nLab heterotic M-theory on ADE-orbifolds




The KK-compactification of M-theory on fibers

S 1G HW×G ADE S^1 \!\sslash\! G_{HW} \times \mathbb{H}\!\sslash\! G_{ADE}

which are, locally, the Cartesian product of

  1. the circle orientifolded by G HW 2G_{HW} \simeq \mathbb{Z}_2 as in Horava-Witten theory;

  2. the quaternions orbifolded by a finite subgroup of SU(2) G ADEG_{ADE}.

(Sen 97, Sec 3, Faux-Lüst-Ovrut 99, Kaplunovsky-Sonnenschein-Theisen-Yankielowicz 99, Faux-Lüst-Ovrut 00a, 00b, 00c)

graphics grabbed from HSS18, Example 2.2.7

For G ADE= 2G_{ADE} = \mathbb{Z}_2 this subsumes M-theory on K3 times S 1G HWS^1 \sslash G_{HW} (Seiberg-Witten 96)

Dual string theory perspectives

Under duality in string theory (specifically: duality between M-theory and type IIA string theory and duality between M-theory and heterotic string theory) M-theory on 𝕊 1 2 HW×𝕋 4G ADE\mathbb{S}^{1}\sslash \mathbb{Z}_2^{HW} \times \mathbb{T}^{4}\sslash G^{ADE} appears through the following string theory-perspectives

  1. HET EHET_{E}-Theory on ADE-singularities

  2. II'-Theory with D6-branes on O8-planes

  3. II'-Theory on ADE-singularities intersecting O8-planes

The following graphics shows how the three perspectives arise from KK-compactification on three different choices of circle-fibers. Indicated also are the M5-branes and their string theoretic images at NS5-branes/D4-branes with geometrically engineer D=6 N=(1,0) SCFTs (see further below).

graphics grabbed from SS19

HET EHET_{E}-Theory on ADE-singularities


Horava-Witten theory, hence heterotic string theory, on ADE-singularities G ADE\mathbb{H} \sslash G_{ADE}

(Witten 99,…)


II'-Theory with D6-branes on O8-planes

from GKSTY 02

If in addition the black NS5-brane sits at an O8-plane, hence at the orientifold fixed point-locus, then in the ordinary /2\mathbb{Z}/2-quotient it appears as a “half-brane” – the half M5-brane – with only one copy of D6-branes ending on it:

graphics grabbed from GKSTY 02

(In Hanany-Zaffaroni 99 this is interpreted in terms of the 't Hooft-Polyakov monopole.)

The lift to M-theory of this situation is an M5-brane intersecting an M9-brane (see at M-theory on S1/G_HW times H/G_ADE):

from GKSTY 02

Alternatively the O8-plane may intersect the black D6-branes away from the black NS5-brane:

from HKLY 15

In general, some of the NS5 sit away from the O8-plane, while some sit on top of it:

from Hanany-Zaffaroni 98

II'-theory on ADE-singularities intersecting O8-planes

(Bergman & Rodriguez-Gomez 12, Sec. 3)


Geometric engineering of D=6D = 6 N=(1,0)N=(1,0) SCFTs on M5-branes

The M5-branes-configurations as above are supposed to geometrically engineer D=6 N=(1,0) SCFTs.

See the references below, for example DHTV 14, Section 6, Gaiotto-Tomasiello 14, HKLY 15.

KK-compactification of M-theory

Dp-D(p+4)-brane bound states and Yang-Mills instantons



The first discussion of this compactification is possibly

in the context of the M-theory lift of gauge enhancement on D6-branes.

The original articles focusing on this situation:

Discussion of heterotic M-theory on smooth K3 originates around

See also

As HET EHET_E-theory with ADE-singularities

As heterotic string theory on orbifold ADE-singularities:

As II'-theory with D6-branes

As type I' string theory with D6-branes:

and in massive type IIA string theory with D6-D8 brane bound states:

As II'-theory with ADE-singularities

As type I' string theory at orbifold ADE-singularities:

  • Oren Bergman, Diego Rodriguez-Gomez, Section 3 of: 5d quivers and their AdS 6AdS_6 duals, JHEP07 (2012) 171 (arxiv:1206.3503)

  • Chiung Hwang, Joonho Kim, Seok Kim, Jaemo Park, Section 3.4.2 of: General instanton counting and 5d SCFT, JHEP07 (2015) 063 (arxiv:1406.6793)

F-theory perspective

The F-theory perspective:

Geometric engineering of D=6,𝒩=(1,0)D=6, \mathcal{N}=(1,0) SCFT

On D=6 N=(1,0) SCFTs via geometric engineering on M5-branes/NS5-branes at D-, E-type ADE-singularities, notably from M-theory on S1/G_HW times H/G_ADE, hence from orbifolds of type I' string theory (see at half NS5-brane):

SU(2)SU(2)-flavor symmetry on heterotic M5-branes

Emergence of SU(2) flavor-symmetry on single M5-branes in heterotic M-theory on ADE-orbifolds (in the D=6 N=(1,0) SCFT on small instantons in heterotic string theory):

Argument for this by translation under duality between M-theory and type IIA string theory to half NS5-brane/D6/D8-brane bound state systems in type I' string theory:

Reviewed in:

  • Santiago Cabrera, Amihay Hanany, Marcus Sperling, Section 2.3 of: Magnetic Quivers, Higgs Branches, and 6d 𝒩=(1,0)\mathcal{N}=(1,0) Theories, JHEP06(2019)071, JHEP07(2019)137 (arXiv:1904.12293)

The emergence of flavor in these half NS5-brane/D6/D8-brane bound state systems, due to the semi-infinite extension of the D6-branes making them act as flavor branes:

Reviewed in:

  • Fabio Apruzzi, Marco Fazzi, Section 2.1 of: AdS 7/CFT 6AdS_7/CFT_6 with orientifolds, J. High Energ. Phys. (2018) 2018: 124 (arXiv:1712.03235)

See also:

Last revised on November 27, 2021 at 13:09:55. See the history of this page for a list of all contributions to it.