String Theory and M-Theory

472 Flux compactifications
Fig. 10.3. According to Maxwell’s theory, an electric current in a solenoid generates
a magnetic field even though no monopoles, electric or magnetic, are present.
The integral of the field strength and its dual over any closed surface in space vanishes.
Similarly, nontrivial flux solutions exist in M-theory, even when no δ-function
sources, corresponding to M2-branes or M5-branes, are present.
this is small
∆ 1 + O
ℓ6 p
v3/4
. (10.59)
In the approximation in which the size of the Calabi–Yau is very large, that
is, when ℓp/v 1/8 → 0, the background metric becomes unwarped.
This analysis shows that nontrivial flux solutions are possible even in the
absence of explicit delta function sources for M2-branes or M5-branes, which
would appear in the equation of motion and Bianchi identity for F4. A rather
similar situation appears in ordinary Maxwell theory, where a magnetic flux
is generated by an electric current running through a loop even though there
are no magnetic monopoles, as illustrated in Fig. 10.3.
According to Eq. (10.50), nonsingular solutions for the warp factor and
the background geometry are possible even in the absence of explicit brane
sources. In fact, a nonsingular background is necessary to justify rigorously
the validity of the supergravity approximation everywhere in space-time.
Nevertheless, the supergravity approximation is valid for singular solutions
provided that the delta-function singularities are treated carefully.
Inclusion of M2-brane sources
If M2-branes filling the external Minkowski space are also present, an additional
integer N (the number of M2-branes) appears on the left-hand side
of Eq. (10.56), resulting in
N +
1
4κ 2 11 TM2
M
F ∧ F = χ
. (10.60)
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