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C-field tadpole cancellation

Contents

Contents

Idea

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

N M2+12(G 4[X (8)]) 2=148(p 2(12p 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}] } } \;\;\;\; \overset{!}{\in} \mathbb{Z} \,,

where

  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

or

then

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

(1)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.

Properties

Integrality on K3×K3K3 \times K3

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

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

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

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

for

(3)G˜ 4G 4+14p 1( TX 8) \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×K3X^8 = K3 \times K3 of K3 with itself:

124χ[K3×K3]12(G 4[K3×K3]) 2, \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 4G_4 by G˜ 4\widetilde G_4 (3) the expression becomes

(4) 124χ12(G 4) 2 =124χ12(14p 1) 212G˜ 4(G˜ 412p 1), \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 K3K3 is a spin manifold so that 12p 1\tfrac{1}{2}p_1 is the Wu class ν 4\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×K3K3 \times K3.

But we have

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

  2. p 1[K3×K3]=2×48 p_1[K3 \times K3] \;=\; - 2 \times 48 (by this Prop.)

Hence

124χ[K3×K3] =12424 2 =24 \begin{aligned} \tfrac{1}{24}\chi[K3 \times K3] & = \tfrac{1}{24} 24^2 \\ & = 24 \;\in\; \mathbb{Z} \end{aligned}

and

132(p 1[K3×K3]) 2 =132×(2×48=3×32) 2 =9×32 \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.