In M-theory compactified on 8-dimensional compact fibers (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
where
is the net number of M2-branes in the spacetime (whose worldvolume appears as points in );
is the field strength/flux of the supergravity C-field
is the first Pontryagin class and the second Pontryagin class combining to I8, all regarded here in rational homotopy theory.
If has
or
then
is the Euler class (see this Prop. and this Prop., respectively), hence in these cases the condition is equivalently
where is the Euler characteristic of .
That (1) should be an integer is a highly non-trivial condition on the manifold .
One case where this is satisfied is for being the product space of K3 with itself.
To see this, one needs the shifted C-field flux quantization-condition
for
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 of K3 with itself:
After replacing by (3) the expression becomes
where the summand over the brace is an integral class (by this Corollary), because is a spin manifold so that is the Wu class (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 .
But we have
(by this Prop.)
(by this Prop.)
Hence
and
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:
Sergei Gukov, Cumrun Vafa, Edward Witten: CFT’s From Calabi-Yau Four-folds, Nucl. Phys. B 584 (2000) 69-108, Erratum-ibid. 608 (2001) 477-478 [arXiv:hep-th/9906070]
Keshav Dasgupta, Govindan Rajesh, Savdeep Sethi, M Theory, Orientifolds and G-Flux, JHEP 9908 (1999) 023 [arXiv:hep-th/9908088, doi:10.1088/1126-6708/1999/08/023]
Lecture notes:
Application in dualities in string theory:
Application in string phenomenology
Discussion via Hypothesis H:
Comm. Math. Phys. 377 (2020) 1961-2025 (doi:10.1007/s00220-020-03707-2, arXiv:1904.10207)
surveyed in:
Last revised on November 2, 2024 at 15:02:52. See the history of this page for a list of all contributions to it.