Hořava-Witten theory


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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).


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

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

Revised on August 6, 2015 16:09:35 by Urs Schreiber (