12.2 Tangent Space
255
where z a,m is the puncture of S a lying in C i . There is no negative index in the
series because the RHS must vanish for w i = 0 that maps to z a = z a,m (puncture
location). The complex coefficients of the series p i,N (i = 1, . . . , n and N ∈ N ∗ )
provide coordinates for the fibre. Thus, coordinates for P g,n are
{x s } = {q , p i,N }.
(12.13)
As usual, derivatives with respect to x s are abbreviated by ∂ s .
When the dependence in the x s must be stressed, the transition functions (12.6)
are denoted by
z a = F ab (z b ; x s ),
z a = f ai (w i ; x s ).
(12.14)
In this coordinate system, each parameter appears in only one transition function and
it looks like one can separate the fibre from the basis. But, this is not an invariant
statement as this would not hold in other coordinate systems. For example, one can
rescale the coordinates to lump all dependence on q in a single circle.
Since both cases are formally identical, it is convenient to fix the orientation of
each C α and to denote by σ α (resp., τ α ) the coordinate on the left (resp., right) of
the contour, such that the transition functions read
on C α :
σ α = F α (τ α ; x s ).
(12.15)
Now that we have coordinates on P g,n , it is possible to construct tangent vectors.
12.2 Tangent Space
A tangent vector V s ∈ T P g,n corresponds to an infinitesimal variation of the
coordinates on the manifold
δx s = V s ,
(12.16)
where is a small parameter, such that functions of x s vary as
V s ∂ s f = f (x s + V s ) − f (x s ).
(12.17)
The transition functions F α provide an equivalent (but redundant) set of coordinates for P g,n . Hence, vectors in T P g,n can also be obtained by considering small
variations of the transition functions F α :
F α −→ F α + δF α .
(12.18)
Considering an overlap circle C α , a deformation of the transition function
σ α = F α (τ α )
(12.19)
255
where z a,m is the puncture of S a lying in C i . There is no negative index in the
series because the RHS must vanish for w i = 0 that maps to z a = z a,m (puncture
location). The complex coefficients of the series p i,N (i = 1, . . . , n and N ∈ N ∗ )
provide coordinates for the fibre. Thus, coordinates for P g,n are
{x s } = {q , p i,N }.
(12.13)
As usual, derivatives with respect to x s are abbreviated by ∂ s .
When the dependence in the x s must be stressed, the transition functions (12.6)
are denoted by
z a = F ab (z b ; x s ),
z a = f ai (w i ; x s ).
(12.14)
In this coordinate system, each parameter appears in only one transition function and
it looks like one can separate the fibre from the basis. But, this is not an invariant
statement as this would not hold in other coordinate systems. For example, one can
rescale the coordinates to lump all dependence on q in a single circle.
Since both cases are formally identical, it is convenient to fix the orientation of
each C α and to denote by σ α (resp., τ α ) the coordinate on the left (resp., right) of
the contour, such that the transition functions read
on C α :
σ α = F α (τ α ; x s ).
(12.15)
Now that we have coordinates on P g,n , it is possible to construct tangent vectors.
12.2 Tangent Space
A tangent vector V s ∈ T P g,n corresponds to an infinitesimal variation of the
coordinates on the manifold
δx s = V s ,
(12.16)
where is a small parameter, such that functions of x s vary as
V s ∂ s f = f (x s + V s ) − f (x s ).
(12.17)
The transition functions F α provide an equivalent (but redundant) set of coordinates for P g,n . Hence, vectors in T P g,n can also be obtained by considering small
variations of the transition functions F α :
F α −→ F α + δF α .
(12.18)
Considering an overlap circle C α , a deformation of the transition function
σ α = F α (τ α )
(12.19)