Contents

Contents

Idea

In M-theory compactified on 8-dimensional compact fibers $X^{(8)}$ (see M-theory on 8-manifolds) tadpole cancellation for the supergravity C-field is equivalently the condition

$N_{M2} \;+\; \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2 \;=\; \underset{ I_8(X^8) }{ \underbrace{ \tfrac{1}{48}\big( p_2 - \tfrac{1}{2}p_1^2 \big)[X^{8}] } } \;\;\;\; \in \mathbb{Z} \,,$

where

1. $N_{M2}$ is the net number of M2-branes in the spacetime (whose worldvolume appears as points in $X^{(8)}$);

2. $G_4$ is the field strength/flux of the supergravity C-field

3. $p_1$ is the first Pontryagin class and $p_2$ the second Pontryagin class combining to I8, all regarded here in rational homotopy theory.

If $X^{8}$ has

or

then

$\tfrac{1}{2}\big( p_2 - \tfrac{1}{4}(p_1)^2 \big) \;=\; \chi$

is the Euler class (see this Prop. and this Prop., respectively), hence in these cases the condition is equivalently

$N_{M2} \;+\; \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2 \;=\; \tfrac{1}{24}\chi[X^8] \;\;\;\; \in \mathbb{Z} \,,$

where $\chi[X]$ is the Euler characteristic of $X$.

References

The C-field tadpole cancellation condition in the special case of vanishing $C$-field flux is due to

and the general condition is due to

Further discussion:

Lecture notes:

Application in dualities in string theory:

Application in string phenomenology

Last revised on April 14, 2019 at 08:38:16. See the history of this page for a list of all contributions to it.