String Theory and M-Theory

464 Flux compactifications
Fig. 10.1. This figure illustrates the Poincaré–Hopf index theorem. A continuous
vector field on a sphere must have at least two zeros, which in this case are located
at the north and south poles, since the Euler characteristic is 2. On the other hand,
a vector field on a torus can be nonvanishing everywhere since χ = 0.
Nonchiral spinors
If η1 and η2 have opposite chirality the complex spinor η = η1 + iη2 is
nonchiral. The two spinors of opposite chirality define a vector field on the
internal manifold
va = η †
1 γaη2. (10.17)
Requiring this vector to be nonvanishing leads to an interesting class of solutions.
Indeed, the Poincaré–Hopf index theorem of algebraic topology states
that the number of zeros of a continuous vector field must be at least equal to
the absolute value of the Euler characteristic χ of the background geometry.
As a result, a nowhere vanishing vector field only exists for manifolds with
χ = 0. An example of this theorem is illustrated in Fig. 10.1. Flux backgrounds
representing M5-branes filling the three-dimensional space-time and
wrapping supersymmetric three-cycles on the internal space are examples of
this type of geometries. Moreover, once the spinor is nonchiral, compactifications
to AdS3 spaces become possible. Compactifications to AdS space
are considered in Chapter 12, so the discussion in this chapter is restricted
to spinors of positive chirality. It will turn out that AdS3 is not a solution
in this case.
Solving the supersymmetry constraints
The constraints that follow from Eq. (10.4) are influenced by the warpfactor
dependence of the metric. As was pointed out in Chapter 8, there
is a relation between the covariant derivatives of a spinor with respect to
a pair of metrics that differ by a conformal transformation. In particular,
in the present case, the internal and external components of the metric are
rescaled with a different power of the warp factor and the vielbeins are given