nLab
Hořava-Witten theory

Idea

There is an observation by Hořava–Witten that suggests that quantum 11-dimensional supergravity on an orbifold of the form $X_{10}\times /(S^1//{\mathbb{Z}}_2))$ induces on its boundary 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

$$[2G_4]=2[c_2]-[\frac{1}{2}{p}_1]$$[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).

The higher Chern-Simons term

$$C_3\mapsto C_3\wedge G_4\wedge G_4$$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 ${\mathbb{Z}}_2$-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

$$C\mapsto G\wedge \star G$$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.

Related concepts

• 11-dimensional supergravity

• M-theory

References

The original articles are

• Petr Hořava, Edward Witten, Heterotic and Type I string dynamics from eleven dimensions, Nucl. Phys. B460 (1996) 506 (arXiv:hep-th/9510209)

  Eleven dimensional supergravity on a manifold with boundary, Nucl. Phys. B475 (1996) 94 (arXiv:hep-th/9603142)

Reviews are in

• Piyush Kumar, Hořava-Witten theory (2004) (pdf)

• Paul Townsend, Four Lectures on M-Theory (arXiv:hep-th/9612121).

Section 3 of

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