nLab duality between M/F-theory and heterotic string theory

Contents

This entry is about M-theory/F-theory compactified on K3-surfaces. For M-theory on MO9-planes see instead at Hořava-Witten theory.

Contents

Idea

F-theoryKK-compactified” on an elliptically fibered K3 with a section is supposed to be equivalent to heterotic string theory KK-compactified on a 2-torus.

More generally, F-theory on a complex $n$-dimensional $X$ fibered $X\to B$ with elliptic K3-fibers is supposed to be equivalent to heterotic string theory on an elliptically fibered Calabi-Yau manifold $Z \to B$ of complex dimension $(n-1)$.

A detailed discussion of the equivalence of the respective moduli spaces is originally due to (Friedman-Morgan-Witten 97). A review of this is in (Donagi 98).

From the abstract of (Donagi 98).

The heterotic string compactified on an $(n-1)$-dimensional elliptically fibered Calabi-Yau $Z \to B$ is conjectured to be dual to F-theory compactified on an $n$-dimensional Calabi-Yau $X \to B$, fibered over the same base with elliptic K3 fibers. In particular, the moduli of the two theories should be isomorphic. The cases most relevant to the physics are $n=2$, $3$, $4$, i.e. the compactification is to dimensions $d=8$, $6$ or $4$ respectively. Mathematically, the richest picture seems to emerge for $n=3$, where the moduli space involves an analytically integrable system whose fibers admit rather different descriptions in the two theories.

Aspects

From M-branes to F-branes to heterotic strings and NS5-branes

from M-branes to F-branes: superstrings, D-branes and NS5-branes

M-theory on $S^1_A \times S^1_B$-elliptic fibrationKK-compactification on $S^1_A$type IIA string theoryT-dual KK-compactification on $S^1_B$type IIB string theoryF-theory on elliptically fibered-K3 fibrationduality between F-theory and heterotic string theoryheterotic string theory on elliptic fibration
M2-brane wrapping $S_A^1$double dimensional reduction $\mapsto$type IIA superstring$\mapsto$type IIB superstring$\mapsto$heterotic superstring
M2-brane wrapping $S_B^1$$\mapsto$D2-brane$\mapsto$D1-brane
M2-brane wrapping $p$ times around $S_A^1$ and $q$ times around $S_B^1$$\mapsto$$p$ strings and $q$ D2-branes$\mapsto$(p,q)-string
M5-brane wrapping $S_A^1$double dimensional reduction $\mapsto$D4-brane$\mapsto$D5-brane
M5-brane wrapping $S_B^1$$\mapsto$NS5-brane$\mapsto$NS5-brane$\mapsto$NS5-brane
M5-brane wrapping $p$ times around $S_A^1$ and $q$ times around $S_B^1$$\mapsto$$p$ D4-brane and $q$ NS5-branes$\mapsto$(p,q)5-brane
M5-brane wrapping $S_A^1 \times S_B^1$$\mapsto$$\mapsto$D3-brane
KK-monopole/A-type ADE singularity (degeneration locus of $S^1_A$-circle fibration, Sen limit of $S^1_A \times S^1_B$ elliptic fibration)$\mapsto$D6-brane$\mapsto$D7-branesA-type nodal curve cycle degenertion locus of elliptic fibration (Sen 97, section 2)SU-gauge enhancement
KK-monopole orientifold/D-type ADE singularity$\mapsto$D6-brane with O6-planes$\mapsto$D7-branes with O7-planesD-type nodal curve cycle degenertion locus of elliptic fibration (Sen 97, section 3)SO-gauge enhancement
exceptional ADE-singularity$\mapsto$$\mapsto$exceptional ADE-singularity of elliptic fibration$\mapsto$E6-, E7-, E8-gauge enhancement

(e.g. Johnson 97, Blumenhagen 10)

Non-reducible heterotic $E_8$-gauge backgrounds

There are some F-theory backgrounds whose supposed dual in heterotic string theory involves an E8-principal connection which is not reducible to $Spin(16)/\mathbb{Z}_2$ (Distler-Sharpe 10, section 5), while in fact the traditional construction of the heterotic worldsheet theory only covers this case. In (Distler-Sharpe 10, section 7-8) it is argued that therefore a more general formulation of heterotic string theory needs to involve parameterized WZW models. See also at heterotic string – Properties – General gauge backgrounds and parameterized WZW models.

F-theory KK-compactified on elliptically fibered complex analytic fiber $\Sigma$

$dim_{\mathbb{C}}(\Sigma)$1234
F-theoryF-theory on CY2F-theory on CY3F-theory on CY4

References

For type IIA and M-theory

The conjectured duality between type IIA string theory KK-compactified on K3 times an n-torus and heterotic string theory on the $(n+2)$-torus is originally due to

Review includes

Further discussion includes

Specifically in relation to the putative K-theory-classification of D-brane charge:

Specifically in M-theory on G2-manifolds:

Specifically in relation to Moonshine:

For F-theory

Discussion for F-theory includes

Review of (Friedman-Morgan-Witten 97) is in

• Ron Donagi, ICMP lecture on heterotic/F-theory duality (arXiv:hep-th/9802093)

• Björn Andreas, $N=1$ Heterotic/F-theory duality, PhD thesis (pdf)

with more details in

The issue with non-reducible $E_8$-gauge connections is highligted in

Last revised on October 26, 2018 at 09:29:09. See the history of this page for a list of all contributions to it.