# nLab Hořava-Witten theory

## Phenomenology

#### Gravity

gravity, supergravity

# Contents

## Idea

There is an observation by Hořava–Witten that 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.

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

Section 3 of

• Adam Falkowski, Five dimensional locally supersymmetric theories with branes, Master Thesis, Warsaw (pdf)

Revised on August 15, 2013 18:54:21 by Urs Schreiber (75.151.240.1)