String Theory and M-Theory

448 String geometry
a possible holonomy group, and in eight dimensions Spin(7) is a possible
holonomy group. The G2 case is of possible physical interest in the context
of compactifying M-theory to four dimensions.
Complex manifolds
A complex manifold of complex dimension n is a special case of a real manifold
of dimension d = 2n. It is defined in an analogous manner using complex
local coordinate systems. In this case the transition functions are required
to be biholomorphic, which means that they and their inverses are both
holomorphic. Let us denote complex local coordinates by z a (a = 1, . . . , n)
and their complex conjugates ¯z ā .
A complex manifold admits a tensor J, with one covariant and one contravariant
index, which in complex coordinates has components
Ja b = iδa b ¯b ¯
, Jā =
b, ¯b −iδā Ja = Jā b = 0. (9.258)
These equations are preserved by a holomorphic change of variables, so they
describe a globally well-defined tensor.
Sometimes one is given a real manifold M in 2n dimensions, and one
wishes to determine whether it is a complex manifold. The first requirement
is the existence of a tensor, J m n, called an almost complex structure, that
satisfies
Jm n Jn p = −δm p . (9.259)
This equation is preserved by a smooth change of coordinates. The second
condition is that the almost complex structure is a complex structure. The
obstruction to this is given by a tensor, called the Nijenhuis tensor
N p mn = Jm q ∂ [qJ n] p − Jn q ∂ [qJ m] p . (9.260)
When this tensor is identically zero, J is a complex structure. Then it is
possible to choose complex coordinates in every open set that defines the
real manifold M such that J takes the values given in Eq. (9.258) and the
transition functions are holomorphic.
On a complex manifold one can define (p, q)-forms as having p holomorphic
and q antiholomorphic indices
Ap,q = 1
p!q! A a1···ap ¯ b1··· ¯ bq dza1 ∧ · · · ∧ dz ap ∧ d¯z ¯ b1 ∧ · · · ∧ d¯z ¯ bq . (9.261)
The real exterior derivative can be decomposed into holomorphic and antiholomorphic
pieces
d = ∂ + ¯ ∂ (9.262)