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 has been argued (Sethi-Vafa-Witten 96, Becker-Becker 96, Dasgupta-Mukhi 97) to be the condition

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

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

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

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

## Properties

### Integrality on $K3 \times K3$

That (1) should be an integer is a highly non-trivial condition on the manifold $X^8$.

One case where this is satisfied is for $X^8$ being the product space of K3 with itself.

To see this, one needs the shifted C-field flux quantization-condition

(2)$[\tilde G_4] \;\in\; H^4(X^8, \mathbb{Z}) \to H^4(X^8, \mathbb{R})$

for

(3)$\tilde G_4 \;\coloneqq\; G_4 + \tfrac{1}{4}p_1(\nabla_{T X^8})$
###### Proposition

With the shifted C-field flux quantization (3), the Dasgupta-Mukhi-expression (1) for the number of M2-branes is indeed an integer on the product space $X^8 = K3 \times K3$ of K3 with itself:

$\tfrac{1}{24}\chi[K3 \times K3] - \tfrac{1}{2} \big( G_4[K3 \times K3]\big)^2 \;\in\; \mathbb{Z} \,,$
###### Proof

After replacing $G_4$ by $\widetilde G_4$ (3) the expression becomes

(4)\begin{aligned} & \tfrac{1}{24}\chi - \tfrac{1}{2} \big( G_4\big)^2 \\ & = \tfrac{1}{24}\chi - \tfrac{1}{2}\big( \tfrac{1}{4} p_1 \big)^2 - \underset{ \in \mathbb{Z} }{ \underbrace{ \tfrac{1}{2} \widetilde G_4 \cdot \big( \widetilde G_4 - \tfrac{1}{2}p_1\big) } } \,, \end{aligned}

where the summand over the brace is an integral class (by this Corollary), because $K3$ is a spin manifold so that $\tfrac{1}{2}p_1$ is the Wu class $\nu_4$ (by this Prop.).

Hence it is now sufficient to show that the first two summands on the right of (4) are both integers, when evaluated on $K3 \times K3$.

But we have

1. $\chi[K3\times K3] \;=\; 24^2$ (by this Prop.)

2. $p_1[K3 \times K3] \;=\; - 2 \times 48$ (by this Prop.)

Hence

\begin{aligned} \tfrac{1}{24}\chi[K3 \times K3] & = \tfrac{1}{24} 24^2 \\ & = 24 \;\in\; \mathbb{Z} \end{aligned}

and

\begin{aligned} \tfrac{1}{32} (p_1[K3 \times K3])^2 & = \tfrac{1}{32} \times ( \underset{ \mathclap{ = 3 \times 32 } } { \underbrace{ 2 \times 48} } )^2 \\ & = 9 \times 32 \;\in\; \mathbb{Z} \end{aligned}

## References

The C-field tadpole cancellation condition is claimed in

referring for proof to the computation in

A comment is also in

Another condition appears in

The formulas of Sethi-Vafa-Witten 96 and Becker-Becker 96 have been plugged together in

Further discussion:

Lecture notes:

Application in dualities in string theory:

Application in string phenomenology

Last revised on November 13, 2019 at 14:55:58. See the history of this page for a list of all contributions to it.