Hořava-Witten theory


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There is an observation by Hořava–Witten 95, Hořava–Witten 96 which suggests that M-theory on an Z/2-orbifold (actually a higher orientifold) of the form X 10×(S 1 2))X_{10} \times (S^1 \slash \mathbb{Z}_2)) in “dual” to heterotic string theory on its boundary (fixed point) “M9-brane”. Therefore one also speaks of “heterotic M-theory” (Ovrut 02).

from Kashima 00

In the above the circle factor is taken to be the circle fiber over the 10d type IIA spacetime.

If instead one considers another of the spatial dimensions to be compactified on S 1 2S^1\sslash \mathbb{Z}_2, then, after T-duality (F-theory) result is supposed to be type I string theory. In this case the intersection of the M2-brane with the M9-brane (the latter now wrapping the M-theory circle fiber) is called the E-string.

More in detail:

One considers the KK-compactification of M-theory on a Z/2-orbifold of a torus, hence of the Cartesian product of two circles

S A 1 × S B 1 radius: R 11 R 10 \array{ & S^1_A &\times& S^1_B \\ \text{radius}: & R_{11} && R_{10} }

such that the reduction on the first factor S A 1S^1_A corresponds to the duality between M-theory and type IIA string theory, hence so that subsequent T-duality along the second factor yields type IIB string theory (in its F-theory-incarnation). Now the diffeomorphism which exchanges the two circle factors and hence should be a symmetry of M-theory is interpreted as S-duality in type II string theory:

IIBSIIB IIB \overset{S}{\leftrightarrow} IIB

graphics taken from Horava-Witten 95, p. 15

If one considers this situation additionally with a /2\mathbb{Z}/2\mathbb{Z}-orbifold quotient of the first circle factor, one obtains the duality between M-theory and heterotic string theory (Horava-Witten theory). If instead one performs it on the second circle factor, one obtains type I string theory.

Here in both cases the involution action is by reflection of the circle at a line through its center. Hence if we identify S 1/S^1 \simeq \mathbb{R} / \mathbb{Z} then the action is by multiplication by /1 on the real line.

In summary:

M-theory on

Hence the S-duality that swaps the two circle factors corresponds to duality between type I and heterotic string theory.

HE KK/ 2 A M KK/ 2 B I T T HO ASA I \array{ HE &\overset{KK/\mathbb{Z}^A_2}{\leftrightarrow}& M &\overset{KK/\mathbb{Z}^B_2}{\leftrightarrow}& I' \\ \mathllap{T}\updownarrow && && \updownarrow \mathrlap{T} \\ HO && \underset{\phantom{A}S\phantom{A}}{\leftrightarrow} && I }

graphics taken from Horava-Witten 95, p. 16

Duality between M-theory and heterotic string theory

Here each of the two copies of the heterotic gauge theory is a “hidden sector” with respect to the other.

The orbifold equivariance condition of the supergravity C-field is that discussed at orientifold (there for the B-field). Therefore it has to vanish at the two fixed fixed points of the 2\mathbb{Z}_2-action. Thereby the quantization condition

[2G 4]=2[c 2][12p 1] [2G_4] = 2 [c_2] - [\frac{1}{2} p_1]

on the supergravity C-field becomes the condition for the Green-Schwarz mechanism of the heterotic string theory on the “boundary” (the orbifold fixed points).

Duality between M-theory and type I string theory

originally suggested in (Horava-Witten 95, section 3)

Further evidence is reviewed in APT 98

Duality between heterotic and type I string theory

via S-duality in the F-theory-picture…


Boundary conditions

The supergravity C-field G^ 4\hat G_4 is supposed to vanish, and differentially vanish at the boundary in the HW model, meaning that also the local connection 3-form C 3C_3 vanishes there. The argument is roughly as follows (similar for as in Falkowski, section 3.1).

The higher Chern-Simons term

C 3C 3G 4G 4 C_3 \mapsto C_3 \wedge G_4 \wedge G_4

in the Lagrangian of 11-dimensional supergravity is supposed to be well-defined on fields on the orbifold and hence is to be 2\mathbb{Z}_2-invariant.

Let ι 11\iota_{11} be the canonical vector field along the circle factor. Then the component of GGG \wedge G which is annihilated by the contraction ι 11\iota_{11} is necessarily even, so the component dx 11ι 11C 3d x^{11}\wedge \iota_11 C_3 is also even. It follows that also dx 11ι 11G 4d x^{11}\wedge \iota_11 G_4 is even.

Moreover, the kinetic term

CGG C \mapsto G \wedge \star G

is to be invariant. With the above this now implies that the components of GG annihiliated by ι 11\iota_{11} is odd, because so is the mixed component of the metric tensor.

This finally implies that the restriction of C 3C_3 to the orbifold fixed points has to be closed.


The original articles are

Reviews are in

The black M2-brane solution in HW-theory, supposedly yielding the black heterotic string at the intersection with the M9-brane is discussed in

  • Zygmunt Lalak, André Lukas, Burt Ovrut, Soliton Solutions of M-theory on an Orbifold, Phys. Lett. B425 (1998) 59-70 (arXiv:hep-th/9709214)

  • Ken Kashima, The M2-brane Solution of Heterotic M-theory with the Gauss-Bonnet R 2R^2 terms, Prog.Theor.Phys. 105 (2001) 301-321 (arXiv:hep-th/0010286)

Explicit discussion of worldvolume CFT of the M2-branes ending on the HW fixed points and becoming heterotic strings is discussed, via the BLG model, in

After KK-reduction to 5d supergravity there is a corresponding 5d mechanism, see the references there.

Disucssion of the duality between heterotic and type I string theory includes

  • I. Antoniadis, H. Partouche, T.R. Taylor, Lectures on Heterotic-Type I Duality, Nucl.Phys.Proc.Suppl. 61A (1998) 58-71; Nucl.Phys.Proc.Suppl. 67 (1998) 3-16

Last revised on May 2, 2018 at 06:59:43. See the history of this page for a list of all contributions to it.