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


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


Duality in string theory

String theory



A duality in string theory. The non-perturbative enhancement of duality between heterotic and type II string theory:

F-theory\,KK-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 nn-dimensional XX fibered XBX\to B with elliptic K3-fibers is supposed to be equivalent to heterotic string theory on an elliptically fibered Calabi-Yau manifold ZBZ \to B of complex dimension (n1)(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 (n1)(n-1)-dimensional elliptically fibered Calabi-Yau ZBZ \to B is conjectured to be dual to F-theory compactified on an nn-dimensional Calabi-Yau XBX \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=2n=2, 33, 44, i.e. the compactification is to dimensions d=8d=8, 66 or 44 respectively. Mathematically, the richest picture seems to emerge for n=3n=3, where the moduli space involves an analytically integrable system whose fibers admit rather different descriptions in the two theories.


Singular locus of the elliptic fibration and 24 D7-branes

In passing from M-theory to type IIA string theory, the locus of any Kaluza-Klein monopole in 11d becomes the locus of D6-branes in 10d. The locus of the Kaluza-Klein monopole in turn (as discussed there) is the locus where the S A 1S^1_A-circle fibration degenerates. Hence in F-theory this is the locus where the fiber of the S A 1×S B 1S^1_A \times S^1_B-elliptic fibration degenerates to the nodal curve. Since the T-dual of D6-branes are D7-branes, it follows that D7-branes in F-theory “are” the singular locus of the elliptic fibration.

Now an elliptically fibered complex K3-surface

T K3 1 \array{ T &\longrightarrow& K3 \\ && \downarrow \\ && \mathbb{C}\mathbb{P}^1 }

may be parameterized via the Weierstrass elliptic function as the solution locus of the equation

y 2=x 3+f(z)x+g(z) y^2 = x^3 + f(z) x + g(z)

for x,y,z 1x,y,z \in \mathbb{C}\mathbb{P}^1, with ff a polynomial of degree 8 and gg of degree twelve. The j-invariant of the complex elliptic curve which this parameterizes for given zz is

j(τ(z))=4(24f) 327g 2+4f 3. j(\tau(z)) = \frac{4 (24 f)^3}{27 g^2 + 4 f^3} \,.

The poles jj\to \infty of the j-invariant correspond to the nodal curve, and hence it is at these poles that the D7-branes are located.

homotopy pasting diagram exhibiting the homotopy Whitehead integral
from SS21

Since the order of the poles is 24 (the polynomial degree of the discriminant Δ=27g 2+4f 3\Delta = 27 g^2 + 4 f^3, see at elliptically fibered K3-surfacesingular points) there are necessarily 24 D7-branes (Sen 96, page 5, Sen 97b, see also Morrison 04, sections 8 and 17, Denef 08, around (3.41), Douglas-Park-Schnell 14).

Under T-duality this translates to 24 D6-branes in type IIA string theory on K3 (Vafa 96, Footnote 2 on p. 6).

Notice that the net charge of these 24 D7-branes is supposed to vanish, due to S-duality effects (e.g. Denef 08, below (3.41)).

(This reminds one of the situation for the third stable homotopy group of spheres…)

For analogous discussion of 24 NS5-branes in heterotic string theory on K3 see Schwarz 97, around p. 50.

For more see at 24 branes transverse to K3.

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 A 1×S B 1S^1_A \times S^1_B-elliptic fibrationKK-compactification on S A 1S^1_Atype IIA string theoryT-dual KK-compactification on S B 1S^1_Btype IIB string theorygeometrize the axio-dilatonF-theory on elliptically fibered-K3 fibrationduality between F-theory and heterotic string theoryheterotic string theory on elliptic fibration
M2-brane wrapping S A 1S_A^1double dimensional reduction \mapstotype IIA superstring\mapstotype IIB superstring\mapsto\mapstoheterotic superstring
M2-brane wrapping S B 1S_B^1\mapstoD2-brane\mapstoD1-brane\mapsto
M2-brane wrapping pp times around S A 1S_A^1 and qq times around S B 1S_B^1\mapstopp strings and qq D2-branes\mapsto(p,q)-string\mapsto
M5-brane wrapping S A 1S_A^1double dimensional reduction \mapstoD4-brane\mapstoD5-brane\mapsto
M5-brane wrapping S B 1S_B^1\mapstoNS5-brane\mapstoNS5-brane\mapsto\mapstoNS5-brane
M5-brane wrapping pp times around S A 1S_A^1 and qq times around S B 1S_B^1\mapstopp D4-brane and qq NS5-branes\mapsto(p,q)5-brane\mapsto
M5-brane wrapping S A 1×S B 1S_A^1 \times S_B^1\mapsto\mapstoD3-brane\mapsto
KK-monopole/A-type ADE singularity (degeneration locus of S A 1S^1_A-circle fibration, Sen limit of S A 1×S B 1S^1_A \times S^1_B elliptic fibration)\mapstoD6-brane\mapstoD7-branes\mapstoA-type nodal curve cycle degeneration locus of elliptic fibration ADE 2Cycle (Sen 97, section 2)SU-gauge enhancement
KK-monopole orientifold/D-type ADE singularity\mapstoD6-brane with O6-planes\mapstoD7-branes with O7-planes\mapstoD-type nodal curve cycle degeneration locus of elliptic fibration ADE 2Cycle (Sen 97, section 3)SO-gauge enhancement
exceptional ADE-singularity\mapsto\mapsto\mapstoexceptional ADE-singularity of elliptic fibration\mapstoE6-, E7-, E8-gauge enhancement

(e.g. Johnson 97, Blumenhagen 10)

Non-reducible heterotic E 8E_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 SemiSpin(16) (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 Σ\Sigma

dim (Σ)dim_{\mathbb{C}}(\Sigma)12345
F-theoryF-theory on CY2F-theory on CY3F-theory on CY4F-theory on CY5

KK-compactification of M-theory


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)(n+2)-torus is originally due to


Further discussion:

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

with more details in

The issue with non-reducible E 8E_8-gauge connections is highligted in

On a subtlety in the application of the Narasimhan-Seshadri theorem in the duality:

  • Herbert Clemens, Stuart Raby, Heterotic/F-theory Duality and Narasimhan-Seshadri Equivalence (arxiv:1906.07238)

See also:

For both

  • Yusuke Kimura, New perspectives in the duality of M-theory, heterotic strings, and F-theory (arXiv:2103.03088)

Last revised on April 4, 2024 at 03:43:31. See the history of this page for a list of all contributions to it.