of Fig. 3 in Ref. [2]. In this picture there are no five-branes in the heterotic dual, only small instantons. Note $C_{*}$ is identified with the heterotic base.

Figure 2

Toric fans for the Hirzebruch surfaces ${\mathbb{F}}_n$ and ${\mathbb{F}}_{n+1}$, as well as the fan for the intermediate space, which we call ${\mathbb{F}}_{n,n+1}$, in which we have blown up ${\mathbb{F}}_n$ but have not yet blown down to ${\mathbb{F}}_{n+1}$. We have also indicated that we wish to consider elliptic fibrations $Y_3^{(·)}$ over these bases.

Figure 3

Possible toric bases of the F-theory threefold after blowing up all intersections of the ${\mathrm{E}}_8$ stacks with the remaining brane locus.

Figure 4

Ray diagram for the result of blowing up the F-theory base ${({\mathbb{P}}^1)}^3$ to remove intersections of the remaining brane locus with the two ${\mathrm{E}}_8$ stacks. Blow-up rays are a different color only for clarity. The ${ϵ}_i$ and ${\rho }_i$ rays are blow-up rays, in the $u–z$ and $v–z$ planes, respectively.

Figure 5

Examples of blow-up distributions in the case of a heterotic base space ${\mathbb{P}}^2$. Blow-up rays are all above/below one of the rays in the base ($u_1$, $u_2$, $u_3$). In the left diagram the blow-ups are distributed democratically, while on the right they are all associated to one base ray.

Figure 6

Horizontal NS5-brane configurations on the heterotic side for a choice of a ${\mathbb{P}}^2$ heterotic base, in the case of (i) a configuration that has a dual geometry described by a toric hypersurface, and (ii) a configuration whose dual geometrical description requires a codimension two complete intersection.

Figure 7

The fan of the Hirzebruch surface ${\mathbb{F}}_2$.

Figure 8

Base ${\widehat{\mathcal{B}}}_3$ of the F-theory fourfold ${\widehat{Y}}_4$ in the stable degeneration limit, in the case where blow-ups have been performed over a ($-2$)-curve in the ${\mathrm{E}}_8$ surfaces, which corresponds to NS5-branes wrapping the ($-2$)-curve in the heterotic base $B_2={\mathbb{F}}_2$. In the ${\mathbb{P}}^1$ fibration, only the reducible fibers over the ($-2$)-curve have been shown, for simplicity.

Figure 9

Schematic depiction of transversely-intersecting M5-branes and the “branches” that remove the intersection, as discussed in the main text. In this example, the heterotic base is $B_2={\mathbb{P}}^1\times {\mathbb{P}}^1$, and two ${\mathbb{P}}^1$ M5-branes sit at for example $u_1=0$ and $v_1=0$ in $B_2$.

Figure 10

Schematic depiction of coincident M5-branes and the branches that remove the coincidence, as discussed in the main text. In this example, the heterotic base is $B_2={\mathbb{P}}_u^2$. In the coincident situation, three ${\mathbb{P}}^1$ M5-branes sit all at for example $u_1=0$ in $B_2$, while in the transverse-intersection case they are at for example $u_1=0$, $u_2=0$, and $u_3=0$ in $B_2$.

Figure 11

Schematic depiction of coincident M5-branes and the branches that remove the coincidence, as discussed in the main text. In this example, the heterotic base is $B_2={\mathbb{P}}_u^1\times {\mathbb{P}}_v^1$. In the coincident situation, two ${\mathbb{P}}^1$ M5-branes sit at for example $u_1=0$ in $B_2$. Deformation to a single smooth brane is impossible, with general deformations giving separated branes.

Figure 12

A schematic depiction of the geometry in the base ${\widehat{\mathcal{B}}}_3$ of the F-theory fourfold ${\widehat{Y}}_4$, when in the dual geometry we pass through a situation with (a) coincident horizontal M5-branes and (b) transversely-intersecting horizontal M5-branes. In each case on the left is the resolved side, in the middle the singular situation, and on the right the deformed side. We have shown only those fibers that have multiple components; the other fibers remain ${\mathbb{P}}^1\mathrm{s}$.

Figure 13

Part of the triangulation of the ray system on the right of Fig. 5 for the F-theory base ${\widehat{\mathcal{B}}}_3$ when the heterotic base $B_2$ is a ${\mathbb{P}}^2$ with a particular NS5-brane configuration, in the situation before and after a removal of one of the blow-up rays. The result corresponds to a situation with two coincident NS5-branes, and gives rise to a curve of singularities.

Figure 14

Part of the triangulation of the ray system in Fig. 4 for the F-theory base ${\widehat{\mathcal{B}}}_3$ when the heterotic base $B_2$ is a ${\mathbb{P}}^1\times {\mathbb{P}}^1$ with a particular NS5-brane configuration, in (center) the nonsingular situation, (left) after a removal of one of the blow-up rays to give coincident NS5-branes, and (right) after removing a 1d face to give transversely-intersecting NS5-branes.

Figure 15

Examples of the correspondence between the positions of the horizontal M5-branes in the bulk and triangulations of the $u_1-v_1$ face of the ray system in Fig. 4 for the F-theory base ${\widehat{\mathcal{B}}}_3$ when $B_2={\mathbb{P}}^1\times {\mathbb{P}}^1$. In the example on the right, there are intersecting and coincident branes. Here we have included only 8 blow-ups rather than the full 24 for simplicity of the diagrams.

Figure 16

Part of the triangulations giving the toric ambient space for the twofold F-theory base in compactification to 6d for a situation in which we have included arbitrarily placed M5-branes through the blow-up $\zeta$. In the triangulation on the left, the blow-up $\zeta$ is performed after ${ϵ}_1$, while on the right $\zeta$ is included first and then ${ϵ}_1$. The two situations are related by a flop, and correspond to different orderings of M5-branes in the 11d bulk.