One can define the tangent vector space as a space of vectors vµ∂x∂µ which lie tangent to a point p on
the manifold. One can take this idea further by considering a collection of points pi on the manifold. The group of all such tangent spaces will form what is called a vector bundle. Formally, this is written as
T M = ∪
p∈MTpM. (1.1.94)
We can obtain a better understanding of the tangent bundle by introducing coordinates. To specify a point in the tangent bundle, we need a coordinate pair (xµ, vµ), where xµ determines the location of point p,
and the vµ characterizes the vector in T
p(M). In this construction, one sees that the tangent bundle is
itself a special kind of manifold - with coordinates, (xµ, vν). When specifying coordinates, one also specifies
from coordinates on the base manifold. Suppose on this base manifold M two coordinate patches, Ui with
coordinates x and Uj with coordinatesex, have an overlapping region. The transition functions then would be
e
xµ=exµ(x). (1.1.95)
Since the tangent vector depends on the coordinate xµ we have:
e vµ ∂
∂xeµ = v
ν ∂
∂xν. (1.1.96)
From this follows that:
e vµ= ∂xe
µ
∂xνv
ν. (1.1.97)
Thus, overlapping regions on the base manifold results in a pair of coordinates (xµ, vµ) and (
e xµ,
e
vµ) which
themselves are overlapping - their tangent spaces are overlapping. We now have a well defined transition between these coordinates, namely eqns: (1.1.95) and (1.1.97). This notion can be generalized for not only tangent spaces, but also vector spaces. The resultant of such a generalization gives us the vector bundle E.
At each point p of the manifold M we have a vector space. Where in the case of tangent space, we have locally a direct product over each coordinate patch of the coordinate x of the base manifold and tangent space:
Ui×Rn. (1.1.98)
Similarly, suppose we have coordinate patch Ui of M, then the vector bundle E looks like:
Ui× V, (1.1.99)
where V is some vector space. In order to clearly express the notion of having a vector space at each point p we make use of the following definition:
E is a vector bundle if it satisfies the following;
• There is a projection map, π : E → M, so that for each point p ∈ M the inverse map π−1(p) is isomorphic to a vector space V
• We can choose an atlas of E so that for each local coordinate chart U , the coordinates of E is given by the pair (x, v) - where x are coordinates over U and v ∈ V .
Over each point p we have a vector space which is isomorphic to V , p = π−1(p). This vector space is called a fiber over p. The vector space can have either real coefficients - real vector bundle - or complex coeffi- cients - complex vector bundle. In particular, a one-dimensional complex fiber is referred to as a line bundle.
In the case of the tangent bundle, the way the coordinates on the base manifold transform, tells us how the coordinates on the tangent bundle transform. On the vector bundle, such information needs to be specified independently. Suppose we have a manifold M, with coordinates (x, v) on Ui and (ex,ev) on Uj.
Just as beforeexµ=
e
xµ(x)4, however now we have to specify how the fiber coordinates transform. We do so
by stipulating that it should depend on some linear transformation of v
e
v = g(x)v, (1.1.100)
where, in the case of a real vector bundle, g(x) ∈ GL(N,R) with N = dimV . In the case of tangent bundles, the linear transformation g(x) was just g = ∂ex
µ
∂xν. For vector bundles g(x) will be something else in general.
One needs to be careful however, as it is not always possible to choose a g(x) in an arbitrary manner. To define a vector bundle consistently, it is necessary and sufficient to satisfy the following condition.
• Choose three overlapping coordinate patches Ui, Uj, Uk as seen below
Then we have two different ways of comparing coordinates from say Ui with Uk. Either we go from Ui
to Uk directly which results in
vk = gk,i(x)vi (1.1.101)
or we can go via Uj giving:
vk = gk,j(x)vj= gk,jgj,i(x)(x)vi.
All of these transformations have to be compatible in order to have a consistent coordinate system. This gives us the following consistency condition, or what is known as the co-cycle condition
gk,i = gk,jgj,i. (1.1.102)
For tangent bundles, we can easily check, for example, that indeed the transition function g = ∂ex
µ ∂xν obeys
the co-cycle condition. In a more general vector bundle, there is no guarantee that this condition is always satisfied. Thus, the existence of a transition function g(x) which satisfies the co-cycle condition defines a unique vector bundle on the manifold. We can go a step further in the generalization process and define a
fiber bundle. Some common examples, which are important in the study of Calabi–Yaus are holomorphic
4If we are dealing with a differentiable manifold the transformation is a smooth function, for a complex manifold it is a
vector bundles and holomorphic line bundles. If the total space E has the structure of a complex manifold, the projection map which we called π is a holomorphic map of complex manifolds and the map
φU : π−1(U ) → U ×C, (1.1.103)
is biholomorphic, then E can be considered to be a holomorphic vector bundles. An example of such is the
trivial bundle, written as in eqn (1.1.98),
M ×Ck, (1.1.104)
but with manifold M. A holomorphic line bundle exists instead when the fiber of the holomorphic vector bundle isC with rank one. An important example is the canonical bundle.
More specifically it is the complex vector bundle KM= Λm,0M, that is, its sections are (m, 0)-forms on
a complex manifold which is called the canonical bundle, or also the holomorphic line bundle. We can also say that the canonical bundle is the determinant line bundle of the holomorphic cotangent bundle, in other words, it is the highest antisymmetric tensor product of the holomorphic cotangent bundle. See [21] for a more in depth review of holomorphic fiber bundles as well as other definitions.