# nLab Hořava-Witten theory

## Phenomenology

#### Gravity

gravity, supergravity

## Spacetimes

black hole spacetimesvanishing angular momentumpositive angular momentum
vanishing chargeSchwarzschild spacetimeKerr spacetime
positive chargeReissner-Nordstrom spacetimeKerr-Newman spacetime

# Contents

## Idea

There is an observation by Hořava–Witten 95, Hořava–Witten 96 which suggests that quantum 11-dimensional supergravity on an $\mathbb{Z}_2$-orbifold (actually a higher orientifold) of the form $X_{10} \times (S^1//\mathbb{Z}_2))$ induces on its boundaryM9-brane” (the $\mathbb{Z}_2$-fixed point manifold) heterotic string theory. Therefore one also speaks of “heterotic M-theory” (Ovrut 02).

from Kashima 00

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 $\mathbb{Z}_2$-action. Thereby the quantization condition

$[2G_4] = 2 [c_2] - [\frac{1}{2} p_1]$

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).

## Properties

### Boundary conditions

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

$C_3 \mapsto C_3 \wedge G_4 \wedge G_4$

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

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

Moreover, the kinetic term

$C \mapsto G \wedge \star G$

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

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

## References

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 $R^2$ 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.

Last revised on April 17, 2018 at 03:14:57. See the history of this page for a list of all contributions to it.