String Theory and M-Theory

9.8 Nonperturbative effects in Calabi–Yau compactifications 405
contribution to the path integral that represents a nonperturbative instanton
correction to the theory. More precisely, fundamental-string instantons
give contributions that are nonperturbative in α ′ , whereas D-branes and
NS5-branes give contributions that are also nonperturbative in gs. 23 If the
internal manifold is a Calabi–Yau three-fold, the values of p for which there
are nontrivial (p + 1)-cycles are p = −1, 1, 2, 3, 5. 24
As was discussed in Chapter 6, type IIA superstring theory contains
even-dimensional BPS D-branes, whereas the type IIB theory contains odddimensional
BPS D-branes. Each of these D-branes carries a conserved
R–R charge. So, in addition to fundamental strings wrapping a two-cycle
and NS5-branes wrapping the entire manifold, one can consider wrapping
D2-branes on a three-cycle in the IIA case. Similarly, one can wrap D1,
D3 and D5-branes, as well as D-instantons, in the IIB case. These configurations
give nonperturbative instanton contributions to the moduli-space
geometry, that need to be included in order for string theory to be consistent.
As explained in Section 9.9, these effects are crucial for understanding
fundamental properties of string theory, such as mirror symmetry. There
are different types of supersymmetric cycles in the context of Calabi–Yau
compactifications, which we now discuss. 25
Special Lagrangian submanifolds
For Calabi–Yau compactification of M-theory, which gives a five-dimensional
low-energy theory, the possible instanton configurations arise from M2branes
wrapping three-cycles and M5-branes wrapping the entire Calabi–
Yau manifold. Let us first consider a Euclidean M2-brane, which has a
three-dimensional world volume. The goal is to examine the conditions under
which a Euclidean membrane wrapping a three-cycle of the Calabi–Yau
manifold corresponds to a stationary point of the path-integral-preserving
supersymmetry. Once this is achieved, the next step is to determine the corresponding
nonperturbative contribution to the low-energy five-dimensional
effective action.
The M2-brane in 11 dimensions has a world-volume action, with global
supersymmetry and local κ symmetry, whose form is similar to the actions
for fundamental superstrings and D-branes described in Chapters 5 and 6.
As in the other examples, in flat space-time this action is invariant under
23 The gs dependence is contained in the tension factor that multiplies the world-volume actions.
24 A p-brane with p = −1 is the D-instanton of the type IIB theory.
25 A vanishing potential for the tensor fields is assumed here. The generalization to a nonvanishing
potential is known.