C-field tadpole cancellation




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

N M2+12(G 4[X (8)]) 2=148(p 212p 1 2)[X 8]I 8(X 8), 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} \,,


  1. N M2N_{M2} is the net number of M2-branes in the spacetime (whose worldvolume appears as points in X (8)X^{(8)});

  2. G 4G_4 is the field strength/flux of the supergravity C-field

  3. p 1p_1 is the first Pontryagin class and p 2p_2 the second Pontryagin class combining to I8, all regarded here in rational homotopy theory.

If X 8X^{8} has



12(p 214(p 1) 2)=χ \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+12(G 4[X (8)]) 2=124χ[X 8], N_{M2} \;+\; \tfrac{1}{2} \big( G_4[X^{(8)}]\big)^2 \;=\; \tfrac{1}{24}\chi[X^8] \;\;\;\; \in \mathbb{Z} \,,

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


The C-field tadpole cancellation condition in the special case of vanishing CC-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.