nLab
Hořava-Witten theory

Context

String theory

Gravity

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

There is an observation by Hořava–Witten that suggests that quantum 11-dimensional supergravity on an 2\mathbb{Z}_2-orbifold (actually a higher orientifold) of the form X 10×/(S 1// 2))X_{10} \times /(S^1//\mathbb{Z}_2)) induces on its boundaryM9-brane” (the 2\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 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).

Properties

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.

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)